hash stringlengths 32 32 | doc_id stringlengths 7 13 | section stringlengths 3 121 | content stringlengths 0 2.2M |
|---|---|---|---|
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.1.3 Open Area Test Site | A fully worked example illustrating the methodology to be used can be found in TR 102 273 [2], part 1, sub-part 2, clause 4. For receiver sensitivity measurement two stages of test are involved. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.1.3.1 Uncertainty contributions: Stage one: Transform Factor | The first stage (determining the Transform Factor) involves placing a measuring antenna as shown in figure 19 and determining the relationship between the signal generator output power level and the resulting field strength (the shaded areas in figure 19 represent components common to both stages of the test). Measuring antenna Measuring antenna cable 1 Receiving device Test antenna cable 2 ferrite beads Signal Attenuator 2 10 dB Test antenna Attenuator 1 10 dB generator Ground plane Figure 19: Stage 1: Transform Factor All the uncertainty components which contribute to this stage of the test are listed in table 5. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 88 Table 5: Contributions for the Transform Factor uj or i Description of uncertainty contributions dB uj36 mismatch: transmitting part uj37 mismatch: receiving part uj38 signal generator: absolute output level 0,00 uj39 signal generator: output level stability uj19 cable factor: measuring antenna cable uj19 cable factor: test antenna cable uj41 insertion loss: measuring antenna cable uj41 insertion loss: test antenna cable 0,00 uj40 insertion loss: measuring antenna attenuator uj40 insertion loss: test antenna attenuator 0,00 uj47 receiving device: absolute level uj16 range length uj44 antenna: antenna factor of the measuring antenna uj45 antenna: gain of the test antenna uj46 antenna: tuning of the measuring antenna uj46 antenna: tuning of the test antenna 0,00 uj22 position of the phase centre: measuring antenna uj14 mutual coupling: measuring antenna to its images in the ground plane uj14 mutual coupling: test antenna to its images in the ground plane uj11 mutual coupling: measuring antenna to the test antenna uj12 mutual coupling: interpolation of mutual coupling and mismatch loss correction factors ui01 random uncertainty The standard uncertainties from table 5 should be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty (uc contributions from the Transform Factor) for the Transform Factor in dB. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.1.3.2 Uncertainty contributions: Stage two: EUT measurement | The second stage (the EUT measurement) is to determine the minimum signal generator output level which produces the required response from the EUT as shown in figure 20 (the shaded areas represent components common to both stages of the test). Test antenna cable 2 ferrite beads Test antenna Signal EUT Attenuator 2 10 dB Ground plane generator Figure 20: Stage 2: EUT measurement All the uncertainty components which contribute to this stage of the test are listed in table 6. Annex A should be consulted for the sources and/or magnitudes of the uncertainty contributions. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 89 Table 6: Contributions from the EUT measurement uj or i Description of uncertainty contributions dB uj36 mismatch: transmitting part uj38 signal generator: absolute output level 0,00 uj39 signal generator: output level stability uj19 cable factor: test antenna cable uj41 insertion loss: test antenna cable 0,00 uj40 insertion loss: test antenna attenuator 0,00 uj20 position of the phase centre: within the EUT volume uj21 positioning of the phase centre: within the EUT over of the axis of rotation of the turntable uj52 EUT: modulation detection uj16 range length uj45 antenna: gain of the test antenna 0,00 uj46 antenna: tuning of the test antenna 0,00 uj55 EUT: mutual coupling to the power leads uj08 mutual coupling: amplitude effect of the test antenna on the EUT uj13 mutual coupling: EUT to its image in the ground plane uj14 mutual coupling: test antenna to its image in the ground plane ui01 random uncertainty The standard uncertainties from table 6 should be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty (uc contribution from the EUT measurement) for the EUT measurement in dB. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.1.3.3 Expanded uncertainty of the receiver sensitivity measurement | The combined uncertainty of the sensitivity measurement is the combination of the components outlined in clauses 4.2.1.3.1 and 4.2.1.3.2. The components to be combined are uc contribution from the Transform Factor and uc contribution from the EUT measurement: u u u c ccontribution fromtheTransform factor ccontribution fromthe EUT measurement = + 2 2 = _ _,_ _ dB Using an expansion factor (coverage factor) of k = 1,96, the expanded measurement uncertainty is ±1,96 × uc = ±__,__ dB (see clause D.5.6.2). |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.1.4 Striplines | For tests in which the results of the verification procedure have been used, the test will have comprised only a single measurement stage. Otherwise, two measurement stages of the test would have been involved. A fully worked example calculation can be found in TR 102 273 [2], part 1, sub-part 2, clause 5. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.1.4.1 Uncertainty contributions: Stage 1: EUT measurement | The first stage involves the measurement set-up as shown in figure 21. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 90 150 termination Load Central axis of stripline Signal Ferrite beads 10dB attenuator Non-conducting, low dielectric constant support stand EUT with volume centre midway between plates Modulation ΩΩΩΩ generator detection Figure 21: Stage 1 schematic: EUT Measurement Table 7 lists the uncertainty contributions involved in this stage of the test. Annex A should be consulted for the sources and/or magnitudes of the uncertainty contributions. Table 7: Uncertainty contributions from the EUT measurement uj or i Description of uncertainty contributions dB uj36 mismatch: transmitting part uj37 mismatch: receiving part uj38 signal generator: absolute output level uj39 signal generator: output level stability uj19 cable factor: signal generator 0,00 uj41 insertion loss: signal generator cable 0,00 uj40 insertion loss: signal generator attenuator 0,00 uj47 receiving device: absolute level 0,00 uj48 receiving device: linearity 0,00 uj32 Stripline: correction factor for the size of the EUT uj24 Stripline: mutual coupling of the EUT to its images in the plates uj55 EUT: mutual coupling to the power leads uj26 Stripline: characteristic impedance uj27 Stripline: non-planar nature of the field distribution uj33 Stripline: influence of site effects uj34 ambient effect uj52 EUT: modulation detection ui01 random uncertainty The standard uncertainties from table 7 should be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty, uc EUT measurement, for the EUT measurement in dB. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.1.4.2 Uncertainty contributions: Stage 2: Field measurement | For tests using the results of the verification procedure As stated above, for tests in which the results of the verification procedure are used, this second stage does not really exist. In terms of its contribution to the overall uncertainty of this test, the verification procedure contributes the full value of its overall uncertainty. So, in this case, the standard deviation of the verification procedure is taken as the contribution uc field measurement. For the Monopole Figure 22 shows schematically the equipment set-up for this stage of the test. The uncertainty contributions resulting are given in table 23. Annex A should be consulted for the sources and/or magnitudes of the uncertainty contributions. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 91 150 termination Load Central axis of Stripline Signal Ferrite beads 10dB attenuators Monopole Receiving ΩΩΩΩ generator device Figure 22: Stage 2 schematic: Monopole field measurement Table 8: Uncertainty contributions from the Monopole field measurement uj or i Description of uncertainty contributions dB uj36 mismatch: transmitting part uj37 mismatch: receiving part uj47 signal generator: absolute output level uj48 signal generator: output level stability uj19 cable factor: signal generator 0,00 uj19 cable factor: monopole cable 0,00 uj41 insertion loss: signal generator cable 0,00 uj41 insertion loss: monopole cable 0,00 uj40 insertion loss: signal generator attenuator 0,00 uj40 insertion loss: monopole attenuator 0,00 uj47 receiving device: absolute level 0,00 uj48 receiving device: linearity 0,00 uj31 Stripline: antenna factor of the monopole uj32 Stripline: correction factor for the size of the EUT uj24 Stripline: mutual coupling of the EUT to its images in the plates uj26 Stripline: characteristic impedance uj27 Stripline: non-planar nature of the field distribution uj33 Stripline: influence of site effects uj34 ambient effect ui01 random uncertainty The standard uncertainties from table 8 should be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty, uc field measurement, for the Monopole field measurement in dB. For the 3-axis probe The uncertainty contributions for this configuration during the test are as given in table 9. Annex A should be consulted for the sources and/or magnitudes of the uncertainty contributions. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 92 Table 9: Uncertainty contributions from the field measurement uj or i Description of uncertainty contributions dB uj36 mismatch: transmitting part uj38 signal generator: absolute output level uj39 signal generator: output level stability uj19 cable factor: signal generator 0,00 uj41 insertion loss: signal generator cable 0,00 uj40 insertion loss: signal generator attenuator 0,00 uj28 Stripline: field strength measurement as determined by the 3-axis probe uj32 Stripline: correction factor for the size of the EUT uj24 Stripline: mutual coupling of the EUT to its images in the plates uj26 Stripline: characteristic impedance uj27 Stripline: non-planar nature of the field distribution uj33 Stripline: influence of site effects uj34 ambient effect uj25 Stripline: mutual coupling of the 3-axis probe to its image in the plates ui01 random uncertainty The standard uncertainties from table 9 should be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty, uc field measurement, for the 3-axis probe field measurement in dB. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.1.4.3 Expanded uncertainty for the Receiver sensitivity measurement | The combined standard uncertainty of the results of the receiver sensitivity measurement is the RSS combination of the components outlined in clauses 4.2.1.4.1 and 4.2.4.1.2 above. The components to be combined are uc EUT measurement and uc field measurement. u = u + u = __,__ c c EUT measurement c field measurement dB 2 2 Using an expansion factor (coverage factor) of k = 1,96, the expanded measurement uncertainty is ±1,96 × uc = ±__,__ dB (see clause D.5.6.2). |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.1.5 Test Fixture | Tests in a test fixture differ to radiated tests on all other types of site in that there is only one stage to the test. All uncertainty contributions for the test can, therefore, be incorporated into one table and these are given in table 10. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.1.5.1 Uncertainty contributions | All the uncertainty contributions for the test are listed in table 10. Table 10: Contributions from the measurement uj or i Description of uncertainty contributions dB uj38 signal generator: absolute output level uj39 signal generator: output level stability uj60 Test Fixture: effect on the EUT uj61 Test Fixture: climatic facility effect on the EUT ui01 random uncertainty ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 93 The standard uncertainties from table 10 should be given values according to annex A. They should then be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty (uc contributions from the measurement) for the EUT measurement in dB. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.1.5.2 Expanded uncertainty of the Maximum usable sensitivity measurement | Tests in a Test Fixture differ to radiated tests on all other types of site in that there is only one stage to the test. However, the Test Fixture measurement could be considered as stage two of a test in which stage one was on an accredited Free-Field Test Site. The combined standard uncertainty of the maximum usable sensitivity measurement is therefore, simply the RSS combination of the value for uc contributions from the measurement derived above and the combined uncertainty of the Free-Field Test Site uc contribution from the Free-Field Test Site. dB __ __, 2 2 = + = site test field free the from ons contributi c t measuremen the from ons contributi c c u u u Using an expansion factor (coverage factor) of k = 1,96, the expanded measurement uncertainty is ±1,96 × uc = ±__,__ dB (see clause D.5.6.2). |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.1.6 Salty Man/Salty lite | |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.1.6.1 Anechoic Chamber | A fully worked example illustrating the methodology to be used can be found in TR 102 273 [2], part 1, sub-part 2, clause 4. The receiver sensitivity measurement involves two stages of testing. 4.2.1.6.1.1 Uncertainty contributions: Stage one: Transform factor measurement The first stage (determining the Transform Factor) involves placing a measuring antenna as shown in figure 23 and determining the relationship between the signal generator output power level and the resulting field strength (the shaded areas in figure 23 represent components common to both stages of the test). Test antenna cable 2 ferrite beads Test antenna Attenuator 2 10 dB Measuring antenna Measuring antenna cable 1 Receiving device Attenuator 1 10 dB Signal generator Figure 23: Stage 1: Transform Factor All the uncertainty components which contribute to this stage of the test are listed in table 11. Annex A should be consulted for the sources and/or magnitudes of the uncertainty contributions. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 94 Table 11: Contributions for the Transform Factor uj or i Description of uncertainty contributions dB uj36 mismatch: transmitting part 0,00 uj37 mismatch: receiving part uj38 signal generator: absolute output level uj39 signal generator: output level stability uj19 cable factor: measuring antenna cable uj19 cable factor: test antenna cable 0,00 uj41 insertion loss: measuring antenna cable uj41 insertion loss: test antenna cable 0,00 uj40 insertion loss: measuring antenna attenuator uj40 insertion loss: test antenna attenuator 0,00 uj47 receiving device: absolute level uj16 range length 0,00 uj02 reflectivity of absorber material: measuring antenna to the test antenna 0,00 uj44 antenna: antenna factor of the measuring antenna uj45 antenna: gain of the test antenna 0,00 uj46 antenna: tuning of the measuring antenna uj46 antenna: tuning of the test antenna 0,00 uj22 position of the phase centre: measuring antenna uj06 mutual coupling: measuring antenna to its images in the absorbing material uj06 mutual coupling: test antenna to its images in the absorbing material 0,00 uj11 mutual coupling: measuring antenna to the test antenna 0,00 uj12 mutual coupling: interpolation of mutual coupling and mismatch loss correction factors 0,00 ui01 random uncertainty The standard uncertainties from table 11 should be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty (uc contributions from the Transform Factor) for the Transform Factor in dB. 4.2.1.6.1.2 Uncertainty contributions: Stage two: EUT measurement The second stage (the EUT measurement) is to determine the minimum signal generator output level which produces the required response from the EUT as shown in figure 24 (the shaded areas represent components common to both stages of the test). Test antenna cable 2 ferrite beads Test antenna Signal Attenuator 2 10 dB EUT generator Figure 24: Stage 2: EUT measurement All the uncertainty components which contribute to this stage of the test are listed in table 12. Annex A should be consulted for the sources and/or magnitudes of the uncertainty contributions. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 95 Table 12: Contributions from the EUT measurement uj or i Description of uncertainty contributions dB uj36 mismatch: transmitting part 0,00 uj37 mismatch: receiving part uj38 signal generator: absolute output level uj39 signal generator: output level stability uj19 cable factor: measuring antenna cable uj19 cable factor: test antenna cable 0,00 uj41 insertion loss: measuring antenna cable uj41 insertion loss: test antenna cable 0,00 uj40 insertion loss: measuring antenna attenuator uj40 insertion loss: test antenna attenuator 0,00 uj47 receiving device: absolute level uj16 range length 0,00 uj02 reflectivity of absorber material: measuring antenna to the test antenna 0,00 uj44 antenna: antenna factor of the measuring antenna uj45 antenna: gain of the test antenna 0,00 uj46 antenna: tuning of the measuring antenna uj46 antenna: tuning of the test antenna 0,00 uj22 position of the phase centre: measuring antenna uj06 mutual coupling: measuring antenna to its images in the absorbing material uj06 mutual coupling: test antenna to its images in the absorbing material 0,00 uj11 mutual coupling: measuring antenna to the test antenna 0,00 uj12 mutual coupling: interpolation of mutual coupling and mismatch loss correction factors 0,00 uj58 Salty man/salty-lite: human simulation uj59 Salty man/salty-lite: field enhancement and de-tuning of the EUT ui01 random uncertainty The standard uncertainties from table 13 should be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty (uc contribution from the EUT measurement) for the EUT measurement in dB. 4.2.1.6.1.3 Expanded uncertainty The combined uncertainty of the sensitivity measurement is the combination of the components outlined in clauses 4.2.1.6.1.1 and 4.2.1.6.1.2. The components to be combined are uc contribution from the Transform Factor and uc contribution from the EUT measurement. u u u c ccontribution fromtheTransformFactor ccontribution fromthe EUT measurement = + 2 2 = _ _ ,_ _ dB Using an expansion factor (coverage factor) of k = 1,96, the expanded measurement uncertainty is ±1,96 × uc = ±__,__ dB (see clause D.5.6.2). |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.1.6.2 Anechoic Chamber with a ground plane | A fully worked example illustrating the methodology to be used can be found in TR 102 273 [2], part 1, sub-part 2, clause 4. The receiver sensitivity measurement involves two stages of testing. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 96 4.2.1.6.2.1 Uncertainty contributions: Stage one: Determination of Transfer Factor The first stage (determining the Transform Factor) involves placing a measuring antenna as shown in figure 25 and determining the relationship between the signal generator output power level and the resulting field strength (the shaded areas in figure 25 represent components common to both stages of the test). Measuring antenna Measuring antenna cable 1 Receiving device Test antenna cable 2 ferrite beads Signal Attenuator 2 10 dB Test antenna Attenuator 1 10 dB generator Ground plane Figure 25: Stage 1: Transform Factor All the uncertainty components which contribute to this stage of the test are listed in table 13. Annex A should be consulted for the sources and/or magnitudes of the uncertainty contributions. Table 13: Contributions for the Transform Factor uj or i Description of uncertainty contributions dB uj36 mismatch: transmitting part uj37 mismatch: receiving part uj38 signal generator: absolute output level 0,00 uj39 signal generator: output level stability uj19 cable factor: measuring antenna cable uj19 cable factor: test antenna cable uj41 insertion loss: measuring antenna cable uj41 insertion loss: test antenna cable 0,00 uj40 insertion loss: measuring antenna attenuator uj40 insertion loss: test antenna attenuator 0,00 uj47 receiving device: absolute level uj16 range length uj02 reflectivity of absorbing material: measuring antenna to the test antenna uj44 antenna: antenna factor of the measuring antenna uj45 antenna: gain of the test antenna uj46 antenna: tuning of the measuring antenna uj46 antenna: tuning of the test antenna 0,00 uj22 position of the phase centre: measuring antenna uj06 mutual coupling: measuring antenna to its images in the absorbing material uj06 mutual coupling: test antenna to its images in the absorbing material uj14 mutual coupling: measuring antenna to its images in the ground plane uj14 mutual coupling: test antenna to its images in the ground plane uj11 mutual coupling: measuring antenna to the test antenna uj12 mutual coupling: interpolation of mutual coupling and mismatch loss correction factors ui01 random uncertainty The standard uncertainties from table 13 should be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty (uc contributions from the Transform Factor) for the Transform Factor in dB. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 97 4.2.1.6.2.2 Uncertainty contributions: Stage two: EUT measurement The second stage (the EUT measurement) is to determine the minimum signal generator output level which produces the required response from the EUT as shown in figure 26 (the shaded areas represent components common to both stages of the test). Test antenna cable 2 ferrite beads Test antenna Signal EUT Attenuator 2 10 dB generator Ground plane Figure 26: Stage 2: EUT measurement All the uncertainty components which contribute to this stage of the test are listed in table 14. Annex A should be consulted for the sources and/or magnitudes of the uncertainty contributions. Table 14: Contributions from the EUT measurement uj or i Description of uncertainty contributions dB uj36 mismatch: transmitting part uj38 signal generator: absolute output level 0,00 uj39 signal generator: output level stability uj19 cable factor: test antenna cable uj41 insertion loss: test antenna cable 0,00 uj40 insertion loss: test antenna attenuator 0,00 uj20 position of the phase centre: within the EUT volume uj21 positioning of the phase centre: within the EUT over the axis of rotation of the turntable uj52 EUT: modulation detection uj16 range length uj01 reflectivity of absorbing material: EUT to the test antenna uj45 antenna: gain of the test antenna 0,00 uj46 antenna: tuning of the test antenna 0,00 uj55 EUT: mutual coupling to the power leads uj08 mutual coupling: amplitude effect of the test antenna on the EUT uj04 mutual coupling: EUT to its images in the absorbing materials uj13 mutual coupling: EUT to its image in the ground plane uj06 mutual coupling: test antenna to its images in the absorbing material uj14 mutual coupling: test antenna to its image in the ground plane uj58 Salty man/salty-lite: human simulation uj59 Salty man/salty-lite: field enhancement and de-tuning of the EUT ui01 random uncertainty The standard uncertainties from table 14 should be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty (uc contribution from the EUT measurement) for the EUT measurement in dB. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 98 4.2.1.6.2.3 Expanded uncertainty The combined uncertainty of the sensitivity measurement is the combination of the components outlined in clauses 4.2.1.6.2.1 and 4.2.1.6.2.2. The components to be combined are uc contribution from the Transform Factor and uc contribution from the EUT measurement. u u u c ccontribution fromtheTransformFactor ccontribution fromthe EUT measurement = + 2 2 = _ _ ,_ _ dB Using an expansion factor (coverage factor) of k = 1,96, the expanded measurement uncertainty is ±1,96 × uc = ±__,__ dB (see clause D.5.6.2). |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.1.6.3 Open Area Test Site | A fully worked example illustrating the methodology to be used can be found in TR 102 273 [2], part 1, sub-part 2, clause 4. The receiver sensitivity measurement involves two stages of testing. 4.2.1.6.3.1 Uncertainty contributions: Stage one: Transfer Factor The first stage (determining the Transform Factor) involves placing a measuring antenna as shown in figure 27 and determining the relationship between the signal generator output power level and the resulting field strength (the shaded areas in figure 27 represent components common to both stages of the test). Measuring antenna Measuring antenna cable 1 Receiving device Test antenna cable 2 ferrite beads Signal Attenuator 2 10 dB Test antenna Attenuator 1 10 dB generator Ground plane Figure 27: Stage 1: Transform Factor All the uncertainty components which contribute to this stage of the test are listed in table 15. Annex A should be consulted for the sources and/or magnitudes of the uncertainty contributions. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 99 Table 15: Contributions for the Transform Factor uj or i Description of uncertainty contributions dB uj36 mismatch: transmitting part uj37 mismatch: receiving part uj38 signal generator: absolute output level 0,00 uj39 signal generator: output level stability uj19 cable factor: measuring antenna cable uj19 cable factor: test antenna cable uj41 insertion loss: measuring antenna cable uj41 insertion loss: test antenna cable 0,00 uj40 insertion loss: measuring antenna attenuator uj40 insertion loss: test antenna attenuator 0,00 uj47 receiving device: absolute level uj16 range length uj44 antenna: antenna factor of the measuring antenna uj45 antenna: gain of the test antenna uj46 antenna: tuning of the measuring antenna uj46 antenna: tuning of the test antenna 0,00 uj22 position of the phase centre: measuring antenna uj14 mutual coupling: measuring antenna to its images in the ground plane uj14 mutual coupling: test antenna to its images in the ground plane uj11 mutual coupling: measuring antenna to the test antenna uj12 mutual coupling: interpolation of mutual coupling and mismatch loss correction factors ui01 random uncertainty The standard uncertainties from table 15 should be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty (uc contributions from the Transform Factor) for the Transform Factor in dB. 4.2.1.6.3.2 Uncertainty contributions: Stage two: EUT measurement The second stage (the EUT measurement) is to determine the minimum signal generator output level which produces the required response from the EUT as shown in figure 28 (the shaded areas represent components common to both stages of the test). Test antenna cable 2 ferrite beads Test antenna Signal generator EUT Attenuator 2 10 dB Ground plane Figure 28: Stage 2: EUT measurement All the uncertainty components which contribute to this stage of the test are listed in table 16. Annex A should be consulted for the sources and/or magnitudes of the uncertainty contributions. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 100 Table 16: Uncertainty contributions from the EUT measurement uj or i Description of uncertainty contributions dB uj36 mismatch: transmitting part uj38 signal generator: absolute output level 0,00 uj39 signal generator: output level stability uj19 cable factor: test antenna cable uj41 insertion loss: test antenna cable 0,00 uj40 insertion loss: test antenna attenuator 0,00 uj20 position of the phase centre: within the EUT volume uj21 positioning of the phase centre: within the EUT over of the axis of rotation of the turntable uj52 EUT: modulation detection uj16 range length uj45 antenna: gain of the test antenna 0,00 uj46 antenna: tuning of the test antenna 0,00 uj55 EUT: mutual coupling to the power leads uj08 mutual coupling: amplitude effect of the test antenna on the EUT uj13 mutual coupling: EUT to its image in the ground plane uj14 mutual coupling: test antenna to its image in the ground plane uj58 Salty man/salty-lite: human simulation uj59 Salty man/salty-lite: field enhancement and de-tuning of the EUT ui01 random uncertainty The standard uncertainties from table 16 should be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty (uc contribution from the EUT measurement) for the EUT measurement in dB. 4.2.1.6.3.3 Expanded uncertainty The combined uncertainty of the sensitivity measurement is the combination of the components outlined in clauses 4.2.1.6.3.1 and 4.2.1.6.3.2. The components to be combined are uc contribution from the Transform Factor and uc contribution from the EUT measurement. u u u c ccontribution fromtheTransformFactor ccontribution fromthe EUT measurement = + 2 2 = _ _ ,_ _ dB Using an expansion factor (coverage factor) of k = 1,96, the expanded measurement uncertainty is ±1,96 × uc = ±__,__ dB (see clause D.5.6.2). |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.2 Co-channel rejection | |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.2.1 Test fixture | Tests in a test fixture differ to radiated tests on all other types of site in that there is only one stage to the test. All uncertainty contributions for the test can, therefore, be incorporated into one table and these are given in table 17. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 101 |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.2.1.1 Uncertainty contributions | All the uncertainty contributions for the test are listed in table 17. Table 17: Contributions from the measurement uj or i Description of uncertainty contributions dB uj60 Test Fixture: effect on the EUT uj61 Test Fixture: climatic facility effect on the EUT ui01 random uncertainty uj38 signal generator A: absolute output level uj38 signal generator B: absolute output level uj39 signal generator A: output level stability uj39 signal generator B: output level stability The standard uncertainties from table 17 should be given values according to annex A. They should then be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty (uc contributions from the measurement) for the EUT measurement in dB. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.2.1.2 Expanded uncertainty | Tests in a Test Fixture differ to radiated tests on all other types of site in that there is only one stage to the test. However, to calculate the measurement uncertainty, the Test Fixture measurement should be considered as stage two of a test in which stage one was on an accredited Free-Field Test Site. The combined standard uncertainty, uc, of the co-channel rejection measurement is therefore, simply the RSS combination of the value for uc contribution from the measurement derived above and the combined uncertainty of the Free-field Test Site uc contribution from the Free-Field Test Site. u u u c c contributions from the measurement c contributions from the free field test site = + = − dB 2 2 _ _ ,_ _ Using an expansion factor (coverage factor) of k = 1,96, the expanded measurement uncertainty is ±1,96 × uc = ±__,__ dB (see clause D.5.6.2). |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.3 Adjacent channel selectivity | |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.3.1 Test fixture | Tests in a test fixture differ to radiated tests on all other types of site in that there is only one stage to the test. All uncertainty contributions for the test can, therefore, be incorporated into one table and these are given in table 18. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.3.1.1 Uncertainty contributions | All the uncertainty contributions for the test are listed in table 18. Table 18: Contributions from the measurement uj or i Description of uncertainty contributions dB uj60 Test Fixture: effect on the EUT uj61 Test Fixture: climatic facility effect on the EUT ui01 random uncertainty uj38 signal generator A: absolute output level uj38 signal generator B: absolute output level uj39 signal generator A: output level stability uj39 signal generator B: output level stability ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 102 The standard uncertainties from table 18 should be given values according to annex A. They should then be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty (uc contributions from the measurement) for the EUT measurement in dB. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.3.1.2 Expanded uncertainty | Tests in a Test Fixture differ to radiated tests on all other types of site in that there is only one stage to the test. However, to calculate the measurement uncertainty, the Test Fixture measurement should be considered as stage two of a test in which stage one was on an accredited Free-Field Test Site. The combined standard uncertainty, uc, of the adjacent channel selectivity measurement is therefore, simply the RSS combination of the value for uc contributions from the measurement derived above and the combined uncertainty of the Free-field Test Site uc contribution from the Free-Field Test Site. u u u c c contributions from the measurement c contributions from the free field test site = + = − dB 2 2 _ _ ,_ _ Using an expansion factor (coverage factor) of k = 1,96, the expanded measurement uncertainty is ±1,96 × uc = ±__,__ dB (see clause D.5.6.2). |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.4 Intermodulation immunity | |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.4.1 Test fixture | Tests in a test fixture differ to radiated tests on all other types of site in that there is only one stage to the test. All uncertainty contributions for the test can, therefore, be incorporated into one table and these are given in table 19. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.4.1.1 Uncertainty contributions | All the uncertainty contributions for the test are listed in table 19. Table 19: Contributions from the measurement uj or i Description of uncertainty contributions dB uj60 Test Fixture: effect on the EUT uj61 Test Fixture: climatic facility effect on the EUT ui01 random uncertainty uj38 signal generator A: absolute output level uj39 signal generator A: output level stability uj38 signal generator B: absolute output level uj39 signal generator B: output level stability uj38 signal generator C: absolute output level uj39 signal generator C: output level stability The standard uncertainties from table 19 should be given values according to annex A. They should then be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty (uc contributions from the measurement) for the EUT measurement in dB. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.4.1.2 Expanded uncertainty | Tests in a Test Fixture differ to radiated tests on all other types of site in that there is only one stage to the test. However, to calculate the measurement uncertainty, the Test Fixture measurement should be considered as stage two of a test in which stage one was on an accredited Free-Field Test Site. The combined standard uncertainty, uc, of the intermodulation immunity measurement is therefore, simply the RSS combination of the value for uc contributions from the measurement derived above and the combined uncertainty of the Free-field Test Site uc contribution from the Free-Field Test Site. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 103 u u u c c contributions from the measurement c contributions from the free field test site = + = − dB 2 2 _ _ ,_ _ Using an expansion factor (coverage factor) of k = 1,96, the expanded measurement uncertainty is ±1,96 × uc = ±__,__ dB (see clause D.5.6.2). |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.5 Blocking immunity or degradation | |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.5.1 Test fixture | Tests in a test fixture differ to radiated tests on all other types of site in that there is only one stage to the test. All uncertainty contributions for the test can, therefore, be incorporated into one table and these are given in table 20. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.5.1.1 Uncertainty contributions | All the uncertainty contributions for the test are listed in table 20. Table 20: Contributions from the measurement uj or i Description of uncertainty contributions dB uj60 Test Fixture: climatic facility effect on the EUT uj61 Test Fixture: effect on the EUT ui01 random uncertainty uj38 signal generator A: absolute output level uj38 signal generator B: absolute output level uj39 signal generator A: output level stability uj39 signal generator B: output level stability The standard uncertainties from table 20 should be given values according to annex A. They should then be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty (uc contributions from the measurement) for the EUT measurement in dB. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.5.1.2 Expanded uncertainty | Tests in a Test Fixture differ to radiated tests on all other types of site in that there is only one stage to the test. However, to calculate the measurement uncertainty, the Test Fixture measurement should be considered as stage two of a test in which stage one was on an accredited Free-Field Test Site. The combined standard uncertainty, uc, of the blocking immunity (or desensitization) measurement is therefore, simply the RSS combination of the value for uc contributions from the measurement derived above and the combined uncertainty of the Free-field Test Site uc contribution from the Free-Field Test Site. u u u c c contributions from the measurement c contributions from the free field test site = + = − dB 2 2 _ _ ,_ _ Using an expansion factor (coverage factor) of k = 1,96, the expanded measurement uncertainty is ±1,96 × uc = ±__,__ dB (see clause D.5.6.2). |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.6 Spurious response immunity to radiated fields | |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.6.1 Anechoic chamber | |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.6.1.1 Uncertainty contributions: Stage 1: Transform Factor | If the first stage involved measuring the Transform Factor (as shown in figure 29) i.e. the relationship between the output level of the signal generator (dBm) and the resulting field strength (dBµV/m) in the vicinity of the turntable, then the shaded areas in figure 29 represent components common to both stages of the test. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 104 Test antenna cable 2 Test antenna Measuring antenna Signal combiner Wanted signal Attenuator 2 10 dB Load Attenuator 1 10 dB Receiving device Measuring antenna cable 1 Figure 29: Stage 1: Transform Factor All the uncertainty components which contribute to this stage of the test are listed in table 21. Annex A should be consulted for the sources and/or magnitudes of the uncertainty contributions. Table 21: Contributions for the Transform Factor uj or i Description of uncertainty contributions dB uj36 mismatch: transmitting part uj37 mismatch: receiving part uj38 signal generator: absolute output level uj39 signal generator: output level stability uj19 cable factor: measuring antenna cable uj19 cable factor: test antenna cable uj41 insertion loss: measuring antenna cable uj41 insertion loss: test antenna cable 0,00 uj40 insertion loss: measuring antenna attenuator uj40 insertion loss: test antenna attenuator 0,00 uj47 receiving device: absolute level uj16 range length 0,00 uj02 reflectivity of absorber material: measuring antenna to the test antenna 0,00 uj44 antenna: antenna factor of the measuring antenna uj45 antenna: gain of the test antenna 0,00 uj46 antenna: tuning of the measuring antenna uj46 antenna: tuning of the test antenna 0,00 uj22 position of the phase centre: measuring antenna uj06 mutual coupling: measuring antenna to its images in the absorbing material uj06 mutual coupling: test antenna to its images in the absorbing material 0,00 uj11 mutual coupling: measuring antenna to the test antenna 0,00 uj12 mutual coupling: interpolation of mutual coupling and mismatch loss correction factors 0,00 ui01 random uncertainty Alternatively, if the 3-axis probe was used, then figure 30 illustrates the test equipment set-up and table 89 lists the uncertainty components that contribute. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 105 Test antenna cable 2 Test antenna Signal combiner Wanted signal Attenuator 2 10 dB Load 3-axis probe Figure 30: Stage 1: 3-axis probe Table 22: Contributions for the 3-axis probe uj or I Description of uncertainty contributions dB uj36 mismatch: transmitting part 0,00 uj38 signal generator: absolute output level 0,00 uj39 signal generator: output level stability uj19 cable factor: test antenna cable uj41 insertion loss: test antenna cable 0,00 uj40 insertion loss: test antenna attenuator 0,00 uj16 range length uj45 antenna: gain of the test antenna 0,00 uj46 antenna: tuning of the test antenna 0,00 uj06 mutual coupling: test antenna to its images in the absorbing material 0,00 uj12 mutual coupling: interpolation of mutual coupling and mismatch loss correction factors 0,00 uj28 field strength measurement as determined by the 3-axis probe ui01 random uncertainty The standard uncertainties from table 21 or table 22 should be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty (uc contributions from the Transform Factor) for the Transform Factor in dB. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.6.1.2 Uncertainty contributions: Stage 2: EUT measurement | In this stage, the wanted signal is set to the level specified in the standard using either the Transform Factor of the 3-axis probe. The unwanted signal is then switched on and the level adjusted until the level of the unwanted signal, as measured on the 3-axis probe, is at the wanted signal level plus the spurious response rejection ratio required. The schematic of the equipment set-up is shown in figure 31. All the uncertainty components that contribute to this stage of the test are listed in table 23. Annex A should be consulted for the sources and/or magnitudes of the uncertainty contributions. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 106 Test antenna cable 2 Test antenna Measuring antenna EUT 3-axis probe Signal combiner Unwanted signal Wanted signal Attenuator 2 10 dB Figure 31: Stage 2: EUT measurement Table 23: Contributions from the EUT measurement uj or I Description of uncertainty contributions dB uj20 position of the phase centre: within the EUT volume uj52 EUT: modulation detection uj28 field strength measurement as determined by the 3-axis probe (unwanted signal measurement) ui01 random uncertainty The standard uncertainties from table 23 should be combined by RSS in accordance with TR 102 273 [2], part 1, sub-part 1, clause 5. This gives the combined standard uncertainty (uc contribution from the EUT measurement) for the EUT measurement in dB. |
9183669bb4288f694ed0630f17da78b6 | 100 028-2 | 4.2.6.1.3 Expanded uncertainty | The combined uncertainty of the spurious response immunity measurement is the combination of the components outlined in clauses 4.2.6.1.1 and 4.2.6.1.2. The components to be combined are uc contribution from the Transform Factor and uc contribution from the EUT measurement. u u u c ccontribution fromtheTransform Factor ccontribution fromthe EUT measurement = + 2 2 = _ _,_ _ dB Using an expansion factor (coverage factor) of k = 1,96, the expanded measurement uncertainty is ±1,96 × uc = ±__,__ dB (see clause D.5.6.2). ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 107 Annex A: Uncertainty contributions This annex contains a list of the uncertainties identified as being involved in radiated tests and gives details on how their magnitudes should be derived. Numerical and alphabetical lists of the uncertainties are given in tables A.20 and A.21. A radiated test, whether a verification procedure or the measurement of a particular parameter, consists of two stages. For a verification procedure the first stage is to set a reference level followed by the second stage which involves a measurement of the path loss between two antennas. For EUT testing, the first stage is to measure the EUT followed by the second stage which involves comparing the result to a known standard or reference. As a result of this methodology there are measurement uncertainty contributions that are common to both stages of any test, some of which cancel themselves out, others are included once whilst yet others have to be included twice. NOTE: For the measurement of some EUT receiver parameters the stages are reversed. Converting data: In the evaluation of any particular contribution it may be necessary to convert given data (e.g. from a manufacturer's information) into standard uncertainty. The following will aid any conversions that may be necessary. Mismatch uncertainties have 'U' shaped distributions. If the limits are ±a the standard uncertainty is: a/√2. Systematic uncertainties e.g. the uncertainty associated with cable loss are, unless the actual distribution is known, assumed to have rectangular distributions. If the limits are ±a the standard uncertainty is: a/√3. The rectangular distribution is a reasonable default model to choose in the absence of any other information. For conversion of % to dB, table A.1 should be used (for more information on the derivation of the table see TR 102 273 [2], part 1, sub-part 1, clause 5). Table A.1: Standard uncertainty conversion factors Converting from standard uncertainties in …: Conversion factor multiply by: To standard uncertainties in …: dB 11,5 voltage % dB 23,0 power % power % 0,0435 dB power % 0,5 voltage % voltage % 2,0 power % voltage % 0,0869 dB Terminology: In this annex the following phases should be interpreted as follows: - "Free Field Test Sites": are Anechoic Chambers, Anechoic Chambers with ground planes and Open Area Test Sites; - "Stripline": refers to the CENELEC EN 55020 [4] design of two plate open Stripline; - "Verification": refers to the measurement in which the test site is compared to its theoretical model; - "Test methods": refers to all radiated tests apart from the verification procedure; - "Transmitting" and "receiving" antennas: are used in the verification procedure only; all other references to antennas (i.e. substitution, measuring and test) are for test methods. REFLECTIVITY Background: The absorber panels in Anechoic Chambers (both with and without ground planes) reflect signal levels which can interfere with the required field distribution. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 108 uj01 Reflectivity of absorbing material: EUT to the test antenna This uncertainty only contributes to test methods on Free Field Test Sites that incorporate anechoic materials. It is the estimated uncertainty due to reflections from the absorbing material. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: If the test is part of a substitution measurement the standard uncertainty is 0,00 dB, otherwise the value from table A.2 should be used. Table A.2: Uncertainty contribution: Reflectivity of absorbing material: EUT to the test antenna Reflectivity of the absorbing material Standard uncertainty of the contribution reflectivity <10 dB 4,76 dB 10 dB ≤ reflectivity < 15 dB 3,92 dB 15 dB ≤ reflectivity < 20 dB 2,56 dB 20 dB ≤ reflectivity < 30 dB 1,24 dB reflectivity ≥ 30 dB 0,74 dB How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj02 Reflectivity of absorbing material: substitution or measuring antenna to the test antenna This uncertainty only contributes to test methods on Free Field Test Sites that incorporate anechoic materials. It is the estimated uncertainty due to reflections from the absorbing material. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: In a substitution type measurement the reflectivity of the absorber material tends to be nullified by the substitution methodology. However, there will always be some differences in the radiation patterns of the EUT and the substitution or measuring antenna and hence the standard uncertainty to allow for this should be taken as 0,50 dB. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj03 Reflectivity of absorbing material: transmitting antenna to the receiving antenna This uncertainty only contributes to the verification procedures on Free Field Test Sites that incorporate anechoic materials. It is the estimated uncertainty due to reflections from the absorbing material. How to evaluate for Free Field Test Sites • Verification: The relevant value for this contribution should be taken from table A.3. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 109 Table A.3: Uncertainty contribution: Reflectivity of absorbing material: transmitting antenna to the receiving antenna Reflectivity of the absorbing material Standard uncertainty of the contribution reflectivity <10 dB 4,76 dB 10 dB ≤ reflectivity < 15 dB 3,92 dB 15 dB ≤ reflectivity < 20 dB 2,56 dB 20 dB ≤ reflectivity < 30 dB 1,24 dB reflectivity ≥ 30 dB 0,74 dB • Test methods: Not applicable. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. MUTUAL COUPLING Background: Mutual coupling is the mechanism which produces changes in the electrical behaviour of an EUT or antenna when placed close to a conducting surface, another antenna, etc. These mechanisms are illustrated in figure A.1. The effects can include de-tuning, gain variations, changes to the radiation pattern and input impedance, etc. EUT Images Images Transmitting dipole Figure A.1: Mutual coupling (Anechoic Chamber illustrated) uj04 Mutual coupling: EUT to its images in the absorbing material This uncertainty contributes to test methods and verification procedures on Free Field Test Sites that incorporate anechoic material. It is the uncertainty which results from the degree of imaging in the absorber/shield of the chamber and the resulting effect on the input impedance and/or gain of the integral antenna. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: The standard uncertainty is 0,50 dB. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 110 How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj05 Mutual coupling: de-tuning effect of the absorbing material on the EUT This uncertainty only contributes to the test methods on Free Field Test Sites that incorporate anechoic materials. It is the uncertainty of any de-tuning effect due to the return loss of the absorbers. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: This value will be 0,00 Hz provided the absorbing panels are more than 1 metre away from the EUT and the return loss of the panels is above 6 dB (testing should not take place for spacings of less than 1 metre). For return losses below 6 dB, the value should be taken as 5 Hz standard uncertainty. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj06 Mutual coupling: substitution, measuring or test antenna to its images in the absorbing material This uncertainty only contributes to test methods on Free Field Test Sites that incorporate anechoic material. It is the uncertainty which results from the degree of imaging in the absorber/shield of the chamber and the resulting effect on the antenna's input impedance and/or gain. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: - for the test antenna only, if it is at the same height for both stages one and two of the test method, then for any absorber depth the uncertainty is 0,00 dB, otherwise the standard uncertainty is 0,50 dB; - for substitution or measuring antennas the standard uncertainty is 0,50 dB. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj07 Mutual coupling: transmitting or receiving antenna to its images in the absorbing material This uncertainty only contributes to verification procedures on Free Field Test Sites that incorporate anechoic material. It is the uncertainty which results from the degree of imaging in the absorber/shield of the chamber and the resulting effect on the antenna's input impedance and/or gain. How to evaluate for Free Field Test Sites • Verification: - for the transmitting antenna the standard uncertainty is 0,50 dB; - for the receiving antenna the standard uncertainty is 0,50 dB. • Test methods: Not applicable. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 111 How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj08 Mutual coupling: amplitude effect of the test antenna on the EUT This uncertainty only contributes to test methods on Free Field Test Sites. It is the uncertainty which results from the interaction (impedance changes, etc.) between the EUT and the test antenna when placed close together. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: This is the uncertainty which results from the interaction (impedance changes, etc.) between the EUT and the test antenna when placed close together. The standard uncertainty should be taken from table A.4. Table A.4: Uncertainty contribution: Mutual coupling: amplitude effect of the test antenna on the EUT Range length Standard uncertainty of the contribution 0,62√((d1 + d2)3/λ) ≤ range length < 2(d1 + d2)2/λ 0,50 dB range length ≥ 2(d1 + d2)2/λ 0,00 dB NOTE: d1 and d2 are the maximum dimensions of the EUT and the test antenna. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj09 Mutual coupling: de-tuning effect of the test antenna on the EUT This uncertainty only contributes to test methods on Free Field Test Sites that incorporate anechoic materials. It is the uncertainty of any de-tuning effect due to mutual coupling between the EUT and the test antenna. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: This value will be 0,00 Hz provided the spacing between the test antenna and EUT is greater than (d1 + d2)2/4λ. For lesser spacing, the value should be taken as 5 Hz standard uncertainty. NOTE 1: d1 and d2 are the maximum dimensions of the EUT and the test antenna. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj10 Mutual coupling: transmitting antenna to receiving antenna This uncertainty only contributes to verification procedures on Free Field Test Sites. It is the uncertainty which results from the change in coupled signal level between the transmitting and receiving antenna when placed close together. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 112 How to evaluate for Free Field Test Sites • Verification: For ANSI dipoles the value of this uncertainty is 0,00 dB since it is included, where significant, in the mutual coupling and mismatch loss correction factors. For non-ANSI dipoles the standard uncertainty can be taken from table A.5. Table A.5: Uncertainty contribution: Mutual coupling: transmitting antenna to receiving antenna Frequency Standard uncertainty of the contribution (3 m range) Standard uncertainty of the contribution (10 m range) 30 MHz ≤ frequency < 80 MHz 1,73 dB 0,60 dB 80 MHz ≤ frequency < 180 MHz 0,6 dB 0,00 dB frequency ≥ 180 MHz 0,00 dB 0,00 dB • Test methods: Not applicable. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj11 Mutual coupling: substitution or measuring antenna to the test antenna This uncertainty only contributes to test methods on Free Field Test Sites. It is the uncertainty which results from the change in coupled signal level between the substitution or measuring and test antenna when placed close together. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: For ANSI dipoles the value of this uncertainty is 0,00 dB since it is included, where significant, in the mutual coupling and mismatch loss correction factors. For non-ANSI dipoles the standard uncertainty can be taken from table A.6. Table A.6: Uncertainty contribution: Mutual coupling: substitution or measuring antenna to the test antenna Frequency Standard uncertainty of the contribution (3 m range) Standard uncertainty of the contribution (10 m range) 30 MHz ≤ frequency < 80 MHz 1,73 dB 0,60 dB 80 MHz ≤ frequency < 180 MHz 0,6 dB 0,00 dB frequency ≥ 180 MHz 0,00 dB 0,00 dB How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj12 Mutual coupling: interpolation of mutual coupling and mismatch loss correction factors This uncertainty contributes to test methods and verification procedures on Free Field Test Sites. It is the uncertainty which results from the interpolation between two values in the mutual coupling and mismatch loss correction factor table (given in the relevant test methods and verification procedures). ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 113 How to evaluate for Free Field Test Sites • Verification: The standard uncertainty can be obtained from table A.7. Table A.7: Uncertainty contribution: Mutual coupling: interpolation of mutual coupling and mismatch loss correction factors Frequency (MHz) Standard uncertainty of the contribution for a spot frequency given in the table 0,00 dB 30 MHz ≤ frequency < 80 MHz 0,58 dB 80 MHz ≤ frequency < 180 MHz 0,17 dB frequency ≥ 180 MHz 0,00 dB • Test methods: The standard uncertainty can be obtained from table A.7. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj13 Mutual coupling: EUT to its image in the ground plane This uncertainty contributes to test methods on Free Field Test Sites that incorporate a ground plane. It is the uncertainty which results from the change in gain and/or sensitivity of an EUT when placed close to a ground plane. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: The standard uncertainty can be obtained from table A.8. Table A.8: Uncertainty contribution: Mutual coupling: EUT to its image in the ground plane Spacing between the EUT and the ground plane Standard uncertainty of the contribution For a vertically polarized EUT spacing ≤ 1,25 λ 0,15 dB spacing > 1,25 λ 0,06 dB For a horizontally polarized EUT spacing < λ/2 1,15 dB λ/2 ≤ spacing < 3λ/2 0,58 dB 3λ/2 ≤ spacing < 3λ 0,29 dB spacing ≥ 3λ 0,15 dB How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj14 Mutual coupling: substitution, measuring or test antenna to its image in the ground plane This uncertainty only contributes to test methods on Free Field Test Sites that incorporate a ground plane. It is the uncertainty which results from the change in input impedance and/or gain of the substitution, measuring or test antenna when placed close to a ground plane. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 114 How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: The standard uncertainty can be obtained from table A.9. Table A.9: Uncertainty contribution: Mutual coupling: substitution, measuring or test antenna to its image in the ground plane Spacing between the antenna and the ground plane Standard uncertainty of the contribution For a vertically polarized antenna spacing ≤ 1,25 λ 0,15 dB spacing > 1,25 λ 0,06 dB For a horizontally polarized antenna spacing < λ/2 1,15 dB λ/2 ≤ spacing < 3λ/2 0,58 dB 3λ/2 ≤ spacing < 3λ 0,29 dB spacing ≥ 3λ 0,15 dB How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj15 Mutual coupling: transmitting or receiving antenna to its image in the ground plane This uncertainty only contributes to verification procedures on Free Field Test Sites that incorporate a ground plane. It is the uncertainty which results from the change in gain of the transmitting or receiving antenna when placed close to a ground plane. How to evaluate for Free Field Test Sites • Verification: For ANSI dipoles the value of this uncertainty is 0,00 dB as it is included, where significant, in the mutual coupling and mismatch loss correction factors. For other dipoles the value can be obtained from table A.10. Table A.10: Uncertainty contribution: Mutual coupling: transmitting or receiving antenna to its image in the ground plane Spacing between the antenna and the ground plane Standard uncertainty of the contribution For a vertically polarized antenna spacing ≤ 1,25 λ 0,15 dB spacing > 1,25 λ 0,06 dB For a horizontally polarized antenna spacing < λ/2 1,15 dB λ/2 ≤ spacing < 3λ/2 0,58 dB 3λ/2 ≤ spacing < 3λ 0,29 dB spacing ≥ 3λ 0,15 dB • Test methods: Not applicable. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 115 RANGE LENGTH Background: The range length over which any radiated test is carried out should always be adequate to enable far field testing. It may also be specified in the relevant deliverable NOTE 2: Range length is defined as the horizontal distance between the phase centres of the EUT and the test antenna. Over a reflective ground plane where a height scan is involved to peak the received signal the distance over which a measurement is performed is not always equal to the range length. Figure A.2 illustrates the difference between range length and measurement distance. EUT Range length Measurement distance Figure A.2: Range length and measurement distance It is important to distinguish clearly between these two terms. uj16 Range length This uncertainty contributes to test methods and verification procedures on Free Field Test Sites. It is the uncertainty associated with the curvature of the phase front resulting from inadequate range length between an EUT and antenna or, alternatively, between two antennas i.e. it should always be equal to or greater than 2(d1 + d2)2/λ. NOTE 3: d1 and d2 are the maximum dimensions of the antennas. How to evaluate for Free Field Test Sites • Verification: If ANSI dipoles are used the value is 0,00 dB, since it is included in the mutual coupling and mismatch loss correction factors, otherwise the value should be taken from table A.11. Table A.11: Uncertainty contribution: Range length (verification) Range length (i.e. the horizontal distance between phase centres) Standard uncertainty of the contribution (d1 + d2)2/4λ ≤ range length < (d1 + d2)2/2λ 1,26 dB (d1 + d2)2/2λ ≤ range length < (d1 + d2)2/λ 0,30 dB (d1 + d2)2/λ ≤ range length < 2(d1 + d2)2/λ 0,10 dB range length ≥ 2(d1 + d2)2/λ 0,00 dB NOTE: d1 and d2 are the maximum dimensions of the antennas. Test methods • For the EUT to test antenna stage the value should be taken from table A.12. For the substitution or measuring antenna to the test antenna stage: If ANSI dipoles are used the value is 0,00 dB, since it is included in the mutual coupling and mismatch loss correction factors, otherwise the value should be taken from table A.12. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 116 Table A.12: Uncertainty contribution: Range length (test methods) Range length (i.e. the horizontal distance between phase centres) Standard uncertainty of the contribution (d1 + d2)2/4λ ≤ range length < (d1 + d2)2/2λ 1,26 dB (d1 + d2)2/2λ ≤ range length < (d1 + d2)2/λ 0,30 dB (d1 + d2)2/λ ≤ range length < 2(d1 + d2)2/λ 0,10 dB range length ≥ 2(d1 + d2)2/λ 0,00 dB NOTE: d1 and d2 are the maximum dimensions of the EUT and the test antenna used in one stage and are the maximum dimensions of the two antennas in the other stage. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. CORRECTIONS Background: In radiated tests the height of the test antenna is optimized in each stage of the test, often the heights for the two stages are different. This leads to different measuring distances and elevation angles and corrections should be applied to take account of these effects. uj17 Correction: off boresight angle in elevation plane This uncertainty only contributes to test methods on Free Field Test Sites that incorporate a ground plane. Where the height of the antenna on the mast differs between the two stages of a particular measurement, two different elevation angles are subtended between the turntable and the test antenna. A correction factor should be applied to compensate. Its magnitude should be calculated using figure A.7 according to the guidance given in the test method. This uncertainty contribution is the estimate of the accuracy of the calculated correction factor and it only applies when the test antenna has a directional radiation pattern in the elevation plane see figure A.3. NOTE 4: Figure A.7 applies to vertically polarized dipoles and bicones and to both polarizations of LPDAs. For horns, or any other type of antenna, figure A.7 is inappropriate and the test engineer should provide specific corrections. Boresight 0 dB -3 dB Antenna radiation pattern Off boresight angle typ. 39 0 Figure A.3: Off boresight correction ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 117 How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: For any antenna: - where the optimized height of the antenna on the mast is the same in the two stages of the test, this value is 0,00 dB; - for vertically polarized dipoles and bicones where the optimized height of the antenna on the mast is different in the two stages of the test, the standard uncertainty of the value is 0,10 dB; - for horizontally or vertically polarized LPDAs where the optimized height of the antenna on the mast is different in the two stages of the test, the standard uncertainty of the value is 0,50 dB; - for any other antenna, after application of a correction specific to that antenna, where the optimized height of the antenna on the mast is different in the two stages of the test, the standard uncertainty of the value is 0,50 dB. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj18 Correction: measurement distance This uncertainty only contributes to test methods on Free Field Test Sites that incorporate a ground plane. Where the height of the antenna on the mast differs between the two stages of a particular measurement, two different path losses result from the different measurement distances involved. A correction factor (see figure A.8) should be applied to compensate. Its magnitude should be calculated according to the guidance given in the test method. This uncertainty contribution is the estimate of the accuracy of the calculated correction factor. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: - where the optimized height of the antenna on the mast is the same in the two stages of the test, this value is 0,00 dB; - where the optimized height of the antenna on the mast is different in the two stages of the test, the standard uncertainty of the value is 0,10 dB. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. RADIO FREQUENCY CABLES Background: There are radiating mechanisms by which RF cables can introduce uncertainties into radiated measurements: - leakage; - acting as a parasitic element to an antenna; - introducing common mode current. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 118 Leakage allows electromagnetic coupling into the cables. Because the electromagnetic wave contains both electric and magnetic fields, mixed coupling occurs and the voltage induced is very dependant on the orientation, with respect to the cable, of the electric and magnetic fields. This coupling can have different effects depending on the length of the cable and where it is in the system. Cables are usually the longest part of the test equipment configuration and as such, leakage can make them act as efficient receiving or transmitting antennas that, as a result, will contribute significantly to the uncertainty of the measurement. The parasitic effect of the cable can potentially be the most significant of the three effects and can cause major changes to the antenna's radiation pattern, gain and input impedance. The common mode current problem has similar effects on an antenna's performance. All three effects can be largely eliminated by routing and loading the cables with ferrite beads as detailed in the test methods. An RF cable for which no precautions have been taken to prevent these effects can, simply by being repositioned, cause different results to be obtained. uj19 Cable factor This uncertainty contributes to test methods and verification procedures. Cable factor is defined as the total effect of the RF cable's influence on the measuring system. How to evaluate for Free Field Test Sites • Verification: In the direct attenuation stage of the procedure (a conducted measurement) all fields are enclosed and hence the contribution is assumed to be zero. However in the radiated attenuation stage, the standard uncertainty for each cable is 0,5 dB provided the precautions detailed in the procedure have been observed. If the precautions have not been observed the contributions have a standard uncertainty of 4,0 dB (justification for these values is given in annex E); • Test methods: The standard uncertainty for each cable is 0,5 dB provided the precautions detailed in the method have been observed. If the precautions have not been observed the contributions have a standard uncertainty of 4,0 dB (justification for these values is given in annex E). Exceptionally, where a cable and antenna combination has not been repositioned between the two stages (as in the case of the test antenna in an Anechoic Chamber) and the precautions detailed in the procedure have been observed, the contribution is assumed to be 0,00 dB. If the combination has not been repositioned but the precautions have not been observed the contribution is 0,5 dB. NOTE 5: Repositioning means any change in the positions of either the cable or the antenna in stage two of the measurement relative to stage one e.g. height optimization over a ground plane. How to evaluate for Striplines • Verification: In the direct attenuation stage of the procedure (a conducted measurement) all fields are enclosed and hence the contribution is assumed to be zero. However in the radiated attenuation stage the standard uncertainty for each cable is 0,5 dB provided the precautions detailed in the procedure have been observed. If the precautions have not been observed the contributions have a standard uncertainty of 4,0 dB (justification for these values is given in annex E). • Test methods: The standard uncertainty for each cable is 0,5 dB provided that the precautions detailed in the method have been observed. If the precautions have not been observed the contribution has a standard uncertainty of 4,0 dB (justification for these values is given in annex E). PHASE CENTRE POSITIONING Background: The phase centre of an EUT or antenna is the point from which the device is considered to radiate. If the device is rotated about this point the phase of the signal, as seen by a fixed antenna, does not change. It is therefore critical to (a) Identify the phase centre of an EUT or antenna and (b) to position it correctly on the test site. uj20 Position of the phase centre: within the EUT volume This uncertainty only contributes to test methods. It is the accuracy with which the phase centre is identified within the EUT. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 119 How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Only applicable in the stage in which the EUT is measured. If the precise phase centre is unknown, the uncertainty contribution should be calculated from: % 100 × ± length range the twice device the of imension d ximum the ma As the phase centre can be anywhere inside the EUT this uncertainty is assumed to be rectangularly distributed (see TR 102 273 [2], part 1, sub-part 1, clause 5.1.2). The standard uncertainty can therefore be calculated and converted to the logarithmic form (see TR 102 273 [2], part 1, sub-part 1, clause 5). How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj21 Positioning of the phase centre: within the EUT over the axis of rotation of the turntable This uncertainty only contributes to test methods. It is the accuracy with which the identified phase centre of the EUT is aligned with the axis of rotation of the turntable. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Only applicable in the stage in which the EUT is measured. The maximum value should be calculated from: % 100 × ± length range tation axis of ro from the ted offset the estima As this error source can be anywhere between these limits this uncertainty is assumed to be rectangularly distributed (see TR 102 273 [2], part 1, sub-part 1, clause 5.1.2). The standard uncertainty can therefore be calculated and converted to the logarithmic form (see TR 102 273 [2], part 1, sub-part 1, clause 5). How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj22 Position of the phase centre: measuring, substitution, receiving, transmitting or test antenna This uncertainty contributes to test methods and verification procedures on Free Field Test Sites. It is the uncertainty with which the phase centre can be positioned. How to evaluate for Free Field Test Sites • Verification: - for the transmitting antenna the maximum value should be calculated from: % 100 × ± length range tation axis of ro from the ted offset the estima ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 120 - for the receiving antenna in an Anechoic Chamber the maximum value should be calculated from: % 100 × ± length range set be can length range the which with ty uncertain the - for the receiving antenna over a ground plane the maximum value should be calculated from: % 100 × ± length range ast p of the m of the to m vertical ection fro mated defl ximum esti the ma As this error source can be anywhere between these limits this uncertainty is assumed to be rectangularly distributed (see TR 102 273 [2], part 1, sub-part 1, clause 5.1.2). The standard uncertainty can therefore be calculated and converted to the logarithmic form (see TR 102 273 [2], part 1, sub-part 1, clause 5). • Test methods: - for the measuring and substitution antennas the maximum value should be calculated from: % 100 × ± length range tation axis of ro from the ted offset the estima - for the test antenna in an Anechoic Chamber the maximum value should be calculated from: % 100 × ± length range set be can length range the which with ty uncertain the - for the test antenna over a ground plane the maximum value should be calculated from: % 100 × ± length range ast p of the m of the to m vertical ection fro mated defl ximum esti the ma As this error source can be anywhere between these limits this uncertainty is assumed to be rectangularly distributed (see TR 102 273 [2], part 1, sub-part 1, clause 5.1.2). The standard uncertainty can therefore be calculated and converted to the logarithmic form (see TR 102 273 [2], part 1, sub-part 1, clause 5). How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj23 Position of the phase centre: LPDA This uncertainty contributes to test methods and verification procedures on Free Field Test Sites. It is the uncertainty associated with the changing position of the phase centre with frequency of the LPDA. How to evaluate for Free Field Test Sites • Verification: The maximum value should be calculated from: % 100 × ± length range the twice device the of imension d ximum the ma As this error source can be anywhere between these limits this uncertainty is assumed to be rectangularly distributed (see TR 102 273 [2], part 1, sub-part 1, clause 5.1.2). The standard uncertainty can therefore be calculated and converted to the logarithmic form (see TR 102 273 [2], part 1, sub-part 1, clause 5). ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 121 • Test methods: For the test antenna the contribution is 0,00 dB. For the substitution or measuring LPDA the maximum value should be calculated from: % 100 × ± length range the twice LPDA the of the length As this error source can be anywhere between these limits this uncertainty is assumed to be rectangularly distributed (see TR 102 273 [2], part 1, sub-part 1, clause 5.1.2). The standard uncertainty can therefore be calculated and converted to the logarithmic form (see TR 102 273 [2], part 1, sub-part 1, clause 5). How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. STRIPLINE Background: The Stripline is an alternative test site to a Free Field Test Site. It is essentially a large open transmission line comprising two flat metal plates between which a TEM wave is generated. The resulting field is assumed to exhibit a planar distribution of amplitude and phase. uj24 Stripline: mutual coupling of the EUT to its images in the plates This uncertainty only contributes to Stripline test methods. It is the uncertainty which results from the imaging of the EUT in the plates of the Stripline. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Not applicable. How to evaluate for Striplines • Verification: Not applicable. • Test methods: The magnitude is dependent on the size of the EUT (which is assumed to be placed midway between the plates). The value of the uncertainty contribution can be obtained from table A.13. Table A.13: Uncertainty contribution: Stripline: mutual coupling of the EUT to its images in the plates Size of the EUT relative to the plate separation Standard uncertainty of the contribution size/separation < 33 % 1,15 dB 33 % ≤ size/separation < 50 % 1,73 dB 50 % ≤ size/separation < 70 % 2,89 dB 70 % ≤ size/separation ≤ 87,5 % (max.) 5,77 dB uj25 Stripline: mutual coupling of the 3-axis probe to its image in the plates This uncertainty only contributes to Stripline test methods. It is the uncertainty which results from the imaging of the 3-axis probe in the plates of the Stripline. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Not applicable. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 122 How to evaluate for Striplines • Verification: Not applicable. • Test methods: The standard uncertainty is 0,29 dB. uj26 Stripline: characteristic impedance This uncertainty only contributes to Stripline test methods. This uncertainty contribution results from the difference between the free space wave impedance (377 Ω) for which the EUT has been developed and that for the Stripline (150 Ω). How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Not applicable. How to evaluate for Striplines • Verification: Not applicable. • Test methods: The standard uncertainty is 0,58 dB. uj27 Stripline: non-planar nature of the field distribution This uncertainty only contributes to Stripline test methods. It is the uncertainty which results from the non-planar nature of the field distribution within the Stripline. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Not applicable. How to evaluate for Striplines • Verification: Not applicable. • Test methods: The standard uncertainty is 0,29 dB. uj28 Stripline: field strength measurement as determined by the 3-axis probe This uncertainty only contributes to Stripline test methods. It is the uncertainty which results from using a 3-axis probe to measure the electric field strength within the Stripline. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Not applicable. How to evaluate for Striplines • Verification: Not applicable. • Test methods: The measurement uncertainty of the 3-axis probe is taken from manufacturer's data sheet and converted to a standard uncertainty if necessary. uj29 Stripline: Transform Factor This uncertainty only contributes to Stripline test methods. It is the uncertainty with which the Transform Factor (i.e. the relationship between the input voltage to the Stripline and the resulting electric field strength between the plates) is determined. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 123 How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Not applicable. How to evaluate for Striplines • Verification: Not applicable. • Test methods: If the verification procedure results are used, the standard uncertainty is the combined standard uncertainty of the verification procedure. uj30 Stripline: interpolation of values for the Transform Factor This uncertainty only contributes to Stripline test methods. It is the uncertainty associated with interpolating between two adjacent Transform Factor for the Stripline. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Not applicable. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Where the frequency of test corresponds to a set frequency in the verification procedure, this contribution to the combined uncertainty is 0,00 dB. For any other frequency, the value of the standard uncertainty is taken as 0,29 dB. uj31 Stripline: antenna factor of the monopole This uncertainty only contributes to Stripline test methods and the verification procedure. It is the uncertainty with which the antenna factor/gain of the monopole is known. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Not applicable. How to evaluate for Striplines • Verification: Not applicable. • Test methods: The standard uncertainty is 1,15 dB. uj32 Stripline: correction factor for the size of the EUT This uncertainty only contributes to Stripline test methods. It is the uncertainty due to the EUT being mounted in the Stripline where the height of the EUT is significant in the E-plane compared to the plate separation. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Not applicable. How to evaluate for Striplines • Verification: Not applicable. • Test methods: For EUT mounted centrally in the Stripline, values can be obtained from table A.14. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 124 Table A.14: Uncertainty contribution: Stripline: correction factor for the size of the EUT Height of the EUT (in the E-plane) is: Standard uncertainty of the contribution height < 0,2 m 0,30 dB 0,2 m ≤ height < 0,4 m 0,60 dB 0,4 m ≤ height ≤ 0,7 m 1,20 dB uj33 Stripline: influence of site effects This uncertainty only contributes to Stripline test methods. It is the uncertainty which results from the possible interaction between the fields of the Stripline and objects in its immediate environment. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Not applicable. How to evaluate for Striplines • Verification: Not applicable. • Test methods: For any method of field strength measurement, it is assumed that, provided none of the absorbing panels placed around the Stripline or the Stripline itself are moved either between the verification procedure and the test or between the measurement on the EUT and the field measurement parts of the test (for Monopole or 3-axis probe). The standard uncertainty of the contribution is 0,00 dB. If, however, the arrangement has been changed, the standard uncertainty of the contribution is 3,00 dB. AMBIENT SIGNALS Background: Ambient signals are localized sources of radiated transmissions that can introduce uncertainty into the results of a test made on an Open Area Test Site and in unshielded Anechoic Chambers and Striplines. uj34 Ambient effect This uncertainty contributes to test methods and verification procedures on Free Field Test Sites and in Striplines. It is the uncertainty caused by local ambient signals raising the noise floor of the receiver at the frequency of test. How to evaluate for Free Field Test Sites • Verification: The values of the standard uncertainties should be taken from table A.15. Table A.15: Uncertainty contribution: Ambient effect Receiving device noise floor (with signal generator OFF) is within: Standard uncertainty of the contribution 3 dB of measurement 1,57 dB 3 dB - 6 dB of measurement 0,80 dB 6 dB - 10 dB of measurement 0,30 dB 10 dB - 20 dB of measurement 0,10 dB 20 dB or more of the measurement 0,00 dB • Test methods: The values of the standard uncertainties should be taken from table A.15. How to evaluate for Striplines • Verification: The values of the standard uncertainties should be taken from table A.15. • Test methods: The values of the standard uncertainties should be taken from table A.15. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 125 MISMATCH Background: When two or more items of RF test equipment are connected together a degree of mismatch occurs. Associated with this mismatch there is an uncertainty component as the precise interactions are unknown. Mismatch uncertainties are calculated in the present document using S-parameters and full details of the method are given in annex D. For our purposes the measurement set-up consists of components connected in series, i.e. cables, attenuators, antennas, etc. and for each individual component in this chain, the attenuation and VSWRs must be known or assumed. The exact values of the VSWRs (which in RF circuits are complex values) are usually unknown at the precise frequency of test although worst case values over an extended frequency band will be known. It is these worst case values which should be used in the calculations. This approach will generally cause the calculated mismatch uncertainties to be worse than they actually are. uj35 Mismatch: direct attenuation measurement This uncertainty only contributes to verification procedures. It results from the interaction of the VSWRs of the components in the direct attenuation measurement. The direct attenuation measurement refers to the arrangement in which the signal generator is directly connected to the receiving device (via cables, attenuators and an adapter) to obtain a reference signal level. See figure A.4. Due to load variations (antennas replacing the adapter in the second stage of the procedure) contributions are not identical in the two stages of the verification procedure. Direct attenuation measurement "In line" adapter Attenuator 2 10 dB Receiving device Attenuator 1 10 dB Signal generator cable 1 cable 2 ferrite beads ferrite beads Figure A.4: Equipment set-up for the direct attenuation measurement How to evaluate for Free Field Test Sites • Verification: The magnitude of the uncertainty contribution due to the mismatch in the direct attenuation measurement, is calculated from the approach described in annex D. • Test methods: N/A. How to evaluate for Striplines • Verification: The magnitude of the uncertainty contribution due to the mismatch in the direct attenuation measurement, is calculated from the approach described in annex D. • Test methods: N/A. uj36 Mismatch: transmitting part This uncertainty contributes to test methods and verification procedures. The transmitting part refers to the signal generator, cable, attenuator and antenna set-up shown in figure A.5. This equipment configuration is used for: - the transmitting part of a Free Field Test Site verification procedure; - the transmitting part of a Stripline verification procedure (where the antenna in the figure is replaced by the Stripline input); - the transmitting part of the substitution measurement in a transmitter test method; - the transmitting part when generating a field in a receiver test method. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 126 Transmitting part Antenna cable ferrite beads Attenuator 10 dB Signal generator Figure A.5: Equipment set-up for the transmitting part How to evaluate for Free Field Test Sites • Verification: The uncertainty contribution due to the mismatch in the transmitting part is calculated from the approach described in annex D. • Test methods: As for the verification. How to evaluate for Striplines • Verification: The uncertainty contribution due to the mismatch in the transmitting part is calculated from the approach described in annex D. • Test methods: As for the verification. uj37 Mismatch: receiving part This uncertainty contributes to test methods and verification procedures. The receiving part refers to the antenna, attenuator, cable and receiving device set-up shown in figure A.6. This equipment configuration is used for: - the receiving part of a Free Field Test Site verification procedure; - the receiving part of a Stripline verification procedure (where the antenna is a Monopole); - the receiving part of the substitution measurement in a transmitter test method; - the receiving part when measuring the field in a receiver test method. Receiving part Antenna cable ferrite beads Receiving device Attenuator 10 dB Figure A.6: Equipment set-up for the receiving part How to evaluate for Free Field Test Sites • Verification: The uncertainty contribution due to the mismatch in the receiving part is calculated from the approach described in annex D. • Test methods: As for the verification. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 127 How to evaluate for Striplines • Verification: The uncertainty contribution due to the mismatch in the receiving part is calculated from the approach described in annex D. • Test methods: As for the verification. SIGNAL GENERATOR Background: The signal generator is used as the transmitting source. There are two signal generator characteristics that contribute to the expanded uncertainty of a measurement: absolute level and level stability. uj38 Signal generator: absolute output level This uncertainty only contributes to test methods. It concerns the accuracy with which an absolute signal generator level can be set. How to evaluate for Free Field Test Sites • Verification: The standard uncertainty is 0,00 dB. • Test methods: The uncertainty contribution should be taken from the manufacturer's data sheet and converted into standard uncertainty if necessary. How to evaluate for Striplines • Verification: The standard uncertainty is 0,00 dB. • Test methods: - for cases where the field strength in a Stripline is determined from the results of the verification procedure, the uncertainty is taken from the manufacturer's data sheet and converted into standard uncertainty if necessary; - where an electric field strength measurement is made in the Stripline this contribution is assumed to be zero. uj39 Signal generator: output level stability This uncertainty contributes to test methods and verification procedures. It concerns the stability of the output level. In any test in which the contribution of the absolute level uncertainty of the signal generator contributes to the combined standard uncertainty of the test i.e. it does not cancel due to the methodology, the contribution from the output level stability is considered to have been included in the signal generator absolute output level, uj38. Conversely, for any level in which the absolute level uncertainty of the signal generator does not contribute to the combined standard uncertainty, the output level stability of the signal generator should be included. The standard uncertainty of the contribution due to the signal generator output level stability is designated throughout all parts of TR 102 273 [2] as uj39. Its value can be derived from manufacturers' data sheet. How to evaluate for Free Field Test Sites • Verification: The uncertainty contribution should be taken from the manufacturer's data sheet and converted into standard uncertainty if necessary. • Test methods: The standard uncertainty of the contribution due to the signal generator output level stability is taken as 0,00 dB as it is covered by the absolute level uncertainty. How to evaluate for Striplines • Verification: The uncertainty contribution should be taken from the manufacturer's data sheet and converted into standard uncertainty if necessary. • Test methods: The standard uncertainty of the contribution due to the signal generator output level stability is taken as 0,00 dB as it is covered by the absolute level uncertainty. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 128 INSERTION LOSSES Test equipment components such as attenuators, cables, adapters, etc. have insertion losses at a given frequency which act as systematic offsets. Knowing the value of the insertion losses allows the results to be corrected by the offsets. However, there are uncertainties associated with these insertion losses which are equivalent to the uncertainty of the loss measurements. uj40 insertion loss: attenuator This uncertainty only contributes to test methods. How to evaluate for Free Field Test Sites • Verification: This value is 0,00 dB. • Test methods: - for the attenuator associated with the test antenna this uncertainty contribution is common to both stage one and stage two of the measurement. Consequently, this uncertainty contribution is assumed to be 0,00 dB due to the methodology. - for the attenuator associated with the substitution or measuring antenna this uncertainty contribution is taken either from the manufacturer's data sheet or from the combined standard uncertainty figure of its measurement. How to evaluate for Striplines • Verification: The value is 0,00 dB. • Test methods: - where the field strength in a Stripline is determined from the results of the verification procedure, for the attenuator associated with the Stripline input this uncertainty contribution is taken either from the manufacturer's data sheet or from the combined standard uncertainty figure of its measurement; - where a monopole or 3-axis probe is used to determine the field strength, for the attenuator associated with the Stripline input this uncertainty contribution is assumed to be 0,00 dB due to the methodology; - where a monopole is used to determine the field strength, for the attenuator associated with the Monopole antenna this uncertainty contribution is taken either from the manufacturer's data sheet or from the combined standard uncertainty figure of its measurement. uj41 Insertion loss: cable This uncertainty only contributes to the test methods. How to evaluate for Free Field Test Sites • Verification: This value is 0,00 dB. • Test methods: - for the cable associated with the test antenna, this uncertainty contribution is common to both stage one and stage two of the measurement. Consequently, it is assumed to be 0,00 dB due to the methodology; - for the cable associated with the substitution or measuring antenna, this uncertainty contribution is taken either from the manufacturer's data sheet or from the combined standard uncertainty figure of its measurement. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 129 How to evaluate for Striplines • Verification: This value is 0,00 dB. • Test methods: - where the field strength in a Stripline is determined from the results of the verification procedure, for the cable associated with the signal generator this uncertainty contribution is taken either from the manufacturer's data sheet or from the combined standard uncertainty figure of its measurement; - where a monopole or 3-axis probe is used to determine the field strength, for the cable associated with the signal generator this uncertainty contribution is assumed to be 0,00 dB due to the methodology; - where a monopole is used to determine the field strength, for the cable associated with the monopole antenna this uncertainty contribution is taken either from the manufacturer's data sheet or from the combined standard uncertainty figure of its measurement. uj42 Insertion loss: adapter This uncertainty only contributes to the verification procedures. How to evaluate for Free Field Test Sites • Verification: This uncertainty contribution is taken either from the manufacturer's data sheet or from the combined standard uncertainty figure of the loss measurement. • Test methods: Not applicable. How to evaluate for Striplines • Verification: This uncertainty contribution is taken either from the manufacturer's data sheet or from the combined standard uncertainty figure of the loss measurement. • Test methods: Not applicable. uj43 Insertion loss: antenna balun This uncertainty contributes to test methods and verification procedures on Free Field Test Sites. How to evaluate for Free Field Test Sites • Verification: The standard uncertainty of the contribution is 0,17 dB. • Test methods: The standard uncertainty of the contribution is 0,17 dB. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. ANTENNAS Background: Antennas are used to launch or receive radiated fields on Free Field Test Sites. They can contribute to measurement uncertainty in several ways. For example, the uncertainty of the gain and/or antenna factor, the tuning, etc. uj44 Antenna: antenna factor of the transmitting, receiving or measuring antenna This uncertainty contributes to test methods and verification procedures on Free Field Test Sites. It is the uncertainty with which the antenna factor is known at the frequency of test. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 130 How to evaluate for Free Field Test Sites • Verification: The antenna factor contributes only to the radiated part of this procedure. For ANSI dipoles the value should be obtained from table A.16. For other antenna types the figures should be taken from manufacturers data sheets. If a figure is not given the standard uncertainty is 1,0 dB. Table A.16: Uncertainty contribution: Antenna: antenna factor of the transmitting, receiving or measuring antenna Frequency Standard uncertainty of the contribution 30 MHz ≤ frequency < 80 MHz 1,73 dB 80 MHz ≤ frequency < 180 MHz 0,60 dB frequency ≥ 180 MHz 0,30 dB • Test methods: The uncertainty contribution should be taken from the manufacturer's data sheet and converted into standard uncertainty if necessary. If no value is given the standard uncertainty is assumed to be 1,0 dB. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj45 Antenna: gain of the test or substitution antenna This uncertainty only contributes to test methods on Free Field Test Sites. It is the uncertainty with which the gain of the antenna is known at the frequency of test. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: For ANSI dipoles the value should be obtained from table A.17. For other antenna types the figures should be taken from manufacturers data sheets. If a figure is not given the standard uncertainty is 1,0 dB. Table A.17: Uncertainty contribution: Antenna: gain of the test or substitution antenna Frequency Standard uncertainty of the contribution 30 MHz ≤ frequency < 80 MHz 1,73 dB 80 MHz ≤ frequency < 180 MHz 0,60 dB frequency ≥ 180 MHz 0,30 dB How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 131 uj46 Antenna: tuning This uncertainty contributes to test methods and verification procedures on Free Field Test Sites. It is the uncertainty with which the lengths of the dipoles arms can be set for any test frequency. How to evaluate for Free Field Test Sites • Verification: The standard uncertainty is 0,06 dB. • Test methods: - in the test antenna case the uncertainty is equal in both stages of the test method so its contribution to the uncertainty is assumed to be 0,00 dB; - in the substitution/measuring antenna case, the standard uncertainty is 0,06 dB. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. RECEIVING DEVICE Background: The receiving device (a measuring receiver or spectrum analyser) is used to measure the received signal level either as an absolute level or as a reference level. It can contribute uncertainty components in two ways: absolute level accuracy and non-linearity. An alternative receiving device (a power measuring receiver) is used for the adjacent channel power test method. uj47 Receiving device: absolute level This uncertainty contributes to test methods where the measurement of field strength is involved and the verification procedures where a range change to the receiving device's input attenuator occurs between the two stages of the procedure. How to evaluate for Free Field Test Sites • Verification: The absolute level uncertainty is not applicable in stage one but should be included in stage two if the receiving device's input attenuator has been changed. This uncertainty contribution should be taken from the manufacturer's data sheet and converted if necessary. • Test methods: Only applicable in the electric field strength measurement stage for a receiving equipment. This uncertainty contribution should be taken from the manufacturer's data sheet and converted if necessary. How to evaluate for Striplines • Verification: The absolute level uncertainty is not applicable in stage one but may be included in stage two if the receiving device's input attenuator has been changed. This uncertainty contribution should be taken from the manufacturer's data sheet and converted if necessary. • Test methods: Only applicable in the electric field strength measurement stage for a receiving equipment. This uncertainty contribution should be taken from the manufacturer's data sheet and converted if necessary. uj48 Receiving device: linearity This uncertainty only contributes to the verification procedures. How to evaluate for Free Field Test Sites • Verification: If the receiving devices input attenuator has been changed the value is 0,00 dB. If not, the value should be calculated from the manufacturer's data sheet e.g. a level variation of 62 dB gives an uncertainty of 0,62 dB at a linearity of 0,1 dB/10 dB. The uncertainty should be converted into standard uncertainty, assuming a rectangular distribution in logs. • Test methods: Not applicable. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 132 How to evaluate for Striplines • Verification: If the receiving devices input attenuator has been changed the value is 0,00 dB. If not, the value should be calculated from the manufacturer's data sheet e.g. a level variation of 62 dB gives an uncertainty of 0,62 dB at a linearity of 0,1 dB/10 dB. The uncertainty should be converted into standard uncertainty, assuming a rectangular distribution in logs. • Test methods: Not applicable. uj49 Receiving device: power measuring receiver This uncertainty only contributes to the transmitter adjacent channel power test method. There are three types of power measuring receiver, they are: - an adjacent channel power meter; - a spectrum analyser; - a measuring receiver with digital filters. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Contributions are the same as for the conducted case, see ETR 028 [5]. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. EQUIPMENT UNDER TEST Background: There are uncertainties associated with the EUT due to the following reasons: - temperature effects: this is the uncertainty caused by the uncertainty in the ambient temperature; - degradation measurement: this contribution is a RF level uncertainty associated with the uncertainty of measuring, 20 dB SINAD, 10-2 bit stream or 80 % message acceptance ratio; - power supply effects. Tis is the uncertainty caused by the uncertainty in the power supply voltage; - mutual coupling to its power leads. uj50 EUT: influence of the ambient temperature on the ERP of the carrier This uncertainty only contributes to the ERP test method. It is the uncertainty in the ERP of the carrier caused by the uncertainty in knowing the ambient temperature. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Only applicable in stage one where the measurement is made on the EUT. The uncertainty caused is calculated using the dependency function (ETR 028 [5], part 2, table C.1: "EUT dependency functions and uncertainties") whose mean value is 4 %/°C and whose standard deviation is 1,2 %/°C. The standard uncertainty of the ERP of the carrier caused by this ambient temperature uncertainty should be calculated using formula (5.3) of ETR 028 [5] and then converted to dB. For example, an ambient temperature uncertainty of ±1°C, results in the standard uncertainty of the ERP of the carrier of: = ) ) C (1,2%/ + ) C ((4,0%/ x ) 3 C) (1 ( 2 2 2 ° ° ° 2,41 %, transformed to dB: 2,41/23,0 = 0,1 dB ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 133 How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable uj51 EUT: influence of the ambient temperature on the spurious emission level This uncertainty contribution only applies to the test methods on Free Field Test Sites. It is the uncertainty in the power level of the spurious emission caused by the uncertainty in knowing the ambient temperature. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Only applicable in stage one where the measurement is made on the EUT. The uncertainty caused is calculated using the dependency function (ETR 028 [5], part 2, table C.1: "EUT dependency functions and uncertainties") whose mean value is 4 %/°C and whose standard deviation is 1,2 %/°C. The standard uncertainty of the spurious emission level caused by this ambient temperature uncertainty should be calculated using formula (5.3) of ETR 028 [5] and then converted to dB. • For example, an ambient temperature uncertainty of ±1°C, results in the standard uncertainty of the spurious emission level of: = ) ) C (1,2%/ + ) C ((4,0%/ x ) 3 C) (1 ( 2 2 2 ° ° ° 2,41 %, transformed to dB: 2,41/23,0 = 0,1 dB How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj52 EUT: degradation measurement This uncertainty only contributes to receiver test methods and is the resulting RF level uncertainty associated with the uncertainty of measuring 20 dB SINAD, 10-2 bit stream or 80 % message acceptance ratio. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: The magnitude can be obtained from ETR 028 [5]. How to evaluate for Striplines • Verification: Not applicable. • Test methods: The magnitude can be obtained from ETR 028 [5]. uj53 EUT: influence of setting the power supply on the ERP of the carrier This uncertainty only applies to the effective radiated power test method and is caused by the uncertainty in setting the power supply level. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Only applicable in stage one where the measurement is made on the EUT. The uncertainty caused is calculated using the dependency function (ETR 028 [5], part 2, table C.1: "EUT dependency functions and uncertainties") whose mean value is 10 %/V and whose standard deviation is 3 %/V. The standard uncertainty of the ERP of the carrier caused by power supply voltage uncertainty should be calculated using formula (5.3) of ETR 028 [5] and then converted to dB. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 134 • For example, a supply voltage uncertainty of ±100 mV results in the standard uncertainty of the ERP of the carrier of: % 0,60 = ) ) (3%/V + ) ((10%/V x 3 ) (0,1V 2 2 2 , transformed to dB: 0,60/23,0 = 0,03 dB How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj54 EUT: influence of setting the power supply on the spurious emission level This uncertainty only applies to the spurious emissions test method and is caused by the uncertainty in setting the power supply level. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: Only applicable in stage one where the measurement is made on the EUT. The uncertainty caused is calculated using the dependency function (ETR 028 [5], part 2, table C.1: "EUT dependency functions and uncertainties") whose mean value is 10 %/V and whose standard deviation is 3 %/V. The standard uncertainty of the spurious emission level caused by power supply voltage uncertainty should be calculated using formula (2) of ETR 028 [5] and then converted to dB. • For example, a supply voltage uncertainty of ±100 mV results in the standard uncertainty of the spurious emission level of: ) ) (3%/V + ) ((10%/V x 3 ) (0,1V 2 2 2 = 0,06 %, transformed to dB: 0,60/23,0 = 0,03 dB How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj55 EUT: mutual coupling to the power leads This uncertainty only contributes to test methods. It is the uncertainty which results from interaction (reflections, parasitic effects, etc.) between the EUT and the power leads. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: The standard uncertainty is 0,5 dB provided that the precautions detailed in the methods have been observed. i.e. routing and dressing of cables with ferrites. If the precautions have not been observed the standard uncertainty is 2,0 dB. How to evaluate for Striplines • Verification: Not applicable. • Test methods: The standard uncertainty is 0,5 dB provided that the precautions detailed in the methods have been observed. i.e. routing and dressing of cables with ferrites. If the precautions have not been observed the standard uncertainty is 2,0 dB. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 135 FREQUENCY COUNTER uj56 Frequency counter: absolute reading This uncertainty only contributes to frequency error test methods performed using a frequency counter. It is the uncertainty of frequency measurement. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: The uncertainty of frequency measurement is taken from the manufacturer's data sheet. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj57 Frequency counter: estimating the average reading This uncertainty only contributes to frequency error test methods performed using a frequency counter. It is the uncertainty with which the average frequency can be estimated. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: The standard uncertainty should be taken as 0,33 × (highest frequency - lowest frequency)/2. How to evaluate for Striplines • Verification: Not applicable. • Test methods: The standard uncertainty should be taken as 0,33 × (highest frequency - lowest frequency)/2. SALTY MAN AND SALTY-LITE Background: The human body has a significant effect on the electrical performance of a body worn EUT. For test purposes the artificial human body should simulate the average human body. Two main types of artificial human bodies are used in testing: Salty man and Salty-lite. uj58 Salty man/Salty-lite: human simulation This uncertainty only contributes to test methods on Free Field Test Sites. It is the uncertainty which results from the differences between the average human being and the artificial one used. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: The standard uncertainty should be taken from table A.18. Table A.18: Uncertainty contribution: Salty man/Salty-lite: human simulation In an Anechoic Chamber the standard uncertainties are: Salty man: 30 MHz to 150 MHz is 0,58 dB Salty man: 150 MHz to 1 000 MHz is 1,73 dB Salty lite: 100 MHz to 150 MHz is 1,73 dB Salty lite: 150 MHz to 1 000 MHz is 0,58 dB On an Open Area Test Site or in an Anechoic Chamber with a ground plane: Salty man: 30 MHz to 150 MHz is 0,58 dB Salty man: 150 MHz to 1 000 MHz is 1,73 dB Salty lite: 70 MHz to 150 MHz is 1,73 dB Salty lite: 150 MHz to 1 000 MHz is 0,58 dB ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 136 How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. uj59 Salty man/Salty-lite: field enhancement and de-tuning of the EUT This uncertainty only contributes to test methods on Free Field Test Sites. It is the uncertainty associated with the variation of the enhanced magnetic field effect produced by the body and the de-tuning of the circuitry of the EUT with spacing away from the outer surface of the salty body. How to evaluate for Free Field Test Sites • Verification: Not applicable. • Test methods: The standard uncertainty of this effect is estimated as 1,00 dB. How to evaluate for Striplines • Verification: Not applicable. • Test methods: Not applicable. TEST FIXTURE Background: A test fixture is a type of test site which enables the performance of an integral antenna EUT to be measured at extreme conditions. uj60 Test Fixture: effect on the EUT Since it is proven on the accredited test site that the test fixture does not have an adverse effect on the EUT (e.g. more than a 0,5 dB change in the received level), it is assumed that the maximum uncertainty introduced by the presence of the test fixture is ±0,5 dB. The corresponding standard uncertainty is 0,29 dB. uj61 Test Fixture: climatic facility effect on the EUT Since it is proven that the climatic facility does not have an adverse effect on the EUT (e.g. more than a 0,5 dB change in the received level), it is assumed that the maximum uncertainty introduced by the presence of the test fixture is ±0,5 dB. The corresponding standard uncertainty is 0,29 dB. RANDOM UNCERTAINTY ui01 Random uncertainty This uncertainty contributes to all radiated tests. It is the estimated effect that randomness has on the final result of a measurement. How to evaluate for Free Field Test Sites • Verification: Random uncertainty should be assessed by multiple measurements of the same measurand and treating the results statistically to derive the standard uncertainty of its contribution. • Test methods: Random uncertainty should be assessed by multiple measurements of the same measurand and treating the results statistically to derive the standard uncertainty of its contribution. How to evaluate for Striplines • Verification: Random uncertainty should be assessed by multiple measurements of the same measurand and treating the results statistically to derive the standard uncertainty of its contribution. • Test methods: Random uncertainty should be assessed by multiple measurements of the same measurand and treating the results statistically to derive the standard uncertainty of its contribution. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 137 Table A.19: Mutual coupling and mismatch loss correction factors (Anechoic Chamber) Frequency (MHz) Range length 3 m Frequency (MHz) Range length 10 m 30 27,1 30 25,8 35 24,3 35 23,3 40 21,7 40 20,8 45 19,0 45 18,2 50 16,1 50 15,4 60 9,7 60 9,1 70 2,2 70 1,7 80 0,7 80 0,2 90 0,6 90 0,1 100 0,6 100 0,1 120 0,3 120 0,1 140 0,4 140 0,1 160 0,3 160 0,2 180 0,2 180 0,1 Table A.20: Mutual coupling and mismatch loss correction factors (over a ground plane) Horizontal polarization Vertical polarization Freq. (MHz) 3 m 10 m Freq. (MHz) 3 m 10 m 30 27,6 26,0 30 25,2 25,4 35 24,6 23,3 35 22,4 22,9 40 21,8 20,7 40 19,8 20, 4 45 19,0 18,1 45 17,2 17,9 50 16,0 15,1 50 14,4 15,1 60 9,5 8,9 60 8,5 9,2 70 2,4 2,8 70 1,6 2,5 80 0,6 0,8 80 0,0 0,4 90 0,2 0,4 90 -0,2 0,1 100 -0,3 0,0 100 -0,6 0,0 120 -2,3 -1,2 120 -0,6 0,0 140 -1,0 -0,7 140 1,1 -0,1 160 -0,3 0,3 160 0,7 0,0 180 -0,3 0,3 180 0,3 0,0 Table A.21: Summary table of all contributions (numerical sort) Description uj01 reflectivity of absorbing material: EUT to the test antenna uj02 reflectivity of absorbing material: substitution or measuring antenna to the test antenna uj03 reflectivity of absorbing material: transmitting antenna to the receiving antenna uj04 Mutual coupling: EUT to its images in the absorbing material uj05 mutual coupling: de-tuning effect of the absorbing material on the EUT uj06 mutual coupling: substitution, measuring or test antenna to its image in the absorbing material uj07 mutual coupling: transmitting or receiving antenna to its image in the absorbing material uj08 mutual coupling: amplitude effect of the test antenna on the EUT uj09 mutual coupling: de-tuning effect of the test antenna on the EUT uj10 mutual coupling: transmitting antenna to the receiving antenna uj11 mutual coupling: substitution or measuring antenna to the test antenna uj12 mutual coupling: interpolation of mutual coupling and mismatch loss correction factors uj13 mutual coupling: EUT to its image in the ground plane uj14 mutual coupling: substitution, measuring or test antenna to its image in the ground plane uj15 mutual coupling: transmitting or receiving antenna to its image in the ground plane uj16 range length ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 138 Description uj17 correction: off boresight angle in the elevation plane uj18 correction: measurement distance uj19 cable factor uj20 position of the phase centre: within the EUT volume uj21 positioning of the phase centre: within the EUT over the axis of rotation of the turntable uj22 position of the phase centre: measuring, substitution, receiving, transmitting or test antenna uj23 position of the phase centre: LPDA uj24 Stripline: mutual coupling of the EUT to its images in the plates uj25 Stripline: mutual coupling of the 3-axis probe to its image in the plates uj26 Stripline: characteristic impedance uj27 Stripline: non-planar nature of the field distribution uj28 Stripline: field strength measurement as determined by the 3-axis probe uj29 Stripline: transfer factor uj30 Stripline: interpolation of values for the transfer factor uj31 Stripline: antenna factor of the monopole uj32 Stripline: correction factor for the size of the EUT uj33 Stripline: influence of site effects uj34 ambient effect uj35 mismatch: direct attenuation measurement uj36 mismatch: transmitting part uj37 mismatch: receiving part uj38 signal generator: absolute output level uj39 signal generator: output level stability uj40 insertion loss: attenuator uj41 insertion loss: cable uj42 insertion loss: adapter uj43 insertion loss: antenna balun uj44 antenna: antenna factor of the transmitting, receiving or measuring antenna uj45 antenna: gain of the test or substitution antenna uj46 antenna: tuning uj47 receiving device: absolute level uj48 receiving device: linearity uj49 receiving device: power measuring receiver uj50 EUT: influence of the ambient temperature on the ERP of the carrier uj51 EUT: influence of the ambient temperature on the spurious emission level uj52 EUT: degradation measurement uj53 EUT: influence of setting the power supply on the ERP of the carrier uj54 EUT: influence of setting the power supply on the spurious emission level uj55 EUT: mutual coupling to the power leads uj56 frequency counter: absolute reading uj57 frequency counter: estimating the average reading uj58 Salty man/Salty-lite: human simulation uj59 Salty man/Salty-lite: field enhancement and de-tuning of the EUT uj60 Test Fixture: effect on the EUT uj61 Test Fixture: climatic facility effect on the EUT ui01 random ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 139 Table A.22: Summary table of all contributions (alphabetical sort) Description uj34 ambient effect uj44 antenna: antenna factor of the transmitting, receiving or measuring antenna uj45 antenna: gain of the test or substitution antenna uj46 antenna: tuning uj19 cable factor uj18 correction: measurement distance uj17 correction: off boresight angle in the elevation plane uj53 EUT: influence of setting the power supply on the ERP of the carrier uj54 EUT: influence of setting the power supply on the spurious emission level uj50 EUT: influence of the ambient temperature on the ERP of the carrier uj51 EUT: influence of the ambient temperature on the spurious emission level uj52 EUT: degradation measurement uj55 EUT: mutual coupling to the power leads uj56 frequency counter: absolute reading uj57 frequency counter: estimating the average reading uj42 insertion loss: adapter uj43 insertion loss: antenna balun uj40 insertion loss: attenuator uj41 insertion loss: cable uj35 mismatch: direct attenuation measurement uj37 mismatch: receiving part uj36 mismatch: transmitting part uj04 Mutual coupling: EUT to its images in the absorbing material uj08 mutual coupling: amplitude effect of the test antenna on the EUT uj05 mutual coupling: de-tuning effect of the absorbing material on the EUT uj09 mutual coupling: de-tuning effect of the test antenna on the EUT uj13 mutual coupling: EUT to its image in the ground plane uj12 mutual coupling: interpolation of mutual coupling and mismatch loss correction factors uj11 mutual coupling: substitution or measuring antenna to the test antenna uj06 mutual coupling: substitution, measuring or test antenna to its image in the absorbing material uj14 mutual coupling: substitution, measuring or test antenna to its image in the ground plane uj10 mutual coupling: transmitting antenna to the receiving antenna uj07 mutual coupling: transmitting or receiving antenna to its image in the absorbing material uj15 mutual coupling: transmitting or receiving antenna to its image in the ground plane uj23 position of the phase centre: LPDA uj22 position of the phase centre: measuring, substitution, receiving, transmitting or test antenna uj20 position of the phase centre: within the EUT volume uj21 positioning of the phase centre: within the EUT over the axis of rotation of the turntable ui01 random uj16 range length uj47 receiving device: absolute level uj48 receiving device: linearity uj49 receiving device: power measuring receiver uj01 reflectivity of absorbing material: EUT to the test antenna uj02 reflectivity of absorbing material: substitution or measuring antenna to the test antenna uj03 reflectivity of absorbing material: transmitting antenna to the receiving antenna uj59 Salty man/Salty-lite: field enhancement and de-tuning of the EUT uj58 Salty man/Salty-lite: human simulation uj38 signal generator: absolute output level ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 140 Description uj39 signal generator: output level stability uj31 Stripline: antenna factor of the monopole uj26 Stripline: characteristic impedance uj32 Stripline: correction factor for the size of the EUT uj28 Stripline: field strength measurement as determined by the 3-axis probe uj33 Stripline: influence of site effects uj30 Stripline: interpolation of values for the transfer factor uj25 Stripline: mutual coupling of the 3-axis probe to its image in the plates uj24 Stripline: mutual coupling of the EUT to its images in the plates uj27 Stripline: non-planar nature of the field distribution uj29 Stripline: transfer factor uj61 Test Fixture: climatic facility effect on the EUT uj60 Test Fixture: effect on the EUT ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 141 Height of antenna on antenna mast 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,2 1,4 1,6 1,8 2 2,2 2,4 2,6 2,8 3 3,2 3,4 3,6 3,8 4 Signal loss (dB) 10 metre range 3 metre range Off boresight angle in the elevation plane corrections Figure A.7: Signal attenuation with increasing elevation offset angle ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 142 Height of test antenna on mast 0 0,5 1 1,5 2 2,5 3 3,5 1 1,2 1,4 1,6 1,8 2 2,2 2,4 2,6 2,8 3 3,2 3,4 3,6 3,8 4 Signal loss (dB) 3 metre range 10 metre range Measurement distance correction Figure A.8: Signal attenuation for antenna height on mast ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 143 Annex B: Maximum accumulated measurement uncertainty The accumulated measurement uncertainties of the test system in use for the parameters to be measured should not exceed those given in table B.1. This is in order to ensure that the measurements remain within an acceptable quality. Table B.1: Recommended maximum acceptable uncertainties RF frequency (see note 1) ±1 x 10-7 (see note 2) RF power (valid to 100 W) (see note 1) ±0,75 dB (see note 2) Maximum frequency deviation - within 300 Hz and 6 kHz of audio frequency (see note 1) - within 6 kHz and 25 kHz of audio frequency (see note 1) ±5 % (see note 2) ±3 dB (see note 2) Deviation limitation (see note 1) ±5 % (see note 2) Audio frequency response of transmitters (see note 1) ±0,5 dB (see note 2) Adjacent channel power (see note 1) ±3 dB (see note 2) Conducted emissions of transmitters (see note 1) ±4 dB (see note 2) Transmitter distortion (see note 1) ±2 % (see note 2) Transmitter residual modulation (see note 1) ±2 dB (see note 2) Audio output power (see note 1) ±0,5 dB (see note 2) Audio frequency response of receivers (see note 1) ±1 dB (see note 2) Amplitude characteristics of receiver limiter (see note 1) ±1,5 dB (see note 2) Hum and noise (see note 1) ±2 dB (see note 2) Receiver distortion (see note 1) ±2 % (see note 2) Sensitivity (see note 1) ±3 dB (see note 2) Conducted emissions of receivers (see note 1) ±4 dB (see note 2) Two-signal measurements (stop band) (see note 1) ±4 dB (see note 2) Three-signal measurements (see note 1) ±3 dB (see note 2) Radiated emissions of transmitters (see note 1) ±6 dB (see note 2) Radiated emissions of receivers (see note 1) ±6 dB (see note 2) Transmitter attack and release time (see note 1) ±4 ms (see note 2) Transmitter transient frequency (see note 1) ±250 Hz (see note 2) Transmitter intermodulation (see note 1) ±5 dB (see note 2) Receiver desensitization (duplex operation) (see note 1) ±0,5 dB (see note 2) NOTE 1: Test methods according to relevant deliverables. NOTE 2: The uncertainty figures are valid for a confidence level of 95 %. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 144 Annex C: Interpretation of the measurement results The interpretation of the results recorded in a test report for the measurements described in the standard should be as follows: 1) the measurement value related to the corresponding limit should be used to decide whether an equipment meets the requirements of the relevant standards; 2) the measurement uncertainty value for the measurement of each parameter should be included in the test reports; 3) the recorded value for the measurement uncertainty should be, for each measurement, equal to or lower than the figures in the appropriate table of "maximum acceptable measurement uncertainties" of the appropriate standard. NOTE: This procedure is usually referred to as "the shared risk approach" and is recommended unless superseded by an appropriate publication of ETSI. Clause D.5.6.2.7.3 shows the way in which double sided limits (e.g. limits stated as "2 W + 1,5 dB") have been handled in ETSI standards, when the tolerance (e.g. +1,5 dB) is smaller than the maximum acceptable measurement uncertainty for that measurement (e.g. +6 dB). ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 145 Annex D: Theoretical support for the evaluation of measurement uncertainties, including mathematical tools and properties of distributions This annex of the present document provides theoretical support for the handling of measurement uncertainties; more precisely, the methods proposed here are based on the usage of random variables (and combinations thereof). The aim of annex D is, therefore, in particular: - to provide guidance on how to use random variables in support of the evaluation of measurement uncertainties (and a theoretical justification for expressions found e.g. in TR 100 028-1 [6], clauses 4 and 5); - to provide methods to handle and to combine random variables. Annex D offers a theoretical background, as complete (self-contained) as practical, in the line of clauses 4 and 5 of TR 100 028-1 [6] of the present document. However, it is expected that the reader is familiar with the definitions and concepts dealt with in clause 4 of TR 100 028-1 [6], and therefore such concepts are not defined again in the present annex. In the following clauses, the reader will also have a chance to get more familiar with: - a number of definitions and with the properties of some usual distributions; - the result of the combination of random variables and how to use all these tools in order to better evaluate the uncertainties relating to a particular test set up. The present annex has evolved in time, and includes contributions from various authors. This may have led to the use of symbols slightly different, according with the targets sought. These specificities have been kept, in order to allow for the internal consistency between certain pieces of text. Different methods may also have been used (some being more general or theoretical than others); they allow the reader to get familiar with different approaches and techniques. Sometimes similar results may have been obtained by different methods … which also helps cross-checking the expressions given. D.1 Probability densities and some of their properties D.1.1 Introduction A random variable X is defined as a variable which takes any value x of a continuum of values at a particular instant in time. It is usual to characterize a random variable X by its probability density function p(x): 0 ≥ ∀ p(x) x (where, x ∀ … means for any x ). D.1.2 Definitions The probability P of the value x of the random variable X lying between x1 and x2 is provided by the probability density function, p ( x ) , as follows: ∫ = 2 1 x x p(x)dx P ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 146 Since x must have its value in the range -∞ to +∞ , and p ( x ) is the corresponding distribution ∫ +∞ ∞ − = 1 dx p(x) . Conversely, P = 0 can be understood as the probability of an event that would not occur, and P = 1 can be understood as the probability of an event that should certainly occur. Small contributions In many clauses of this annex, for example in clause D.3 , p ( x ) (also noted as f ( x ) ) is used in relation to small contributions. In this case, the probability Pf of the random variable F having a value x such that x1 < x < x2 is ∫ = 2 1 x x f f(x)dx P . Similarly, we can consider ∫ ∞ − = x f dt f(t) x P ) ( , and therefore (by differentiation) dPf = f ( x ) dx . Note concerning signs: It is also to be noted that, according to the usual conventions (see above), p ( x ) and P are always positive, while, according to the conventions used with integrals: ∫ ∫ − = = 1 2 2 1 x x x x p(x)dx p(x)dx P . As a result, when writing ∫ = 2 1 x x f f(x)dx P , one has to make sure that x1 < x2 . Should we have x1 > x2 then the integration limits have to be inverted … or absolute values have to be used. This has a direct effect on calculations such as those found in clauses D.3, for example in clause D.3.2 (i.e. discussions concerning the signs). Mean value (or 1st moment) The mean of a random variable X defined by its probability density function p is given by: ∫ +∞ ∞ − = dx p(x) x xm the term xm has been used, in particular, in annex E. However, at a later stage, in the present annex, the mean value of random variable X has also been called mx or mX . The mean is also called 1st moment. For further proposals concerning notation, please see also clause D.10.6. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 147 Second moment The second moment of a probability density function p(x) about the origin is: dx p(x) x xm ∫ +∞ ∞ − = 2 2 and xm 2 is called sometimes the mean square value. The expression " xm² " has been used, in particular, in annex E. However, at a later stage, in the present annex, the second moment corresponding to random variable X has often been referred to as sx² or sX ². Variance It is usual to take the 2nd moment about the mean as a measure of dispersion. This is often termed the variance (σ2) of the probability density function, hence: ∫ +∞ ∞ − − = dx p(x) ) x (x m 2 2 σ Standard deviation In the present document, σσσσ is often called "standard deviation" , and to show it relates to X , it has been written as σσσσx or σσσσX . Relations between some of these properties Using mx , sx and σσσσx …. the previous expression can be written as: dx x p m x x x ) ( ) ( 2 2 ∫ +∞ ∞ − − = σ dx x p x ) ( 2 ∫ +∞ ∞ − = - dx x p m x x ) ( 2∫ +∞ ∞ − + dx x p mx ) ( 2 ∫ +∞ ∞ − dx x p m dx x p x m dx x p x x x x ) ( ) ( 2 ) ( 2 2 2 ∫ ∫ ∫ +∞ ∞ − +∞ ∞ − +∞ ∞ − + + = σ and therefore: σσσσx ² = sx² - 2 mx mx + mx². Finally we get: σσσσx² = sx² - mx² an expression which will be used quite often in the present annex. Notations In documentation relating to the theory of probabilities, where only one probability density is addressed at the time, it can be handy to use notations such as p ( x ) … However, when discussing uncertainties, where a significant number of physical parameters are handled simultaneously, it can be practical to use notations linking in an obvious manner, these physical parameters with corresponding random variables (i.e. mapping), in which case notations such as those proposed in clause D.3.10.6 may seem more convenient. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 148 D.1.3 Means and standard deviations of usual distributions The term distributions has been used in this clause instead of probability density. In many of the following drawings the mean value of the distributions shown is 0. However, this has no effect on the value of the standard deviations. D.1.3.1 Rectangular distributions In the example above, the mean value is 0 (but a rectangular distribution could, as well, be centred around some other value C: in which case, the mean value would have been C ); The standard deviation is 3 A (independent of the mean value …): [ ] 3 A 3 A ) A ( A A 6 1 3 x A 2 1 dx A 2 1 x 2 3 3 A A 3 A A 2 2 = = − − = = = − −∫ σ σ In the case where the mean is C and not 0 , in the interval (C - A ) to (C + A ) , x occurs with equal probability, i.e. p(x)=1/(2A). In this annex, this interval has some times been called "spread" or "foot print". Example of usage of rectangular distributions: unknown systematic error distributions are assumed, in the present document, to be rectangularly distributed. Power ranges (e.g. expressed in dBs) provide good examples of rectangular distributions centred around non-zero values ( C non zero). D.1.3.2 Triangular distributions Triangular distributions can be found as the result of additive combinations of identical triangular distributions. The additive combination of two random variables generates, as shown in clause D.3.3, a random variable having a probability density equal to: ∫ +∞ ∞ − − = dx x f x z g z h ) ( ) ( ) ( , where g ( y ) and f ( x ) are the original probability densities. x p x ( ) A A A A ∈− + → ∉− + → = [ , ] [ , ] 0 x x p x( ) p x( ) = A + A _ 1 A 2 1 A 2 ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 149 D.1.3.2.1 Additive combination of two rectangular distributions having the same spread In the special case, where the distributions f and g are rectangular distributions, corresponding to the same parameter A (see the definition in clause D.1.3.1 above), it can be interesting to track the values of x and y = z – x , corresponding to where there are discontinuities in the definition of the probability densities … as a result, h ( z ) can be split as follows: - when z < – A – A = – 2 A then both g and f = 0 for all values of x and, therefore, h ( z ) = 0 - when z > A + A = 2 A then both g and f = 0 for all values of x and, therefore, h ( z ) = 0 - when z is negative and greater than – 2A , the zone to be integrated is shopped between the intervals where either f or g are equal to zero: ∫ + − = A z A dx A A z h 2 1 2 1 ) ( [ ] ) 2 ( 4 1 ) ( 4 1 4 1 2 2 2 A z A A A z A x A A z A + = + + = = + − ; - when z is positive and smaller than 2A , the zone to be integrated is also shopped between the intervals where either f or g are equal to zero: ∫ + − = A A z dx A A z h 2 1 2 1 ) ( [ ] ) 2 ( 4 1 ) ( 4 1 4 1 2 2 2 A z A A A z A x A A A z + − = + + − = = + − ; - when z is zero, the zone to be integrated is common to f and g : ∫ + − = A A dx A A h 2 1 2 1 ) 0 ( [ ] A A A A A A x A A A 2 1 ) 2 ( 4 1 ) ( 4 1 4 1 2 2 2 = = + = = + − ; this value is, in fact common to both expressions found above when z 0. The final result is, therefore, a triangular distribution spreading between -2A and +2A, with a maximum value of 1/2A (the same as the value corresponding to the original rectangular distributions). The result of the combination is, therefore, a distribution "smoothed". Should the original distributions be different, the same "smoothing" mechanism would be observed (see also the clause on trapezoidal distributions, D.1.3.3.1). In the above example, centred distributions have been used. Should there have been an offset, the triangular distribution would have had an offset equal to the sum of both offsets (as shown in clause D.3.3). Examples of additive combination of rectangular distributions are also provided in clause D.3.3.5.2. D.1.3.2.2 Properties of triangular distributions Assume a triangular distribution spreading from –A to +A with a maximum of 1/A (note a change in the definition of A in relation to that found in clause D.1.3.2.1, above): The mean value is 0 (for distribution symmetrical around the y'y axis); a triangular distribution could, as well, be centred around some other value C: in which case, the mean value would have been C. The calculation of the variance shows a method which can be used extensively: 1/A 0 -A ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 150 [ ] [ ] 12 A 4 A 3 A ) A ( 0 A 3 1 ) A ( 0 A 4 1 3 x A 1 4 x A 1 dx A 1 A x x 2 2 2 3 4 2 0 A 3 0 A 4 2 0 A 2 2 2 = − = − − + − − = + = + = − − −∫ σ Finally, noting that the distribution is symmetrical: reapplying this method for the other part, gives the same result. Hence, for both parts, 6 A 12 A 2 2 2 = = σ σ D.1.3.3 Trapezoidal distributions D.1.3.3.1 Symmetrical trapezoidal distributions Triangular distributions may be found as the result of the additive combination of two identical rectangular distributions. The additive combination of two distributions with a different spread (different parameters "A" with "B" < "A" ), under similar assumptions would result in a trapezoidal distribution: The discontinuities in the slope correspond to: 4 points (- A - B) (- A + B) (+ A - B) (+ A + B) and the corresponding spread ("foot print") is: from (- A - B) to (+ A + B). In the above drawing, the rectangle in yellow colour corresponds to the original distribution of parameter A . As a result, it is clear that rectangular distributions ARE NOT STABLE in relation to additive combinations (it is shown in clause D.3.3.5.1.1 that normal distributions (Gaussian) are). Combining rectangular distributions (spreads A and B symmetrical around zero) ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 151 The properties of trapezoidal distributions corresponding to an additive combination can be easily found using the general properties given in clause D.3.3.3 of this annex: - the mean value is the sum of the means of the original distributions (zero in the drawing above); - the square of the standard deviation is the sum of the squares of the original standard deviations (RSSing). These two properties are valid as well when the original distributions are not centred, as it could have been shown also by direct calculations… D.1.3.3.2 Non symmetrical trapezoidal distributions Such distributions may be found as the result of very simple operations on distributions (e.g. results corresponding to inverse functions (see clause D.3.7), results of the linearization of the result of transforms operated on distributions such as the conversion into dBs and vice-versa). See clause D.3.8.4.2.4. Many other distributions presented in this clause are symmetrical around some axis … This is not the case here! As shown on the drawing, p(x) = 0 outside [A, B]. See also other clauses in D.3.8 and annex E. When such distributions are obtained as the result of some operation, the properties of the mean and standard deviation can be found using the general properties found in the various clauses of clause D.3 (e.g. D.3.3 in the case of additive combinations). The values of the first moments can also be evaluated directly, using the definitions found in clause D.1.2 (similar calculations have been performed a number of times in clause D.3). D.1.3.4 Gaussian distributions −∞ +∞ p x ( ) p x x ( ) exp = − 1 2 2 2 2 σ ππππ σ ( ) Mean value = 0 (in the case of the figure above); Standard deviation = σ A B ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 152 A more general expression is: 2 2 2 ) ( 2 1 σ π σ c x e y − − = , for Gaussian Curves symmetrical around C, in which case the Mean value is C. The normal (or Gaussian) distribution is stable in respect to additive combinations (see clause D.3.3.5.1.1)… and the additive combination of an infinity of identical rectangular distributions converges into the normal distribution … This property is used extensively in clause D.5.6.2. In order to identify the correct coefficients for the equation corresponding to this distribution, let us start from a general form: 2 Bx Ae y − = and then write two basic properties: dx Ae Bx 2 1 − +∞ ∞ −∫ = (property of any probability density) dx Ae x Bx 2 2 2 − +∞ ∞ −∫ = σ (by definition, in the case when the curve is centred and the mean is 0). The first integral can be calculated as follows: S dy Ae dx Ae By Bx = = − +∞ ∞ − − +∞ ∞ − ∫ ∫ 2 2 , and S = 1 … Therefore: dy dx e A dy dx Ae Ae dy Ae dx Ae S y x B By Bx By Bx ) ( 2 2 2 2 2 2 2 2 + − +∞ ∞ − +∞ ∞ − − − +∞ ∞ − +∞ ∞ − − +∞ ∞ − − +∞ ∞ − ∫ ∫ ∫ ∫ ∫ ∫ = = = which can be written in polar co-ordinates: θ ρ ρ ρ π π d d e A S B ) ( 0 2 2 2 − +∞ + −∫ ∫ = with - π ≤ θ ≤ + π and 0 ≤ ρ < +∞ . I A d e A d d e A S B B 2 ) ( 0 2 ) ( 0 2 2 2 2 2 2 π ρ ρ π θ ρ ρ ρ π π ρ = = = − +∞ + − − +∞ ∫ ∫ ∫ , where: [ ] [ ] + = − − = − = = ∞ + − − +∞ ∫ B e B e B d e I B B 2 1 0 2 1 2 1 0 0 ) ( 0 2 2 ρ ρ ρ ρ . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 153 As a result: 1 1 2 1 2 2 2 2 2 2 = = = = B A B A I A S π π π and: B A = 2 π , while noting that B A dx e Bx π = = − +∞ ∞ −∫ 1 2 . The expression: B A dx e Bx π = = − +∞ ∞ −∫ 1 2 is used again later (in clause D.3.3.5.1.1). The second integral can then be used to provide the relation between A, B and σ : dx Ae x Bx2 2 2 − +∞ ∞ −∫ = σ Integrating by parts: [ ] ∫ ∫ ∞ ∞ − ∞ ∞ − ∞ ∞ − − = vdu uv udv let us call dx xe dv Bx2 − = x u = . We then have: 2 2 1 Bx e B v − − = dx du = and finally: [ ] dx e B dx e B e B x vdu uv udv A Bx Bx Bx 2 2 2 2 1 0 2 1 2 1 2 − +∞ ∞ − − +∞ ∞ − ∞ + ∞ − − +∞ ∞ − ∞ + ∞ − +∞ ∞ − + = − − − = − = = ∫ ∫ ∫ ∫ σ BA dx e B A Bx 2 1 2 1 2 2 = = − +∞ ∞ −∫ σ and B 2 1 2 = σ or 2 2 1 σ = B . Knowing that: B A = 2 π , π σ π σ π 2 1 2 1 2 = = = B A ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 154 The expression of the normal distribution is, therefore: 2 2 2 2 1 σ π σ x e y − = . It is interesting to perform this calculation in detail here, since in one way or another, similar types of calculation will be found over and over as soon as normal probability densities are handled. The drawing and the subsequent calculations addressed the case where the distribution is centred. Like rectangular distributions, normal distributions may have some offset, in which case the mean is not zero (i.e. equal to the offset value). D.1.3.5 Oblique pseudo-Gaussian distributions Such non-symmetric distributions can be obtained as the result of transformations on Gaussian distributions…e.g. in (approximations of) transformations from dBs to linear (or vice versa): see clause D.3.8. As shown in clause D.5.6.2 , the shape of a distribution has a direct effect on the relation between "expansion factors" and "confidence levels". D.1.3.6 'U' shaped distributions 1 A π A A A A ∈− + → ∉− + → = [ , ] [ , ] 0 x x p x( ) p x( ) = 1 A − 2 2 π x A − A + p x ( ) Mean value = 0; Standard deviation 2 A = . EXAMPLE: the "U" shaped distribution is used when sine functions are involved. This occurs with mismatch errors, temperature regulators and other sinusoidal cyclic variations. The equation of such distributions is: 2 2 1 x A y − = π , with -A < x < +A. Its basic properties are discussed in the following clauses. D.1.3.6.1 Can this be the expression of a probability density? First, it is clear that y ( x ) is positive. Second, let us evaluate: ∫ − − = A A 2 2 dx x A 1 P π . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 155 Integrating by substitution, [ ] 1 2 2 1 1 d cos A cos A d ) sin 1 ( A cos A d sin A A cos A P 2 2 2 2 2 2 2 2 2 2 2 2 2 = − − = = = − = − = − − − − ∫ ∫ ∫ π π π θ π θ θ θ θ θ θ θ θ θ π π π π π π π π So the two basic requirements are met. The expression given can, therefore, be a valid expression for a density of probability function. D.1.3.6.2 Variance ∫ − − = A A 2 2 2 2 dx x A 1 x 1 π σ Integrating by parts, Obtaining the terms du and v by substitution, = = − = = = − A x sin v x A 1 dx dv x 2 dx du x u 1 2 2 2 θ i2 A dx A x sin x 2 A xdx 2 A x sin A x sin x 2 A A 1 2 A A 1 A A 1 2 2 − = − = − = ∫ ∫ − − − − − − π π π σ ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 156 Integrating i by parts, 2 2 1 1 x A A x sin x v A x sin dx dv 1 dx du x u − + = = = = − − ∫ ∫ − − − − − − − − = − + − = A A A A 2 2 2 2 2 1 A A 1 2 dx x A i A dx x A A x sin x A x sin x i π Integrating the last term, [ ] 2 A 2 2 sin 2 A d 2 2 cos 1 A d cos A dx cos A sin A A dx x A 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 A A 2 2 π θ θ θ θ θ θ θ θ π π π π π π π π π π = + = + = = − = − − − − − − − ∫ ∫ ∫ ∫ Therefore, 4 A 2 A 2 A i A i 2 2 2 2 π π π π − = − − = Therefore, 2 A 2 A 4 A 2 A 2 A 2 2 2 2 2 = = − − = σ π π π π π θ … the standard deviation quoted above … ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 157 D.1.3.7 Maximum value of the standard deviation for bound distributions In order to validate long calculations (e.g. approximations of Logs) it cal be useful to have, a priori, an idea of maximum values to be found. In the case of bound distributions, the maximum values are easy to find. Let ( ) p x be a distribution where ( ) 0 p x = outside [ , ] A A − + (it has been taken centred for the simplification of the presentation). As stated a number of times, already: ( ) 1 p x dx +∞ −∞ = ∫ (property of any probability density), and 2 2 ( ) xs x p x dx +∞ −∞ = ∫ (by definition of sx , m and σσσσ ), and finally 2 2 2 xs m σ = + where ( ) m x p x dx +∞ −∞ = ∫ . The second moment can also be written as: 0 2 2 2 2 0 ( ) ( ) ( ) xs x p x dx x p x dx x p x dx +∞ +∞ −∞ −∞ = = + ∫ ∫ ∫ . Noting that ( ) 0 p x = outside [ , ] A A − + , we get: 0 2 2 2 0 ( ) ( ) A x A s x p x dx x p x dx + − = + ∫ ∫ . The expression 2 0 ( ) A x p x dx + ∫ is maximum for all covered contributions from p ( x ) as far away as possible from 0 and therefore close to A …resulting in: 2 2 0 0 ( ) ( ) A A A p x dx A p x dx + + = ∫ ∫ . Likewise, the maximum for the negative contribution is: 0 0 2 2 ( ) ( ) A A A p x dx A p x dx − − = ∫ ∫ . Combining the two parts we get, at the maximum: 0 0 2 2 2 2 2 0 0 ( ) ( ) ( ) ( ) A A x A A s x p x dx x p x dx A p x dx A p x dx + + − − = + = + ∫ ∫ ∫ ∫ . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 158 And noting that ( ) 1 p x dx +∞ −∞ = ∫ we have finally: 0 2 2 2 2 2 0 ( ) ( ) ( ) A A x A A s A p x dx A p x dx A p x dx A + + − − = + = = ∫ ∫ ∫ . Noting that 2 2 2 xs m σ = + , 2 2 2 xs m σ = − and in order to have a maximum standard deviation, m should be minimal (a centred symmetrical distribution would have had a mean equal to 0 ). So, finally, at the maximum: 2 2 2 xs A σ = = . D.1.3.8 Standard deviation for bound distributions (summary table) The values of the standard deviations of usual distributions having a "footprint" from –A to + A can be summarized as follows: Distribution Maximum value at Maximum value reached Standard deviation Triangular Centre 1 A 6 A Rectangular Centre 1 2A 3 A U-Shaped Maximum at the edges Minimum in Centre Maximum unlimited Minimum 1 A π 2 A Maximum value for bound Distributions (see clause D.1.3.7) Edges Unlimited 1 A A Gaussian has an unlimited "footprint" and cannot therefore be compared directly … For completeness, however, its characteristics have been recalled below, with the same format: Distribution Maximum value at Maximum value reached Standard deviation Gaussian Centre 1 2 A π 1 A Or: (another Gaussian) Centre 1 2A same as rectangular above 1,57 2 A A π ≈ D.2 Uncertainties and probability densities This clause of the present document is intended to show basic methodologies and the relations between measurement uncertainties and random variables. It uses definitions and intuitive approaches corresponding to both the definitions and clauses 4 and 5 of TR 100 028-1 [6]. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 159 D.2.1 Examples of very simple systems and corresponding naïve (direct) analysis These examples are intended to establish a link between the various concepts (random variables, probability densities, uncertainties, etc.). In order to keep the text of these examples as simple as possible, simplifying assumptions have been made. It is understood that all effects other than those to be highlighted are considered negligible. Methods to cover complete system analysis are given in clause D.5 of this annex. D.2.1.1 Ohm's law D.2.1.1.1 Relations between Random Variables under Ohm's law D.2.1.1.1.1 Establishing the Relations between Random Variables For the purpose of this example, a current generator G is connected (in series) with a resistor having a resistance R. V is the voltage across the resistor. Generator G is providing current i. I is considered as a random variable characterized by its value i at a certain time and by its probability density i (x): by definition, the probability P of having the random variable I having a value i such that i1 < i < i2 is ∫ = 2 1 i i i(x)dx P , and dP = i(x) dx. For each value of I, Ohm's law provides the value v of the random variable V : for any value i , v = R i. Under these circumstances, V can be considered as a random variable for which the probability density, v ( y ) , is also known. The way to evaluate v ( y ) is quite simple: when the value of I is i = i1 or i2 , the value of V is v = v1 or v2 where vk = R ik ( for k = 1 or 2 ). The probability P of having i1 < i < i2 is also that of having v1 < v < v2 , which is also, by definition of v(y): ∫ = 2 1 v v v(y)dy P , which can also be written dP = v(y) dy . Therefore, the two values of dP can be related and : dP = v(y) dy = i(x) dx. When the voltage across the resistor is y , the intensity is x = y / R. In the same way, the effect corresponding to dx is dy = R dx … and dx = ( 1 / R ) dy. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 160 Replacing, we get: dP = v(y) dy = i(x) dx = i ( y / R ) (1 / R ) dy , which, in turn, gives: v ( y ) = (1 / R ) i ( y / R) , the relation between the probability densities corresponding to the random variables I and V. In this example, great care has been taken to clearly designate the random variables and the values they can take… Obviously, some more synthetic presentation could have been used … as long as it is always clear for the reader what the various symbols do represent! Other types of presentations may be found later in this annex. The multiplication of a random variable by a constant has been presented in a more systematic manner in clause D.3.2. D.2.1.1.1.2 Verifications concerning Ohm's law When providing the definitions and "basic" characteristics of probability densities characterizing random variables, 2 criteria had been expressed. A probability density, p ( x ) ,in general, and in this case, the probability density associated with V , v ( y ) shall be such that: - 0 ≥ v(y) - ∫ +∞ ∞ − = 1 dy v(y) It is therefore wise to verify the 2 properties, which, in practise, could help detecting problems occurred during the calculations. Obviously, if 0 ≥ ∀ i(x) x , then 0 ≥ v(y) . Concerning the second relation, verifications can be done on specific situations (for a probability density i(x) ) or in a more generic manner: ∫ +∞ ∞ − = dy v(y) ∫ +∞ ∞ − dy R) i(y R / ) / 1( By introducing t = y / R ( which gives dt = dy / R , and dy = R dt ) , this equation may be transformed into: ∫ +∞ ∞ − = dt i(t)R R) / 1( ∫ +∞ ∞ − = i(t)dt R R ) / ( 1 ∫ +∞ ∞ − = i(t)dt . Which ensures that v ( y ) can be a proper probability density function characterizing some random variable (hopefully V, should the above calculations be correct!). D.2.1.1.2 Uncertainties and the usage of Ohm's law The set up discussed in clause D.2.1.1.1 could have been used in order to measure the value of the resistor, having in hand a current generator (G) and a voltmeter. For this purpose, G would have been expected to deliver a known current i0 and the voltage v0 found, would have been supposed to provide the value of the resistor, R0 : R0 = v0 / i0. Unfortunately, G does not provide exactly i0 , but it provides i , related to a random variable, I , of which only the probability density, i ( x ) is known. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 161 In order to simplify the discussion, the voltmeter is supposed to provide the true value of v , the voltage across the resistor. In order to simplify also the discussion, the value of the resistor is also expected not to change during the measurement (it had been called R0 to reflect this characteristic). The uncertainty of the measurement of the resistor is, in this case, the result of the uncertainties relating to i. In fact, in a practical case, the value measured by the voltmeter would have been mapped to a value in Ohms, using the sought relation between R0 and i0 : R0 = v / i0 = v ( 1 / i0 ). Therefore the statistical properties of the voltage measured across the resistor v ( y ) would have been mapped (multiplication by a constant factor , k = ( 1 / i0 ) ) to the results of the reading of the value of the resistance. Finally, the measured value of the resistance can be considered as a random variable, R , linked to the voltage measured, the random variable V , by R = k V. The properties of V have been calculated above; its probability density is v ( y ) , and: v ( y ) = (1 / R0 ) i ( y / R0). Similarly, noting that R = k V (in the same way as V = R I , see also clause D.3.1 ), the probability density r ( z ) of R can be expressed using function of v ( y ) : r ( z ) = ( 1 / k ) v ( z / k) and finally r ( z ) = ( 1 / k ) v ( z / k ) = ( 1 / k ) ( 1 / R0 ) i ( z / k R0 ) = ( 1 / k R0 ) i ( z / k R0 ) The statistical properties of R (probability density r ( z ) ) are known as soon as the statistical properties of I , depending on the generator, are known … In short, the measurement uncertainty of the measurement is directly depending upon I (and i ( x )): by definition, the error made in the measurement of the value of the resistance is ε , with ε = z – R0. Therefore, the probability of the error having a particular value ε relates directly to r ( z ) and, in turn, to i ( x ) … ε ε ε ε = z – R0 with r( z ) = [ ( 1 / k R0 ) i ( z / k R0 ) ] . The error , ε , can, beyond its probability density ε (t) be characterized by other statistical properties such as its mean value or its standard deviation. The value of such parameters can be calculated from the expression given above, using the general relations given in clause D.3, but it can be also calculated directly, as shown below (see clause D.2.1.1.3). The expression of the error, above, also shows that there may be some influence of the value of the measurand on the estimation of the uncertainty. This is further developed in clause D.4 where influence quantities are addressed. D.2.1.1.3 Examples concerning Ohm's law using particular distributions D.2.1.1.3.1 Rectangular distributions and the corresponding interpretation of uncertainties The properties of a rectangular distribution defined by a parameter A have been given in clause D.1.3. As a follow on from the example of the measurement of the resistor where: r ( z ) = r ( R0 + ε)ε)ε)ε) = ( 1 / k R0 ) i ( z / k R0 ) special cases can be further discussed. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 162 Let us assume that the probability density i ( x ) is rectangular, centred around i0 and having a value 1/2a between i0 – a and i0 + a ( a is given, for instance, in mA): r ( z ) , the probability density of having a particular value as "the measured value" will also be given by a rectangular distribution centred around ( z / kR0 ) = i0 z = i0 ( kR0 ) = R0; with boundaries for z / kR0 = + a z = + a R0 / i0 ; and having a density ( 1 / 2a )( 1 / k R0 ) = i0 / ( 2a R0 ) . As a result, the "measurement error" can also be considered as a random variable, of which the probability of having a value, ε , ε , ε , ε , corresponds to a probability density function: - centred around 0 - having a rectangular shape with boundaries at + a R0 / i0 - and a density i0 / (2a R0). The interpretation of these results could be two fold: - worst case approach R0 = (v / i0 ) + a R0 / i0 - statistical approach the value of the resistor is R0 and the probability of error has a standard deviation of a R0 / i0 divided by square root of 3 (providing the "measurement uncertainty" for some particular confidence level … See also clause D.5.6) (see also D.1.3.1 concerning the standard deviation of a rectangular distribution). The confidence level can be subsequently improved, by multiplying the value of the measurement uncertainty indicated above (multiplication by 1,96 in the case of normal distributions … as indicated in TR 100 028-1 [6], clause 4.1, in order to change the confidence level from 68,3 % to 95 %) …(see also clause D.5.6). It is clear from the above that the multiplication of the above value by square root of 3 would return back the full span of the distribution (100 % confidence). In this case the span of the worst case approach and that of the statistical approach can both be easily calculated. D.2.1.1.3.2 Gaussian distributions and the corresponding interpretation of uncertainties Calculations similar to the above could be performed directly. However, it looks more practical to use the results obtained in D.3, in order to find the parameters of the uncertainty. In fact, it is possible to cut it short to: - random variable I "standard deviation" ( the input given … ) : σσσσI - random variable V = RI σσσσV = R0 σσσσI - random variable R = k V σσσσR = k σσσσV - random variable "measurement uncertainty" σσσσ = σσσσR = ( R0 / i0 ) σσσσI The above presentation is, in fact independent of the distribution addressed … ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 163 One difference with clause D.2.1.1.3.1 is that in the case of Gaussian distributions, the "standard deviation", σσσσ , appears explicitly in the equation of the probability density function, 2 2 2 2 1 σ π σ x e y − = while it has to be calculated from parameter "A", in the case of rectangular distributions … Another difference is that, if random variable I has a Gaussian distribution, there is not, per say, a genuine "worst case" situation, since there is a non-zero probability of i taking any value, which would result in the measured value of the resistor … having also any value! (the value of the random variable in the case of rectangular distributions has lower and upper bounds, but not in the case of normal distributions). D.2.1.2 A basic voltmeter In order to penetrate further in the area of measurement uncertainties, let us consider how one could build a voltmeter. For the sake of the discussion, in order to build a single-scale voltmeter, two basic components could be assembled: - a resistor of value R1 - a micro-Amperemeter. In order to simplify the discussion: - the resistor could have been taken from a set of resistors given with a certain uncertainty (e.g. 2 % resistors) - the micro-Amperemeter can be considered not to introduce any further uncertainty. As an example, the micro-Amperemeter could have a full scale deflexion for 50 µA and an internal resistance of 2 kilo Ohms (electro-mechanical) or infinite internal resistance (electronic device). The usage of a resistor R1 of 200 kilo Ohms would cater for a full scale of 10 V. V = R I and therefore dV = I dR + R dI (i.e. "differentiation"). Or, noting that V = R I and dividing by V both sides: I dI R dR V dV + = (i.e. "logarithmic differentiation"). The micro-Amperemeter was not supposed to contribute for the uncertainty, therefore dI = 0 , and: dV = I dR or R dR V dV = . Should dR be the random variable characterizing the resistor (i.e. by its probability density), it could be considered as having a rectangular distribution (plus or minus 2 % of 200 000, which is plus or minus 4 kilo Ohms). Conversely, the random variable to be considered could have been R dR and as a result, the distribution would also have been rectangular, expressed in percentage: plus or minus 2 %. Obviously, both expressions are equivalent. For the voltmeter, the performance could have been expressed in percent ("relative uncertainty"): - plus or minus 2 %. This would have corresponded to an "absolute uncertainty" of 200mV on a full scale deflexion. This presentation shows that a meter can be considered as a perfect device (providing some reading) coupled to some other set of components "responsible" for the uncertainty. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 164 When using this voltmeter to evaluate some voltage, resistor R1 could as well be incorporated in the rest of the test set up … This presentation has been further suggested in clause D.5. D.2.1.2.1 Building a multi-range voltmeter In the same way as resistor R1 could have been used for a single-scale voltmeter, a set of resistors having different values could have been used in support of several ranges, e.g.: - resistor R2 2 MegaOhms could be used for a 100 Volt range; and - resistor R3 20 MegaOhms could be used for a 1000 Volt range. Should all the resistors be 2 %, then the performance of the Voltmeter would have been 2 % in all ranges. However, it is clear that the real value of each resistor Rn is not known, nor any of the actual ratios such as (resistor Rn ) / (resistor Rp ) . As a result, readings in the different scales of this voltmeter can be considered to have measurement uncertainties statistically independent. D.2.1.2.2 Correlations between measurements with different voltmeters Having in hand sets of resistors with the various values Rn … Rp allows for the building of several voltmeters with the same design, (i.e. as described above). Assuming that in each set of resistors, the actual resistance values are different, while respecting the 2 % uncertainty (rectangular distribution) clause (for a resistor the usual term would be 2 % tolerance), all the voltmeters would provide statistically independent readings in each of the scales, but always within the 2 % uncertainty (rectangular distribution). It can be interesting, however, to go a little further. Some measurements use substitution methods (see clause D.5). In this case, it can be important to know the statistical independence of the uncertainties relating to the various evaluations. When using the same voltmeter and the same range : uncertainty values are not statistically independent. When using the same voltmeter and different ranges : uncertainty values are statistically independent. When using another voltmeter : uncertainty values are statistically independent. As a result, great care has to be taken when translating the test set up into the calculation of the uncertainty as two test set up and procedures almost identical can result in different calculations (see also clause D.3.4). Another situation can be found in the "Example clauses" of the present document: two attenuators are used in a test set up and are to be measured. The uncertainties corresponding to these two devices are to be treated in a different manner if the evaluation of their characteristics is statistically independent (i.e. measured with different instruments) or not (i.e. measured with the same instrument, same range, etc …). In the case of the Voltmeters "built" above, it is quite clear when uncertainties are independent or not (there is only one source of uncertainty) … in real life, the situation may be less clear … but, in any case, care should be taken in order to avoid clear mistakes … which may be a real problem, since such mistakes are almost impossible to be detect afterwards (it really depends on how the individual measurements were performed and several different results may be equally likely). As it is indicated in clause D.3.4, in general, the contribution of independent contributions are more favourable in terms of uncertainties: in case of doubt, it is therefore better to make the measurements which could have introduced some correlation with different instruments, in order to make it crystal clear that no correlations were introduced. Extreme care has therefore to be exercised in the case of substitution measurements where the effect may be totally opposite (the "aim of the game", in the case of substitution measurements, is to have two measurements correlated, as much as possible, in order to discard the majority of the contributions … by making "a difference" between two "consecutive" measurements); see also clause D.5. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 165 D.2.1.3 Adding voltages This clause was intended to: - provide an example of addition of random variables (see D.3.3 for the corresponding theoretical approach) - give some practical support in order to continue the discussion started on D.2.1.2.2. Two resistors in series can be used as a voltage splitter. When the two resistors are supposed to be identical, the voltage across them is supposed to be identical. Such a set up could be used to increase the range of the home built voltmeter discussed above. However, in order to measure the voltage across one of these two identical resistors, Voltmeter(s) can be used in different ways. More precisely, the measurement can be made using one or two ("identical") voltmeters. As a result, in order to have an idea whether the uncertainties are correlated or not, several questions may be asked, e.g.: "Was the voltmeter used for both resistors the same, and what are the possible correlations between uncertainties ?". Clause D.2.4 addresses the question "independent or not" , which is fundamental, but is often forgotten. D.2.1.4 The Wheatstone Bridge This clause is intended to show ways of handling more complex systems … It also shows that the statement that "all measurements are based on linear operations" is not correct at all times. As a result, there are days when other operations than RSSing may have to be performed. Such bridges are often used to measure the value of an unknown resistor X using a set of calibrated resistors. Assume the bridge is built using 3 calibrated resistors P , Q , R (used as a reference) and a meter g , powered by e . Appropriate bridges can also be used for the evaluation of capacitors and other impedances. i g P Q R X e ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 166 D.2.1.4.1 Fully balanced Bridge When the bridge is balanced, the current crossing g is zero. Under these circumstances: Q R P X = . By logarithmic differentiation we get: dX dP dR dQ X P R Q = + − . This expression can interpreted as follows: small variations of P , Q and R , dP , dQ , and dR will result in small variations dX of X . These small variations can be due differences between the value noted on the resistor and the actual value of the component. Such errors will, in turn, generate and error in the measurement: | dX | will be the difference between the calculated value and the true value. Hard luck, the difference between the value noted on the resistor and the actual value of the component is generally not known (should it be known, then the true value should have been used!), and some idea of it is covered by the term uncertainty … In the worst case approach, the more unfavourable values of each contribution are to be used. As a result, the uncertainty on X , dX is given by: dX dP dR dQ X P R Q = + + . Should the uncertainty on all resistors be the same, then : 3 dX dP X P = . However, the probability that all components of the uncertainty are "pushing" the result in the same direction is small, if the various components do not have correlated properties. It can therefore be assumed that the "worst case" approach is, indeed, providing very conservative results. As done in other clauses before, it can be interesting, here also, to introduce the concept of random variables. A very simplistic approach would have been to say that 3 dX dP X P = is relating two random variables: - one related to the characteristic of the source of uncertainty dP P , - one related to the uncertainty of the measurement dX X ; these two random variables being related by the relation 3 dX dP X P = . The knowledge of the properties of the distribution of the source uncertainty would then immediately provide the sought results. Clause D.3.2 provides the relations between distributions obtained by multiplication by a constant, and associated properties. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 167 Using such results would have provided expressions such as: 2 2 9 X P σ σ = , which relate the standard deviations of the 2 distributions involved. However, this approach would provide, still, a conservative view of the situation. In order to take full advantage of the usage of the concept of random variables, then the previous expression should have been used.: dX dP dR dQ X P R Q = + − . A direct mapping with random variables: - 3 related with the characteristic of the sources of uncertainty , e.g. dP P , and - one related with the uncertainty of the measurement dX X , would have provided a linear relationship between these random variables. The knowledge of the properties of the distribution of the source uncertainties would then immediately provide the sought results. Clauses D.3.3 , D.3.4 and D.3.5 provide the relations between distributions, when obtained by linear operations and associated properties. Using such results would have provided expressions such as: 2 2 2 2 X P Q R σ σ σ σ = + + , which relate the standard deviations of the 4 distributions involved. Should the uncertainty on all resistors be the same, then this expression would become: 2 2 3 X P σ σ = . This expression recalls the expression found above, except that a factor of 3 has been introduced (or a factor of 3 between the standard deviations). Clause D.5 offers global approaches based on the principles indicated here. The calculations above were based on differentiation. However, the calculations could have been performed directly on P , Q and R , instead, using: Q R P X = . In such case, instead of using the relations supporting linear expressions, clauses such as D.3.6 and D.3.7 should have been used… and, heroically, right results should have been obtained, at least once the particulars of each distribution would have been given. D.2.1.4.2 Bridge not fully balanced When the bridge is not fully balanced, the current across g is not zero any more and its value can be found as follows. Using Thévenin's theorem, solve for i, ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 168 Remove g and find VTh, ( ) ( ) ( )( ) + + + − + = + − + = Q P X R X R P Q P X e Q P P X R X e VTh Remove e, replace with a short-circuit and find RTh looking back, P Q R X e VTh P Q R X RTh ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 169 ≡ ( ) ( ) ( )( ) Q P X R X R PQ Q P RX Q P PQ X R RX RTh + + + + + = + + + = Hence, ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) X R PQ Q P RX Q P X R G PR XQ e X R PQ Q P RX Q P X R G PX PR XQ XP e G Q P X R X R PQ Q P RX Q P X R X R P Q P X e G R V i R R R R Th Th + + + + + + − = + + + + + + − − + = + + + + + + + + + − + = + = . This expression is clearly more complex; however, by differentiation, it is easy to get some linear expression out of it... RTh P X R Q ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 170 This expression is very interesting due to the fact that, this time, the current i can be mapped into a random variable corresponding to the uncertainty of the test equipment. However, the approach used above (see clause D.2.1.4.1) can still be used: - identifications of the appropriate variables (including those referring to test equipment) - differentiation (makes it more easy, but is not really necessary) - mapping from electrical parameters to random variables - combination of the various random variables (corresponding to the various contributions to the uncertainty) - calculation of the sought results using the properties of these combinations, i.e. calculation of the combined uncertainty of the measurement considered. This is the basis of clause D.5 … D.2.1.5 Influence of temperature This clause is intended to discuss the effect of "influence quantities", and in this case, temperatures. It is also intended to highlight how the effects of these influence quantities can affect the uncertainties in different manners du to the possible correlation between the various effects. The equations above relate to 3 "known" (reference) resistors; each one may have its own reaction to temperature, but they may be "identical", as well.. In the case of a Wheatstone bridge, one can think of a rather small test set up. In this case, it can be assumed that the temperature is the same for all three resistors: so possibly similar equations (the reference resistors may be "identical") and correlated effects. However, bridges could also be used to measure high currents and clumsy EUTs. Dissipation of heat is not necessarily to be excluded, and is not necessarily the same in all 3 reference resistors. In some situations, it can also happen that each "reference" resistor is in a different environment. As a result temperatures may have to be taken as different or "independent" (and the effect of temperature on each resistor may also be different). The theoretical material needed to solve these situations can be found in clauses D.3.6 and D.4. It is however clear in this example that the experimental conditions may have a direct influence on the equations to be used. In this case, like in many others, the operator performing the experiments has to have an understanding of the work to be done and select the right equations, since he is the only one able to determine which variables are independent and which are not. It implies that the usage or predetermined calculations, examples or spread sheets has always to be handled with care. D.2.2 Modelling instruments In a measurement set up, in particular for the evaluation of radio equipment, can usually be found: - power supplies, signal generators, etc … (see discussion in D.1.1.1.1) - instruments allowing to evaluate some electrical signal (e.g. powermeters, voltmeters, etc…). It was already suggested in D.2.1.2 that a Voltmeter could be artificially split in two parts. More generally, most usual instruments (e.g. meters) can be considered as being composed of: - a perfect device (providing some reading) - coupled to some other set of components "responsible" for the uncertainty. These components could as well be incorporated in the rest of the test set up … and be analysed together with the "original test set up". This is one of the basis for the presentation which has been proposed in clause D.5. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 171 D.2.3 Comparison with worst case methods Among all the types of distributions referred to in the present document, only the "Normal distributions" provide a non- zero probability p ( x ) for all the values of x. All the others are "bound" (for values below a lower value of x and for values above some other value of x , p ( x ) = 0 ). It is clear that a probability density corresponding to a random variable obtained by a linear combination of random variables (see clause D.3.4) which have a bound probability density, is also bound. In such case, it is possible to consider either a probabilistic/statistical approach or a worst case approach for the evaluation of the measurement uncertainties. In the case of non bound distributions, obviously, no worst case approach is possible! This is further discussed in clause D.5.6. D.2.4 Independent or not …that is the question! D.2.4.1 Different effects All through out this annex, the fact that "events and random variables are independent or not", has been addressed. This is due to the fact that the probability of having simultaneously two events is the product of the probabilities of having each event, if and only if these events are independent: Prob (A and B) = Prob (A) x Prob (B) , when A and B are independent events. In the following clauses, this property is often written for small contributions, where the probability of events is given using probability densities: f ( x ) dx x g ( y ) dy (corresponding to having both f ( x ) dx AND g ( y ) dy ). Should C and D correspond to a single event (referred to under two different names), it is obvious that: Prob (C and D) = Prob (C) = Prob (D) which is fundamentally different from the above. D.2.4.2 Making the right choices It is therefore extremely important to identify among all the sources of uncertainty which are independent and which are not. For example, has some particular source of uncertainty (e.g. a cable or an attenuator) been used more than only once in the measurement ? If some component has been used twice, and if it can be considered that the resulting contribution to the uncertainty has not changed, then the corresponding contribution, in the calculation of the combined uncertainty is 2 σσσσ as opposed to σσσσ multiplied by square root of 2 …a value to be used when two "independent" sources of uncertainty are considered (e.g. when 2 different cables having the same characteristics have been used, instead of just only one). Through out the present document, random variables associated to parameters such as temperature or supply voltage have been addressed (relating for instance to "influence quantities"). It can be accepted, for example, that the same voltage being delivered by two independent power supplies correspond to two independent random variables … … while the room temperature of a small room could be considered as a unique random variable … unless there were good reasons to believe that the temperature in the room was not homogeneous, in which case, the effect of the temperature on various pieces of equipment of a particular test set up could be handled as relating to different and independent random variables. In many situations, only the person making the measurement is in a position to know which of the random variables concerned were independent and not. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 172 As a result, it is important to identify such situations and to handle the calculations accordingly. The effects resulting from such mis-evaluation are further addressed in clause D.3.4.6: as shown in clause D.3.4.6.2 taking for independent uncertainty sources which are not, results in an under-estimation of the combined uncertainty. D.3 Combination of distributions Clause D.2 has highlighted a number of situations where operations on random variables had to be performed, and, in particular operations on 2 random variables ("combinations" of random variables). In the present clause, a systematic approach has been used, in order to provide the equations (and formulas) and the properties of a number of usual (and simple) operations on random variables, including combinations thereof. If for some particular problem the usage of other combinations is needed, an attempt could be made to use the tools developed below or methods based on the approaches shown below, in order to complete the corresponding calculations (see, in particular, clauses D.3.9, D.3.10, D.3.11 and the table in D.3.12). In this clause, results corresponding to some usual combinations have been presented in a systematic manner. However, the end of the clause provides more general results. As a consequence, the calculations corresponding to usual combinations have either been obtained directly, or as an application of more general methods, in order to show examples of how to use them … the results being independent of the method used, it was not felt necessary to show (all the time) how to use more than one method for each calculation! For information, typing and searching was done at the same time … however, using the text editor is much more time consuming than writing the equations by hand. After some time, the typing was therefore lagging substantially behind the searching, with implies that new thoughts may have been imported in clauses left behind. It is expected that the reader will not suffer from this effect. It is also expected that both forward and backward cross-references will help the reader. There may also be differences in the notations (symbols) used, compared with those of annex D.2: it was felt that, in order to make the text easier to read, in clause D.2, notations should be closer to their usage from the physicist point of view, while, for D.3, priority should be given to notations making the mathematical expressions easier to read and to handle … it is expected, anyhow, that when reaching D.4 , the reader is expected to be familiar enough with all the concepts, so that the notations (symbols) chosen will have little importance! As a result a further proposal is made in clause D.3.10.6. In order to implement this proposal, 2 different character sets have to be used. After discussions within ETSI, the set "Monotype Corsiva" has been chosen. It has been used to designate the name of random variables. It has to be noted, however, that the tools used to draft the present document do not seem to allow the use of this character set in "equation boxes". Finally, it has to be noted that this clause was written in a way to be as simple and clear as practical. It has not the mathematical accuracy that could be expected in a mathematical book, in particular functions are expected to be "good" functions…so it may be easy to find special cases and functions for which the general findings do not exactly apply. To avoid such risks, it would have been necessary, in particular, to define probabilistic spaces and functions in a more formal way, which could have been considered out of the scope of the present document. D.3.1 Addition of a constant to a random variable This clause deals with: α + = F H , where F is a random variable and H the result of the addition to F of a constant αααα. Results in this clause could have been established directly; but it was felt as interesting to use this clause as an example of application of general expressions found in clause D.3.9. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 173 D.3.1.1 Evaluation of the corresponding distribution Clause D.3.9 provides the general expression of h ( z ) , the probability density of H , when some operation ( g ) has been performed on a random variable, F . The resulting probability density is given as: )) ( (' )) ( ( 1 1 z g g z g f h(z) − − = , where z = g ( x ) and (z) g x 1 − = (the reciprocal of g … has sometimes been expressed using the notation " °" , giving x = g° ( z ) as a result of keyboard limitations …but it is more usually expressed as (z) g x 1 − = ). In this particular case: g | x z = x + αααα | F H = F + αααα g' | x 1 (the derivative function of g) g-1 | z x = z - αααα (the reciprocal function of g ).... As a result: )) ( (' )) ( ( 1 1 z g g z g f h(z) − − = 1 ) ( α − = z f , or ) ( α − = z f h(z) . In the expression above g' > 0 … so there is no special care to be taken in relation to the absolute values found with the expressions discussed in this clause. The relation between the probability densities corresponding to the random variables F and H , is therefore: ) ( ) ( α − = z f z h . D.3.1.2 Verification It is obvious that: - ∫ +∞ ∞ − = 1 dz h(z) since the transformation is a simple translation; - and the sign of h is that of f (positive). The two criteria are, therefore, met. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 174 D.3.1.3 Means and standard deviations As a result of the general expression found in clause D.3.9, dx x f x g mh ) ( ) ( ∫ +∞ ∞ − = dx x f x ) ( ) (∫ +∞ ∞ − + = α dx x f x ) ( ) (∫ +∞ ∞ − = dx x f ) ( ) ( ∫ +∞ ∞ − + α dx x f x ) ( ∫ +∞ ∞ − = + α α α α 1 and the mean value, mh, is therefore: mh = mf + αααα .... In the same way, dx x f x g sh ) ( ) ( 2 2 ∫ +∞ ∞ − = dx x f x ) ( ) ( 2 ∫ +∞ ∞ − + = α dx x f x ) ( 2 ∫ +∞ ∞ − = + dx x f x ) ( 2∫ +∞ ∞ − α + dx x f ) ( 2 ∫ +∞ ∞ − α and therefore: sh² = sf² + 2 α α α α mf + αααα² . As indicated in clause D.1.2 (definitions) σσσσh² = sh² - mh² and, similarly, σσσσf² = sf² - mf² ; therefore , σσσσh² = sh² - mh² = sf² + 2 α α α α mf + αααα² - mh²…= sf² + 2 α α α α mf + αααα² - ( mf + αααα )² = sf² - mf² = σσσσf² and the "standard deviation" σσσσh is, finally: σσσσh = σσσσf (the standard deviation is unchanged). D.3.1.4 Examples of usage The conclusion of the paragraph above is that "the standard deviation is unchanged". As a result, in the examples found in the present document, practical situations where this clause would have been used, may have been overlooked! D.3.1.5 Examples of conversion An area where "radio" people often make conversions is the level in dBs. Some prefer dBm other dBµV, etc … and the conversion between such values is by the addition of a constant (a topic covered by the present clause). D 3.2 Multiplication of a random variable by a constant factor This clause deals with F H λ = , where F is a random variable and H the result of the multiplication of F by a constant factor λλλλ . It is supposed that λλλλ is not equal to 0 (zero). This clause is, in fact, very important: it shows how to handle multiplications by positive or negative expressions, a topic which will be discuss a number of times, later, in this annex. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 175 D.3.2.1 Evaluation of the corresponding distribution When F is a random variable characterized by the fact that the probability of F having a particular value x is given by the probability density f ( x ) , then, by definition: the probability Pf of having the random variable F having a value x such that: x1 < x < x2 is ∫ = 2 1 x x f f(x)dx P . Similarly, we can consider ∫ ∞ − = x f dt f(t) x P ) ( , and therefore (by differentiation) dPf = f ( x ) dx . Should H be the random variable resulting from the multiplication of F by λλλλ , then, with the current notations, its probability density is h ( z ) , to be evaluated. For each value x of F , the value z of the random variable H is : z = λλλλ x . D.3.2.1.1 Case λλλλ positive In the following, λλλλ is supposed to be a positive constant. The way to evaluate h ( z ) is very simple: when the value of F is x = x1 or x2 , the value of H is z = z1 or z2 where zk = λλλλ xk ( for k = 1 or 2 ). The probability P of having x1 < x < x2 is therefore also that of having z1 < z < z2 , which is also, by definition of h ( z ): ∫ = 2 1 z z h(z)dz P . This property can also be written as dP = h ( z ) dz (by differentiation, as it was done for Pf , above). Therefore, the two values of dP can be related and : dP = h ( z ) dz = f ( x ) dx . When the value of H is z , the value of x is x = z / λλλλ . In the same way, when λλλλ is positive , dx is corresponding to dz = λλλλ dx … and dx = ( 1 / λλλλ ) dz . Replacing, we get: dP = h ( z ) dz = f ( x ) dx = f ( z / λλλλ ) (1 / λλλλ ) dz , which, in turn, gives: h ( z ) = ( 1 / λλλλ ) f ( z / λλλλ ) , the relation between the probability densities corresponding to the random variables F and H. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 176 D.3.2.1.2 Case λλλλ negative Doing the same calculation as above, while noting that: - multiplying inequalities by negative numbers swaps the inequality signs, - and that, in the case of intervals, the leftmost value is expected to be smaller than the rightmost value, - and that, finally, in this particular case (λλλλ is now supposed to be negative) the correspondence between k = 1 and 2 have to be swapped for x and z , we get: h ( z ) = - ( 1 / λλλλ ) f ( z / λλλλ ) D.3.2.1.3 Conclusion Combining the two results found above we get the final result: ) ( 1 ) ( λ λ z f z h = . D.3.2.2 Verifications It is clear, noting: - the case where λλλλ is positive, - and also that 1 λ is positive when λλλλ is negative, that in all cases: 0 ≥ h(z) . What then for the other requirement ? ∫ +∞ ∞ − =1 dz h(z) ? When λλλλ is positive , replacing h by its expression using f and then by substitution writing that x = z / λλλλ (and therefore dx = dz / λλλλ ) we get: 1 1 ) ( 1 ∫ ∫ ∫ ∫ +∞ ∞ − +∞ ∞ − +∞ ∞ − +∞ ∞ − = = = = dx f(x) dx f(x) dz z f dz h(z) λ λ λ λ . When λλλλ is negative then the use of ε can be useful. As indicated in clause D.3.10.3, for λλλλ negative the value of εεεε is -1 (by definition | ε | = 1 and ε has the sign of λλλλ ). The change of variable indicated above inverts upper and lower bounds in the integration. As a result we get: ( ) 1 1 ∫ ∫ ∫ ∫ ∫ +∞ ∞ − −∞ ∞ + −∞ ∞ + −∞ ∞ + +∞ ∞ − = = − = = = dx f(x) dx f(x) dx f(x) dx f(x) dz h(z) ε λε λ . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 177 This type of calculation will be found a number of times in this annex (e.g. in clause D.10). D.3.2.3 Means and standard deviations Once the definition has been written and simple calculations completed (exactly as above), it can be found that the mean value, mh , is: mh = λλλλ mf (whether λλλλ is positive or negative). As an example, let's make the calculation for λλλλ < 0 (and calling y the variable): dy ) y f( y dy h(y) y mh λ λ 1 − = = ∫ ∫ +∞ ∞ − +∞ ∞ − Should x be defined as x = y / λλλλ , we get dx = dy / λλλλ , and f h m dx f(x) x dx f(x) x dy ) y f( y m λ λ λ λ λ + = + = − = − = ∫ ∫ ∫ +∞ ∞ − −∞ ∞ + +∞ ∞ − 1 Similarly, it can be easily shown that "standard deviation" σσσσh is such that: σσσσh² = λλλλ² σσσσf² . For positive values of λλλλ , without risk, it can be written that σσσσh = λλλλ σσσσf . However, in order to avoid problems with negative values, when λλλλ is negative, it can be as easy to use the expression above (σσσσh² = λλλλ² σσσσf² ); after all, for the purpose of RSSing, which is what has been done all over the present document, the expression needed is σσσσh². D.3.2.4 Examples of usage Properties related to multiplications by constants have already been used in clause D.2.1.4 (relating to the Wheatstone bridge)… D.3.2.5 Examples of conversions In the radio world, a wide range of units is often used: e.g. µV, mV, V … A multiplicative factor of 1000 is therefore often found. This factor may also be found when handling the corresponding standard deviations. (It is not surprising, but cannot be taken for granted before any evidence is given! The usage of units in a probabilistic environment is also discussed in clause D.3.10.7). D.3.3 Sums (additions) of random variables This clause deals with G F H + = , where F and G are independent random variables and H is a combination (additive) thereof. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 178 D.3.3.1 Evaluation of the corresponding distribution When F is a random variable characterized by the fact that the probability of F having a particular value x is given by the probability density f ( x ) , then, by definition: the probability Pf of the random variable F having a value x such that: x1 < x < x2 is ∫ = 2 1 x x f f(x)dx P . Similarly, we can consider ∫ ∞ − = x f dt f(t) x P ) ( , and therefore (by differentiation) dPf = f ( x ) dx . When G is also a random variable, characterized by the fact that the probability of G having a particular value y is given by the probability density g ( y ) , then, by definition: the probability Pg of the random variable G having a value y such that y1 < y < y2 is ∫ = 2 1 y y g g(y)dy P . Similarly, we can consider ∫ ∞ − = y g dt g(t) y P ) ( , and therefore (by differentiation) dPg = g( y ) dy . Should H be the random variable resulting from the addition of F and G , then its probability density h ( z ) , is to be evaluated. For each value x of F and y of G , the value z of the random variable H is : z = x + y . The way to evaluate h ( z ) is simple: the probability of having the value of F within a very small interval [x , x + dx] is f ( x ) dx ; similarly, the probability of having the value of G within a small interval [y1 , y2] is g( y ) ( y2 - y1 ) = g ( y ) Dy where Dy = y2 - y1 , and where it is assumed that g( y1 ) = g ( y2 ) = g ( y ) ( is a small interval); under both circumstances, we get the value of H within [z1 , z2] where zi = x + yi (neglecting dx , very small compared with Dy ) and the probability of such an event (the contribution of dx in h ( z ) ) is g ( y ) Dy f ( x ) dx (the probability of having both events is the product of the probability of having each event, when the events are independent). ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 179 When Dz = z2 - z1 , by definition, h ( z ) Dz is the probability of having the value of H within [ z1, z2 ] , and is therefore, the sum of the probabilities of all the individual contributions, corresponding to all values of x : ∫ +∞ ∞ − = dx x f Dy y g Dz z h ) ( ) ( ) ( . Since Dz = z2 - z1 = x + y2 – ( x + y1) = y2 – y1 = Dy , we have Dz = Dy and noting that y = z – x , the integral above becomes ∫ +∞ ∞ − − = dx x f Dz x z g Dz z h ) ( ) ( ) ( which can be simplified into ∫ +∞ ∞ − − = dx x f x z g z h ) ( ) ( ) ( . This expression provides the value of h ( z ) as a function of f ( x ) and g ( y ) … which is the relation between the probability densities corresponding to the random variables F , G and H. NOTE: The result given above, could also have been found using the concept of substitutions discussed in clause D.3.10.3 … In this case, the probability of having simultaneously two independent events is the product of the two corresponding probabilities; therefore, it could have been written that: ∫ +∞ ∞ − = dx x f y g z h ) ( ) ( ) ( , while y x z + = . Using the properties of substitutions given in clause D.3.10.3, y could have been replaced as follows: x z y y x z − = ⇒ + = , and noting that the corresponding derivative function is 1 (see D.3.10.3), as a result we find: ∫ +∞ ∞ − − = dx x f x z g z h ) ( ) ( ) ( . D.3.3.2 Verifications When providing the definitions and characteristics of probability densities characterizing random variables, 2 criteria had been expressed. The probability density associated with H , h ( z ) shall be such that: - 0 ≥ h(z) - ∫ +∞ ∞ − = 1 dz h(z) It is therefore wise to verify the 2 properties, which, in practise, could help detecting problems occurred during the calculations. Obviously, when 0 ≥ ∀ f(x) x and 0 ≥ ∀ g(y) y then 0 ( ≥ z) h . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 180 Concerning the second relation, verifications can be done as follows: ∫ +∞ ∞ − = dz h(z) = − ∫ ∫ +∞ ∞ − +∞ ∞ − dz dx f(x) x z g ) ( dx dz x z g x f ∫ ∫ +∞ ∞ − +∞ ∞ − − ] ) ( [ ) ( By introducing t = z – x ( dt = dz , where x is considered as a constant) , this equation may be transformed into: ∫ +∞ ∞ − = dz h(z) ∫ ∫ +∞ ∞ − +∞ ∞ − = dx dt t g x f ] ) ( [ ) ( ∫ +∞ ∞ − = dx x f ]1[ ) ( 1 ∫ +∞ ∞ − = f(x)dx . Which ensures that h ( z ) can be a proper probability density function characterizing some random variable (hopefully H, should the calculations in D.3.3.1 be correct!). D.3.3.3 Means and standard deviations The method used in the calculations of clause D.3.5.3 (which were fully expanded) can also be used in this case … with the change of variable : t = z – x ; and the results are two fold: - the mean value, mh , is: mh = mf + mg - and "standard deviation"σσσσh is: σσσσh 2 = σσσσf 2 + σσσσg 2 (Similar calculations have been fully expanded in cases where great care was needed. See other usual operations (e.g. multiplications) in clause D.3.) D.3.3.4 Examples This last expression is certainly the expression which has been more often used in the present document: it is the basis for "RSSing" … D.3.3.5 Adding several distributions The corresponding effects are very different from case to case …as shown in clauses D.1.3.2 and D.1.3.3, the addition of two rectangular distributions can generate either trapezoidal or triangular distributions. The addition of several rectangular distributions is further addressed in clause D.3.3.5.2. Clause D.3.3.5.2.2 provides an interesting result relating to the addition of an infinite number of rectangular distributions. D.3.3.5.1 Adding Normal distributions D.3.3.5.1.1 Using the expressions giving the probability density D.3.3.5.1.1.1 Case where two identical Normal distributions are added Let us consider two Normal (Gaussian) distributions having the same standard deviation and no offset: 2 2 2 1 1 2 x y e σ σ π − = ; and ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 181 2 2 2 2 1 2 x y e σ σ π − = ; corresponding to two independent random variables. Clause D.3.3.1 provides: ∫ +∞ ∞ − − = dx x f x z g z h ) ( ) ( ) ( . as the distribution corresponding to the sum of the two independent random variables. With appropriate notations, we get: 2 2 2 2 ( ) 2 2 1 1 ( ) 2 2 z x x h z e e dx σ σ σ π σ π − +∞ − − −∞ = ∫ ; Simplifying : 2 2 2 2 ( ) 2 2 2 1 ( ) 2 z x x h z e e dx σ σ π σ − +∞ − − −∞ = ∫ ; and 2 2 2 2 ( ) 2 2 2 1 ( ) 2 z x x h z e dx σ σ π σ − +∞ − + −∞ = ∫ ; or 2 2 2 1 ( ) 2 2 1 ( ) 2 z x x h z e dx σ π σ +∞ − − + −∞ = ∫ . The calculation of the squares provides: 2 2 2 1 2 2 2 2 1 ( ) 2 z zx x h z e dx σ π σ +∞ − − + −∞ = ∫ . Reorganizing, and noting the beginning of a square starting with x2 - z x : 2 2 2 2 2 1 2( ) 2 4 4 2 2 1 ( ) 2 z z x z x z h z e dx σ π σ +∞ − − + + − −∞ = ∫ or 2 2 2 1 2( ) 2 2 2 2 1 ( ) 2 z z x h z e dx σ π σ +∞ − − + −∞ = ∫ . Reassembling differently we get 2 2 2 2 2( ) 1 2 2 2 2 2 1 ( ) 2 z x z h z e e dx σ σ π σ − +∞ − − −∞ = ∫ and, separating what is "constant" (in relation to the integral) 2 2 2 2 2( ) 1 2 2 2 2 2 1 ( ) 2 z x z h z e e dx σ σ π σ − +∞ − − −∞ = ∫ . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 182 This expression is composed clearly of a first part, which looks like the expression of some Gaussian, multiplied by some coefficient K where 2 2 2( ) 2 2 z x K e dx σ − +∞ − −∞ = ∫ . Noting that 2 Bx e dx B π +∞ − −∞ = ∫ (as shown in clause D.1.3.4) and that a simple variable change ( X = x – z / 2) in the integral providing K can give: 2 2 2 2 X K e dX σ +∞ − −∞ = ∫ , it comes that 2 1 B σ = and K B π σ π = = . Replacing in the expression of h (z ) we get: 2 2 2 2 2( ) 1 2 2 2 2 2 1 ( ) 2 z x z h z e e dx σ σ π σ − +∞ − − −∞ = ∫ = 2 2 1 2 2 2 1 2 z e K σ π σ − 2 2 2 2 1 2( 2) 2 2 2 1 1 2 2 2 z z e e σ σ σ π π σ σ π − − = = . So we finally have: 2 2 2( 2) 1 ( ) ( 2) 2 z h z e σ σ π − = which is the expression of a Normal distribution having 2 σ as its standard deviation. This calculation shows that, under these specific conditions (i.e. the two distributions are identical and have no offset), the distribution corresponding to the addition of two Normal distributions is another Normal distribution having 2 σ as its standard deviation. It can be noted that the value found for the standard deviation ( 2 σ ) is consistent with the general expression given in D.3.3.3 … D.3.3.5.1.1.2 Case where two identical Normal distributions with different offsets are added Let us consider two Normal (Gaussian) distributions having the same standard deviation and different offsets: ( ) 2 2 1 2 1 2 1 σ π σ x x e y − − = and ( ) 2 2 2 2 2 2 1 σ π σ x x e y − − = , corresponding to two independent random variables. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 183 As above, clause D.3.3.1 provides: ∫ +∞ ∞ − − = dx x f x z g z h ) ( ) ( ) ( as the distribution corresponding to the sum of the two independent random variables. With corresponding notations, we get, calculating as above: ( ) ( ) 2 2 1 2 2 2 ( ) 2 2 1 1 ( ) 2 2 z x x x x h z e e dx σ σ σ π σ π − − − +∞ − − −∞ = ∫ . Simplifying: ( ) ( ) 2 2 1 2 2 2 ( ) 2 2 2 1 ( ) 2 z x x x x h z e e dx σ σ π σ − − − +∞ − − −∞ = ∫ and ( ) ( ) 2 2 1 2 2 2 ( ) 2 2 2 1 ( ) 2 z x x x x h z e dx σ σ π σ − − − +∞ − + −∞ = ∫ or ( ) ( ) 2 2 1 2 2 1 2 2 1 ( ) 2 z x x x x h z e dx σ π σ +∞ − −− + − −∞ = ∫ . The calculation of the squares provides: 2 2 2 2 2 1 1 1 2 2 2 1 2 2 2 2 2 1 ( ) 2 z x x zx xx zx x xx x h z e dx σ π σ +∞ − + + − + − + − + −∞ = ∫ . Reorganizing: 2 2 2 2 1 2 1 1 2 1 2 1 2 2 2 2 2 2 2 1 ( ) 2 x zx xx xx zx z x x h z e dx σ π σ +∞ − − + − − + + + −∞ = ∫ . And calculating, as above: 2 2 2 2 1 2 1 1 2 2 1 2( ) 2 2 2 1 ( ) 2 x zx xx xx zx z x x h z e dx σ π σ +∞ − − + − − + + + −∞ = ∫ , and 2 2 2 2 1 2 1 1 2 2 1 2( ) 2 2 2 1 ( ) 2 x zx xx xx zx z x x h z e dx σ π σ +∞ − − + − − + + + −∞ = ∫ , or reorganizing 2 2 2 2 2 1 2 1 2 1 2 1 1 2 2 ( ) 1 2 2( ( ) ) ( ) 2 4 4 2 2 1 ( ) 2 x x z x x x x z x x z zx z x x h z e dx σ π σ − − +∞ − + − − + − − − − + + + −∞ = ∫ 2 2 2 2 2 2 2 1 2 1 2 1 1 2 2 1 1 2 2 ( ) 1 1 2 ( 2 2 2 ) 2 2 2 2 2 1 ( ) 2 x x z x x x z zx x x zx zx z x x h z e dx σ π σ − − +∞ − + − + + − − + − + + + −∞ = ∫ 2 2 2 2 2 2 2 1 2 1 2 1 1 2 2 1 1 2 2 ( ) 1 1 1 1 2 ( ) 2 2 2 2 2 2 2 1 ( ) 2 x x z x x x z zx x x zx zx z x x h z e dx σ π σ − − +∞ − + − + + − − + − + + + −∞ = ∫ ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 184 2 2 2 2 2 2 2 1 2 1 2 1 1 2 2 1 1 2 2 ( ) 1 1 1 1 2 2 2 2 2 2 2 2 1 ( ) 2 x x z x x x z zx x x zx zx z x x h z e dx σ π σ − − +∞ − + − − − + + − − + + + −∞ = ∫ 2 2 2 2 1 2 1 2 1 2 2 1 2 ( ) 1 1 1 1 2 2 2 2 2 2 2 1 ( ) 2 x x z x x x z x x zx zx h z e dx σ π σ − − +∞ − + + + + + − − −∞ = ∫ 2 2 2 2 1 2 1 2 1 2 2 1 2 ( ) 1 1 2 2 2 2 2 2 2 2 1 ( ) 2 x x z x x x z x x zx zx h z e dx σ π σ − − +∞ − + + + + + − − −∞ = ∫ [ ] 2 2 1 2 1 2 2 ( ) 1 1 2 ( ) 2 2 2 2 1 ( ) 2 x x z x z x x h z e dx σ π σ − − +∞ − + + − + −∞ = ∫ . As in the calculation above, it is easy to split this integral in several parts; and using the above methods and results we get: [ ] 2 1 2 2 1 2 2 2 ( ) 2 1 1 2 ( ) 2 2 2 2 1 ( ) 2 x x z x z x x h z e e dx σ σ π σ − − + +∞ − − + − −∞ = ∫ … and finally : ( ) 2 1 2 2 ( ) 2( 2) 1 ( ) ( 2) 2 z x x h z e σ σ π − + − = which is the expression of a Normal distribution having 2 σ as its standard deviation and an offset equal to 1 2 x x + . This calculation shows that, under these specific conditions (i.e. same standard deviation and different offsets), the distribution corresponding to the addition of two Normal distributions is another Normal distribution having 2 σ as its standard deviation and an offset equal to the sum of the offsets. The values of the resulting standard deviation and offset are consistent with the general expression given in D.3.3.3 … D.3.3.5.1.1.3 Case of two Normal distributions having different standard deviations Let us consider two Normal (Gaussian) distributions having different standard deviations and no offset: 2 2 1 2 1 1 1 2 x y e σ σ π − = and 2 2 2 2 2 2 1 2 x y e σ σ π − = , corresponding to two independent random variables. Clause D.3.3.1 provides ∫ +∞ ∞ − − = dx x f x z g z h ) ( ) ( ) ( as the distribution corresponding to the sum of the two independent random variables. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 185 With corresponding notations, we get: 2 2 2 2 1 2 ( ) 2 2 1 2 1 1 ( ) 2 2 z x x h z e e dx σ σ σ π σ π − +∞ − − −∞ = ∫ . Simplifying : 2 2 2 2 1 2 ( ) 2 2 1 2 1 ( ) 2 z x x h z e e dx σ σ π σ σ − +∞ − − −∞ = ∫ and 2 2 2 2 1 2 ( ) 2 2 1 2 1 ( ) 2 z x x h z e dx σ σ π σ σ − +∞ − + −∞ = ∫ or 2 2 2 2 2 1 2 2 1 2 1 ( ) 2 1 2 1 ( ) 2 z x x h z e dx σ σ σ σ π σ σ +∞ − − + −∞ = ∫ . The calculation of the squares provides: 2 2 2 2 2 2 2 2 1 2 2 2 1 2 1 2 ( ) 2 1 2 1 ( ) 2 z zx x h z e dx σ σ σ σ σ σ π σ σ +∞ − − + + −∞ = ∫ . Reorganizing, and noting again the beginning of a square starting with x2 : 2 4 4 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 1 2 1 2 1 2 1 2 2 1 ( ) 2 ( ) ( ) ( ) 1 2 1 ( ) 2 z x z z x z h z e dx σ σ σ σ σ σ σ σ σ σ σ σ σ σ π σ σ +∞ − + − + − + + + + −∞ = ∫ or 2 2 4 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 1 2 1 2 1 2 1 ( ) 2 ( ) ( ) 1 2 1 ( ) 2 z z x z h z e dx σ σ σ σ σ σ σ σ σ σ σ π σ σ − + − − + +∞ + + −∞ = ∫ 2 2 2 2 2 4 2 2 2 2 2 2 1 2 2 1 2 2 2 2 2 2 2 1 2 1 2 1 2 ( ) 1 ( ) 2 ( ) ( ) 1 2 1 ( ) 2 z z z x h z e dx σ σ σ σ σ σ σ σ σ σ σ σ σ π σ σ + − − + − + +∞ + + −∞ = ∫ Reassembling differently, simplifying and separating what is constant, we get: 2 2 2 2 2 2 2 2 1 1 2 2 2 2 2 2 2 2 2 2 1 2 1 2 1 2 1 2 ( ) 1 2 ( ) 2 ( ) 1 2 1 ( ) 2 z z x h z e e dx σ σ σ σ σ σ σ σ σ σ σ σ σ π σ σ + +∞ − − − + + −∞ = ∫ or 2 2 2 2 2 1 2 2 2 2 2 2 2 2 1 2 1 2 1 2 ( ) 1 2 ( ) 2 ( ) 1 2 1 ( ) 2 z z x h z e e dx σ σ σ σ σ σ σ σ σ π σ σ + +∞ − − − + + −∞ = ∫ . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 186 This expression is composed clearly of a first part, which looks like the expression of some Gaussian, multiplied by some coefficient K where 2 2 2 2 1 2 2 2 2 2 2 1 2 1 2 ( ) 2 ( ) z x K e dx σ σ σ σ σ σ σ + +∞ − − + −∞ = ∫ . Noting that 2 Bx e dx B π +∞ − −∞ = ∫ (as shown in D.1.3.4) and that a simple variable change ( X = x – z / 2) in the integral providing K can give: 2 2 2 1 2 2 2 1 2 ( ) 2 X K e dx σ σ σ σ + +∞ − −∞ = ∫ , it comes that 2 2 1 2 2 2 1 2 ( ) 2 B σ σ σ σ + = and 2 2 1 2 2 2 2 2 1 2 1 2 2 2 1 2 2 ( ) ( ) 2 K B π π πσ σ σ σ σ σ σ σ = = = + + . Replacing in the expression of h (z ) we get: 2 2 2 1 2 1 2 2 2 ( ) 1 2 2 2 1 2 1 2 1 2 ( ) 2 ( ) z h z e σ σ πσ σ π σ σ σ σ − + = + which (hopefully!) can be simplified as: 2 2 2 1 2 1 2 ( ) 2 2 1 2 1 ( ) (2 )( ) z h z e σ σ π σ σ − + = + . So we finally get 2 2 2 1 2 1 2 ( ) 2 2 1 2 1 ( ) (2 )( ) z h z e σ σ π σ σ − + = + which is the expression of a good Gaussian (Normal) distribution having 2 2 1 2 ( ) σ σ + as its standard deviation. This calculation shows that, under these specific conditions (i.e. no offset and different standard deviations), the distribution corresponding to the addition of two Normal distributions is another Normal distribution having 2 2 1 2 ( ) σ σ + as its standard deviation. The value of 2 2 1 2 ( ) σ σ + for the standard deviation is consistent with the more general expression given in D.3.3.3 … D.3.3.5.1.1.4 Case of two different Normal distributions Anyone willing to calculate the general case (and willing also to possibly crash his word processor a number of times (which has occurred while typing clause D.3.3.5.1.1 , a clause with less than 300 k bytes, with Microsoft ™ Word 97 (on Windows 95), with or without Math Type version 4 installed, with a diagnostic like "unable to save file: not enough space on disk" while there were more than one hundred Mbytes on the hard disk)…) could try and write the corresponding equations … and would probably find (one day) the correct result. However, it could be quite useless …and painful. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 187 In fact, the calculations above show the structure of the complete calculation: - playing simultaneously with different standard deviations and offsets can only (as already seen above) generate terms in x2 , xz and z2 ; - as above, the expression could have been split into two parts, etc … - so at the end, the result would have been some Gaussian like shape with complicated coefficients. So, finally, we could only get an expression which could have been written as: ( ) 2 2 2 2 1 s s x s s e y σ π σ − − = . Similarly to what has been indicated previously, clause D.3.3 provides the general expressions of both the standard deviation and the offset of the distribution ( ys ) corresponding to the sum of the independent random variables. Therefore the values of s and s σ can be calculated directly from the offsets and standard deviations corresponding to the random variables being added as follows: with the notations used in this clause 1 2 s x x = + and 2 2 1 2 s σ σ σ = + . The corresponding distribution would therefore be ( ) 2 1 2 2 2 1 2 ( ) 2( ) 2 2 1 2 1 2 x x x sy e σ σ σ σ π − + − + = + . D.3.3.5.1.1.5 Conclusion The conclusion is that, as already announced in clause D.1.3.3.1, Normal distributions are "stable" when additions are performed on independent random variables having both Normal distributions. It is obvious that Normal distributions are also stable when the associated random variable is multiplied by a constant. Multiplying one random variable by -1 and then adding another would correspond to a subtraction. Since Normal distributions are stable when these two operations are performed, it becomes obvious that Normal distributions are also stable when random variables are subtracted. It can, therefore, be stated that Normal distributions are stable in relation to multiplication by a constant, addition or subtraction of the corresponding independent random variables. Obviously, the addition of any number of Normal distributions would also correspond to a Normal distribution … The actual shape of the distribution resulting from the combinations of independent random variables corresponding to different distributions, one Normal and the other rectangular, is not provided in the present version of the document, and could be a topic for further work. D.3.3.5.1.2 Example of application It takes me an average of : 21 minutes to go to my office; the distribution is Gaussian and the standard deviation is 10 minutes; it takes me an average of : 25 minutes to go from my office to the airport; and the standard deviation is 10 minutes. (the distribution is also Gaussian). I need to go to my office, pick up the last version of TR 100 028 (all parts), go to the airport and jump into a plane. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 188 The departure time slot is in exactly in one hour. What is the probability of missing my time slot ? Using the above, the reply is fully strait forward: - the time needed to go to the airport is the sum of the time to go to the office plus the time to go to the airport; - the corresponding random variables are, therefore to be added; - there is no indication that these variables are inter-related, so it will be assumed that they are independent; - the distribution corresponding to the addition of two Gaussian distributions is, as shown above, also a Gaussian; - and the average (mean value of the resulting Gaussian) is the sum of the averages, i.e. 21 + 25 = 46 minutes; - while the resulting standard deviation is equal to the original deviation (both deviations were equal to 10 minutes) multiplied by the square root of two, i.e. 14 minutes; - the security margin is 1 hour – the average duration (46 minutes) i.e. 14 minutes (therefore equal to 1 standard deviation in our case); - as seen in TR 100 028-1 [6] and fully developed in clause D.5 , the probability of being within plus or minus one standard deviation is 68,3 %; but if I arrive earlier, there is no problem … so the probability of being in time is 50 % plus one half of 68,3 % i.e. 50 % + 34 % = 84 %; - …and 16 % is the probability of missing the departure time slot! Obviously, bringing the original of TR 100 028 (all parts) in time is extremely important… so a good security margin should have been included. Clause D.5.6.2 shows that in the case of Gaussians (Normal distributions) the usage of an expansion factor of 1,96 provides a probability of 95 % of being within the new limits. In our case, once again, being earlier is not a problem … so the multiplication by this "expansion" factor would have provided a probability of 50 % + 47,5 % = 97,5 % of being in time, which, in turn would correspond to a probability of 2,5 % of missing the departure time slot. In this case, the security margin should have been 14 * 1,96 = 28 minutes , and I should have left 14 minutes before, in order to reduce to 2,5 % the probability of missing the departure time slot. In this particular case, increasing the security margin by 14 minutes would have reduced the probability of missing the slot from 16 % to 2,5 % (… general considerations on single sided limits can be found in clause D.5.6.2.8). Further reductions of the risk can be envisaged, but no one is sure of not having an engine problem or a tire puncture… In the case where Normal distributions are considered, it is impossible to reduce that probability to zero … that is why regular Airlines always count on their passengers' understanding …when they are late (passengers may understand, but not necessarily the rest of the World … that is why some ETSI Chairman, trusting regular Airlines may have found someone else sitting in the Chair when reaching the meeting room! (and possibly, someone not intending to give up the Chair for the remainder of the meeting!)). Such problems would not occur with finite distributions: if both distributions would have been rectangular (and would have had the same parameter), then their combination would have been a triangular distribution (see D.1.3.2 ). Under such circumstances, the problem above would also have been easy to solve, and the resulting values would, obviously, have been different… providing, this time, a chance for a worst case analysis and 100 % certainty: with finite distributions, it is also possible to implement a worst case approach, and be sure not to arrive late. As shown above, Gaussians are stable in relation to the addition; should there have been another action to complete before reaching the airport, it would have been possible to add its contribution in the same way. As shown in the following clauses, in the case of rectangular distributions, the shape of the resulting distribution depends on the number of contributions added. The increase of the security margin being specific of the shape of the distribution … in the case of addition of rectangular distributions, there would have been a need to evaluate the expansion factor for each particular number of contributions added. This could, obviously have been done, and implemented using a table. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 189 However, the fact that Gaussians are stable in relation to additions avoids the need to have a table of that nature when handling Normal distributions; but, on the other hand, many calculations on rectangular distributions are much more simple. D.3.3.5.2 Adding several rectangular distributions The case where two rectangularly distributed distributions are combined has already been addressed in clause D.1 (e.g. in clauses D.1.3.2 and D.1.3.3 ): the result obtained was respectively a triangular and a trapezoidal distribution (respectively in the case of identical parameters and of different parameters). In order to simplify the presentation, only distributions with a mean value of zero will be considered here below, in the remainder of clause D.3.3.5.2. However, noting that mh = mf + mg (see clause D.3.3.3), it would be very easy to generalize. D.3.3.5.2.1 Adding several rectangular distributions having the same parameter An examples using dice can be found in clause 4.1.3, in TR 100 028-1 [6]. This example, shows the result obtained when successively throwing up to 6 dice. Even though, this case addresses discrete probabilities, the results are comparable to those found with the combination of up to 6 rectangular distributions having the same parameter. As seen on the corresponding figures, the shapes tend to the shape of a Gaussian when the number of combinations increase. It has, however, to be noted that even if a sum having an infinite number of terms would tend towards the Normal distribution, in practical cases, there is only a finite number of contributions and: - there is still an upper and a lower bound (having the values + n A ) - so there is still the possibility of working on the basis of worst case methods. It is quite easy to see (although somewhat lengthy) that the resulting distributions have the following properties: - 1 single variable rectangular shape 1 horizontal line degree 0 - 2 random variables triangular shape 2 oblique lines degree 1 - 3 random variables parabolic segments smoothed curves (no angles) degree 2 - 4 random variables … pieces of curves of degree 3 … - N random variables … pieces of curves of degree N - 1 … NOTE: Clause D.3.3 provides the expression of the resulting distributions as integrals and not necessarily as explicit functions. However, some of the properties indicated above can be found using such type of expressions. Likewise, it is easy to see that: - 1 single variable rectangular shape p ( x ) has discontinuities - 2 random variables triangular shape p ( x ) has no discontinuities , p' ( x ) has discontinuities - 3 random variables parabolic segments p ( x ) has no discontinuities , p' ( x ) has no discontinuities p'' ( x ) has discontinuities - N random variables etc … Adding another distributions to the Nth combination is like smoothing the Nth combination, while expanding its spread by A (at each end of the "foot print" of the distribution). This process obviously generates a distribution slowly reaching infinity. A slow convergence into a normal distribution appears as a possibility: not many functions offer, as the exponentials do, an infinity of "good" derivative functions … ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 190 D.3.3.5.2.2 Adding several rectangular distributions having different parameters In practical situations, it is often found that there is a major contributor for the uncertainties, and then a number of smaller … So it can be interesting to understand what may happen when a family of rectangularly distributed distributions (having a different parameter) are added together; let us take an example: - distribution 1 defined by A1 = A - distribution 2 defined by A2 = q A1 - …/… - distribution n defined by An = q An -1 Like in the previous example, the result of the N first distributions (starting by the wider ones) is then smoothed by the N+1 th … and so on. For q << 1, the result is quite simple to be presented: - Sum of the first 1 distribution rectangle with spread A1 = A - Sum of the first 2 distributions trapezoidal shape with spread A1 + A2 - Sum of the first 3 distributions smoothed trapezoidal shape with spread A1 + A2 + A3 - Sum of the first n distributions smoothed trapezoidal shape with spread Sn = A1 + A2 +…+ An The spread corresponding to n distributions can be easily calculated: Sn = A1 + A2 +…+ An Sn = A + A q + A q2 +…+ A qn-1 ( ) q q A q q A S n n n − − = + + = − 1 1 .... 1 0 For q = (1/10), and a few distributions, this expression can be simplified: ( ) A q A q A Sn 1,1 1 1 1 ≈ + ≈ − = More exactly, Sn = 1,11111 A … A similar calculation can also be made in respect to the standard deviations … 3 p p A = σ and ( ) 3 3 2 1 2 2 2 − = = p p p q A A σ . ( ) 2 2 2 ) 1 ( 2 ) 2 )( 2 ( 2 0 2 2 1 1 3 .... 3 q q A q q q q A n n n − − = + + + = − ∑σ As above, and for q = (1/10), and a few distributions, this expression can be simplified: ( ) 01 ,1 3 1 3 1 1 3 2 2 2 2 2 2 A q A q A n ≈ + ≈ − = ∑σ . In a word, the standard deviation of the sum is almost equal to the standard deviation of the biggest contribution … Interesting also to note that the standard deviation of the sum, multiplied by square root of 3 is almost equal to the total span of the sum of the distributions … ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 191 Should this situation be found, by multiplying the RSS of all the contributions by square root of 3 ( = 1,732 …), the new value would provide a worst case approach for the measurement uncertainty (or a measurement uncertainty with a 100 % confidence)… The usual factor of 1,96 (providing a confidence level of 95 % in the case of a Normal distribution) would therefore be much larger than the factor needed in this particular case to provide a confidence level 100 % …the worst case. D.3.4 Linear combinations of random variables This clause deals with G F H µ λ + = Where F and G are independent random variables and H a combination thereof, and λ, µ are (positive) constants. D.3.4.1 Evaluation of the corresponding distribution D.3.4.1.1 Using a direct method Holding the breath for a while, and using the step by step approach used in clause D.3.3.1 , it would be possible to reach the result. However, the discussion relating to the effect of the various signs would split the work in a number of cases … making it even longer. Therefore, the following clause provides a way much more elegant to reach the results. D.3.4.1.2 Using the "Building blocs" method As opposed to the "direct method", with the method using "building blocs", several of the above properties are applied successively in order to reach the sought result. [ F f(x) ] [ λF ( |1/λ| ) f ( x/λ ) ] [ G g(y) ] [µG ( |1/µ| ) g ( y/µ ) ] By a double direct substitution (using D.3.3.1 above) we get: G F H µ λ + = h(z) = ∫ +∞ ∞ − − dx x z g ) x f( )) (( ) 1 ( µ λ λµ . D.3.4.2 Verification Should h ( z ) be a distribution, ∫ +∞ ∞ − = 1 dz h(z) applies … The other property (h ( z ) > 0 ) is obviously met. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 192 D.3.4.3 Means and standard deviations The method used in clause D.3.6.3 can also be used in this case … As a result the mean value, mh , is: mh = λλλλ mf + µµµµ mg and "standard deviation"σσσσh is then: σσσσh 2 = λλλλ2 σσσσf 2 + µµµµ2σσσσg 2 D.3.4.4 Examples In clause 6.5.5 of TR 100 028-1 [6], a theoretical analysis of 3rd order intermodulation is given. It provides a linear combination of terms. The calculations provided in clause D.3.4 allows for the explanation of the usage of coefficients 1 , 2 and 1 / 3 found in the components corresponding to the intermodulation, in relation with the RSS evaluation. D.3.4.5 Extrapolation This clause covers the case of: n nF F F H λ λ λ + + + = .... 2 2 1 1 where F1, F2, … Fn are independent random variables and H the combination thereof, and where λ1, λ2, … λn are constants. D.3.4.5.1 Extrapolation in the general case The expression of the distribution may be somewhat awkward. However, it is quite easy to group step by step the various random variables and to establish, as a result that: the mean value, mh , is: mh = λλλλ1 mf1 + λλλλ2 mf2+ … + λλλλn mfn and "standard deviation"σσσσh is then given by: σσσσh 2 = λλλλ1 2 σσσσf1 2 + λλλλ2 2 σσσσf2 2 + … +λλλλn 2 σσσσfn 2 D.3.4.5.2 Extrapolation in a particular case (RSSing) When all λλλλk are equal to 1 … this relation does simplify into the RSS … (the core of the "BIPM method"!). Therefore, RSSing is valid for the additive combination of independent random variables, where all coefficients λλλλk are equal to 1. D.3.4.5.3 Using differentiation When the equations of a system can be expressed as V = V ( x1 , … , xn ) , and it is possible to evaluate dV as dV = λλλλ1 dx1 + … + λλλλn dxn or (dV / V) = λλλλ1 dx1 + … + λλλλn dxn ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 193 then the above expression: σσσσh 2 = λλλλ1 2 σσσσf1 2 + λλλλ2 2 σσσσf2 2 + … +λλλλn 2 σσσσfn 2 provides the statistical properties of dV or dV / V as soon as the statistical properties of dx1 … dxn are know (e.g. the n σσσσdxn. ): σσσσdV 2 = λλλλ1 2 σσσσdx1 2 + λλλλ2 2 σσσσdx2 2 + … +λλλλn 2 σσσσdxn 2 or σσσσdV/V 2 = λλλλ1 2 σσσσdx1 2 + λλλλ2 2 σσσσdx2 2 + … +λλλλn 2 σσσσdxn 2 as appropriate. This relates immediately the uncertainties corresponding to the various elements of a measurement (i.e. the various contributions to the uncertainty), xi to the uncertainty of the result (i.e. the combined uncertainty). Further proposals concerning methodologies to relate systems (e.g. a measurement set up), random variables and uncertainties can be found in clause D.5. D.3.4.6 Case of non independent random variables This clause covers the case where: G F H µ λ + = F and G are non-independent random variables and H is a combination thereof, while λ and µ are constants. Under such circumstances, F can be written as k G . Therefore, H = (λλλλ k + µµµµ) G and: h ( z ) = ( 1/(λλλλ k + µµµµ ) ) g ( z / (λλλλ k + µµµµ) ) . As a result the mean value, mh , is (using D.3.2): mh = (λλλλ k + µµµµ) mg and "standard deviation"σσσσh is then: σσσσh = ( λλλλ k + µµµµ ) σσσσg or σσσσh 2 = ( λλλλ k + µµµµ ) 2 σσσσg 2 . These results are very different from those found above, when the random variables were independent. D.3.4.6.1 Comparison between results If F and G had been wrongly handled as independent random variables, σσσσh 2 = λλλλ2 σσσσf 2 + µµµµ2σσσσg 2 which, having, in reality σσσσf = k σσσσg would have given σσσσh 2 = λλλλ2 k2 σσσσg 2 + µµµµ2σσσσg 2 = (λλλλ2 k2 + µµµµ2 ) σσσσg 2 instead! This shows how important it is to assess, before any attempt to use "the RSS" method to identify which are the independent random variables…which may be quite difficult, if the system has not been analysed globally. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 194 Great care has therefore to be exercised while using the complete developed examples of calculation found in the main body of the present document, in order to identify, for a particular test set up, which are the independent random variables, and which are those which, for one or another reason, are in fact linked together (e.g. is the room temperature the same for all components, or not; has one particular instrument been used twice in the same configuration, or was it another instrument of the same type…or another configuration). Therefore, the calculations may differ from one test set up to another test set up even if they look almost identical…(see also clause D.2.4). D.3.4.6.2 Conclusions As ( a + b ) 2 = a2 + b2 + 2 a b , when a and b are positive, ( a + b ) 2 > a2 + b2 . This implies that taking random variables for independent when they are not, may lead to uncertainty values smaller than they are in reality (under estimation of the uncertainties). D.3.5 Subtraction of random variables This clause deals with: G F H − = , where F and G are independent random variables and H a combination (subtraction) thereof. D.3.5.1 Evaluation of the corresponding distribution When F is a random variable characterized by the fact that the probability of F having a particular value x is given by the probability density f ( x ) , then, by definition: the probability Pf of having the random variable F having a value x such that x1 < x < x2 is ∫ = 2 1 x x f f(x)dx P . Similarly, we can consider ∫ ∞ − = x f dt f(t) x P ) ( , and therefore (by differentiation) dPf = f ( x ) dx. When G is also a random variable, characterized by the fact that the probability of G having a particular value y is given by the probability density g ( y ) , then, by definition: the probability Pg of having the random variable G having a value y such that y1 < y < y2 is ∫ = 2 1 y y g g(y)dy P . Similarly, we can consider ∫ ∞ − = y g dt g(t) y P ) ( , and therefore (by differentiation) dPg = g( y ) dy. Should H be the random variable resulting from the subtraction of F and G , then its probability density h ( z ) , is to be evaluated. For each value x of F and y of G , the value z of the random variable H is : z = x - y . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 195 A way to evaluate h ( z ) is as follows: the probability of having the value of F within a very small interval [x , x + dx] is f ( x ) dx ; the probability of having the value of G within a small interval [y1 , y2] is g( y ) ( y2 - y1 ) = g ( y ) Dy where Dy = y2 - y1 , and where it is assumed that g( y1 ) = g ( y2 ) = g ( y ) (the interval is small). The interval within which z remains has to be looked at with attention … y1 < y2 , therefore - y1 > - y2 and x - y1 > x - y2 implying that z1 > z2 . Under both of the above circumstances, we get the value of H within [z2 , z1] where zi = x - yi (neglecting dx , very small compared with Dy ) and the probability of such an event (the contribution of dx in h(z) ) is f(x) dx g(y) Dy (the probability of having both events is the product of the probability of having each event, when the events are independent). When Dz = z1 – z2 , by definition, h ( z ) Dz is the probability of having the value of H within [z2, z1] and is, therefore, the sum of the probabilities of all the individual contributions, corresponding to all values of x : ∫ +∞ ∞ − = dx x f Dy y g Dz z h ) ( ) ( ) ( . Since Dz = z1 – z2 = x – y1 – ( x – y2) = y2 - y1 = Dy , we have Dz = Dy and noting that y = x –z , the integral above becomes ∫ +∞ ∞ − − = dx x f Dz z x g Dz z h ) ( ) ( ) ( which can be simplified into ∫ +∞ ∞ − − = dx x f z x g z h ) ( ) ( ) ( This equation provides the value of h ( z ) as a function of f ( x ) and g ( y ) … which is the relation between the probability densities corresponding to the random variables F , G and H. D.3.5.2 Verifications When providing the definitions and characteristics of probability densities characterizing random variables, 2 criteria had been expressed. The probability density associated with H , h ( z ) shall be such that: - 0 ≥ h(z) - ∫ +∞ ∞ − = 1 dz h(z) It is therefore wise to verify the 2 properties, which, in practise, could help detecting problems occurred during the calculations. Obviously, when 0 ≥ ∀ f(x) x and 0 ≥ ∀ g(y) y then 0 ( ≥ z) h . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 196 Concerning the second relation, verifications can be done in a generic manner (i.e. not depending on specific distributions): ∫ +∞ ∞ − = dz h(z) = − ∫ ∫ +∞ ∞ − +∞ ∞ − dz dx f(x) z x g ) ( dx dz z x g x f ∫ ∫ +∞ ∞ − +∞ ∞ − − ] ) ( [ ) ( By introducing t = x- z ( dt = - dz , where x is considered as a constant) , this equation may be transformed into: = ∫ ∫ +∞ ∞ − −∞ ∞ + = − dx dt t g x f ] )1 () ( [ ) ( ∫ ∫ +∞ ∞ − +∞ ∞ − = dx dt t g x f ] ) ( [ ) ( ∫ +∞ ∞ − = dx x f ]1[ ) ( 1 ∫ +∞ ∞ − = f(x)dx . Which ensures that h ( z ) can be a proper probability density function characterizing some random variable (hopefully H, should the above calculations be correct!). D.3.5.3 Means and standard deviations The method for evaluating the mean and the standard deviation for a number of operations discussed in clause D.3 is very similar. D.3.5.3.1 Mean value In the case of a subtraction of random variables, it has been shown that the resulting density of probability is: ∫ +∞ ∞ − − = dx x f z x g z h ) ( ) ( ) ( . The general expression of mh being: ∫ +∞ ∞ − = dz z h z mh ) ( , it comes that dz dx x f z x g z mh ∫ ∫ +∞ ∞ − +∞ ∞ − − = ) ( ) ( ∫ ∫ +∞ ∞ − +∞ ∞ − − = dx x f dz z x g z mh ) ( ] ) ( [ . For each particular value of x , the internal integral can be easily calculated by a simple change in variable: t = x - z . Under these circumstances, dz = - dt and: dx x f m x dx x f dt t g t x dx x f dz z x g z m g h ) ( ) ( ) ( ] ) ( ) ( [ ) ( ] ) ( [ − = − = − = ∫ ∫ ∫ ∫ ∫ +∞ ∞ − +∞ ∞ − +∞ ∞ − +∞ ∞ − +∞ ∞ − and g f h m m m − = . As a result the mean value, mh , is: mh = mf - mg which is valid independently of the distributions addressed (i.e. should they be normal or not). ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 197 D.3.5.3.2 Standard deviation In the present case, we have: ∫ +∞ ∞ − − = dx x f z x g z h ) ( ) ( ) ( . The general expression of sh 2 being: ∫ +∞ ∞ − = dz z h z sh ) ( 2 2 , it comes that: dz dx x f z x g z sh ∫ ∫ +∞ ∞ − +∞ ∞ − − = ) ( ) ( 2 2 ∫ ∫ +∞ ∞ − +∞ ∞ − − = dx x f dz z x g z sh ) ( ] ) ( [ 2 2 . For each particular value of x , the internal integral can be easily calculated by a simple change in variable: t = x - z . Under these circumstances, dz = - dt and: ∫ ∫ ∫ ∫ +∞ ∞ − +∞ ∞ − +∞ ∞ − +∞ ∞ − − = − = dx x f dt t g t x dx x f dz z x g z sh ) ( ] ) ( ) ( [ ) ( ] ) ( [ 2 2 2 ∫ ∫ ∫ ∫ +∞ ∞ − +∞ ∞ − +∞ ∞ − +∞ ∞ − + − = − = dx x f dt t g t xt x dx x f dt t g t x sh ) ( ] ) ( ) 2 ( [ ) ( ] ) ( ) ( [ 2 2 2 2 ∫ ∫ ∫ ∫ +∞ ∞ − +∞ ∞ − +∞ ∞ − +∞ ∞ − + − = + − = dx x f dt t g t t g xt t g x dx x f dt t g t xt x sh ) ( ] )) ( ) ( 2 ) ( ( [ ) ( ] ) ( ) 2 ( [ 2 2 2 2 2 2 2 2 2 2 2 ) ( ] ) 2 )1( ( [ g g f f g g h s m m s dx x f s m x x s + − = + − = ∫ +∞ ∞ − . Noting the relation 2 2 2 m s − = σ (or 2 2 2 s m σ = + ), we then get: 2 2 2 2 2 2 ( ) 2 ( ) h h f f f g g g m m m m m σ σ σ + = + − + + 2 2 2 2 2 2 2 g g g f f f h h m m m m m + + − + = + σ σ σ and replacing mh by its value ( ) 2 2 2 2 2 2 2 g g g f f f g f h m m m m m m + + − + = − + σ σ σ and, after lots of sweat and tears …. and noting that ( ) 2 2 2 2 f g f f g g m m m m m m − = − + we get (simplifying): 2 2 2 g f h σ σ σ + = , ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 198 which is valid independently of the distributions addressed (i.e. should they be normal or not). (alternatively, it could have been written: 2 h 2 h 2 h 2 h 2 h 2 h m s m s + = ∴ − = σ σ Hence, ( ) 2 g 2 f 2 g 2 g 2 f 2 f 2 h 2 g 2 f 2 g 2 f 2 h 2 g g f 2 f 2 g g f 2 f 2 h 2 g g f 2 f 2 g f 2 h m s m s s s m m s m m 2 s m m m 2 m s m m 2 s m m σ σ σ σ σ σ + = − + − = + = + + + − = + − + + − = − + which provides the same result …) D.3.5.4 Examples The fact that RSSing is used for both additions and subtractions of random variables may have hidden the use of subtractions in the numerous examples found in the present document. Substitution measurements are favoured for radio equipment. This is certainly an area where subtractions may have to be performed. D.3.5.5 Subtracting several distributions In order to avoid problems with the signs, operations involving several distributions have to be done more carefully than in the case of additions, e.g. handling one operation at the time (step by step approach). D.3.6 Multiplication of random variables This clause deals with: G F H = . where F and G are independent random variables and H is a combination (multiplication) thereof. Problems may be found, when the value of F or G is zero … (or too often equal to zero, creating possible convergence problems). Should this occur, then in that particular case, careful attention should be devoted to the situation. As written above, the operation is symmetrical in relation to F and G. However, the expression found below is not. By exchanging the role of F and G (or the role of x and y ) another expression may be found, which, in some cases could be more friendly for a particular usage. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 199 D.3.6.1 Evaluation of the corresponding distribution When F is a random variable characterized by the fact that the probability of F having a particular value x is given by the probability density f ( x ) , then, by definition: the probability Pf of the random variable F having a value x such that: x1 < x < x2 is ∫ = 2 1 x x f f(x)dx P . Similarly, we can consider ∫ ∞ − = x f dt f(t) x P ) ( , and therefore (by differentiation) dPf = f ( x ) dx. When G is also a random variable, characterized by the fact that the probability of G having a particular value y is given by the probability density g ( y ) , then, by definition: the probability Pg of the random variable G having a value y such that: y1 < y < y2 is ∫ = 2 1 y y g g(y)dy P . Similarly, we can consider ∫ ∞ − = y g dt g(t) y P ) ( , and therefore (by differentiation) dPg = g( y ) dy. Should H be the random variable resulting from the multiplication of F and G , then its probability density h ( z ) , is to be evaluated. For each value x of F and y of G , the value z of the random variable H is : z = x y . In fact, in the following, the situation is slightly different when x < 0 (the situation is comparable with that discussed in the case where λ was negative, in clause D.3.2). The way to evaluate h ( z ) is quite simple, and is given in the following. The probability of having the value of F within a very small interval [x , x + dx] is f ( x ) dx ; the probability of having the value of G within a small interval [y1 , y2] is g( y ) ( y2 - y1 ) = g ( y ) Dy (where Dy = y2 - y1 , and where it is assumed that g( y1 ) = g ( y2 ) = g ( y ) , Dy being considered as small ); when both events occur, then, the value of H is within [z1 , z2] where zi = x yi (neglecting dx , considered to be very small compared with Dy ) and the probability of such an event (which provides the contribution of dx in h( z ) ) is f( x ) dx g( y ) Dy . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 200 Case where x > 0. When Dz = z2 - z1 , by definition, h ( z )Dz is the probability of having the value of H within [z1, z2] and is, therefore, the sum of the probabilities of all the individual contributions, corresponding to all positive values of x : ∫ +∞ = 0 ) ( ) ( ) ( dx x f Dy y g Dz z h . Since Dz = z2 - z1 = x y2 – x y1 = x ( y2 – y1 ) = x Dy , we have Dz = x Dy and noting that y = z / x ( x non zero!) , the integral above becomes ∫ +∞ = 0 ) ( ) / )( / ( ) ( dx x f x Dz x z g Dz z h . Case where x < 0. When Dz = z2 - z1 , by definition, h ( z )Dz is the probability of having the value of H within [z1, z2] (where [z1, z2] is an interval and therefore z1 < z2 ) and is, therefore, the sum of the probabilities of all the individual contributions, corresponding to all negative values of x : ∫ ∞ − = 0 ) ( ) ( ) ( dx x f Dy y g Dz z h . Since Dz and Dy are intervals, Dz = z2 - z1 = | x | y2 – | x | y1 = | x | ( y2 – y1 ) = - x Dy , we have Dz = - x Dy and noting that y = z / x ( x non zero!) , the integral above becomes ∫ ∞ − − = 0 ) ( ) / )( / ( ) ( dx x f x Dz x z g Dz z h . Taking into account both positive and negative contributions of x , and simplifying by Dz , the two expressions above can be combined into ∫ +∞ ∞ − = dx x f x z g x z h ) ( ) ( ) 1 ( ) ( This relation provides the value of h ( z ) as a function of f ( x ) and g ( y ) … which is the sought relation between the probability densities corresponding to the random variables F , G and H. NOTE 1: When F or G take zero as a value, then the value of H is also zero, independently of the other random variable … NOTE 2: In the expression above, f and g have roles slightly different, which is not the case with H = F G. As a result, the expression (obtained by permutation): ∫ +∞ ∞ − = dy y z f y g y z h ) ( ) ( ) 1 ( ) ( should be as much appropriate as the expression given above, but could be more convenient in some cases i.e. when the variable y can be mapped to a physical variable which never reaches zero. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 201 D.3.6.2 Verifications When providing the definitions and characteristics of probability densities characterizing random variables, 2 criteria had been expressed. The probability density associated with H , h ( z ) shall be such that: - 0 ≥ h(z) - ∫ +∞ ∞ − = 1 dz h(z) It is therefore wise to verify the 2 properties, which, in practise, could help detecting problems occurred during the calculations. Obviously, when 0 ≥ ∀ f(x) x and 0 ≥ ∀ g(y) y then: 0 ( ≥ z) h . This situation is close to that when lambda was negative …in clause D.3.2. The verifications can be done in a generic manner, but with the help of the function ε (see clause D.3.10.3): ∫ +∞ ∞ − = dz h(z) = ∫ ∫ +∞ ∞ − +∞ ∞ − dz dx f(x) x x z g ) / )( / ( ε dx dz x z g x f x ] ) / ( [ ) ( ) / ( ∫ ∫ +∞ ∞ − +∞ ∞ − ε By introducing t = z / x ( dt = dz / x , x being considered as a constant, within the inside integral ) , this expression may be split into 2 parts and then transformed into: ∫ ∫ +∞ +∞ ∞ − = 0 ] ) ( [ ) ( ) ( ) / 1( dx dt t g x x f x ∫ +∞ = 0 ]1[ ) ( dx x f I , when x > 0 and ε = 1. And ∫ ∫ ∞ − −∞ ∞ + = − 0 ] ) ( [ ) ( ) ( ) / 1 ( dx dt t g x x f x ∫ ∞ − = 0 ]1[ ) ( dx x f J when x < 0 and ε = - 1. Finally, it can be noted that I + J = 1 , which ensures that h ( z ) can be a proper probability density function characterizing some random variable (hopefully H, should the above calculations be correct!). D.3.6.3 Means and standard deviations It has been indicated above that, using the function ε: ∫ +∞ ∞ − = dx x f x z g x z h ) ( ) ( ) 1 ( ) ( ε . The general expression of mh being: ∫ +∞ ∞ − = dz z h z mh ) ( , ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 202 it comes that: dz dx x f x z g x z mh ∫ ∫ +∞ ∞ − +∞ ∞ − = ) ( ) ( ) (ε ∫ ∫ +∞ ∞ − +∞ ∞ − = dx x f dz x z g z x mh ) ( ] ) ( )[ (ε . For each particular value of x, the internal integral can be easily calculated by a simple change in variable: y = ( z / x ). Under these circumstances, dz = x dy and , splitting again into 2 parts: dx x f xxm x dx x f xdy y g xy x dx x f dz x z g z x m g h ) ( ] )[ 1 ( ) ( ] ) ( )[ 1 ( ) ( ] ) ( )[ 1 ( 0 0 0 ∫ ∫ ∫ ∫ ∫ +∞ +∞ ∞ − +∞ +∞ ∞ − +∞ = = = + dx x f xxm x dx x f xdy y g xy x dx x f dz x z g z x m g h ) ( ] )[ 1 ( ) ( ] ) ( )[ 1 ( ) ( ] ) ( )[ 1 ( 0 0 0 ∫ ∫ ∫ ∫ ∫ ∞ − −∞ ∞ + ∞ − +∞ ∞ − ∞ − = − = − = − and reassembling the 2 parts it comes that : f g g h m m dx x f x m m ∫ +∞ ∞ − = = ) ( . As a result the mean value, mh , is: mh = mf mg which is valid independently of the distributions addressed (i.e. should they be Normal or not). A similar method can be used for the standard variation: ∫ +∞ ∞ − = dz z h z sh ) ( 2 2 Therefore, dx dz x z g z x f x dxdz x f x z g x z sh ∫ ∫ ∫ ∫ ∞ ∞ − ∞ ∞ − ∞ ∞ − ∞ ∞ − = = 2 2 2 ) ( ) ( ε ε Integrating by substitution, disassembling on ε and reassembling (as above): ( ) ( ) 2 2 2 2 2 3 2 2 ) ( ) ( ) ( ) ( ) ( g f g h s s dx x f x s dx dy y g y x f x x dx x dy y g xy x f x s = = = = ∫ ∫ ∫ ∫ ∫ ∞ ∞ − ∞ ∞ − ∞ ∞ − ∞ ∞ − ∞ ∞ − ε ε ε Noting, 2 2 2 m s − = σ ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 203 Then, ) m )( m ( m s s m s 2 g 2 g 2 f 2 f 2 h 2 h 2 g 2 f 2 h 2 h 2 h + + = + ∴ = + = σ σ σ σ The expression above, σσσσh 2 + mh 2= (σσσσf 2 + mf 2 )( σσσσg 2 + mg 2 ) recalls to a certain extent that found for the addition of random variables … D.3.6.4 Examples The results found above are the basis for the handling of influence quantities, in clause D.4.1. D.3.6.5 Extrapolations Independently of the distributions handled, a step by step method based on the properties shown above would provide, for K G F H = : mh = mf mg mk ... and σσσσh 2 + mh 2= (σσσσf 2 + mf 2 )( σσσσg 2 + mg 2 )( σσσσk 2 + mk 2 ) … A similar expression will be found in clause D.4.2.2. D.3.7 Inversions and divisions Again, it would be possible to find the sought results either by direct methods or by application of clauses D.3.9 and D.3.11 … or with D.3.10 … The latter approach has been preferred: rather than starting from scratch (as done for the multiplication in clause D.3.6), a step by step approach using results already established ("the building bloc approach") was used to establish the properties relating to: Y = 1 / X Y = 1 / X Y = 1 / X Y = 1 / X and H = F / G H = F / G H = F / G H = F / G .... D.3.7.1 Evaluation of distributions corresponding to inversions (The notations proposed in clause D.3.10.6 have been used). This clause deals with Y = 1 / X Y = 1 / X Y = 1 / X Y = 1 / X (using the character set Monotype Corsiva) , where X is a random variable and Y is its transformed by the inversion g , where g is obviously a function of one variable which is monotonous (therefore clauses D.3.9 and possibly D.3.10.3 apply). X is a random variable characterized by the fact that the probability of X having a particular value x is given by the probability density X ( x ). By definition, the probability P of having the values x taken by the random variable X such that x1 < x < x2 is ∫ = 2 1 x x dx X(x) P . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 204 Similarly, we can consider ∫ ∞ − = x X dt X(t) x P ) ( , and, by differentiation : dPX = X ( x ) dx . Y is the random variable which probability density is Y ( y ) (to be evaluated). g | x y = g (x) = 1 / x g' | x y' = g' (x) = - 1 / x2 As a result from clause D.9.1: )) ( (' )) ( ( 1 1 z g g z g f h(z) − − = or, with the notations used here: )) ( (' )) ( ( 1 1 y g g y g X Y(y) − − = where g-1 | y x = 1 / y . Therefore we have: y y X y g y X Y(y) 2 ) 1 ( ) 1 (' ) 1 ( − = = . Finally, the sought probability density is : ) 1 ( 1 2 y X Y(y) y = D.3.7.2 Verification in the case of the inversion Obviously Y is positive. Should Y be a distribution then 1 ) 1 ( ) ( 2 = = ∫ ∫ ∞ + ∞ − ∞ + ∞ − dy y y X dy y Y would be true. This integral can be easily calculated using the variable x such that: x = 1 / y dx = - ( dy ) / y2 and, as a result, ∫ ∫ ∫ +∞ ∞ − +∞ ∞ − + = 0 0 ) ( ) ( ) ( dy y Y dy y Y dy y Y replacing Y ( ) by its expression ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 205 dy y y X dy y y X dy y Y ∫ ∫ ∫ ∞ + ∞ − ∞ + ∞ − + = 0 2 0 2 ) 1 ( ) 1 ( ) ( or after the substitution dx y y x X dx y y x X dy y Y ) ( ) ( ) ( ) ( ) ( 2 0 2 0 2 2 − + − = ∫ ∫ ∫ ∞ + −∞ +∞ ∞ − . dx x X dx x X dy y Y ) ( ) ( ) ( 0 0 ∫ ∫ ∫ ∞ + −∞ +∞ ∞ − − − = (by simplification) 1 ) ( ) ( ) ( = = − = ∫ ∫ ∫ +∞ ∞ − −∞ ∞ + +∞ ∞ − dx x X dx x X dy y Y and Y fulfils the 2 requirements indicated; so it can be a valid expression for a probability density. The method used for the verification can be extended to support also the calculation of the mean, below. D.3.7.3 Means and standard deviations in the case of the inversion D.3.7.3.1 Mean value By definition, the mean is: dy y y Y my ) ( ∫ +∞ ∞ − = . Replacing Y by its value provides: 1 ) 1 ( ) ( 2 = = ∫ ∫ ∞ + ∞ − ∞ + ∞ − dy y y y X dy y y Y . This integral can be easily calculated using the variable x such that: x = 1 / y dx = - ( dy ) / y2 and, as a result, ∫ ∫ ∫ +∞ ∞ − +∞ ∞ − + = 0 0 ) ( ) ( ) ( ydy y Y ydy y Y ydy y Y and replacing Y ( ) by its expression gives ydy y y X ydy y y X ydy y Y ∫ ∫ ∫ ∞ + ∞ − ∞ + ∞ − + = 0 2 0 2 ) 1 ( ) 1 ( ) ( dx x y y x X dx x y y x X ydy y Y ) 1 )( ( ) ( ) 1 )( ( ) ( ) ( 2 0 2 0 2 2 − + − = ∫ ∫ ∫ ∞ + −∞ +∞ ∞ − ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 206 dx x x X dx x x X dy y y Y ) 1 )( ( ) 1 ( ) ( ) ( 0 0 ∫ ∫ ∫ ∞ + −∞ +∞ ∞ − − − = dx x x X dx x x X dy y y Y my ) ( ) 1 ( ) ( ) ( ∫ ∫ ∫ +∞ ∞ − −∞ ∞ + +∞ ∞ − = − = = . This expression looks like moment ( - 1) of the probability density X … not that much friendly! NOTE: this expression could have been obtained directly using the results of clause D.9.3. However, since this expression is somewhat different from expressions found in other clauses of the present annex, it was felt wise to obtain it also directly. D.3.7.3.2 Comment concerning the mean value As indicated above, dx x x X my ) ( ∫ +∞ ∞ − = . Should the distribution X correspond to a constant x0 , then, the above expression could be simplified: 0 0 0 1 )1( 1 ) ( 1 ) ( x x dx x X x dx x x X my = = = = ∫ ∫ +∞ ∞ − +∞ ∞ − and we would also have 0x mx = . In this case (only) we would get: x y m x m 1 1 0 = = …. An expression that we could have expected. D.3.7.3.3 Standard deviation The results found in clause D.3.7.3.1 support a calculation of the standard variation using the results of clause D.3.9.3 which provides directly (by substituting the names of the variables): dx x x X dx x X x g my y 2 2 2 2 ) ( ) ( ) ( ∫ ∫ +∞ ∞ − +∞ ∞ − = = + σ . As in the case of the mean, should the distribution X correspond to a constant x0 , then, the above expression could also have been simplified: 2 0 2 0 2 0 2 2 ) 1 ( ) ( ) 1 ( ) ( x dx x X x dx x x X my y = = = + ∫ ∫ +∞ ∞ − +∞ ∞ − σ and we would also have 0x mx = and 0 1 x my = . In this case (only) we would get: 2 0 2 0 2 2 2 ) 1 ( ) 1 ( x x m y y y = + = + σ σ …. And 0 2 = y σ which is fair for a constant! ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 207 D.3.7.4 Examples of inversions Ohm's law can be expressed as v = r i , as well as i = v / r . D.3.7.4.1 Evaluation of the distribution To simplify the calculations, in the following, v = 1 (clause D.3.2 indicates how to handle a multiplication by a constant, so it is very simple to introduce another value and to derive the corresponding result when necessary). Using the notations of clause D.10.6 , we can therefore consider the case where R is a rectangular distribution. In this case, the probability density I is given by clause D.3.7.1 i.e.: ) 1 ( 1 2 y X Y(y) y = where y = 1 / x . The relation between the relevant variables is as follows: y i x r . And with the appropriate names of variables and notations, we get: ) 1 ( 1 2 i R i I(i) = , where R is a rectangular distribution with a spread from r1 to r2 or 2 A (as defined in clause D.1.3.1). When r1 < ( 1 / i ) < r2 then A i I(i) 2 1 1 2 = ; otherwise I ( i ) = 0. The corresponding distribution is therefore represented by a chunk of curve between two vertical lines (corresponding to 1 / r1 and 1 / r2 ), looking like a somewhat trapezoidal distribution. D.3.7.4.2 Evaluation of the mean value The general expression for the mean value is provided in clause D.3.7.3.1 , and as a result, in the case of a rectangular distribution: )] ( [ 2 1 2 1 2 1 2 1 2 1 ) ( ) ( r Log r r r r r r i A dr Ar dr r r R dr r r R m = = = = ∫ ∫ ∫ +∞ ∞ − and )] ( [ 1 2 2 1 r r Log A mi = . Noting that if r0 is the middle of [r1 , r2 ] , we have r2 = r0 + A and r1 = r0 – A , mi can be expressed as: ] / 1 / 1 [ )] ( [ 0 0 2 1 0 0 2 1 r A r A Log A r A r Log A A mi − + − + = = . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 208 When A is small compared to r0 … we can use Log ( 1 + x ) equivalent to x and, therefore: 0 0 1 2 2 0 0 2 1 )] / ( ) / [( r r A A A m r A r A i = = ≈ − − + … not very surprising (but gives confidence!) : when v = 1 and r = r0 … i = ( v / r ) = 1 / r0 ! The approximation used for the expression of mi although precise enough for the purpose of this clause, has to be enhanced for the needs of clause D.3.7.4.3. As a result, a better approximation of Log ( 1 + x ) has to be used: ) ( 3 2 ) 1( 3 3 2 x x x x x Log ε + + − = + . And, therefore, )) / 1( ) / 1( ( 2 1 0 0 2 1 0 0 ] / 1 / 1 [ r A Log r A Log A A m r A r A Log i − − + = = − + ))] ) (( ) ( 3 1 ) ( 2 1 ) (( )) ) (( ) ( 3 1 ) ( 2 1 ) [(( 2 1 3 0 3 0 2 0 0 3 0 3 0 2 0 0 r A r A r A r A r A r A r A r A A mi ε ε + − + − − − − + + − = and, after another crash of Word 97 ™ with loss of information … another attempt to type in the text provides: )] ) (( 2 3 1 1[ 1 ))] ) (( ) ( 3 2 ) ( 2 [( 2 1 3 0 0 2 0 2 0 3 0 3 0 0 r A A r r A r r A r A r A A mi ε ε + + = + + = , another expression of the mean, which will be used in the next clause. It can be noted, that the offset relating to the mid-point is equal to: 2 0 2 0 3 1 1 r A r . The value of this offset was not visible with a first order approximation. D.3.7.4.3 Evaluation of the standard deviation What would then be the standard deviation ? Its value, in the general case is provided by: dx x x X my y 2 2 2 ) ( ∫ +∞ ∞ − = + σ . When R is rectangularly distributed, we get: ) 1 1 ( 2 1 2 1 2 1 ) ( ) ( 2 1 2 2 2 2 2 ] 1 [ 2 1 2 1 2 1 r r A A dr r A dr r r R dr r r R m r r r r r r r i i − = = = = = + − ∫ ∫ ∫ ∞ + ∞ − σ ) 1 ( ) ( 2 1 ) 1 1 ( 2 1 2 1 2 1 1 2 2 1 2 2 r r r r r r A r r A mi i = − = − = + σ . When writing r2 = r0 + A and r1 = r0 – A , as above, and using approximations, we get: ( ) ( ) 2 0 2 0 2 0 2 0 0 0 2 0 2 1 2 2 / 1 ) / 1( 1 ) / 1( ) / 1( 1 1 r r A r A r r A r A r r r mi i + ≈ − = − + = = + σ , ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 209 and replacing the mean by its approximate value: )] ) (( 3 1 1[ 1 3 0 0 2 0 2 0 r A A r r A r mi ε + + = )] ) (( 3 2 1[ 1 )] ) (( 3 1 1[ 1 3 0 0 2 0 2 2 0 2 3 0 0 2 0 2 2 0 2 r A A r r A r r A A r r A r mi ε ε + + = + + = and: ( ) ] 1[ 1 / 1 )] ) (( 3 2 1[ 1 2 0 2 2 0 2 0 2 0 3 0 0 2 0 2 2 0 2 r A r r r A r A A r r A r i + = + ≈ + + + ε σ or, finally: 3 1 1 ) 3 2 1( 1 2 0 2 2 0 2 0 2 2 0 2 r A r r A r i = − ≈ σ . NOTE: It can be noted, that the use of a first order approximation for the mean would provide a wrong result: ( ) 2 0 2 0 2 0 2 / 1 1 r r A r i + ≈ + σ or 2 0 2 0 2 2 1 r r A i ≈ σ . This value would have been in excess of the correct value found above. D.3.7.4.4 Comments concerning the standard deviation The result found above (in clause D.3.7.4.3) is not surprising: it recalls the expression of the standard deviation of a rectangular distribution having, as a footprint, the extremes values of the intensity corresponding to the extreme values of the footprint of R . It can also be noted that the "simplification" v = 1 , results in the loss of the term expressed in Volts, and, therefore, a checked based in units (see clause D.3.10.7) becomes difficult. As a result, it can be wise to reintroduce this constant v . Using the results of clause D.3.2 , we get: 0r v mi ≈ and 2 0 2 2 0 2 2 3 1 r v r A i ≈ σ . With these values, should a footprint of i have been defined by its spread of + B , then, we would have had: 0 0 r A i B = , when requiring corresponding extreme values. For a rectangular distribution i of spread of + B , then we would have had (see clause D.1.3.1): 2 0 2 0 2 2 0 2 0 2 2 2 3 1 3 3 i r A r i A B iB = = = σ where 2 0 2 2 0 r v i = , and therefore, 2 0 2 2 0 2 2 3 1 r v r A iB = σ . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 210 The two expressions 2 iB σ and 2 i σ : - resulting respectively from a rectangular distribution (i + B) - and from the inverse of a rectangular distribution (r + R) have obviously the same structure and, with the approximations made, the same coefficient. Therefore, in order to find differences due to the differences in the shapes of the corresponding distributions it would have been necessary to use approximations at an higher order, so that the influence of the approximations made in the calculations of the standard deviation … would not have hidden the effects! However, this example shows the method to handle this type of problems and type of results which can be expected when using the methodology developed in this clause. D.3.7.5 Evaluation of the distribution corresponding to divisions (The notations proposed in clause D.3.10.6 have, once again, been used). This clause deals with H = F / G (using the character set Monotype Corsiva) Where F and G are independent random variables and H is the result of the division of F by G . Let Y be the inverse of G … H can therefore be considered as the product of F by Y and clauses D.3.6 and D.3.7.1 apply... Y = 1 / G H = F * Y When F is a random variable characterized by the fact that the probability of F having a particular value f is given by the probability density F ( f ) , by definition, the probability P of having the values f taken by the random variable F such that f1 < f < f2 is ∫ = 2 1 f f df F(f) P . Similarly, we can consider ∫ ∞ − = f F dt F(t) f P ) ( , and therefore (by differentiation) dPF = F ( f ) df . When G is a random variable characterized by the fact that the probability of G having a particular value g is given by the probability density G ( g ) , by definition, the probability P of having the values g taken by the random variable G such that g1 < g < g2 is ∫ = 2 1 g g dg G(g) P . Similarly, we can consider ∫ ∞ − = g G dt G(t) g P ) ( , and therefore (by differentiation) dPG = G ( g ) dg . H is the random variable which probability density is H ( h ) (to be evaluated). ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 211 By definition, Y is the inverse of G and, therefore, its probability density is (see clause D.7.1): ) 1 ( 1 2 y G Y(y) y = . The probability density of the product of random variables is, according to D.3.6.1: ∫ +∞ ∞ − = dx x f x z g x z h ) ( ) ( ) 1 ( ) ( . With the variables and notations used in this clause, | h z H h | f x F f | g y Y y and we get: ∫ +∞ ∞ − = df f F f h Y f h H ) ( ) ( ) 1 ( ) ( or, substituting Y ( ) by its value: ∫ +∞ ∞ − = df f F h f G f h f h H ) ( ) ( 1 ) 1 ( ) ( 2 . After simplification we get: ( ) ( ) ( ) ∫ ∫ +∞ ∞ − +∞ ∞ − = = df f F h f G h f df f F h f G h f f h H ) ( ) ( ) ( ) ( ) 1 ( ) ( 2 2 2 , or using ε as proposed in D.10.3 ( ) ∫ +∞ ∞ − = df f F h f G h f h H ) ( ) ( ) ( 2 ε D.3.7.6 Verification in the case of divisions Obviously H is positive. Should H be a distribution then ( ) 1 ) ( ) ( ) ( 2 = = ∫ ∫ ∫ +∞ ∞ − +∞ ∞ − +∞ ∞ − dh df f F h f G h f dh h H ε would be true. Reordering the terms we get: ( ) df f F f dh h f G h dh h H ) ( ] ) ( [ ) ( 2 ∫ ∫ ∫ +∞ ∞ − +∞ ∞ − +∞ ∞ − = ε . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 212 The internal integral is now easy to calculate using a new variable z and considering f as a constant: z = f / h dz = - ( f dh ) / h2 and, as a result, - when f (and ε ) is positive ( ) ( ) ( ) ∫ ∫ ∫ +∞ ∞ − +∞ ∞ − + = 0 2 0 2 2 ) ( 1 ) ( 1 ) ( dh h f G h dh h f G h dh h f G h ε ( ) ( ) ( ) ∫ ∫ ∫ ∞ + −∞ +∞ ∞ − − + − = 0 2 2 0 2 2 2 )1 ( ) ( 1 )1 ( ) ( 1 ) ( dz f h z G h dz f h z G h dh h f G h ε ( ) ( ) f dz z G f dz f h z G h dh h f G h 1 ) ( 1 )1 ( ) ( 1 ) ( 2 2 2 = + = − = ∫ ∫ ∫ +∞ ∞ − −∞ ∞ + +∞ ∞ − ε - when f (and ε ) is negative ( ) ( ) ( ) ∫ ∫ ∫ +∞ ∞ − +∞ ∞ − − + − = − 0 2 0 2 2 ) ( 1 ) ( 1 ) ( 1 dh h f G h dh h f G h dh h f G h ( ) ( ) ( ) ∫ ∫ ∫ ∞ − +∞ +∞ ∞ − − − + − − = 0 2 2 0 2 2 2 )1 ( ) ( 1 )1 ( ) ( 1 ) ( dz f h z G h dz f h z G h dh h f G h ε ( ) ( ) f dz z G f dz f h z G h dh h f G h 1 ) ( 1 ) ( 1 ) ( 2 2 2 = + = = ∫ ∫ ∫ +∞ ∞ − −∞ ∞ + +∞ ∞ − ε . In both cases the result is expressed in the same way, so finally we have: ( ) 1 ) ( ) ( 1 ) ( ] ) ( [ ) ( 2 = = = = ∫ ∫ ∫ ∫ ∫ +∞ ∞ − +∞ ∞ − +∞ ∞ − +∞ ∞ − +∞ ∞ − df f F df f F f f df f F f dh h f G h dh h H ε and H fulfils the 2 requirements indicated; so it can be a valid expression for a probability density. D.3.7.7 Means and standard deviations in the case of divisions D.3.7.7.1 Corresponding evaluation The mean and the standard deviation are provided, in the case of a multiplication, in clause D.3.6.3: with the notations of that clause: g f h m m m = and ) )( ( 2 2 2 2 2 2 g g f f h h m m m + + = + σ σ σ . With the present notations, we get for the mean: y f h m m m = and substituting m y by its value, as given in clause D.3.7.3 dx x x X my ) ( ∫ +∞ ∞ − = , ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 213 the expression of the mean becomes, with the appropriate variables dg g g G m m f h ) ( ∫ +∞ ∞ − = . With the present notations, we get for the standard deviation: ) )( ( 2 2 2 2 2 2 y y f f h h m m m + + = + σ σ σ and substituting ) ( 2 2 y y m + σ by its value, as given in clause D.3.7.3 dx x x X my y 2 2 2 ) ( ∫ +∞ ∞ − = + σ , the expression providing the standard deviation becomes, with the appropriate variables dg g g G m m f f h h 2 2 2 2 2 ) ( ) ( ∫ +∞ ∞ − + = + σ σ . D.3.7.7.2 Comments Clause D.3.6.3 provides: f g h m m m = . Using once again Ohm's law, we have v = r i , and i r v m m m = As a result, of a quick calculation, it could have been tempting to write: r v i m m m = . However, the result provided above, in clause D.3.7.7.1 is: dg g g G m m f h ) ( ∫ +∞ ∞ − = , which, with the notations corresponding to Ohm's law, i = v / r become dr r r R m m v i ) ( ∫ +∞ ∞ − = … so what ? Would normally, dr r r R mr ) ( 1 ∫ +∞ ∞ − = ? The example provided in clause D.3.7.4 does not suggest it. So ? A key can be found in the definitions. In clause D.3.7.5, it is indicated: " F and G are independent random variables and H is the result of the division of F by G ". So, in this case, the independent random variables are VVVV and RRRR … while in the other case, the independent random variables were RRRR and I I I I . The importance of clearly identifying which random variables are independent and which are not, had already been stressed in clauses such as D.2.4 or D.3.4.6. When this is not done carefully, there is a clear risk of getting wrong results. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 214 D.3.7.8 Examples in the case of divisions In clause D.3.7.7.2 above, an example with Ohm's law was already discussed. D.3.8 Using Logs and dBs This clause deals with ) (F Log H = and dBs Where F is a random variable and H its Logarithm. It is supposed that F has only positive values. In clause D.3.8.1 a direct method has been used. In clause D.3.8.4 the method used is based on the results of clause D.3.9 (using functions). Substitutions (see clause D.10.3) could also have been used. D.3.8.1 Evaluation of the corresponding distribution When F is a random variable characterized by the fact that the probability of F having a particular value x is given by the probability density f ( x ) , then, by definition: the probability Pf of having the random variable F having a value x such that x1 < x < x2 is ∫ = 2 1 x x f f(x)dx P . Similarly, we can consider ∫ ∞ − = x f dt f(t) x P ) ( , and therefore (by differentiation) dPf = f ( x ) dx . In the following, x is supposed within the definition range of the function Log i.e. x is supposed positive. Should H be the random variable corresponding to H = Log ( F ) (using log e ) , then, with the current notations, its probability density h ( z ) , is to be evaluated. For each value of F , the value z of the random variable H is : z = Log ( x ) . The way to evaluate h ( z ) is very simple: when the value of F is within [ x , x + dx ] , event having a probability f ( x ) dx the value of H is within [ Log ( x ) , Log ( x + dx ) ] , event having a probability h ( z ) dz . This means that these two events have the same probability, and, therefore: f ( x ) dx = h ( z ) dz . When the value of F is x , the value of z is z = Log ( x ) . We will also have, dz = ( 1 / x ) dx , and ze x = . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 215 Replacing, we get: dP = h ( z ) dz = f ( x ) dx h ( z ) (1 / x ) dx = f ( x ) dx , which, in turn, gives: h ( z ) = x f ( x ) , or ) ( z z e f e h(z) = the relation between the probability densities corresponding to the random variables F and H (when using log e (caution: dB calculations utilize log 10 )). D.3.8.2 Verifications When providing the definitions and characteristics of probability densities characterizing random variables, 2 criteria had been expressed. The probability density associated with H , h ( z ) shall be such that: - 0 ≥ h(z) - ∫ +∞ ∞ − = 1 dz h(z) It is therefore wise to verify the 2 properties, which, in practise, could help detecting problems occurred during the calculations. Obviously, ze is positive and f is such that 0 ≥ ∀ f(x) x , therefore 0 ( ≥ z) h . Concerning the second relation, verifications can be done in a generic manner: ∫ +∞ ∞ − = dz h(z) ∫ +∞ ∞ − dz ) z f( z ) exp( )) (exp( By introducing t = exp (z ) dt = t dz , this equation may be transformed into: ∫ +∞ ∞ − = dt t f(t t ) / 1( ) 1 ∫ +∞ ∞ − = f(t)dt . Which ensures that h ( z ) can be a proper probability density function characterizing some random variable (hopefully H , should the above calculations be correct!). D.3.8.3 Mathematical support for calculations with Logs and dBs ) a ( log ) N ( log ) N ( log ) N ln( ) N ( log x e N ) N log( ) N ( log x 10 N b b a e x 10 x = = = ⇒ = = = ⇒ = 1 m logm = ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 216 x ) e ( log ))' x ( (log a a ⇒ x e ) x ln( )) e ( (log ) x ( log ) x ln( 10 10 = = ) a ln( e a z z = …and … Log ( 1 + x ) = x – ( x 2 / 2 ) + ( x 3 / 3 ) … log10 = (Log x) / (Log 10) (Log x )' = 1 / x D.3.8.4 Using dBs In order to write this clause, a direct calculation could have been performed … Using the various elementary operations described in the clauses above, it would also have been possible to chain a number of those elementary operations (the method using "building blocks) … and reach the sought result! However, the more elegant way is probably to combine all operations in one single transformation, using the results found in clause D.3.9 (below). As it has already been noted, annex E also refers to conversions … and the results are consistent! When thinking in dBs and linear terms, before any further action, the first thing to do is to try and understand the situation, and to settle on the best strategy. Are the uncertainties (probability densities) relating to the various elements of the test set up expressed in dB or in linear terms? If the uncertainties are given in dBs (e.g. the attenuation of a 10 dB attenuator given as +0,1 dB …) then dBs have to be used, at least for a while … as shown in clauses D.3.8 (below) and also annex E, a rectangularly shaped distribution based on an uncertainty of +0,1 dB flat in dBs, will convert into some part of a curve if transformed into linear terms (and vice-versa). Even if the edges of the rectangular distribution are converted correctly (in order to save time, approximations may be used, but they may introduce errors of significance (see the note at the end of clause D.3.7.4.3 ) the fact that the transformed curves are not flat any more, means that values such as an average and a standard deviation do not correspond easily … which can be noticed looking at the equations! In such cases, it could be wise to think also in terms of medians … So, the real question is to find if the shape of the distribution corresponding to the uncertainties being addressed is more easily described in linear terms or in dBs. When this decision is made, then the expressions in the present clause allow for conversions to be performed. RSSing standard deviations is correct when random variables are added (as shown in clause D.3.4) … but when mixing random variables otherwise, the complete and correct calculations may have to be completed. When values of x are small, ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 217 Log (1 + x ) can be taken as x (property used to establish the conversion tables (see table 1 in TR 100 028-1 [6])). When x becomes greater, then the approximation becomes less and less acceptable and it is to the person carrying the tests to choose the best route. In clause D.3.7.4.3 an expression at a higher order: ) ( 3 2 ) 1( 3 3 2 x x x x x Log ε + + − = + was successfully used. The general expression is, in fact: ) ( )1 ( ... 3 2 ) 1( 1 3 2 n n n x n x x x x x Log ε + − + + + − = + + The following graph illustrates the approximation Log (1 + x ) = x …and… the clauses below provide all the information required to perform complete conversions when this approximation is no longer acceptable … D.3.8.4.1 Transformation of linear terms into dBs First of all, it has to be noted that dBs are defined in two different manners which have to be listed here: - as relative values (e.g. in the case of attenuators) - as values relative to some reference (e.g. dBm, dB µV , etc.); both references to power and voltages are used, providing therefore two sets of coefficients ( 10 and 20 ), which have to be handled separately (see, for instance, table 1 in TR 100 028-1 [6]). This may have an influence in the way to write and to handle the conversions with dBs, and the approximations thereof … D.3.8.4.1.1 Converting powers into dBs The method provided in clause D.3.9 has been used in order to perform a conversion into dB W. Noting: the power in linear terms as x (i.e. in Watts) … so x is a positive value! and the corresponding value in dB (i.e. dB relative to 1 Watt) as z , we have z = 10 log ( x ) . as indicated in clause D.3.9, we have )) ( (' )) ( ( 1 1 z g g z g f h(z) − − = , where: g | x ) 10 ( ) ( 10 ) log( 10 Log x Log x z = = g' | x ) 10 ( 10 Log x g -1 | z 10 10 10 10 z Log z e x = = As a result, )) ( (' )) ( ( 1 1 z g g z g f h(z) − − = ) 10 (' ) 10 ( 10 10 z z g f = or 10 ) 10 ( ) 10 ( ) 10 ( ) ( 10 10 z z f Log z h = . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 218 The moments can now easily be calculated with the expressions also given in D.3.9 , as soon as f is also given: dx x f x g m ) ( ) ( ∫ +∞ ∞ − = dx x f x ) ( ) log( 10 ∫ +∞ ∞ − = . (noting that log is "base" 10) Similarly, dx x f x g s ) ( ) ( 2 2 ∫ +∞ ∞ − = dx x f x ) ( ) ) log( 10 ( 2 ∫ +∞ ∞ − = … (noting that log is "base" 10). In many clauses of this annex e.g. in clauses D.3.1 and D.3.2, it had been possible to express the mean value after the specific operation as an explicit function of the original mean. The same in respect to the standard deviation. Clearly, in this case, as already found in clause D.3.7 (inversions and divisions), there appears not to be a simple relation, independent of the actual distribution, between these parameters. D.3.8.4.1.2 Converting a rectangular distribution into dBs As a example, should it be intended to convert a rectangular distribution (foot-print defined by parameters A and B … with a definition of A and B different from that used in clause D.1.3 ), then we would have: ) ( 10 ) 10 ( ) 10 ( 10 )) /( 1 )( 10 ( ) 10 ( ) ( 10 10 A B Log A B Log z h z z − = − = within the corresponding interval and zero outside … (noting that Log is "base" e). See also annex E. D.3.8.4.1.3 Converting voltages in dBs In this case, we have z = 20 log ( x ) . as indicated in clause D.3.9, we have )) ( (' )) ( ( 1 1 z g g z g f h(z) − − = , where: g | x ) 10 ( ) ( 20 ) log( 20 Log x Log x z = = g' | x ) 10 ( 20 Log x g -1 | z 20 20 10 10 z Log z e x = = As a result, )) ( (' )) ( ( 1 1 z g g z g f h(z) − − = ) 10 (' ) 10 ( 20 20 z z g f = or 20 ) 10 ( ) 10 ( ) 10 ( ) ( 20 20 z z f Log z h = . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 219 The moments can be, once again, calculated with the expressions given in D.3.9, as soon as f is also known: dx x f x g m ) ( ) ( ∫ +∞ ∞ − = dx x f x ) ( ) log( 20 ∫ +∞ ∞ − = (noting that log is "base" 10). Similarly, dx x f x g s ) ( ) ( 2 2 ∫ +∞ ∞ − = dx x f x ) ( ) ) log( 20 ( 2 ∫ +∞ ∞ − = … (noting that log is "base" 10). D.3.8.4.2 Transformation of dBs into linear terms The reverse operation can also be made … D.3.8.4.2.1 Converting powers As noted in clause D.3.8.4.1 , dBs can be expressed in relation to some reference. This is where the term x0 is coming from. g | x 10 )) log( 10 ( ) log( 10 0 0 10 Log x x x x e z + + = = g' | x 10 )) log( 10 ( 0 ) 10 10 ( Log x x e Log + g-1 | z x = 10 (log ( z / xo ) = 10 ( log ( z ) – log ( xo ) ) )) log( ) (log( 10 0x z x − = As a result, we get )) ( (' )) ( ( 1 1 z g g z g f h(z) − − = 10 )) log( 10 ))) log( ) (log( 10 ( ( 0 0 0 ) 10 10 ( ))) log( ) (log( 10 ( Log x x z e Log x z f + − − = . When the value is expressed in dB in the appropriate reference, xo = 1 and log (xo ) is 0; the above expression simplifies in: 10 ) 10 )) (log( 10 ( ) 10 10 ( ))) (log( 10 ( ) ( Log z e Log z f z h = 10 ) log( 10 ))) (log( 10 ( 10 Log z e Log z f = ) ( 10 )) log( 10 ( 10 z Log e Log z f = 10 )) log( 10 ( 10 Log z z f = and finally, we have: 10 )) log( 10 ( 10 ) ( Log z z f z h = . The moments can now easily be calculated with the expressions also given in D.3.9 , as soon as f is also given: dx x f x g m ) ( ) ( ∫ +∞ ∞ − = dx x f e Log x ) ( 10 ) 10 ( ∫ +∞ ∞ − = . Similarly, dx x f x g s ) ( ) ( 2 2 ∫ +∞ ∞ − = dx x f e Log x ) ( ) ( 2 10 ) 10 ( ∫ +∞ ∞ − = … ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 220 D.3.8.4.2.2 Converting Voltages Should dB Volts (or dBµV) have been used, the corresponding conversion relations would have been: 10 )) log( 20 ( ) log( 20 0 0 10 Log x x x x e z + + = = , as the general expression or, when the value of x0 is 1 : 10 ) 20 ( 20 10 Log x x e z = = , in which case: 10 )) log( 20 ( 20 ) ( Log z z f z h = dx x f e m Log x ) ( 10 ) 20 ( ∫ +∞ ∞ − = dx x f e s Log x ) ( ) ( 2 10 ) 20 ( 2 ∫ +∞ ∞ − = . D.3.8.4.2.3 Converting rectangular distributions In annex E, conversions of rectangular distributions have been also studied. In such a case, the above relation becomes: 10 )) log( 20 ( 20 ) ( Log z z f z h = 10 ) 2 / 1( 20 Log z A = in the converted interval, zero, outside …After further simplification: z Log A z h 1 10 10 ) ( = or zero, outside the appropriate interval. (The corresponding probability density had been called p2 (x) in clause E.1.1.) An approach using spread sheets has also been proposed. Further details concerning this approach can be found in … D.3.8.4.3 Examples It was stressed earlier that the term dB may, in fact, cover different situations from the mathematical point of view. It has also been emphasized in particular in clause D.2 (and will be covered again in clause D.5) that in the mapping of physical parameters, random variables may be associated either with the variable itself or with small variations of it. The following clauses address these two different cases. D.3.8.4.3.1 Evaluation of uncertainties In this case, it can be expected that only small variations are considered. Therefore, multiplicative constants such as x0 appearing in the relations are equal to one ( Log ( x0 ) = 0 ). ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 221 D.3.8.4.3.2 Evaluation of link budgets In this case, it can be expected that the statistics of the various components are interesting per se, and not only its small variations. Among the parameters to be considered (and to be mapped to random variables), can be quoted: - transmitter power (e.g. a mobile Base Station) - cable attenuation (plus attenuation of couplers, if any) - transmitter antenna characteristics - attenuation due to the propagation - receiver antenna characteristics - cable attenuation (if any … the situation can be different in the case of mobile communications or fixed links) - receiver sensitivity. In this situation, it is likely that a great variety of types of dBs have to be used together (dB m, dB µV…). Therefore, constant such as x0 appearing in the relations may have to be considered carefully. Beyond these "radio" characteristics, can also be quoted: - effect of temperature - effect of power supply voltages. The corresponding effects on the link budget can be handled thanks to the methods provided in clause D.4. D.3.8.4.3.3 Usage in the case of evaluation of link budgets and interference In this case, it can be necessary to handle simultaneously two links: - the link being considered - the interfering signal. Under such circumstances, it may happen that the corresponding standards use different expressions (e.g. dB W in one standard and dB m in the other) and therefore, constant such as x0 appearing in the relations may have to be considered with extreme care. Using different references for the expressions in dB, can be considered, in fact, as having additive offsets (which could be handled in accordance with clause D.1) or as having to multiply by some constant (which could be handled in accordance with clause D.2). D.3.9 Combination using deterministic functions of one variable This clause deals with ) (F g H = Where F is a random variable and H its transformed by g , where g is a deterministic function of one variable. Only the case where g is monotonous is addressed here, and it is supposed that F takes values within the definition of g (which can be expected, noting that g is monotonous …). ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 222 D.3.9.1 Evaluation of the corresponding distribution When F is a random variable characterized by the fact that the probability of F having a particular value x is given by the probability density f ( x ) , then, by definition: the probability Pf of having the random variable F having a value x such that x1 < x < x2 is ∫ = 2 1 x x f f(x)dx P . Similarly, we can consider ∫ ∞ − = x f dt f(t) x P ) ( , and therefore (by differentiation) dPf = f ( x ) dx . In the following, x is supposed within the definition range of g . Should H be the random variable corresponding to g ( F ) ( H = g ( F ) ), then, with the current notations, its probability density is h ( z ) , to be evaluated. For each value of F , the value z of the random variable H is : z = g ( x ) . The way to evaluate h ( z ) is, again, quite simple: when the value of F is within [ x , x + dx ] , event having a probability f ( x ) dx the value of H is within [ g ( x ) , g ( x + dx ) ] , event having a probability h ( z ) dz . This means that these two events have the same probability, and, therefore: f ( x ) dx = h ( z ) dz . When the value of F is x , the value of z is z = g ( x ) . We will also have, dz = g' ( x ) dx , where, for the moment, g' ( x ) is supposed to be > 0 and (z) g x 1 − = . In order to have a reciprocal function, g' has to be monotonous (no changes of the sign). Replacing, we get: dP = h ( z ) dz = f ( x ) dx h ( z ) g'( x ) dx = f ( x ) dx , which, in turn, gives: h ( z ) g' ( x ) = f ( x ) , or )) ( (' )) ( ( 1 1 z g g z g f h(z) − − = the relation between the probability densities corresponding to the random variables F and H , valid when g' > 0 … (see D.3.9.2). Should g' ( x ) be < 0 , then as in the case of a multiplication by a negative constant (see clause D.3.2.1), the effects on inequalities and intervals have to be taken into account. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 223 The final result is, therefore, )) ( (' )) ( ( 1 1 z g g z g f h(z) − − = . NOTE: an equivalent result has been found in clause D.3.10.3 relating to "substitutions"; the method used to derive the corresponding relation was different. D.3.9.2 Verifications When providing the definitions and characteristics of probability densities characterizing random variables, 2 criteria had been expressed. The probability density associated with H , h ( z ) shall be such that: - 0 ≥ h(z) - ∫ +∞ ∞ − = 1 dz h(z) It is therefore wise to verify the 2 properties, which, in practise, could help detecting problems occurred during the calculations. It is obvious that the fact that f is such that 0 ≥ ∀ f(x) x , makes it always true that 0 ( ≥ z) h … The second property is less obvious. So, g will be considered to be so that g' > 0 . The verification can be done in a generic manner: ∫ +∞ ∞ − = dz h(z) dz z g g z g f )) ( (' )) ( ( 1 1 − − +∞ ∞ −∫ By introducing t = g-1 (z ) z = g ( t ) and dz = g' ( t) dt , this equation may be transformed into: ∫ +∞ ∞ − = dt t g t g t f ) (' ) (' ) ( 1 ∫ +∞ ∞ − = f(t)dt . Which ensures that h ( z ) can be a proper probability density function characterizing some random variable. When g' < 0 , then, when replacing z by t , the limits of integration are inverted, which compensates for the negative sign introduced. This phenomenon is similar to that found in the case of the multiplication by a negative constant and has also been presented in detail in the case of multiplications (see clause D.3.6.2 ). D.3.9.3 Means and standard deviations The mean value of F has been defined as: ∫ +∞ ∞ − = f(x)dx x m f . What will then be the first two moments of h ( z ) ? Can they be simply expressed as a function of the two first moments of f , mf and sf ??? ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 224 The calculations below apply to the case when g' < 0; when g' < 0 , then, when replacing z by x , the limits of integration are inverted, which compensates for the negative sign introduced. This phenomenon is similar to that found in the case of the multiplication by a negative constant and has also been presented in detail in the case of multiplications (see clause D.3.6.2 ). ∫ +∞ ∞ − = = h(z)dz z mh dz z g g z g f z )) ( (' )) ( ( 1 1 − − +∞ ∞ −∫ noting that z = g( x ) and dz = g'( x ) dx = = ∫ +∞ ∞ − dx x g x g x f x g mh ) (' ) (' ) ( ) ( dx x f x g ) ( ) ( ∫ +∞ ∞ − . What then concerning the second moment ??? ∫ +∞ ∞ − = = h(z)dz z sh 2 2 = − − +∞ ∞ −∫ dz z g g z g f z )) ( (' )) ( ( 1 1 2 = ∫ +∞ ∞ − dx x g x g x f x g ) (' ) (' ) ( ) ( 2 dx x f x g ) ( ) ( 2 ∫ +∞ ∞ − Should g be a rather simple expression, it is clear that the corresponding expressions of m , s and σ should be very simple also … Example, g | x λλλλ x (i.e. z = g( x ) = λλλλ x) then, mh = λ λ λ λ mf , sh = λλλλ sf and σh² = sh² - mh² = (λλλλ sf )² - (λ λ λ λ mf )² = λλλλ² σσσσf² which had been found directly in clause D.3.2 However, it is clear that outside simple cases such as the linear case handled above, it is not often the case that resulting mean and standard deviation can be expressed explicitly using the mean and the standard deviation of the original distribution… see, in particular, clause D.3.8, where Logs and dBs are handled. D.3.9.4 Examples Conversions of linear terms to dBs and vice-versa have been performed in this annex using this method… see clause D.3.8.4 In annex E a direct method had been used. The comparison is interesting. D.3.10 Further theoretical material and reciprocals A systematic review of the effect of mathematical operations on probability densities has been provided in the previous clauses. The corresponding properties have often been given based on calculations "as simple (and basic) as practical". The purpose of the present clause is to provide also some material more theoretical … which could have been used, as well, to establish some of the results provided in this annex. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 225 D.3.10.1 Integrals and derivatives In the present annex, a number of calculations have been performed using the probability density. Similar results might also have been obtained starting from expressions such as: ∫ = = < < 2 1 ) ( 2 1 x x p(x)dx P x x x y probabilit (where P is the probability of the value x of the random variable XXXX (X X X X using the character set "Monotype Corsiva") lying between x1 and x2 , expressed using the probability density function p ( x ) ) (see clause D.1.2). It has to be stressed that , with these conventions, 2 1 x x < . This fact has been used extensively in clause D.3, in particular when multiplying the extremities of intervals by negative numbers (see, in particular, clauses D.3.2 and D.3.6). Should integrals be used, it is important to recall that if ∫ ∞ − = X p(x)dx X P ) ( then the derivative function P' is such that: p(X) X P = ) (' . This may have to be kept in mind, when thinking in terms of cumulative probabilities rather than probability densities. D.3.10.2 Substitutions and integrals Calculations based on changes of variables ("substitutions") have been used a significant number of times in the annex. However, for the sake of completeness, it can be useful to express it in a more formal way: Take, for example, ∫ = = = 2 1 x x x x p(x)dx P ; let see the effect of a substitution with: x = k ( t ) ; dt dk t k dt dx = = ) (' ∫ = = = ) 2 ( ) ( 1 ) (' ) ( x g t x g t dt t k ) t p(k P , where g is the reciprocal of k . It is interesting to compare this expression with that obtained in clause D.3.10.3 below. It can also be interesting to consider P as a function of T in the same way as it was considered in clause D.3.10.1: ∫ = = = = T X g t g t dt t k ) t p(k T P ) ( ) 0 ( ) (' ) ( ) ( and note that now ) (' ) ( ) (' T k ) T p(k T P = , which shows the effect of the substitution. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 226 D.3.10.3 Substitutions and distributions The expression: ∫ +∞ ∞ − =1 dx f(x) has already been used a number of times as a requirement for f to be a valid probability density (distribution). What happens when a variable change is performed ? Let's consider ) (t k x = where k is monotonous , and where k' exists and k' > 0. dt dk t k dt dx = = ) (' and (by "substitution") the integral above becomes: ∫ +∞ ∞ − =1 ) (' )) ( ( dt t k t k f . Should ) t f(k ) ( be considered as a function e of t , then we have: ∫ +∞ ∞ − =1 ) (' ) ( dt t k t e and ) (' ) ( t k t e is therefore a valid candidate for a probability density … Since f is a "good" probability density (and, therefore, has only positive values), and since k' was supposed to be positive, then ) (' ) ( t k t e is also positive … and a second necessary criterion is met. Noting that when 2 functions ( f and g ) are reciprocal the corresponding derivative functions have inverse expressions: ( ' 1 ) (' g t k = ) it is clear that the expression above is similar to that already found in clause D.3.9.1 … where z had been used instead of t … The fact that k is supposed to be monotonous (and that therefore there are no changes of sign of k' ) is required so that there is an inverse (reciprocal) function ( g ) … What happens then if k' < 0 ? When making the substitution on the integral, the upper bound and lower bounds get inverted, due to the fact that −∞ → ⇒ +∞ → < t x t k , 0 ) (' . As a result ∫ −∞ ∞ + =1 ) (' ) ( dt t k t e and ∫ +∞ ∞ − = − 1 ) (' ) ( )1 ( dt t k t e or , noting that k' < 0 ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 227 ∫ +∞ ∞ − =1 ) (' ) ( dt t k t h . Therefore, in both cases ( k' positive or negative): ∫ +∞ ∞ − =1 ) (' ) ( dt t k t h is the result of the substitution of x by t = k ( t ) in the probability density (distribution). This rule, concerning the change of variables, is different from that to be used for functions … so extreme care has to be developed when performing substitutions with these mathematical objects … however, the rule is quite simple: When ) (t k x ⇒ then ) (' ) ( ) (' )) ( ( ) ( t k t e t k t k f x f = ⇒ , where f is the probability density of the random variable X (of which x is a possible value) and h is the probability density of the random variable T (of which t is a possible value). With the notations proposed in clause D.10.6 , the above expression would become: ) (t k x ⇒ then ) (' ) ( ) (' )) ( ( ) ( ) ( t k t e t k t k X t T x X = = ⇒ , where X and T are probability densities characterizing respectively the probability of occurrence of the values x and t . NOTE 1: The expressions above are quite similar to those found in clause D.3.10.2, with the difference that the absolute value of k' is used instead of simply k' .This is the result of the constraint 2 1 x x < found in the definition of P. NOTE 2: It is essential for k to be monotonous (no changes of sign for k' ).If not, there is no inverse function. A way to overcome (by hand …) this limitation is shown in clause D.3.10.8. NOTE 3: Rather than handling absolute values, it is often easier to multiply the relevant expression: - by the value ε; − the value of ε would be +1 for a positive k' and -1 for a negative k' . This convention has been extensively used in clauses D.6 and D.7. D.3.10.4 Example of application: the inverse See clause D.3.7.1. D.3.10.5 Reciprocals Besides the interest in terms of completeness, reciprocal operations are often performed in calculations relating to radio equipment, for example, conversions into dBs and vice-versa. It can, therefore, be useful to keep in mind the corresponding relations. Using the notions proposed below in clause D.10.6 … Assume: - 2 random variables X and Y - taking values such as x and y - with density probabilities X and Y or X ( x ) and Y ( y ) ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 228 where: y = g ( x ) or x = k ( y ) ( k being the inverse of g ) where g is supposed to be strictly monotonous (and so will be k , its inverse …). Then ) (' 1 ) (' x g y k = ( g being strictly monotonous, then g' cannot be 0…). From clause D.3.9.1 or D.3.10.3 above we get (changing the names appropriately): )) ( (' )) ( ( )) ( (' ) ( ) (' ) ( ) (' )) ( ( ) ( y k g y k X y k g y e y k y e y k y k X y Y = = = = and )) ( (' )) ( ( )) ( (' ) ( ) (' ) ( ) (' )) ( ( ) ( x g k x g Y x g k x d x g x d x g x g Y x X = = = = , where d ( x ) = Y ( g ( x ) ) and e ( x ) = X ( k ( y ) ) . As a final note, it is clear that the knowledge of the probability density of one of the random variables gives "directly" the density probability of the other. D.3.10.6 Notations Beyond the fact that different clauses in the present annex have been written by different authors, a reader may have also noted different notations due to the intention of the clause: some clauses are more related to physics, in which case the variables used tend to look like the usual expressions used for physical values (i, r, v), while others are more related to mathematical calculations … At this point in the annex, considering that the reader is familiar with the concepts, and that only very seldom the name of the random variable concerned is quoted … the following notations could be suggested: - name of the random variable : V (character set Monotype Corsiva) - values taken by the random variable : v - density probability : V or V ( v ) (rather than p ( v ) or pV (v) as could have been expected, in view of D.1 , where "p" recalls the word probability). Resulting therefore in expressions like: ∫ +∞ ∞ − =1 ) ( dv v V … where there are certainly too many "v" , but can be more clear when a considerable number of random variables are concerned. The difficulty with the notations is that there are, in fact 3 items interrelated, and 2 practical ways to type (lower case and upper case). So it is either necessary: - to use more than 1 character set (which the equation box mechanism does not seem to handle), or - to use conventions such as those of C++ where f ( ) may be a function and at the same time f may be a variable; - or to use different letters for items related, which can be confusing when a significant number of items are used. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 229 In the present annex, standard deviations have often been called σr , where r indicates the random variable being considered. For practical reasons, in other clauses of the present document, u has been used instead. However, u can recall "uncertainty" … but, in many cases, u is in fact the standard deviation σ of the contribution being considered. D.3.10.7 Units D.3.10.7.1 Some properties In the present annex, units have been dropped in a number of situations. Therefore, it can be useful to recall that: - probabilities are numbers without unit … 0 ≤ P ≤ 1 - values such as A in the definition of rectangular distributions have the unit of the item concerned; for example, when referring to Volts, A would be expressed in Volts (e.g. +2 V) - as a result, density probabilities are expressed in the inverse of the corresponding physical unit for example, V(v) would be expressed in (Volts)-1 , (e.g. V(v) = (1 / (2 A)) (V)-1) - an integration (e.g. using dx where x is a length) adds one dimension - a differentiation reduces dimensions by 1. A careful handling is therefore required when, for instance, handling mA instead of A , in practical examples. D.3.10.7.2 Example Take a resistor … V = R I . Clause D.3.6 provides the probability density corresponding to the product of probability densities: ∫ +∞ ∞ − = dx x f x z g x z h ) ( ) ( ) 1 ( ) ( , or with the units corresponding to this example, and the notations of D.3.10.6: R F , x I G , y V H , z ∫ +∞ ∞ − = dr r R r v I r v V ) ( ) ( ) 1 ( ) ( . With: - dr expressed in ( A V ) or ( Ω ) - ) (r R expressed in ( A V ) –1 or ( Ω ) –1 - ) (I expressed in A-1 - ) 1 ( r expressed in ( A V ) –1 or ( Ω ) –1 ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 230 finally, it becomes clear that V ( v ) is expressed in ( V ) –1 , which would have been expected for a density probability relating to Volts. It was also noted in clause D.3.6 that an equivalent expression would have been: ∫ +∞ ∞ − = di i I i v R i v V ) ( ) ( ) 1 ( ) ( which would also have provided a result expressed in ( A V ) –1 or ( Ω ) –1. It is worth looking at both expressions. The former evaluation of V ( v ) is most probably more friendly than the latter: r can be expected to be always > 0 … while i can often be positive or negative or null. D.3.10.8 Application of the substitution method in difficult situations One operation could have been also found in clause D.3: raising to the square. It could have been useful for finding powers out of voltages or currents. At first sight, one could have said that there was no need: the multiplication is already dealt with in clause D.3.6. But in that clause the two input random variables are supposed to be independent … which is certainly not the case for the square! Next idea could have been to use clause D.3.9 (functions of one variable). But it is not possible to use it because, in that clause, g is supposed to be monotonous! One way out could be to use the principles of the substitution (as set in clause D.3.7. 3 ), analysing the implications carefully at each step … D.3.10.9 From the time domain to density probabilities This is an area were further work could be useful … to be incorporated in a future edition of the present document. D.3.11 Combinations using deterministic functions of two variables This clause deals with ) , ( K F g H = Where F and K are independent random variables and H the result of g , where g is a deterministic function of two variables. It is supposed that F and K take values within the definition of g . Problems could be expected, should F or K take (too often) particular values (such as zero …). Should this occur, then in that particular case, careful attention should be devoted to the situation. A careful discussion shows similar situations as for clause D.3.9 in relation to the signs. In order to avoid to have too much text, the discussion has been simplified. D.3.11.1 Evaluation of the corresponding distribution When F is a random variable characterized by the fact that the probability of F having a particular value x is given by the probability density f ( x ) , then, by definition: the probability Pf of having the random variable F having a value x such that ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 231 x1 < x < x2 is ∫ = 2 1 x x f f(x)dx P . Similarly, we can consider ∫ ∞ − = x f dt f(t) x P ) ( , and therefore (by differentiation) dPf = f ( x ) dx . When K is also a random variable, characterized by the fact that the probability of K having a particular value y is given by the probability density k ( y ) , then, by definition: the probability Pk of having the random variable K having a value y such that y1 < y < y2 is ∫ = 2 1 y y k k(y)dy P . Similarly, dPk = k( y ) dy . H is the random variable resulting from the effect of g on F and K , and its probability density h ( z ) , is to be evaluated. For each value x of F and y of K , the value z of the random variable H is : z = g( x , y) . The way to evaluate h ( z ) is relatively simple (very similar to a number of calculations completed above) , and is given in the following. The probability of having the value of F within a very small interval [x , x + dx] is f ( x ) dx ; the probability of having the value of K within a small interval [y1 , y2] is k( y ) ( y2 - y1 ) = k ( y ) Dy (where Dy = y2 - y1 , and where it is assumed that k ( y1 ) = k ( y2 ) = k ( y ) , Dy being considered as small ); when both events occur, then, the value of H within [z1 , z2] where zi = g ( x , yi ) (neglecting dx , considered to be very small compared with Dy ) and the probability of such an event (which provides the contribution of dx in h( z ) ) is f( x ) dx k( y ) Dy . When Dz = z2 - z1 , by definition, h ( z )Dz is the probability of having the value of H within [z1, z2] and is, therefore, the sum of the probabilities of all the individual contributions, corresponding to all values of x : ∫ +∞ ∞ − = dx x f Dy y k Dz z h ) ( ) ( ) ( . Having dy y g dx x g dz ∂ ∂ + ∂ ∂ = , we can write Dy y g ∂ ∂ = = = ) y , x ( g - ) y , x ( g z - z Dz 1 2 1 2 , ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 232 and we get Dy y g Dz ∂ ∂ = which makes dx x f y g Dz y k Dz z h ) ( ) ( ) ( ∫ +∞ ∞ − ∂ ∂ = . As already noted in clause D.3.9 , expressions such as the one above are valid when 0 > ∂ ∂ y g . Otherwise, the intervals have to be inverted and to cover all cases it is necessary to write: dx x f y g Dz y k Dz z h ) ( ) ( ) ( ∫ +∞ ∞ − ∂ ∂ = . y g ∂ ∂ is, in all cases, expected to be monotonous (no changes of the sign allowed). Noting that, solving g we can write ) , ( x z y γ = ( with, may be some restrictions), the integral above becomes dx x f y g Dz x z k Dz z h ) ( )) , ( ( ) ( ∫ +∞ ∞ − ∂ ∂ = γ , which can, in turn, be simplified into dx x f y g x z k z h ) ( )) , ( ( ) ( ∫ +∞ ∞ − ∂ ∂ = γ This integral provides the value of h ( z ) as a function of f ( x ) , k ( y ) … which gives a relation between the probability densities corresponding to the random variables F , K and H . D.3.11.2 Verifications When providing the definitions and characteristics of probability densities characterizing random variables, 2 criteria had been expressed. The probability density associated with H , h ( z ) shall be such that: - 0 ≥ h(z) - ∫ +∞ ∞ − = 1 dz h(z) It is usually wise to verify the 2 properties, which, in practise, could help detecting problems occurred during the calculations. The fact that 0 ≥ ∀ f(x) x and 0 ≥ ∀ k(y) y makes it clear that 0 ( ≥ z) h … Concerning the second item, the situation is close to that found when lambda was negative in clause D.3.2 … and in the clauses above … ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 233 The verification can be done in a generic manner: ∫ +∞ ∞ − = dz h(z) dz dx x f y g x z k ) ( )) , ( ( ∫ ∫ +∞ ∞ − +∞ ∞ − ∂ ∂ γ and in the positive case, dx dz y g x z k x f ] )) , ( ( [ ) ( ∫ ∫ +∞ ∞ − +∞ ∞ − ∂ ∂ = γ . As done previously, the integral inside is handled considering x as a constant, and by introducing a change in the variable: ) , ( x z y γ = . We have dx x dz z dy ∂ ∂ + ∂ ∂ = γ γ so dz z dy ∂ ∂ = γ and this expression may be transformed into: dx dy z y g y k x f ] ) ( [ ) ( ∫ ∫ +∞ ∞ − +∞ ∞ − ∂ ∂ ∂ ∂ = γ . To simplify this relation (which we always succeeded in the practical cases above), let us see the relations between both partial derivations (is this English ? ) … We have both dx x dz z dy ∂ ∂ + ∂ ∂ = γ γ and dy y g dx x g dz ∂ ∂ + ∂ ∂ = . Therefore: dy y g dx x g dz ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ = dx x dz z y g dx x g γ γ which is true for any value of dz and any value of dx … which, in turn, implies that ∂ ∂ ∂ ∂ = z y g γ 1 . As a result : ∫ +∞ ∞ − dz h(z) dx dy z y g y k x f ] ) ( [ ) ( ∫ ∫ +∞ ∞ − +∞ ∞ − ∂ ∂ ∂ ∂ = γ dx dy y k x f ] 1 ) ( [ ) ( ∫ ∫ +∞ ∞ − +∞ ∞ − = = ∫ +∞ ∞ − = dx x f ]1[ ) ( 1 ∫ +∞ ∞ − = f(x)dx . Which ensures that h ( z ) (under the conditions stated above) could be a proper probability density function. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 234 D.3.11.3 Means and standard deviations As found in clause D.3.9.3 , even though the expression of h ( z ) is rather complicated, the first two moments have a quite friendly expression: will there be a similar situation here ? ∫ +∞ ∞ − = = dz h(z) z mh dz dx x f y g x z k z ) ( )) , ( ( ∫ ∫ +∞ ∞ − +∞ ∞ − ∂ ∂ γ dx dz y g x z k z x f ] )) , ( ( [ ) ( ∫ ∫ +∞ ∞ − +∞ ∞ − ∂ ∂ = γ Let us try and make the same change of variable as in the case of the verification above (see clause D.3.11.2) ) , ( x z y γ = … we then get, in line with the expressions found above: dx x f dy y k y x g mh ) ( 1 ) ( ) , ( ∫ ∫ +∞ ∞ − +∞ ∞ − = . This can be written as: dy y k dx x f y x g mh ) ( ) ( ) , ( ∫ ∫ +∞ ∞ − +∞ ∞ − = , which means, that, in other words, the mean value obtained corresponds to the 2D average of the points obtained weighted by the original probabilities of occurrence. In fact f ( x ) dx is a probability of occurrence in a one-D space, k (y ) dy is a probability of occurrence in another one-D space, and f ( x ) dx k ( y ) dy is the probability of occurrence of the couple ( x , y ) in the two-D space, product of the two original spaces. What then concerning the second moment ??? In the same way, ∫ +∞ ∞ − = = dz h(z) z sh 2 2 dz dx x f y g x z k z ) ( )) , ( ( 2 ∫ ∫ +∞ ∞ − +∞ ∞ − ∂ ∂ γ the same change of variable as above gives: dy y k dx x f y x g sh ) ( ) ( ) , ( 2 2 ∫ ∫ +∞ ∞ − +∞ ∞ − = , which is an expression extremely similar to those found above, e.g. in the case of the effect of a function having only one variable (see clause D.3.9). It is nice to find such a simple expression, when the expression of h ( z ) has lead us through rather delicate calculations … ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 235 D.3.11.4 Examples Should g ( x , y ) be a rather simple expression, it is clear that the corresponding expressions of m , s and σσσσ should be very simple also … Examples can be found in the clause dealing with subtractions and divisions of distributions, in clauses D.3.5 and D.3.7 of annex D. D.3.11.5 Generalization to spaces of dimension N The results found above in relation to the mean and to the variance could be extended to spaces of dimension N, the expression of the distribution looking somewhat more complex. However, for the purpose of the evaluation of measurement uncertainties according to the present document, the more important relation is that leading to the standard deviations…which looks very friendly. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 236 D.3.12 Combination of distributions – Summary table Operations relating to random variables Equations (1) Resulting distribution Mean value Standard deviation Clause Addition of a constant value H=F+α h(z)=f(z-α) mh=mf+α σh=σf D.3.1 Multiplication by pos. const. H=(λ)F h(z)=(1/λ)f(z/λ) mh=λmf σh=λσf D.3.2 Multiplication by neg. const. H=(-λ)F h(z)=-(1/λ)f(z/λ) mh=λmf σh 2=λ2σf 2 D.3.2 One random variable Inverse function H= 1 / F h(z)= f(1/z) / z2 mh=∫ (f(z) / z) dz σh 2+mh 2=∫ (f(z) / z2) dz D.3.7 Sum H=F+G h(z)=∫g(z-x)f(x)dx mh=mf+mg σh 2=σf 2+σg 2 (2) D.3.3 independent variables H=λF+µG h(z)=∫(1/λµ)f(x/λ)g((z-x)/µ)dx mh=λmf+µmg σh 2=λ2σ f 2 + µ2σ g 2 D.3.4 non independent variables H=λF+µG where F=kG h(z)=(1/(λk+µ))g(z/(λk+µ)) mh=(λk+µ)mg σh 2=(λk+µ)2σg 2 D.3.4.6 Subtraction H=F-G h(z)=∫g(x-z)f(x)dx mh=mf-mg σh 2=σf 2+σg 2 D.3.5 Multiplication H=FG h(z)=∫(1/ | x | )g(z/x)f(x)dx mh=mf mg σh 2+mh 2=(σf 2+mf 2)(σg 2+mg 2) D.3.6 Two random variables Division H=F/G h(z)=∫ g(x/z) ( | x | / z2 ) f(x)dx mh= mf (∫ (g(z) / z) dz) σh 2+mh 2=(σf 2+mf 2)(∫(g(z)/z2)dz) D.3.7 ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 237 Using Logs H=Log(F) h(z)=ez f ( ez ) mh= ∫ Log(x) f(x) dx σh 2= ( ∫ Log2(x) f(x) dx ) - mh 2 D.3.8 Linear terms dB H=10 log(F) h(z)=10z/10(Log(10)f(10z/10)/10) mh=∫ 10 log(x)f(x)dx σh 2=( ∫ (10log(x))2f(x) dx) - mh 2 D.3.8.4.1 Powers dB linear terms H= 10 (F/10) h(z)=10(f(10log(z)))/(zLog10) mh=∫ e(x/10) Log10 f(x)dx σh 2=(∫(e(x/10)Log10)2f(x)dx)-mh 2 D.3.8.4.2 Linear terms dB H=20 log(F) h(z)=10z/20(Log(10)f(10z/20)/20) mh=∫ 20 log(x)f(x)dx σh 2=( ∫ (20log(x))2f(x) dx) - mh 2 D.3.8.4.1 Using Logs Volts dB linear terms H= 10 (F/20) h(z)=20(f(20log(z)))/(zLog10) mh=∫ e(x/20) Log10 f(x)dx σh 2=(∫(e(x/20)Log10)2f(x)dx)-mh 2 D.3.8.4.2 One variable H=g(F) h(z)=(f(g-1(z)))/ | g'(g-1(z) | ) mh= ∫ g(x) f(x) dx σh 2= ( ∫ g2(x) f(x) dx ) - mh 2 D.3.9 Using a function Two variables H=g(F, K) h(z)=∫((k(ϒ(z,x))/ | δg/δy |)f(x)dx mh=∫∫g(x,y)f(x)dx k(y)dy σh 2=(∫∫g2(x,y)f(x)dx k(y)dy)-mh 2 D.3.11 Substitutions t replaces x in a distribution x k(t) X(x) T(t) = X(k(t)) |k'(t)| See D.9.3 See D.9.3 D.3.10.3 Reciprocals y = g ( x ) x = k ( y ) See D.3.10.5 See D.3.10.5 D.3.10.5 NOTE: In the above table, the symbol ∫ stands for: In the table above, the effect of the sign of a multiplicative constant has been highlighted. Great care is recommended with regard to possible effects on the validity of these expressions due to signs and possible zeros of expressions used above. Functions like g are supposed to be monotonous; for more details, please refer to the appropriate clause of the annex. (1) The equations are related to independent variables, unless otherwise stated. (2) TR 100 028 uses extensively this formula. ∫ +∞ ∞ − ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 238 D.4 Influence quantities D.4.1 Theoretical approach The basic concept addressed in this clause is the introduction of a factor K relating parameters ("quantities") not very well controlled … such as temperature or voltage, which may have some influence on the measurement considered to their effect. This factor is to be multiplied by the parameter whose influence is being considered. The situation can therefore be interpreted using the product of two random variables, and the properties found in clause D.3.6 can therefore be used. This will introduce expressions such as those found in clause D.3.6.3: therefore the mean value in terms of effect, mh , is: mh = mf mk and "standard deviation"σσσσh is such that: σσσσh 2 + mh 2= (σσσσf 2 + mf 2 )( σσσσk 2 + mk 2 ) , where f relate to the random variable (parameter) being addressed (e.g. temperature) and k to random variable corresponding to the multiplicative factor K. D.4.2 Examples D.4.2.1 Effect of the temperature Suppose the temperature can have an effect modelled as K dT , where dT is supposed to be a random variable, with a rectangular distribution, and K is known by its average value mk and its standard deviation σσσσk . As indicated above, we have then: σσσσh 2 + mh 2= (σσσσdt 2 + mdt 2 )( σσσσk 2 + mk 2 ) . However, dT can be defined such that its average value, mdt , be 0. Noting that we also have: mh = mdt mk when mdt = 0 , we also have mh = 0 . In this case, the expression of σσσσh can be simplified: σσσσh 2 = (σσσσdt 2 ) ( σσσσk 2 + mk 2 ) . This expression recalls equation 5.2 (when m dt = 0) found in clause 5.4 of TR 100 028-1 [6] of the present document: " The standard uncertainty to be converted is uj 1. The mean value of the influence quantity is A and its standard uncertainty is uj a. The resulting standard uncertainty uj converted of the conversion is: ) +u A ( u = u a j j converted j 2 2 2 1 (5.2) ". ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 239 Further information concerning the values of influence quantities may be found in table C.1. When building similar tables it is of primary importance to address how terms such as the term K dT are to be incorporated in the general set of equations describing the measurement (see clause D.5). D.4.2.2 Effect of the temperature on a resistor As in the clause above, suppose the temperature can have an effect modelled as K dT , where dT is supposed to be a random variable, with a rectangular distribution, and K is known by its average value mk and its standard deviation σσσσk . A general expression of the value of a resistor could be: R = R0 ( 1 + K dT ) , where R0 and R are respectively the resistance for temperatures defined by dT = 0 and for any other value of dT . The above expression can also be written as: R = R0 + R0 K dT and be interpreted as an operation involving 4 random variables R0 , R , K and dT . In this case, R0 can be considered as the result of an appropriate combination of distributions, providing the measurement uncertainty for the measurement of the resistor (see clause D.5). From the properties found in clause D.3 , it comes that: σσσσR 2 = σσσσR0 2 + σσσσR0KdT 2 and σσσσR0KdT 2 + mR0KdT 2= (σσσσR0 2 + mR0 2 ) (σσσσdt 2 + mdt 2 )( σσσσk 2 + mk 2 ) . As indicated in the previous clause, it is possible to choose values so that some of the average values are 0 , and to simplify the expressions accordingly; furthermore, when R0 is considered as providing the probability density for the resistor (together with the measurement uncertainty) we get: σσσσR0KdT 2 = (σσσσR0 2 + mR0 2 ) (σσσσdt 2 )( σσσσk 2 + mk 2 ) . Therefore, σσσσR 2 = σσσσR0 2 + (σσσσR0 2 + mR0 2 ) (σσσσdt 2 )( σσσσk 2 + mk 2 ) . Hopefully σσσσR0 2 << mR0 2 so finally we get an approximation: σσσσR 2 = σσσσR0 2 + mR0 2 σσσσdt 2 ( σσσσk 2 + mk 2 ) or σσσσR 2 = σσσσR0 2 + R0m 2 σσσσdt 2 ( σσσσk 2 + mk 2 ) where R0m represents the measured value of the resistor. Should R0m be equal to 1 then σσσσR 2 = σσσσR0 2 + σσσσdt 2 ( σσσσk 2 + mk 2 ) an expression which is, similar to those implicitly found in the main body of the present document. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 240 D.5 Global approaches D.5.1 Using directly the random variables in a measurement D.5.1.1 Introduction The method to calculate the density probability of any (well behaved) combination of two random variables has been given in clause D.3.11 and the expression of the first moments of the probability density of any (also supposed well behaved) function (deterministic) of N variables has been given in clause D.3.11.5. Clause D.3 provides similar results for usual operations and combinations of random variables. Therefore, it should be possible to calculate step by step any (well behaved) combination of random variables. As a result, as soon as a system (e.g. a measurement set up) can be mapped to such a mathematical model, it is possible to evaluate its outputs as a function of its inputs (e.g. in terms of results of measurements and of uncertainties). D.5.1.2 Writing the equations Let us therefore consider a system with: - a set of inputs I1 … Ij … In - a set of outputs R1 … Rk … Rp where the outputs Rk have been expressed as functions of the various inputs Ij using a set of p functions of n variables g1 ( I1 , … Ij , … In ) … gk … gp ( I1 , … Ij , … In ) . When each input Ij is considered as a random variable, and all inputs are considered as a set of n independent random variables I1 … Ij … In , then, the set of p outputs, R1 … Rk … Rp , can be considered as a set of random variables of which the statistical/probabilistic properties are known and determined by the equation found in clauses D.11 and D.11.5, as soon as g1 ( I1 , … Ij , … In ) … gk … gp ( I1 , … Ij , … In ) and the statistical/probabilistic properties of the inputs (i.e. I1 … Ij … In) are given. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 241 D.5.1.3 Number of equations Some rather simple measurements (e.g. "conducted power") can be modelled using only one equation. To model a substitution measurement (see clause D.5.3) it can be user friendly to use a set of two of such equations. D.5.1.4 Mapping variables As already proposed in clause D.2.1 , the characteristics of the output signal of a generator can be represented by a random variable, G , where the uncertainties relating to the generator's output signal characterize G . For example, the probability density of G could be a rectangular distribution centred around 10 mV , having a zero value outside [9 , 11] ( values given in mV). As also addressed in clause D.2.1.2 and D.2.1.4 , a model for measuring instruments can be constructed as follows: - a meter providing the corresponding reading, considered perfect (fully deterministic) - and a random variable associated with it , for example V , covering the uncertainties relating to the actual reading of the meter which characterize V ( V could be thought of as corresponding to the internal noise of the instrument). As a result, the "inputs" of the system can be classified in several groups containing, in particular: - actual physical inputs to the system (e.g. signals from generators) - random variables associated with measuring equipment (e.g. voltmeters and other instruments) - random variables relating to the environment (e.g. temperatures, supply voltages) which may affect the results via the influence quantities (see clause D.4). D.5.1.5 Conclusions Based on such a model, the outputs such as Rk can be interpreted as random variables characterizing the sought output(s) of the measurement (e.g. an output power), where the statistical/probabilistic properties of Rk provide the corresponding measurement uncertainty (probability of finding a specific value as the result of the measurement). Clause D.5.6 also addresses the interpretation of the results obtained (outputs Rk of the system). Examples where this approach was used, can be found in clauses D.2. D.5.2 Using random variables together with differentiation in a measurement The methodology presented in clause D.5.1 is based on the handling of a set of p functions of n variables. In the case of radio systems, these equations may be somewhat bulky. In the case of the evaluation of measurement uncertainties of a particular measurement, the input variables (corresponding to random variables in the methodology addressed in clause D.5.1) can be understood as having a very small probability of being far away for the setting sought for that measurement. Should Ij be such setting, then it could equally be interesting to consider small variations around Ij , dIj. In this case, it can be more convenient to consider Ij as a constant and dIj as the random variable to be further handled in the statistical/probabilistic analysis. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 242 In order to continue the evaluation of the measurement uncertainties, with this approach, the set of functions which had been used in clause D.5.1, - g1 ( I1 , … Ij , … In ) - gk ( I1 , … Ij , … In ) - gp ( I1 , … Ij , … In ) has to be differentiated which provides a set of p relations: n n j j dI I g dI I g dI I g dg ∂ ∂ + + ∂ ∂ + + ∂ ∂ = 1 1 1 1 1 1 .... .... n n k j j k k k dI I g dI I g dI I g dg ∂ ∂ + + ∂ ∂ + + ∂ ∂ = .... .... 1 1 n n p j j p p p dI I g dI I g dI I g dg ∂ ∂ + + ∂ ∂ + + ∂ ∂ = .... .... 1 1 . In fact, for a particular measuring point, this is a set of p linear equations (of n variables ) which can be mapped in a quite friendly manner to the expressions found in clause D.3.4.5 , as already suggested in clause D.3.4.5.3. The expression of σσσσ as given in clause D.3.4.5.3 was: σσσσdV 2 = λλλλ1 2 σσσσdx1 2 + λλλλ2 2 σσσσdx2 2 + … + λλλλn 2 σσσσdxn 2 and translates with the present set of equations into: 2 2 1 2 2 1 2 1 2 1 1 2 1 .... .... dIn n dIj j dI g I g I g I g σ σ σ σ ∂ ∂ + + ∂ ∂ + + ∂ ∂ = 2 2 2 2 2 1 2 1 2 .... .... dIn n k dIj j k dI k gk I g I g I g σ σ σ σ ∂ ∂ + + ∂ ∂ + + ∂ ∂ = 2 2 2 2 2 1 2 1 2 .... .... dIn n p dIj j p dI p gp I g I g I g σ σ σ σ ∂ ∂ + + ∂ ∂ + + ∂ ∂ = . Another advantage of this approach is that for the determination of the set of p linear equations of n variables, there is no real need to have an explicit expression of the outputs as: - g1 ( I1 , … Ij , … In ) - gk ( I1 , … Ij , … In ) - gp ( I1 , … Ij , … In ) which is required for the approach proposed in D.5.1. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 243 It is, in the present approach (D.5.2 ), sufficient to find the expressions relating inputs and outputs, differentiate, and then resolve the linear equations in order to obtain: - dg1 - dgk - dgp . It has finally to be noted that, in this approach, the output random variables can be matched directly to the estimation of the errors corresponding to measured values (probability of having the error within a certain interval), as opposed to clause D.5.1 where the output random variables would correspond to the probabilities of having a value of the measurement itself within a particular interval. More precisely, the difference in interpretation (between D.5.1 and D.5.2) differs by a constant, which is the measured value. Therefore, calculations on sigmas ( σσσσ ) are the same when using either the approach given in D.5.1 or that given in D.5.2 … D.5.3 Examples of application to particular cases D.5.3.1 Using random variables together with differentiation in a measurement, case of multiplicative functions In the case where the equations are multiplicative, the set of functions can be written as: - g1 ( I1 , … Ij , … In ) = A1 ( I1 )b1 … ( Ij ) bj … ( In ) bn - gk ( I1 , … Ij , … In ) = A2 ( ) …. - gp ( I1 , … Ij , … In ) = … . Then it becomes more convenient to use other type of expressions: - either n n n j j j I dI b I dI b I dI b g dg + + + + = .... .... 1 1 1 (logarithmic differentiation) - or … to transform the expressions into dBs. The handling and understanding of these situations is similar to that of D.5.2 … with the exception that the random variables (and corresponding sigmas) can be mapped now to relative values, as opposed to absolute values in the approach given in D.5.2. It has to be noted, however, that in approaches D.5.1 and D.5.2 random variables (and sigmas) have a unit (mA, Volts, etc) while in D.5.3 random variables (and sigmas) are relative, and have no real units (noting that values expressed in dBs are some kind of relative values). D.5.3.2 Substitution measurements Substitution measurements are often used in radio. It is expected by doing so, to reduce the influence of some parts of the set up, and their contribution in the uncertainty. The methodology presented in clause D.5.1 is based on the handling of a set of p functions of n variables. In the case of substitution measurements, the test set up for the measurement of radio systems can be modelled using two of these equations: - one equation corresponding to the test set up "before" the substitution, - one equation corresponding to the test set up "after" the substitution. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 244 The set of equations can therefore look like: - g1 ( I1 , … Ij , … In ) - g2 ( I1 , … Ij , … In ) The practical handling and understanding of this set of two equations is similar to that corresponding to D.5.1 or D.5.2 (using differentiation) … with the exception that the random variables involved in the two equations are not necessarily independent …and that the aim of this method is to reduce the number of terms to be taken into account. This is usually done by calculating the equation corresponding to the difference (subtraction) of the two equations of the set. It is therefore basic to identify: - which inputs are in reality identical and appear in a way that they can be discarded (no contribution for the uncertainty, e.g. a cable which is used twice in the same conditions) - which inputs (mapped to contributions of the uncertainty) are independent - which inputs (mapped to contributions of the uncertainty) are not independent. As a result of this analysis, some of the contributions are to be combined by RSSing, others disappear, others have to be combined in other ways (e.g. by linear combination as indicated in clause D.3.4.6) … Substitution methods are often used for radio measurements because they are expected to provide better results. However, the analysis required for the evaluation of the corresponding uncertainties requires certainly more care than the analysis required in the case of direct measurements. NOTE: This analysis has not necessarily been completed in all examples included in the present edition of the present document. D.5.4 Empirical approach to find a model of the system When the equations are difficult to reach or to handle, it is possible for a complete system or for a part thereof (see clause D.5.5 , below) to try and find the equivalent of the partial derivatives (the coefficients needed in the linear equations addressed in clauses D.5.2 and possibly in D.5.3) by practical means. Having the measurement set up operational for the measurement being considered, and having performed that measurement once, it is then possible to make "small" variations of the settings of the various instruments, in particular concerning the generators. Such small differences (matching mathematically the dIj ) shall be: - small enough so that the system being analysed can be considered as linear within that range ( + dIj ) - big enough to be large compared with the uncertainties of the measurement ("measurement noise") - small enough so that equipment remains within the same operating range (e.g. the same scale for a voltmeter) - made preferably both sides of the original setting ( Ij ) , in order to obtain directly + dIj . The direct observation of the outputs of the system, would allow for a model to be established, providing the effect of the corresponding inputs (i.e. providing the values of the various coefficients corresponding to the k j g I ∂ ∂ of clause D.5.2). In order to evaluate the random uncertainties in the set up, each time an input value is changed, it should be, for a while brought back to its initial value ( Ij ) , and the measurement performed again. In this way, there is a great number of evaluations of the measurand under nominal conditions, which gives a good visibility of the randomness associated with the set up. The knowledge of the dispersion of the results can be very helpful in order to choose how small should be the variations ("step sizes") in the settings of the various instruments (it is important to avoid taking noise for the effect of variations of the inputs!). ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 245 Example of sequence of such steps: - ( I1 , … Ij , … In ) - ( I1 + δδδδ , … Ij , … In ) - ( I1 , … Ij , … In ) - ( I1 - δδδδ , … Ij , … In ) - ( I1 , … Ij , … In ) - similar sequence for I2 - etc …until … In . With 4 points per input variable … there are 4 n points to be measured. More points may be necessary if the effects are not linear. Obviously, this procedure is supposed to cover only those parameters for which small variations are possible. This procedure can be very useful when the mathematical expression providing the effect of such inputs is difficult to obtain. The evaluation of the effect of small variations of one variable (input) could be completed with the evaluation of the effects of changing simultaneously two or more inputs (e.g. for verification purposes, in particular for identification of variables which may interact) … as long as the interpretation of the corresponding results is fruitful. Methods given in D.5.2 and D.5.3 could then be used, based on these empirical values found, or on an appropriate mix of values empirical and/or theoretical. D.5.5 Splitting into sub-systems The aim of defining sub-systems is 3 fold: - to keep equations within manageable sizes, - to provide "building blocs" which could be used several times, without further mathematical work (i.e. subsets common to different measurements), - to support and simplify methods such as substitution methods, where parts of the set up are expected to be used twice. When looking at the present document and its previous versions, it becomes clear that one of the major problems the present document had to cope with is the need, in radio measurements, to handle simultaneously electrical signals whose levels cover several orders of magnitude. Therefore, in some cases it is more practical to handle dBs, in others to handle linear terms. Clauses of annex D.3.8 and annex E show that besides very simple approximations (based on Log (1+x) = x) conversions in either directions are somewhat awkward and subject to discussion (e.g. to start with, questions such as "what are the basic shapes of the uncertainties, and in which domain" have to be answered). The usage of sub-systems could, in some cases help this problem: an attempt could be made to isolate, in some sub- systems, parts to be handled in dBs, and, in other sub-systems, parts to be handled in linear terms, in an attempt to reduce the number of conversions (in particular conversions of uncertainties having values too large for simple approximations to be acceptable). However, it has to be stated once again that all the analysis performed in clause D.3 (combination of random variables) were based on calculations on independent random variables. Therefore, to be in a position to use the tools developed so far, great care has to be taken so that there are not two variables inter-related in two different subsystems. It can also be noted that empirical methods were proposed in clause D.5.4, in order to establish a model for a complete systems or parts thereof. Such possibilities may have also to be taken into account when tying to split systems into subsystems. In the case of automated uncertainty evaluation systems, splitting in sub-systems could lead to concepts having a flavour of subroutines or even a flavour of object oriented systems. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 246 D.5.6 Presentation and interpretation of results obtained (outputs) The last paragraph(s) of clauses D.5.1 and D.5.2 have provided for an interpretation of the results obtained when combining in an appropriate manner the statistical/probabilistic properties of the "inputs" to the system being considered. The purpose of clause D.5.6 is to provide a more general view on the topic and to go one step further, into the area of confidence levels. Therefore, this clause starts with a classical approach, the "worst case" approach, and continues with the "probabilistic approach", which corresponds, in fact, to the "main stream" of the present document. D.5.6.1 Worst case approach This clause can be understood as part of an introductory clause to clause D.5.6. In the "worst case approach", each contribution to the uncertainty is expected to be bound (which would not be the case for a probability density having a normal distribution). In this approach, the evaluation of the uncertainty is based on the analysis of the situation where each variable would have had a value contributing to the "worst case" scenario. In the case where all contributions correspond to rectangular distributions and are to be combined using an addition, then the "worst case approach" would provide the extreme points of the "foot-print" of the combined uncertainty (found in accordance with clauses D.3 and D.5), i.e. the interclause of the curve representing the distribution of combined uncertainty with the xx' axis (the horizontal axis). D.5.6.2 Probabilistic approach The " probabilistic approach" would rather focus on other properties of the combined uncertainty (e.g. its standard deviation or the shape of the corresponding distribution) than on "foot-prints", which is the focus of the "worst case approach". D.5.6.2.1 Preliminary comments (and choice of scenario) Clause D.5.6 and more particularly clause D.5.6.2 are intended to establish the relation between the results found when combining the various contributions to the uncertainty ("combined uncertainty") and the value to be provided as the result of the evaluation of the corresponding uncertainty. As shown in clause D.5.6.1, in the case of the approach called "worst case approach", this is quite straight forward. It can be a little more complex in the case of the " probabilistic approach": the "worst case approach" leads to the calculation of the value of a set of extreme points, while the "probabilistic approach" requires the understanding of the under-laying phenomena (and not only the RSSing of all the contributions). The "probabilistic approach" triggers also new problems such as those related to the co-existence of expressions in linear terms and in dBs (in the case of the "worst case", should this happen, it is only necessary to calculate the two extreme points, so mixing dBs and linear terms is not a real problem, it only means that there are a few conversions to be performed). Looking more in depth, it could be expected that the individual contributions to the measurement uncertainty are relatively small so that their conversions (dB into linear terms and vice-versa) are not a real problem (they can be performed using linear approximations). It is nevertheless important to make sure that the shape of the corresponding distribution has been correctly chosen (should the corresponding distribution have a rectangular shape, should it be rectangular in terms of dBs or in linear terms ?). In the case of results of complete measurements, however, the combined uncertainty value may be quite large (see the table in annex B providing "the maximum uncertainty" values). For such high values (up to several dBs) significant differences may result from the way in which the conversions are handled (see, for example, clause D.3.8.4 and annex E). The example provided in clause D.3.7.4 shows clearly how much care is to be devoted to approximations… ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 247 As a result, the following strategy can be proposed: - to use rather simple conversion methods in order to perform the conversions relating to the various contributions (small values) - to use more accurate methods when the values become higher (in particular final results of a measurement or final result of some "sub-system" (see the presentation of the sub-system concept in clause D.5.5)). Among possible methods to make the conversions, can be quoted those presented in this annex (see D.3.8.4), those in annex E (presented differently, but equivalent (as indicated in clause D.3.8.4)); spread sheets can also be used, etc. Attention has also to be drawn, again, to the fact that, during such conversions, familiar distributions, simple to describe in mathematical terms, are transformed in less familiar distributions (often having asymmetrical shapes and more complex to describe in mathematical terms) where the first moments (mean value, standard deviation) do not necessarily convey the expected information in a handy way ...and are not necessarily the images of the corresponding points (moments) before the conversion… D.5.6.2.2 Summary of the methodology The approach proposed in a number of detailed examples (given in annex D and in the main body of the present document as well) can be summarized as follows. 1) All the contributions for the uncertainty have to be identified (and the relations between the various parameters established). 2) The statistical/probabilistic properties (e.g. the standard deviations of the various contributions) have to be identified and appropriately combined together (see clauses D.5.1 and D.5.2 ). If the combination corresponds to mere additions, then the situation is covered by the "BIPM method" and an RSSing of the various components can be performed. 3) Assuming that the appropriate combination of all contributions would result in a Gaussian shaped distribution, then the "combined uncertainty", characterized by its standard deviation, would be equal to the standard deviation of that Gaussian distribution. This Gaussian would then represent, in fact (more precisely, in the case of the method given in clause D.5.2) the probability of error of the measurement (i.e. the uncertainty). NOTE 1: In the case where the method provided in clause D.5.1 is used, the interpretation is similar, except that the resulting Gaussian would then correspond to measured values. Its mean value would then correspond to the result of the measurement (it could provide the "measured value"). 4) A random variable E , the error of the measurement, corresponding to the above Gaussian distribution can be considered. It is characterized (similarly to what has been written a number of times in the present annex) by the fact that its value x has a probability of occurrence given by the corresponding probability density e ( x ): by definition, the probability Pe of the random variable E (the "error") having a value x such that x1 < x < x2 is ∫ = 2 1 x x e e(x)dx P . Similarly, we can consider ∫ ∞ − = x e dt e(t) x P ) ( , and therefore (by differentiation) dPe = e ( x ) dx . 5) When a certain set x1 , x2 is given, these bounds together with the shape of the Gaussian provide the probability of the error of the measurement being within those bounds. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 248 The equation of such a Gaussian is 2 2 2 2 1 σ π σ x e y − = , where σσσσ (sigma) is the standard deviation of the Gaussian (and is equal to the combined uncertainty of the measurement), as shown in clause D.1. When x = + σσσσ (sigma, the standard deviation), the corresponding values y1 and y2 are known, and the surface between the curve and the axis xx' (between + σσσσ (sigma)) can be found: this surface provides the probability of the error being between + σσσσ (sigma), which is ∫ + − = σ σ e(x)dx Pe or ∫ + − − = σ σ σ π σ dx e P x e 2 2 2 2 1 . This probability is equal to 68,3 % and provides the linkage to the confidence level. 6) As defined in TR 100 028-1 [6], clause 4.1.1: absolute error = measured value - true value . Therefore, when the probability of the absolute error being within + σσσσ is 68,3 % , then, the probability of the result of the measurement being within + σσσσ of the true value is also 68,3 % . 7) In order to have another (usually greater) confidence level, Pe' another set (therefore with wider values) x1' , x2' has to be found … so that 2 1 ' ' ' x e x P e(x)dx = ∫ . The value of 1,96 has been given in the main body of the present document, as the multiplicative factor ("expansion factor") to be used in order to reach a confidence level of 95 %: - when x1 = -1,96 x σσσσ - and x2 = +1,96 x σσσσ , 95 ,0 2 1 96 ,1 96 ,1 2 2 2 = ∫ + − − σ σ σ π σ dx e x , which is the sought confidence level. This is true for any normal distribution (it is true for any Gaussian, independently of the value of σσσσ ), but true for normal distributions only. An expansion factor of 2 can also be used: 2 2 2 2 2 1 0,9545 2 x e dx σ σ σ σ π + − − = ∫ . An expansion factor of 2 provides therefore a confidence level of 95,45 %. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 249 NOTE 2: The values of 2 2 2 1 2 x k k e dx σ σ σ σ π + − −∫ i.e. the values of the confidence levels corresponding to an expansion factor k can be found easily in tables (such tables are often appended to books relating to probabilities (and providing properties of the Gaussians)). D.5.6.2.3 Normal and non-Normal distributions The principles given above are valid in all cases. However, it is obvious that all numerical values, and in particular the actual values corresponding to "expansion factors " (i.e. 1,96 or 2 in the case of Gaussian distributions), are depending on the shape of the probability density resulting of the combination (i.e. the density probability of the error in case D.5.2) for a particular measurement. An interesting example can be found in clause D.3.3.5.2.2. Should the final probability density curve have a shape significantly different from a Gaussian, then the multiplicative factor (the "expansion factor") to get the 95 % confidence level would have to be re-evaluated, taking into account the actual probability density …(this kind of difficulty had already been identified in TR 100 028-1 [6], clause 6.6.5.1, where the direct usage of the expansion factor would have led to negative bit error ratios! ) That is why in clause D.3, not only the two first moments of the various combinations were evaluated, but were also provided the equations corresponding to the resulting probability densities themselves. D.5.6.2.4 Confidence levels for non-Normal distributions When having the expression of the resulting distribution e ( x ) , then the confidence level is given by the same expression as for normal distributions: ( ) k k e x dx σ σ + −∫ = confidence level corresponding to the expansion factor k . However, for unusual expressions of e ( x ) , it is unlikely to find the corresponding values in tables … the corresponding calculations will therefore have to be made on a case by case basis. Further comments 1) In one of the examples given in annex D (in clause D.3.3.5.1), it is shown that the result of the additive combination of two Gaussian shaped uncertainties (i.e. random variables) is also a Gaussian shaped uncertainty (i.e. random variable). In this respect Gaussians are stable (rectangular distributions are not: the combination of two identical rectangular distributions is a triangular distribution, as shown in clause D.1.3.2). 2) Converting dBs into linear and vice-versa, tends to generate asymmetric distributions … and this may have to be duly taken into account. An attempt to give some properties of asymmetrical distributions has been made in clause D.1.3.3 (trapezoidal) and D.1.3.5 , but calculations with such expressions are not always that easy. Handling such expressions is an area where approximations can be used extensively. Symmetrical expansion factors can be used in all cases, but when distributions are asymmetric, it can also be thought of using asymmetric expansion factors (one for expanding the lower bound and another for expanding the upper bound)… Another proposal had been made in the first days of ETR 028 [5]: to calculate both a "sigma plus" and a "sigma minus" … as if the final error distribution was composed of 2 half Gaussian distributions: 2 2 2 2 1 σ π σ x e y − = with two values for sigma, one when x is positive and another when x is negative. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 250 … one trouble with such a representation is that the 2 distributions do not necessarily fit together in 0: π σ 2 1 ) 0 ( = y , which shows that y ( 0 ) depends on σσσσ (sigma). Therefore e ( 0 + ) ≠≠≠≠ e (0 -) and e ( 0 + ) dx ≠≠≠≠ e (0 -) dx finally ∫ + + = ε ε 0 ) ( e(x)dx P ≠≠≠≠ ∫ − −= 0 ) ( ε ε e(x)dx P which does mean that the probability of having a range of very small positive errors is significantly different from that of having a very small range of negative errors … not very satisfactory! The way to handle the uncertainties in the present version of the present document seems more satisfactory. 3) It can also be noted that a finite sum of distributions having a finite footprint has also a finite footprint. As a result, in such a situation, there should be an expansion factor providing a 100 % confidence. 4) clause D.3.3.5.2 has highlighted a case where a non finite sum of rectangular shaped distributions has provided a finite footprint. In such case, there should also be an expansion factor providing for a 100 % confidence level. 5) In the case where a "worst case" (see clause D.5.6.1) value exists …then there should also be an expansion factor providing a 100 % confidence level. D.5.6.2.5 Practical conclusions As a result, and in order to avoid extensive discussion, results could be presented: - as a "1.96 x σσσσ (sigma)" value - or as a "95 % confidence level" value, with a note stating that the two values are equivalent in the case of normal distributions. This should replace text such as: "The expanded uncertainty is ± 1,96 × 1,06 dB = 2,07 dB at a 95 % confidence level", which has also been used for cases where there is no evidence that the distribution concerned is normal (the number (and relative weight) of contributions combined in many evaluations of the measurement uncertainty may not be sufficient for the central limit theorem to be valid). NOTE: As shown above, the method to be used when changing the confidence level can be justified by the properties of the distribution obtained when combining the various contributions in order to obtain the combined uncertainty, in particular, when a Gaussian distribution is obtained. There is no need to use the t-Student theory (which is valid only when normal distributions are handled)…and which relates to statistics (e.g. series of measurements). D.5.6.2.6 Implications Corresponding changes in text should therefore be introduced in a numbers of places (including in a number of clauses of the present document). In a report relating to measurements, should be found: - the measured value; - the uncertainty value found; ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 251 - a statement indicating that: - this uncertainty value corresponds to "a confidence level of 95 %" or - this uncertainty value corresponds to "1.96 x σσσσ (sigma)" (where 95 % and 1,96 are the values used in the main body of the present document) - and a note indicating that "1.96 x σσσσ (sigma) is equivalent to a confidence level of 95 % in the case where distributions are normal". NOTE: An expansion factor of 2 is also acceptable. It corresponds to a confidence level of 95,45 %. In this case, the statements above should be amended accordingly. D.5.6.2.7 Examples (excerpts from available standards) ETSI has been drafting technical standards in support of a variety of radio equipment, and also a number of standards to be harmonized under Directives, such as the R&TTE Directive. The following excerpts were taken from: - Part 1 (corresponding to "the radio product standard"); and - Part 2 (corresponding to "the candidate harmonized standard") of the standard corresponding to one particular product. This material, provided as an example, shows how the words proposed above (in clause D.5.6.2.6) have been used in recent standards prepared by ETSI. A third example shows how double sided limits have been handled in TR 100 028-1 [6] of a standard relating to integral antenna equipment (in the clause relating to limits). D.5.6.2.7.1 Excerpts from a "Part 1" " 11 Measurement uncertainty Table D.1: Absolute measurement uncertainties: maximum Values Parameter Uncertainty Radio Frequency ±1 x 10-7 RF Power (up to 160 W) ±0,75 dB Radiated RF power ±6 dB Adjacent channel power ±5 dB Conducted spurious emission of transmitter Valid up to 12,75 GHz ±4 dB Conducted spurious emission of receiver, Valid up to 12,75 GHz ±7 dB Two-signal measurement, Valid up to 4 GHz ±4 dB Three-signal measurement ±3 dB Radiated emission of the transmitter, valid up to 4 GHz ±6 dB Radiated emission of receiver, valid up to 4 GHz ±6 dB Transmitter attack time ±20 % Transmitter release time ±20 % Transmitter transient frequency (frequency difference) ±250 Hz Transmitter intermodulation ±3 dB Receiver desensitization (duplex operation) ±0,5 dB Valid up to 1 GHz for the RF parameters unless otherwise stated. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 252 For the test methods, according to the present document, the measurement uncertainty figures shall be calculated in accordance with TR 100 028 and shall correspond to an expansion factor (coverage factor) k = 1,96 or k = 2 (which provide confidence levels of respectively 95 % and 95,45 % in the case where the distributions characterizing the actual measurement uncertainties are normal (Gaussian)). Table D.1 is based on such expansion factors. The particular expansion factor used for the evaluation of the measurement uncertainty shall be stated. " NOTE: the table of "Absolute measurement uncertainties" is included here just for completeness. The "standard table" can be found in annex B of the present document. D.5.6.2.7.2 Excerpts from a "Part 2" " 5.2 Interpretation of the measurement results The interpretation of the results recorded in a test report for the measurements described in the present document shall be as follows: - the measured value related to the corresponding limit will be used to decide whether an equipment meets the requirements of the present document; - the value of the measurement uncertainty for the measurement of each parameter shall be included in the test report; - the value of the measurement uncertainty shall be, for each measurement, equal to or lower than the figures in table D.2. For the test methods, according to the present document, the measurement uncertainty figures shall be calculated in accordance with TR 100 028 and shall correspond to an expansion factor (coverage factor) k = 1,96 or k = 2 (which provide confidence levels of respectively 95 % and 95,45 % in the case where the distributions characterizing the actual measurement uncertainties are normal (Gaussian)). Table D.2 is based on such expansion factors. The particular expansion factor used for the evaluation of the measurement uncertainty shall be stated. Table D.2: Absolute measurement uncertainties: maximum values Parameter Uncertainty Radio Frequency ±1 X 10-7 RF Power conducted (up to 160 W) ±0,75 dB Conducted RF Power variations using a test fixture ±0,75 dB Radiated RF power ±6 dB Adjacent channel power ±5 dB Average sensitivity (radiated) ±3 dB Two-signal measurement, valid up to 4 GHz (using a test fixture) ±4 dB Two-signal measurement using radiated fields (see note) ±6 dB Three-signal measurement (using a test fixture) ±3 dB Radiated emission of the transmitter, valid up to 4 GHz ±6 dB Radiated emission of receiver, valid up to 4 GHz ±6 dB Transmitter transient frequency (frequency difference) Transmitter transient time ±250 Hz ±20 % Values valid up to 1 GHz for the RF parameters unless otherwise stated. NOTE: For blocking and spurious response rejection measurements. " ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 253 NOTE: the table of "Absolute measurement uncertainties" is included here just for completeness. The "standard table" can be found in annex B of the present document. D.5.6.2.7.3 Excerpts from a "Part 1" showing words used for double sided limits The following piece of text shows one way to adapt the "shared risk approach" to the case where the measurement uncertainties are larger than the allowed tolerances. Should such a case happen, the direct implementation of the "shared risk approach" could have resulted in a situation where good equipment might have failed the test. " 5.1.2.1 Effective radiated power under normal test conditions The maximum effective radiated power under normal test conditions shall be within df of the rated maximum effective radiated power. …/… The allowance for the characteristics of the equipment (±1,5 dB) shall be combined with the actual measurement uncertainty in order to provide df, as follows: df 2 = dm 2 + de 2; where: - dm is the actual measurement uncertainty; - de is the allowance for the equipment (±1,5 dB); - df is the final difference. All values shall be expressed in linear terms. In all cases the actual measurement uncertainty shall comply with clause 10. Furthermore, the maximum effective radiated power shall not exceed the maximum value allowed by the administrations. Example of the calculation of df: - dm = 6 dB (value acceptable, as indicated in the table of maximum uncertainties, table 8); = 3,98 in linear terms; - de = 1,5 dB (fixed value for all equipment fulfilling the requirements of the present document); = 1,41 in linear terms; - df 2 = [3,98]2 + [1,41]2; therefore df = 4,22 in linear terms, or 6,25 dB. This calculation shows that in this case df is in excess of 0,25 dB compared to dm, the actual measurement uncertainty (6 dB). " Comment: In the present document, it was chosen to combine the two components in linear terms. It could have been decided, as well, to do the operation in dBs. See the corresponding discussion in clause D.5.6.2.1. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 254 D.5.6.2.8 Confidence levels and single sided limits The confidence level has been related to ∫ = 2 1 x x e e(x)dx P , the probability of the value x of the random variable E being so that x1 < x < x2 . In the case where L is a limit value (single sided), and V the true value of the measurand, then the probability of having good equipment failing the test is such as: fail L V P e(x)dx ∞ + − = ∫ or V L fail P e(x)dx − − ∞ = ∫ as appropriate (depending on the relative position of the sought value, V , in relation to L ). In the particular case when the distribution is, in fact, a normal distribution, and when the true value of the measurand is at 1.96 x σσσσ (sigma) from the limit L , then the expression of the probability of having good equipment failing the test is such as: 2 2 2 1,96 1 2 x fail P e dx σ σ σ π +∞ − = ∫ = 0,5 ( 1 – 0, 95) = 0,025. It can be noted, however, that, as already suggested, in the case of radio measurements, finite sums of finite distributions are often found. Therefore, it is far from being sure that the Gaussian model is suitable for the discussion of effects far away from the area -σ σ σ σ to +σ σ σ σ , such as the probability of failing good equipment … It is quite likely that, in many cases, by increasing the expansion factor, the "worst case" value is reached, while, with the Gaussian model, there is always a (remote) probability to fail a good unit. The safe approach to calculate the probability of failing good equipment is certainly to calculate the actual distribution first, and then to use expressions such as those given in the beginning of the present clause, in order to calculate the appropriate probabilities. D.5.6.3 Conclusions Clause D.5.6 has provided an overview of the usual ways of addressing uncertainties: - the "worst case" approach and - the "probabilistic" approach. It has also covered the relations between these approaches as well as methods and caveats relating to the evaluation of the corresponding "confidence levels". Finally, it has also proposed methods to calculate correctly the probability of failing good equipment. D.5.7 Summary Clause D.5 has provided a set of approaches and methods that should cover the evaluation of measurement uncertainties and their confidence levels in a most situations (and can also cover applications far beyond the scope of the present document). The majority of the clause in D.5 address however, implicitly, the case where differentiation is used (clause D.5.2). But most concepts are usable also without differentiation (clause D.5.1); in some cases a slight transposition may have to be performed by the reader (trying to cover fully and individually, in this clause all possible combinations of methods and approaches could have resulted in an unnecessarily bulky clause…). ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 255 Clause D.5 provide, in fact, the basis for the various clauses of the present document (i.e. the "examples"), even though, in the majority of cases only the handling of the "sigmas" (standard deviations) has been described (while forgetting quite often to provide the underlying physical equations and to discuss which variables are independent and which are not)… an area which could be enhanced in future editions. D.6 Conclusions Annex D has provided general methods based upon the analysis of complex systems and a number of tools (e.g. in clause D.3) allowing to evaluate the measurement uncertainties related to the various measurement set up. It has in particular provided support for a number of clauses of both Part 1 and 2 of the present document, as well as highlighted precautions in order to avoid fundamental errors while using the examples developed over the various clauses (e.g. special attention to the independence (or possible inter-dependence) of the various associated random variables). When drafting this annex, the new situation in Europe, originated by the implementation of the R&TTE was also in mind: it is likely that in the future, with concepts such as self-declaration or self-certification, many more partners will have to make and understand radio measurements … and to handle the corresponding measurements uncertainties (hopefully in the same way). Therefore, new text was written in an attempt to make the present document as much self contained as practical, including all the theoretical elements allowing for any laboratory to understand what is to be done and obtain correct values, while giving any one a chance to try and find solutions well adapted to his own measurement set up … It is also expected that many other types of systems might be analysed using the methods developed in this annex. It can be noted, for example, that a number of mobile systems use adaptive techniques, such as power control. Such techniques are usually, in one way or another, based upon measurements (made by the mobiles and/or by base or monitoring stations). The methods presented in this annex could certainly be helpful also when evaluating the influence of the measurement uncertainties relating to such (simple) measurements, on the performance of the modern mobile systems where such features are implemented. Among possible effects of such uncertainties can be quoted loss of system capacity, signalling overhead … or even system oscillations … Measurement uncertainties (as well as dispersion of equipment characteristics) may also have to be taken into account in studies relating to the compatibility between systems, systems lay out, etc … ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 256 Annex E: Mathematical transforms This annex shows how direct methods can be used to transform distributions. Other methods (more general methods) for transforming (or converting) distributions are presented in clause D.3.9. E.1 Principles of derivation of formulas when transforming from log to linear When transforming from one co-ordinate system to another the following apply: 1) The probability of an event being within an interval is the same no matter which scale on the co-ordinate system you look at: A B dB % A/ B/ A/% dB A B/% dB B p x 1 ( ) p x 2 ( ) ∫ ∫ = / / B A B A )dx (x p (x)dx p 1 1 2 1 2) this also means that: 1 1 1 2 1 = = ∫ ∫ +∞ ∞ − +∞ ∞ − )dx (x p (x)dx p 3) based on this, the converted distribution can now be derived. E.1.1 A rectangular distribution in logarithmic terms converted to linear terms In this example a rectangular distribution in logarithmic terms is converted to linear terms: -A +A 1 2A x1 x2 0 dB p x ( ) A A x A x = − ≤ ≤ = 1 2 0 for for all other values of p x ( ) p x ( )| | ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 257 The probability of x being in the interval between x1 and x2 is: − = ∫ 1 2 2 1 2 1 2 1 2 1 x A x A dx A x x ; ( ) 1 2 2 1 x x A − = . In log terms. Therefore in linear terms this becomes: ( ) ( ) 1 2 10 10 1 2 2 1 20 2 20 1 x x A dx x p x x − = ∫ ; − = 20 2 20 2 1 2 10 10 x x P P ; where ( ) ( ) ∫ = x p x P 2 2 or in other words A x P x 2 10 2 20 2 2 = . Substituting 10 2 Log K P / = gives: A x x K Log K / x / 2 20 10 2 2 20 10 2 = = ; A K / 10 = ; ( ) ( ) ( ) x Ln A Ln x Log A 10 10 10 10 = ; ( ) x = dx x dLn As 1 ; ( ) ( ) x A Ln x p 1 10 10 2 = . 1 A 20 10 A 20 10 − P2 (x) From p2(x) the mean value xm and the standard deviation can be found. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 258 General formula: dx x xp xm ) ( 2 ∫ = ; ∫ ∫ = = C B C B m K dx x dx x K x 1 ; [ ] ( ) C-B =K Kx x C B m = . where ( ) 20 20 10 10 10 10 A A ; C ; B A Ln K = = = − . Then the standard deviation σ can be found. The general formula is: ( ) ( )dx x p x x s m ∫ +∞ ∞ − − = 2 2 ; ( ) dx x K x x s C B m ∫ − = 1 2 2 ; ( ) dx x K x x x x C B m m ∫ − + = 2 2 2 ; dx K x Kx x Kx C B m m ∫ − + = 2 2 ; ( ) C B m m Kx x Kx x Ln Kx − + = 2 2 2 2 ; ( ) ( ) ( ) ( ) − − − + − B C x B C B Ln C Ln x K m m 2 2 1 2 2 2 ; ( ) ( ) ( ) 1 = − B Ln C Ln As K . Therefore: ( ) ( ) 2 2 2 2 2 1 2 B C K B C K x x s m m − + − − = ; and ( ) B C K xm − = hence: ( ) ( ) ( ) 2 2 2 2 2 2 2 2 1 2 B C K B C K B C K s − + − − − = ; ( ) ( )2 2 2 2 2 1 B C K B C K − − − = ; therefore: ( ) ( )2 2 2 2 5 0 B C K B C K , s − − − = . This procedure can (in principle) be applied to any conversion of any distribution. See also clause D.3.9 where a general approach is provided. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 259 E.1.2 A triangular distribution in logarithmic terms converted to linear terms In the same way as with the rectangular distribution the conversion from logs to linear terms are made: dB x1 x2 A 1 A + A − p x A A x x A p x A A x A x p x x 1 2 1 2 1 1 0 1 0 0 ( ) ( ) ( ) = + > ≥− = − ≥ ≥ = ||| ||| ||| ||| for for for all other values of p x 1 ( ) In the negative interval: ( ) 2 1 2 1 2 1 2 2 2 1 2 1 x x x x x x A x A x dx A x A dx x p + = + = ∫ ∫ ; − = + − + 20 2 20 2 2 2 1 1 2 2 2 2 1 2 10 10 2 2 x x P P A x A x A x A x ; 2 2 20 2 2 10 A x A x P x + = . Solution: ( ) ( ) ( )2 2 1 y Log K y Log K + ; A x = x K Log K x 20 10 1 20 1 = ; A K 20 1 = ; 2 2 2 20 2 2 10 A x = Log K x ; 2 2 2 2 2 2 20 A x = x K ; 2 1 2 2 2 2 1 2 20 K = A K = . Logs converted to Ln: ( ) 10 20 1 A Ln K = ; ( ) ( ) ( ) ( )2 2 1 1 2 2 1 y Ln K y Ln K y P + = ; ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 260 ( ) ( ) y y Ln K y K dy y dP 2 1 1 1 + = ; ( ) y y Ln K y K 2 1 1 1 + for 1 10 20 ≤ ≤ − y A ; and ( ) y y Ln K y K 2 1 1 1 − for 20 10 1 A y ≤ ≤ ; 20 20 10 10 A A and C B = = − . Mean value: ( ) ( ) ∫ ∫ − + + = C B m xdx dx x Ln K x K xdx dx x Ln K x K x 1 2 1 1 1 2 1 1 1 1 ; ( ) ( ) ( ) ( ) ∫ ∫ − + + = C B dx x Ln K K dx x Ln K K 1 2 1 1 1 2 1 1 ; ( ) ( ) ∫ ∫ ∫ − + = C B C B dx x Ln K dx x Ln K K 1 2 1 1 2 1 1 ; [ ] ( ) [ ] ( ) [ ]C B C B x x xLn K x x xLn K x K 1 2 1 1 2 1 1 − − − + = ; ( ) ( ) ( ) ( ) ( ) 1 1 2 1 2 1 2 1 2 1 1 k C C C Ln K B B B Ln K K B C K − − − − − + − = ; ( ) − − − − − − − = 1 1 1 1 2 1 2 1 1 2 1 2 1 1 k c K k B K K B C K ; ( ) C K C K B K B K K B C K 2 1 1 2 1 1 2 1 1 2 + − × + + − − = ; ( ) 2 2 1 − + = C B K xm . Standard deviation: ( ) ∫ +∞ ∞ − − = p(x)dx x x s m 2 2 ; ( ) ( ) ( ) ( ) dx x x Ln K x K x x dx x x Ln K x K x x C m B m ∫ ∫ − − + + − = 1 2 1 1 2 1 2 1 1 2 1 1 ; ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ − − − + − = C m B m C B m dx x x Ln K x x dx x x Ln K x x x K x x 1 2 1 2 1 2 1 2 1 2 1 ; ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ − + − − + + − + = C m m B m m C B m m dx x x Ln K x x x x dx x x Ln K x x x x x x x x x K 1 2 1 2 2 1 2 1 2 2 2 2 1 2 2 1 2 ; ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 261 ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ − + − − + + − + = C m m B m m C B m m dx x Ln x x xLn x x Ln x K dx x Ln x x xLn x x Ln x K dx K x x K x K x 1 2 2 1 1 2 2 1 1 1 1 2 2 2 2 ; ( ) ( ) − = ∫ 2 2 4 1 2 1 x x Ln x x xLn ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) C m m B m m C B m m x x xLn x x Ln x x Ln x K x x xLn x x Ln x x Ln x K x x x x Ln x K 1 2 2 2 2 1 1 2 2 2 2 1 2 2 1 2 2 1 2 1 2 1 2 2 1 2 1 2 1 2 2 1 − − − + − − − − + + − + = ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) − + − − − + − − + − − − − − − + − − − + − = m m m m m m m x C C CLn x C Ln C C Ln x K B B BLn x B Ln B B Ln x x K B C x B C B Ln C Ln m K 2 4 1 2 2 1 2 1 2 1 2 2 1 2 1 2 1 1 2 2 1 2 1 2 2 1 2 2 2 2 1 2 2 2 2 1 2 2 2 1 ; ( ) ( ) ( ) ( ) ( ) − = = = − 1 1 1 1 1 1 K B , Ln K C , Ln B Ln c Ln K ; ( ) ( ) 2 2 2 2 1 2 4 1 2 1 4 m m m x C B x C B x K + + − + + − = ; and ( ) ( ) 2 2 2 2 1 2 4 1 2 1 4 m m m x C B x C B x K s + + − + + − = . E.1.3 A rectangular distribution in linear terms converted to logarithmic terms: In this example a rectangular distribution in linear terms is converted in to logarithmic terms: x1 x2 p x ( ) B C 1 C B 1 B A 1 C A 1 K A 1 2 1 Linear ( ) ∫ ∫ = 2 1 2 1 20 20 2 1 Log x Log x x x dy y p dx K ; ( ) ( ) ( ) 1 2 2 2 1 1 2 2 20 20 Log x p Log x p x K x K − = − . In other words: ( ) ( ) x Log p X K 20 2 1 = , the solution: ( ) x K =K x p 2 10 3 2 where = = 20 1 2 K ( ) 1 3 20 3 1 1 1 2 10 x K K x K x Log K = = Now ( ) ( )x Ln K x K e K K x p K K 10 3 3 2 1 3 2 2 10 = = = . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 262 Then ( ) 20 10 2 Ln K = . Now: ( ) = − = 20 10 1 2 1 Ln , K B C K ( ) ( ) dx x dp x p 2 2 = x K e K K 2 2 1 = 2 1 3 K K K = Check: ( ) 1 2 = ∫ +∞ ∞ − x p ( ) ( ) ( ) ( ) A Log A Log x K e K K A Log A Log dx x K e K + − = ∫ + − 1 20 1 20 2 2 3 1 20 1 20 2 3 ( ) ( ) ( ) A Log K A Log K e e K K − + − = 1 20 1 20 2 3 2 2 ( ) ( ) ( ) ( ) − = − × × + × × A Log Ln A Log Ln e e A 1 20 20 10 1 20 20 10 2 1 ( ) ( ) ( ) 1 1 1 2 1 = − − + = A A A Mean Value: A A, B C − = + = 1 1 ∫ Log C Log B x K dx e xK 20 20 3 2 Log C Log B x K x K e K xe K K 20 20 2 2 2 3 2 2 1 1 − = Log C Log B x K K x e K K 20 20 2 2 3 1 2 − = ( ) ( ) − − − = 2 2 2 3 1 20 1 20 K B Log B K C Log C K K ( ) ( ) ( ) ( ) [ ] 1 20 1 20 2 2 2 2 3 − − − = B Log K B C Log K C K K ( ) ( ) ( ) ( ) [ ] 1 1 2 1 − − − = B Ln B C Ln C K K xm ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 263 Standard deviation ( ) ( )dx x p x x s m ∫ − = 2 2 ( ) ( ) ( ) ∫ = + = − − + = E A Log D A Log x K m m dx e K x x x x s 1 20 1 20 3 2 2 2 2 2 E D x K E D x K E D x K m K x e K K m K K x x e K K e K K x − − − − + = 2 2 3 2 2 2 2 2 3 2 3 2 1 2 2 2 2 2 2 Now ∫ + = K x e K xe Kx Kx 1 1 and ∫ + − = 2 2 2 2 2 1 K K x x e K e x Kx Kx and 1 2 3 K K K = ( ) ( ) − − − − − − + + + + = D x K D D A E x K E E A K x K x A K s m m m m 2 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 1 ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 264 E.2 Conversion factors 0 10 20 30 40 50 60 70 80 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 Triangular log to linear U-Distribution log to linear Rectangular log to linear Gaussian log to linear Figure E.1: Standard deviations ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 265 Figure E.1 shows that if the standard deviation of a distribution in logarithms is smaller than 2,5 dB to 3,0 dB (resembling errors in the region of 5 dB to 6 dB), the following formula is a good approximation: ujlin = 11,5 × ujlog. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 266 Annex F: Influence quantity dependency functions Table F.1 is a list of influence quantity dependency functions and uncertainties that are dependant on the equipment under test only. They are nevertheless necessary for the calculation of the absolute measurement uncertainty. The table contains three types of parameters: - reflection coefficients for the calculation of mismatch uncertainty; - dependency factors for the conversion from influence quantity uncertainty to uncertainty related to the measurand; - additional uncertainty caused by influence quantities. The test laboratory making the measurements may, by means of additional measurements, estimate its own influence quantity dependencies, but if this is not carried out the values stated in table F.1 should be used. Table F.1 is based on measurements on a variety of equipment types. Each dependency is expressed as a mean value with a standard deviation reflecting the variation from one EUT to another. Some dependencies related to the general test conditions (supply voltage, ambient temperature, etc.) theoretically influence the results of all the measurements, but in some of the measurements they are so small that they are considered to be negligible. The table is divided into sub tables relating to the measurement examples described in clause 7 of TR 100 028-1 [6] (transmitter examples) and clause 4 of the present document (receiver examples). The corresponding clause numbers are shown in brackets. Table F.1: EUT-dependency functions and uncertainties Mean Standard deviation Frequency error (see clause 7.1.1 of TR 100 028-1 [6]) Temperature dependency 0,02 0,01 ppm/°C Carrier power (see clause 7.1.2 of TR 100 028-1 [6]) Reflection coefficient Temperature dependency Time-duty cycle error Supply voltage dependency 0,5 4,0 % 0 10 0,2 1,2 %/°C 2 % (p) 3 % (p)/V Frequency deviation (see clause 7.1.9 of TR 100 028-1 [6]) Temperature dependency 0,02 0,01 ppm/°C Adjacent channel power (see clause 7.1.3 of TR 100 028-1 [6]) Deviation dependency Filter position dependency Time-duty cycle error 0,05 15 0 0,02 % (p)/Hz 4 dB/kHz 2 % (p) Conducted spurious emissions (see clause 7.1.4 of TR 100 028-1 [6]) Reflection coefficient Time-duty cycle error Supply voltage dependency 0,7 0 10 0,1 2 % (p) 3 % (p)/V Intermodulation attenuation (see clause 7.1.5 of TR 100 028-1 [6]) Reflection coefficient Time-duty cycle error Supply voltage dependency 0,5 0 10 0,2 2 % (p) 3 % (p)/V Transmitter attack/release time (see clauses 7.1.6 and 7.1.7 of TR 100 028-1 [6]) Time/frequency error gradient Time/power level gradient 1,0 0,3 0,3 ms/kHz 0,1 ms/% Measured usable sensitivity (see clause 4.1.1 of the present document) Reflection coefficient Temperature dependency Noise gradient (below the knee point) Noise gradient (above the knee point) Noise gradient (direct carrier modulation) 0,2 2,5 0,375 1,0 1,0 0,05 1,2 %/°C 0,075 % level/% SINAD 0,2 % level/% SINAD 0,2 % level/% SINAD ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 267 Mean Standard deviation Amplitude characteristic (see clause 4.1.8 of the present document) Reflection coefficient RF level dependency 0,2 0,05 0,05 0,02 %/% level Two signal measurements (see clauses 4.1.2, 4.1.3, 4.1.4 and 4.1.6 of the present document) Reflection coefficient (in band) Reflection coefficient (out of band) Noise gradient Deviation dependency Absolute RF level dependency 0,2 0,8 0,7 0,05 0,5 0,05 0,1 0,2 % level/% SINAD 0,02 %/Hz 0,2 %/% level Intermodulation response (see clause 4.1.5 of the present document) Reflection coefficient Noise gradient (unwanted signal) Deviation dependency Capture ratio dependency 0,2 0,5 0,05 0,1 0,05 0,1 % level/% SINAD 0,02 %/Hz 0,03 %/% level Conducted spurious emission (see clause 4.1.7 of the present document) Reflection coefficient Supply voltage dependency 0,7 10 0,1 3 %/V Desensitization (Duplex) (see clause 5.2 of the present document) Reflection coefficient Temperature dependency Noise gradient (below the knee point) Noise gradient (above the knee point) Noise gradient (direct carrier modulation) 0,2 2,5 0,375 1,0 1,0 0,05 1,2 %/°C 0,075 % level/% SINAD 0,2 % level/% SINAD 0,2 % level/% SINAD Spurious response rejection (Duplex) (see clause 5.1 of the present document) Reflection coefficient (pass band) Reflection coefficient (stop band) Noise gradient Deviation dependency Absolute RF level dependency 0,2 0,8 0,7 0,05 0,5 0,05 0,1 0,2 % level/% SINAD 0,02 %/Hz 0,2 %/% level ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 268 Annex G: Mismatch uncertainties G.1 Introduction Mismatch uncertainties are calculated in the present document using S-parameters. A two-port network connects a generator and a load with reflection coefficients ρG and ρL respectively. Input and output wave amplitudes a1 and a2, b1 and b2 exist at the planes shown in figure G.1. The performance of this two-port network can be specified in terms of four complex quantities known as S-parameters where: b1 = S11a1 + S12a2 b2 = S21a1 + S22a2 Load Two-port ρG ρL 1 a a2 b2 1 b Figure G.1: Two-port network The corresponding matrix of the network can be described by an S-parameter (S for scattering) matrix: = 22 21 12 11 S S S S S Where S11 is the complex reflection coefficient at port 1 when port 2 is perfectly terminated (and vice versa). S21 is the complex transmission coefficient (or gain) from port 1 to port 2 when both ports are perfectly terminated (and vice versa). For passive, linear networks S21 = S12. From the definition of S parameters it is easy to see that mismatch loss is covered by the transmission coefficients. In other words it is of no importance whether the attenuation of a network is caused by power dissipation in the network or by reflection at the input. To illustrate this consider an ideal filter (ideal means it is lossless). All of the filtering is due to reflections at the input, as in an ideal filter, no power can be dissipated inside itself. Therefore if a loss (or gain) has been measured, the mismatch loss has already been taken into account and only the mismatch uncertainty remains. Therefore no correction due to mismatch loss is required. G.1.1 Cascading networks If two networks are cascaded (see figure G.2) the resulting network S-parameter matrix is a combination of the two original S-parameters. First each individual S-parameter matrix must be transformed to a T-matrix (T for transformation) = S t de - S -S S T 11 22 21 1 1 Where det S is the determinant of S. Then the resulting T matrix is calculated. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 269 For example: Network C Network A Network B Figure G.2: Cascading networks S-parameters: = 22 21 12 11 A A A A A S S S S S = 22 21 12 11 B B B B B S S S S S Which gives: = 22 21 12 11 A A A A A T T T T T = 22 21 12 11 B B B B B T T T T T The T-matrix for the resulting (combined) network (c) is then: TC = TATB = 22 21 12 11 22 21 12 11 B B B B A A A A B A T T T T T T T T T T = 22 22 12 21 21 22 11 21 22 12 12 11 21 12 11 11 B A B A B A B A B A B A B A B A T + T T T T + T T T T + T T T T + T T T From the resulting TC back to S parameters: = 12 21 11 1 1 - T t T -de T T S . From these general methods some useful formulas can be derived: Applying the methods on the two A and B, TA is found: = A A A A A S t de - S - S S T 11 22 21 1 1 ; + = 21 12 22 11 11 22 21 1 1 A A A A A A A S S S - S S - S S . In the same way TB is found: + = 21 12 22 11 11 22 21 1 1 B B B B B B B S S S S S - S S . The combination therefore is: = 22 21 12 11 22 21 12 11 B B B B A A A A B A T T T T T T T T T T ; ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 270 + + = 21 12 22 11 11 22 21 12 22 11 11 22 21 21 1 1 1 B B B B B B A A A A A A B A S S S - S S - S S S S - S S - S S S ; ( ) ( ) ( )( ) − − − + − = 22 11 12 21 22 11 12 21 22 11 22 11 12 21 11 11 22 11 21 12 22 22 11 22 21 21 1 1 B B B B A A A A B A A A A A B A B B B B A B B A B A S S S S S S S S S - S S S S S S S S S S S S - S S - S S S . Which gives: 21 21 11 22 11 1 B A B A S S S S Tc − = ( ) 21 21 22 11 1 21 11 11 21 B A A A A A B A S S S S S S S S Tc − + = ( ) 21 21 22 11 21 12 22 22 12 B A B B B B A B S S S S S S S S Tc − − − = ( )( ) 21 21 22 11 12 21 22 11 12 21 22 11 22 B A B B B B A A A A B A S S S S S S S S S S S S Tc − − + − = − − = = 12 21 11 22 21 12 11 1 det 1 tc Tc tc tc Sc Sc Sc Sc Sc ( ) ( ) 11 22 22 11 12 21 11 11 21 21 22 11 12 21 11 11 11 22 21 21 11 21 11 1 1 B A A A A A B A B A A A A A B A B A B A S S S S S S S S S S S S S S S S S S S S tc tc Sc − − + × − + × − = = 11 22 22 11 11 12 21 11 11 11 1 B A A A B A A B A S S S S S S S S S Sc − − + = ( ) 11 22 12 21 11 11 22 11 11 1 1 B A A A B B A A S S S S S S S S Sc − + − = 11 22 12 21 11 11 11 1 B A A A B A S S S S S S Sc − + = (1) 11 22 21 21 11 21 1 1 B A B A S S S S tc Sc − = = (2) Sc11 is the input reflection coefficient of the combined network and Sc21 is the forward transmission coefficient. For symmetry reasons Sc22 and Sc12 can be derived directly from Sc11 and Sc21: 11 22 21 12 22 22 22 1 B A B B A B S S S S S S Sc − + = (3) 11 22 12 12 12 1 B A B A S S S S Sc − = (4) From formula it can be seen that now the reflection coefficient in the connection between the two networks becomes part of the total transfer function: the denominator 1 - SA22 SB11. This causes the mismatch uncertainty as only the magnitudes of SA22 and SB11 are known, the phase of the product is unknown. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 271 The two worst case values of the term 1 - SA22 SB11 are: 1 + |SA22|×|SB11| and 1 - |SA22|×|SB11|. The magnitude of the denominator is the magnitude of the sum of two vectors as shown in figure G.3 (where the circle of radius SA22SB11 is normally much smaller than 1). 1 Imaginary Real 1 + S S A22 B11 S S A22 B11 Figure G.3: Vector summation As can be seen from figure G.3 the denominator can be anywhere in the circle with the radius |SA22|×|SB11|. It can also be seen that there are angles for which the argument of the denominator is 1. The magnitude of the denominator is: ( ) ( )2 2 sin cos 1 φ φ a a + + = φ φ φ 2 2 2 2 sin cos 2 cos 1 a a a + + + where: a = |SA22|×|SB11| ( ) φ φ φ cos 2 cos sin 1 2 2 2 a a + + + (as sin2φ + cos2φ = 1) φ cos 2 1 2 a a + + (since a<< 1: a2 ≈ 0 and 1+ 2a cosφ ≈ (1+ a cosφ)2: ( )2 cos 1 φ a + = 1+ a cosφ The mismatch error magnitude is a cosφ where φ is unknown (random). This function has the U distribution described in clause B.2.3. From the formula for Sc11 and Sc22 it can also be seen that the resulting input (or output) reflection coefficient is a combination of the reflection coefficient of network A and a contribution from the reflection coefficient of network B connected at the far end of the network. For a passive linear network (like attenuators, cables and passive filters) S12 = S21. In other words the transmission coefficient and therefore the attenuation is the same in both directions. In this case the resulting input reflection coefficient is S11 (which is the input reflection coefficient when the output is perfectly terminated) plus the reflection coefficient of the network connected to the output times the transmission coefficient squared (and with the mismatch in the connector at the far end expressed by the denominator of the second term of the formula). This also shows that if two components with poor VSWRs are connected together, it does not minimize the mismatch uncertainty to use a perfect cable between the two components. The resulting input reflection coefficient of the cable and the component is merely the reflection coefficient of the component phase shifted by the length of the cable. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 272 From the formulas for Sc21 and Sc12 it can be seen that the resulting transmission coefficient (S21/S12) of the combined network is the individual transmission coefficients multiplied and combined with the mismatch in the connection between the two networks (as expressed by the denominator). G.1.2 Mismatch uncertainty calculations Having discussed the individual uncertainty components of the test equipment an analysis is required, when they are connected together, to determine the combined standard uncertainty contribution. From the formulas derived in this annex the uncertainties due to mismatch can be assessed. A measurement set-up where absolute RF levels are important parts of the measurement often consist of some RF modules connected in series, see figure G.4 (Cables, attenuators, filters, combiners, amplifiers, etc.). A B C D RF source SB21 SB22 SB12 SA12 SA11 ρG RF load ρL SA22 SB11 SA21 Figure G.4: Typical network For each individual component in this chain, transmission coefficients and reflection coefficients (or VSWRs) must be known or assumed. Often the transmission coefficients are well known from data or measurements. The exact values of the reflection coefficients VSWRs (which in RF circuits are complex values) are normally not known as they do not have direct influence on the measured results. Even if the magnitude is known, generally, the phase is unknown. More often worst case values are known. This will generally cause the calculated mismatch uncertainties to be more conservative (or worse) than they actually are. The uncertainty due to mismatches of the RF level at the RF load (which can be an antenna, a detector, an EUT) in a network like the one shown in figure G.5 can be calculated in the following ways: The simplest case for assessing the uncertainty due to mismatch is a generator connected to a load through a coupling network. Generator Coupling network Load Figure G.5: Generator to load through a coupling network For the purpose of the calculations the generator is modelled as a perfect generator (output reflection coefficient = 0) connected to a network with an output reflection coefficient equal to the actual generator output reflection coefficient. (Also the network only has a forward transmission of 1,0 and a backwards coefficient of 0,0). In the same way the load is modelled as a network connected to a perfect matched load. Also with a forward transmission coefficient of 1,0 and a backwards coefficient of 0,0. The set-up of figure G5 now appears as shown in figure G.6. 1,0 1,0 0,0 0,0 0 0 0 0 Generator network Perfect generator Coupling network network Load Perfect load S11 ρG S12 S21 ρL S22 Figure G.6: Perfect generator to perfect load through a coupling network ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 273 The S matrices for each component in figure G.6 is: Generator network: G ρ 0,1 0,0 0,0 (SG) Coupling network: 22 21 12 11 S S S S (S) Load network: 0,0 0,1 0,0 L ρ (SL) The total transmission from the generator to the load can then be characterized by the combined network of the 3 components. As the input and output reflection coefficients of the combined network is zero, the forward and reverse transmission coefficients of the network fully describes the RF signal flow between the generator and the load, including all mismatch uncertainties. The forward transmission coefficient is calculated as follows: The S-parameter matrix for the combined network is: SG S SL: S/ = SG S: Using formulas (1), (2), (3) and (4) the resulting matrix is: 11 22 12 21 11 11 11 1 S S S S S S S G G G G / − + = 0 1 0 1 0 11 11 = × + × × + = S S G ρ (formula 1) 11 21 11 22 21 21 21 1 1 1 S S S S S S S G G G / ρ − × = − = 11 21 1 S S G ρ − = (formula 2) 11 22 12 21 22 22 22 1 S S S S S S S G G / − + = 11 12 21 22 1 S S S G S G G ρ − + = (formula 3) 0 1 0 1 11 12 11 12 12 12 = − × = − = S S S S S S G G G / ρ ρ (formula 4) − + − = 11 12 21 22 11 21 1 1 0 0 S S S S S S S G G G / ρ ρ ρ Now only S21// needs to be calculated: 11 22 21 21 21 1 L / L / // S S S S S − = ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 274 L G G G S S S S S S ρ ρ ρ ρ × − + − × − = 11 21 12 22 11 21 1 1 1 1 11 21 12 22 11 21 1 1 1 S S S S S S G L G L G ρ ρ ρ ρ ρ − + − − = ( )( ) 21 12 22 11 21 1 1 S S S S S L G L G ρ ρ ρ ρ + − − = (5) From the formula it can be seen that there are three mismatch contributions: One at each end of the coupling network (characterized by the brackets in the denominator of (5)) and one caused by direct interaction between the generator and the load. It is also seen that this direct interaction is depending on the transmission coefficients of the network. The greater the attenuation the less the interaction. If the coupling network between the source and the load consists of more than one component there will be more contributions to the mismatch uncertainty, unless the coupling network has been measured as one component. Mismatch uncertainty at the connections between the individual components in the network. For all network consisting of two components A and B, figure G.7. Generator Coupling network A Coupling network B Load Figure G.7: Generator to load through two coupling networks The input and output reflection coefficients are calculated using formulas (1) and (3): 11 22 21 12 11 11 11 1 b a a a b a S − + = (6) 11 22 21 12 22 22 22 1 b a b b a b S − + = (7) and the transmission coefficients are calculated using Formulas (2) and (4): 11 22 21 21 21 1 b a b a S − = (8) 11 22 12 12 12 1 b a b a S − = (9) = 22 21 12 11 a a a a A = 22 21 12 11 b b b b B For the purpose of calculating mismatch uncertainties the derived S-parameters are put into formula (5): ( ) 11 22 21 12 12 21 11 22 21 12 22 22 11 22 21 12 11 11 11 22 21 21 1 1 1 1 1 1 b a b b a a b a b b a b b a a a b a b a b a L G L G − + − − − − − − − = ρ ρ ρ ρ (10) From formula (10) it can be seen that there are 4 mismatch uncertainty contributions: Mismatch uncertainty between A and B: ±a22b11 ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 275 Mismatch uncertainty at the generator: ± − + 11 22 21 12 11 11 1 b a a a b a G ρ Mismatch uncertainty at the load: ± − + 11 22 21 12 22 22 1 b a b b a b L ρ Mismatch uncertainty due to direct interaction between the generator and the load: ± 11 22 21 12 12 21 1 b a b b a a L G − ρ ρ . In the 3 later cases the denominator form of 1- a22b11 can be ignored as the average is 1. Therefore it does not contribute to the mismatch uncertainty. Furthermore the two formulas with brackets consist of components which are not correlated. These components must be treated individually. This gives the following contributions: Mismatch uncertainty between A and B: ± a22 × b11 Mismatch uncertainty at the generator: ± ρG × a11 and ± ρG × b11 × a12 × a21 Mismatch uncertainty at the load: ± ρL × b22 and ± ρL × a22 × b12 × b21 Mismatch uncertainty due to the direct interaction between the generator and the load: ± ρG × ρL × a12 × a21 × b12 × b21 G.2 General approach A general method for the calculation of the total mismatch uncertainty of a network consisting of any number N of components between the generator and the load is as follows: Each individual component is characterized by its S-parameter matrix: = 22 21 12 11 Si Si Si Si Si ρi ρ1, i(n) The generator reflection coefficient is S (0)22 and the load reflection coefficient is S (n + 1)11; the mismatch uncertainty is the combination of all possible products of the form: Si22 × Sj11 × S (i + 1)12 × S (i + 1)21 × S (i + 2)12 × ....... × S (j-2)12 × S (j-2)21 × S (j-1)12 × S (j-1)21 (0 (i (n) and (1 (j (n + 1) and i (j-2) G.3 Networks comprising power combiners/splitters In some tests power combiners/splitters are involved either to combine the signals from several signal sources or to split the signals to several detectors or measuring instruments. Under these circumstances there may be mismatch uncertainty contributions from the other branches of the splitters/divider as well as those from the branch of interest. If there is a high isolation between some of the ports, this can normally be ignored. It plays, however, a vital part where isolation between input ports is needed. (i.e. between generators to avoid third order intermodulation). Consider the network shown in figure G.8. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 276 Port 1 Port 2 Port 3 Load Generator Load 3 port combiner Figure G.8: Three port combiner The 3 port combiner is characterized by the S-matrix = 33 32 31 23 22 21 13 12 11 S S S S S S S S S S Based on the general formula B = S × A, where: = 3 2 1 b b b B where bn is the output signal from port n, = 3 2 1 a a a A where an is the input signal to port n, and each port n is connected to a reflection coefficient ρn, the transfer function from the generator connected to port 1 to the load connected to port 3 can be derived. For a linear and symmetrical network (where Sin = Sni for all S) the transfer function (formula 5) is: 2 1 11 32 1 12 13 3 2 2 1 2 12 2 22 1 11 3 1 2 13 3 33 1 11 2 1 2 12 2 22 1 11 31 1 11 32 1 12 31 12 2 )) 1( ( ) ) 1 )( 1 )(( ) 1 )( 1 (( ) ) 1 )( 1 (( )) 1( ( ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ × − + × × × − × × − × − × − × × − × − × − × × − × − × − + × − + × × × S S S S S S S S S S S S S S S S S S S As can be seen in the following the 3. port (in this case port 2) adds to the mismatch uncertainty between the generator and the load connected to port 3. If all reflection coefficients except S22 and ρ2 are 0,0 formula 5 is reduced to the following: (formula 6) ) S ( ) S ( S S S 2 22 2 22 31 32 12 2 1 1 ρ ρ ρ × − × − + × × = )) 1( 1( 2 22 31 32 12 2 31 ρ ρ × − × × + S S S S S (6) If the denominator second order uncertainty is disregarded in formula 6 an additional mismatch uncertainty contribution appears: 31 32 12 2 S S S × × ρ . As can be seen S22 does not directly contribute. This mismatch component has a u-shaped distribution like the conventional mismatch uncertainty contributions. If all reflection coefficients except ρ1 and ρ2 are 0,0 formula 5 is reduced to the following: (formula 7) ) 1( ) 1( ) ( 2 1 2 12 2 1 2 12 31 32 1 12 31 12 2 ρ ρ ρ ρ ρ ρ × × − × × − + + × × × S S S S S S S = ) 1( 2 1 2 12 31 32 12 2 ρ ρ ρ × × − + × × S S S S = ) 1( ) 1( 2 1 2 12 31 32 12 2 31 ρ ρ ρ × × − × × + S S S S S (7) In the nominator we see the term already found in formula 6. In addition to this there is a contribution from the denominator: 2 1 2 12 ρ ρ × × S . ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 277 In the same way if only ρ2 and ρ3 are different from 0,0: ) 1( 3 2 2 32 31 32 12 2 ρ ρ ρ × × − + × × S S S S = ) 1( ) 1( 3 2 2 32 31 32 12 2 31 ρ ρ ρ × × − × × + S S S S S (8) giving the mismatch uncertainty contribution: 3 2 2 32 ρ ρ × × S . From these 3 additional mismatch contributions it can be concluded that in networks comprising combiners or splitters, all other ports than the ports in the main path can contribute to the mismatch uncertainty in the main path. If all other ports are connected to perfect terminations, they do not contribute, and the network can be regarded as one path. If, however, the other ports (n) are connected to reflection coefficients ρn different from 0,0, these reflection coefficients contributes to the total reflection coefficient at both the input and the output of the combiner, thereby combining to the total mismatch uncertainty in the main path. But in addition there is a contribution which is not the usual combination of two reflection coefficients: io no in n S S S × × ρ , where port i is the input port, port o is the output port, and port n is any of the other ports. It contains only one reflection coefficient and some transmission coefficients. As the transmission coefficients can be very high (close to 1 or even higher if amplifiers are involved) this contribution can be dominating. It can cause much bigger mismatch uncertainty than the sum of the rest of the components, and it can cause lack of isolation between ports, where isolation is needed. It should be noted that there are such mismatch uncertainty contributions from all ports except the two ports in the main path. Imagine an ideal 3 port hybrid combiner with a transfer function of ∞ dB between the two input ports and 3 dB from each port to the output. If the output of the hybrid combiner is connected to a load with reflection coefficient 0,1 the effective isolation between the two input ports is: 0dB 17 1414 ,0 2 2 2 1,0 ≈ = × × . Therefore the matching of the unused ports is very important. In these cases the mismatch uncertainty between the input port and the output port (e.g. port 1 to port 3 of a combiner) must then be calculated as follows: 1) all the "normal" mismatch uncertainty contributions must be found; 2) the reflection coefficients connected to port 2 must be taken into account; 3) in addition to this there is an extra uncertainty component. NOTE 1: This uncertainty component is not a normal mismatch component, it is calculated from: ρ2×S21×S32/S31. Where ρ2 is the reflection coefficient of the network connected to port 2 of the combiner. If a resistive combiner - for instance with an attenuation of 6 dB between the ports - is involved, this last contribution can be a dominant one if ρ2 is big. NOTE 2: This contribution is in the numerator of the transfer function, whereas the "normal" uncertainty contributions come from the denominator. The formula shown is consistent with the fact that if S31 approaches zero this uncertainty will grow to be greater than one, and the combiner will act as a reflection measuring bridge. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 278 EXAMPLE: A 6 dB resistive combiner has a signal generator (1) connected to port 1 and a second signal generator (2) connected to port 2 (both input ports). The combiner port 3 (the output port) is connected to an EUT. The signal generator and combiner reflection coefficients are 0,2 and the EUT has a reflection coefficient of 0,8. The mismatch uncertainty is calculated as follows: The standard uncertainty of the mismatch between the signal generator 1 and combiner input: uj generator 1 and combiner = % 828 ,2 % 2 100 2,0 2,0 = × × The standard uncertainty of the mismatch between the combiner output and the EUT: uj combiner and EUT = % 31 , 11 % 2 100 8,0 2,0 = × × The standard uncertainty of the mismatch between the signal generator 1 and the EUT: uj generator 1 and EUT = % 828 ,2 % 2 100 5,0 8,0 2,0 2 = × × × The standard uncertainty of the mismatch between the signal generator 1 and signal generator 2: uj generator 1 and generator 2 = % 707 ,0 % 2 100 5,0 2,0 2,0 2 = × × × The standard uncertainty of the mismatch between the signal generator 2 and the combiner: uj generator 2 and combiner = % 828 ,2 % 2 100 2,0 2,0 = × × The additional component is calculated as: % 071 ,7 % 2 5,0 100 5,0 5,0 2,0 = × × × × The combined standard uncertainty of the mismatch is: % 50 , 14 % 071 ,7 828 ,2 828 ,2 707 ,0 828 ,2 31 , 11 828 ,2 2 2 2 2 2 2 2 = + + + + + + An extreme situation would be if all the components - except the load on port 2 - were exactly 50 Ω; in this case the only mismatch component would be the additional component (7 %). Figure G.9 shows the distribution where all reflection coefficients are 0,1 and all transfer functions are 0,5 (simulated 200 000 000 times). The standard deviation based on the simulation is found to be 3,6871 %. The calculated standard deviation is 3,7541 %. (The difference is due to that some second order components are disregarded in the calculation.). ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 279 0 2 4 6 8 10 -2 -4 -6 -8 -10 p(x) 3,6871 U95: min = -5,9600 and max = 6,0800 % Figure G.9: Distribution from the simulation The formulae shown are also applicable to non symmetrical networks. Instead of the squared terms the products of the transfer coefficients in both directions must be used. EXAMPLE: S4 S5 Load 1 S1 Generator S2 S3 S6 Load 2 Figure G.10: Example path between the generator and load = 050 ,0 79433 ,0 79433 ,0 050 ,0 1 S = 060 ,0 89125 ,0 89125 ,0 060 ,0 2 S = 07 ,0 70795 ,0 70795 ,0 70795 ,0 07 ,0 70795 ,0 70795 ,0 707095 ,0 07 ,0 3 S = 080 ,0 0,1 0,1 080 ,0 4 S = 1,0 94406 ,0 94406 ,0 1,0 5 S = 1,0 5,0 5,0 1,0 6 S ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 280 ρG = 0,2 = S (0)22; ρL1 = 0,333; ρL2 = 0,2 All possible contributions are: Contributions in the main path between % 707 ,0 % 2 100 05 ,0 20 ,0 1 = × × = S of input and generator j u % 212 ,0 % 2 100 06 ,0 05 ,0 2 1 = × × = S of input and S of output j u % 297 ,0 % 2 100 07 ,0 06 ,0 3 2 = × × = S of input and S of output j u % 396 ,0 % 2 100 08 ,0 07 ,0 4 3 = × × = S of input and S of output j u % 566 ,0 % 2 100 10 ,0 08 ,0 5 4 = × × = S of input and S of output j u % 35 ,2 % 2 100 333 ,0 10 ,0 1 5 = × × = load and S of output j u % 535 ,0 % 2 100 794 ,0 06 ,0 20 ,0 2 2 = × × × = S of input and generator j u % 157 ,0 % 2 100 891 ,0 07 ,0 05 ,0 2 3 1 = × × × = S of input and S of output j u % 170 ,0 % 2 100 708 ,0 08 ,0 06 ,0 2 4 2 = × × × = S of input and S of output j u % 495 ,0 % 2 100 0,1 10 ,0 07 ,0 2 5 3 = × × × = S of input and S of output j u % 68 ,1 % 2 100 944 ,0 333 ,0 08 ,0 2 1 4 = × × × = load and S of output j u % 495 ,0 % 2 100 891 ,0 794 ,0 07 ,0 20 ,0 2 2 3 = × × × × = S of input and generator j u % 113 ,0 % 2 100 708 ,0 891 ,0 08 ,0 05 ,0 2 2 4 1 = × × × × = S of input and S of output j u % 284 ,0 % 2 100 0,1 708 ,0 10 ,0 08 ,0 2 2 5 2 = × × × × = S of input and S of output j u % 47 ,1 % 2 100 944 ,0 0,1 333 ,0 07 ,0 2 2 1 3 = × × × × = load and S of output j u % 284 ,0 % 2 100 708 ,0 891 ,0 794 ,0 08 ,0 20 ,0 2 2 2 4 = × × × × × = S of input and generator j u ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 281 % 141 ,0 % 2 100 0,1 708 ,0 891 ,0 10 ,0 05 ,0 2 2 2 5 1 = × × × × × = S of input and S of output j u % 631 ,0 % 2 100 944 ,0 0,1 708 ,0 333 ,0 06 ,0 2 2 2 1 2 = × × × × × = load and S of output j u % 355 ,0 % 2 100 0,1 708 ,0 891 ,0 794 ,0 10 ,0 20 ,0 2 2 2 2 5 = × × × × × × = S of input and generator j u % 418 ,0 % 2 100 944 ,0 0,1 708 ,0 891 ,0 333 ,0 05 ,0 2 2 2 2 1 1 = × × × × × × = load and S of output j u % 053 ,1 % 2 100 944 ,0 0,1 708 ,0 891 ,0 794 ,0 333 ,0 20 ,0 2 2 2 2 2 1 = × × × × × × × = load and generator j u Contributions from the network connected to the 3rd port of S3: Contributions: % 212 ,0 % 2 100 708 ,0 10 ,0 06 ,0 2 6 2 = × × × = S of input and S of output j u % 284 ,0 % 2 100 708 ,0 08 ,0 10 ,0 2 4 6 = × × × = S of input and S of input j u % 141 ,0 % 2 100 708 ,0 891 ,0 1,0 05 ,0 2 2 6 1 = × × × × = S of input and S of output j u % 106 ,0 % 2 100 50 ,0 708 ,0 20 ,0 06 ,0 2 2 2 2 = × × × × = load and S of output j u % 354 ,0 % 2 100 0,1 708 ,0 10 ,0 10 ,0 2 2 5 6 = × × × × = S of input and S of input j u % 142 ,0 % 2 100 708 ,0 50 ,0 08 ,0 20 ,0 2 2 2 4 = × × × × = S of input and load j u % 354 ,0 % 2 100 708 ,0 891 ,0 794 ,0 10 ,0 20 ,0 2 2 2 6 = × × × × × = S of input and generator j u % 070 ,0 % 2 100 50 ,0 708 ,0 891 ,0 20 ,0 05 ,0 2 2 2 2 1 = × × × × × = load and S of output j u % 052 ,1 % 2 100 944 ,0 0,1 708 ,0 333 ,0 10 ,0 2 2 2 1 6 = × × × × × = load and S of input j u % 177 ,0 % 2 100 0,1 708 ,0 50 ,0 10 ,0 20 ,0 2 2 2 2 5 = × × × × × = S input and load j u % 177 ,0 % 2 100 50 ,0 708 ,0 891 ,0 794 ,0 20 ,0 20 ,0 2 2 2 2 2 = × × × × × × = load and generator j u ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 282 % 526 ,0 % 2 100 944 ,0 0,1 708 ,0 50 ,0 333 ,0 20 ,0 2 2 2 2 1 2 = × × × × × × = load and load j u Contributions from the 3rd port: % 01 ,5 % 2 708 ,0 100 708 ,0 10 ,0 2 6 = × × × = S from on contributi j u % 50 ,2 % 2 708 ,0 100 708 ,0 50 ,0 20 ,0 2 2 2 = × × × × = load from on contributi j u The root sum of the squares of all these components is 6,90 %. As can be seen from the calculations the major contributions to the mismatch uncertainty is from the reflection coefficients connected to the 3 rd port of the network. This means that the matching of that port is of great importance to keep the uncertainty low. Alternatively the total insertion loss and the reflection coefficients at the generator and at load 1 should be measured with S6 and load 2 connected. This would minimize the mismatch uncertainty. These formulations can now be applied to the actual circuits encountered during testing. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 283 Annex H: Bibliography "A designers guide to shielding", Hewlett Packard: RF and microwave measurement symposium and exhibition. "Analysis of trials on Artificial Human Body", I. L. Gallan and P. R. Brown Interference technology international consultants ltd. Contract ref MC/078. "Calculation of site attenuation from antenna factors" A. A. Smith Jr, RF German and J B Pate. IEEE transactions EMC. Vol. EMC 24 pp 301-316 August 1982. "Computer simulation and measurements of electromagnetic fields close to phantom Humans", Electricity and magnetism in biology and medicine by Martin Blank, Ed 1993 San Francisco Press. "Control of errors on Open Area Test Sites", A. A. Smith Jnr. EMC technology October 1982 pg 50-58. "Fundamentals of RF and Microwave Power Measurements", Hewlett Packard: Application note 64-1 August 1977. "Getting better results from an Open Area Test Site", Joseph DeMarinus. "Guide to the evaluation and expression of the uncertainties associated with the results of electrical measurements", Ministry of Defence :00-26/Issue 2:September 1988. "Measurement uncertainty generally", Statens Tekniske Provenaevn, The Danish Accreditation Committee (STP). June 1988 (Danish) "Specifications for equipment's for use in the Land Mobile Service" CEPT Recommendation T/R 24-01. "Techniques for measuring narrowband and broadband EMI signals using spectrum analysers", Hewlett Packard RF and microwave measurement symposium and exhibition. "The expression of uncertainty in electrical measurement", B3003, November 1987 National Measurement Accreditation Service (NAMAS). "The gain resistance product of the half-wave dipole", W. Scott Bennet Proceedings of IEEE vol. 72 No. 2 December 1984 pp 1824-1826. "Uncertainties of Measurement for NATLAS electrical testing laboratories. NAMAS policy and general notes", National Testing Laboratory Accreditation Scheme (NATLAS), NIS20 July 1986 (English) "Use of Simulated Human Bodies in pager receiver sensitivity measurements", K.Siwiak and W.Elliott III. Southcom/92 conference, Orlando 1992. pp 189/92. "Usikkerhed på måleresultater" (Per Bennich, Institute for Product Development: "Uncertainty of measured results"). October 1988. (Danish) "Calculation of site attenuation from antenna factors" A. A. Smith Jr, RF German and J B Pate. IEEE transactions EMC. Vol. EMC 24 pp 301-316 August 1982. "Standard site method for determining antenna factors", A. A. Smith Jr. IEEE transactions EMC. Vol EMC 24 pp 316-322 August 1982. IEC 60050-161: "International Electrotechnical Vocabulary. Chapter 161: Electromagnetic compatibility" "Advanced National certificate mathematics", PEDOE, Hodder and Stoughton Volumes I and II. "Antenna engineering handbook", R. C. Johnson, H. Jasik. "Antenna theory", C. Balanis, J. E. Wiley 1982. "Antennas and radio wave propagation", R. E. Collin, McGraw Hill. "Antennas", John D. Kraus, Second edition, McGraw Hill. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 284 Chambers Science and Technology dictionary, 1988. Published by Chambers Cambridge. "Digital Communications", John G. Proakis, McGraw-Hill International Editions, second edition. ETSI ETR 027: "Radio Equipment and Systems (RES); Methods of measurement for private mobile radio equipment". "Guide to the Expression of Uncertainty in Measurement" (International Organisation for Standardisation, Geneva, Switzerland, 1995). "Radiowave propagation and antennas for personal communications", K. Siwiak, Artech House Publications. "The new IEEE standard dictionary of electrical and electronic terms". Fifth edition, IEEE Piscataway, NJ USA 1993. "The telecommunications factbook and illustrated dictionary", Kahn, Delmar publications Inc. New York 1992. "Vocabulary of metrology", British Standard Institution (BSI): PD 6461: Part 2: September 1980. "Wave transmission", F. R. Conner, Arnold 1978. ETSI ETSI TR 100 028-2 V1.4.1 (2001-12) 285 History Document history Edition 1 March 1992 Publication as ETR 028 Edition 2 March 1994 Publication as ETR 028 V1.3.1 March 2001 Publication V1.4.1 December 2001 Publication |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 1 Scope | The present document provides a method to be applied to all the applicable deliverables, and supports TR 100 027 [2]. It covers the following aspects relating to measurements: a) methods for the calculation of the total uncertainty for each of the measured parameters; b) recommended maximum acceptable uncertainties for each of the measured parameters; c) a method of applying the uncertainties in the interpretation of the results. The present document provides the methods of evaluating and calculating the measurement uncertainties and the required corrections on measurement conditions and results (these corrections are necessary in order to remove the errors caused by certain deviations of the test system due to its known characteristics (such as the RF signal path attenuation and mismatch loss, etc.)). |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 2 References | For the purposes of this Technical Report (TR) the following references apply: [1] Guide to the Expression of Uncertainty in Measurement (International Organization for Standardization, Geneva, Switzerland, 1995). [2] ETSI TR 100 027: "Electromagnetic compatibility and Radio spectrum Matters (ERM); Methods of measurement for private mobile radio equipment". [3] ETSI TR 102 273 (all parts): "Electromagnetic compatibility and Radio spectrum Matters (ERM); Improvement of Radiated Methods of Measurement (using test sites) and evaluation of the corresponding measurement uncertainties". [4] ITU-T Recommendation O.41: "Psophometer for use on telephone-type circuits". [5] Void. [6] ETSI ETR 028: "Radio Equipment and Systems (RES); Uncertainties in the measurement of mobile radio equipment characteristics". [7] EN 55020: "Electromagnetic immunity of broadcast receivers and associated equipment". [8] ETSI TR 100 028-2: "Electromagnetic compatibility and Radio spectrum Matters (ERM); Uncertainties in the measurement of mobile radio equipment characteristics; Part 2". |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 3 Definitions, symbols and abbreviations | |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 3.1 Definitions | For the purposes of the present document, the following terms and definitions apply: accuracy: This term is defined, in relation to the measured value, in clause 4.1.1; it has also been used in the rest of the document in relation to instruments. AF load: resistor of sufficient power rating to accept the maximum audio output power from the EUT NOTE: The value of the resistor should be that stated by the manufacturer and should be the impedance of the audio transducer at 1 000 Hz. In some cases it may be necessary to place an isolating transformer between the output terminals of the receiver under test and the load. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 11 AF termination: any connection other than the audio frequency load which may be required for the purpose of testing the receiver (i.e. in a case where it is required that the bit stream be measured, the connection may be made, via a suitable interface, to the discriminator of the receiver under test) NOTE: The termination device should be agreed between the manufacturer and the testing authority and details should be included in the test report. If special equipment is required then it should be provided by the manufacturer. antenna: part of a transmitting or receiving system that is designed to radiate or to receive electromagnetic waves antenna factor: quantity relating the strength of the field in which the antenna is immersed to the output voltage across the load connected to the antenna NOTE: When properly applied to the meter reading of the measuring instrument, yields the electric field strength in V/m or the magnetic field strength in A/m. antenna gain: ratio of the maximum radiation intensity from an (assumed lossless) antenna to the radiation intensity that would be obtained if the same power were radiated isotropically by a similarly lossless antenna bit error ratio: ratio of the number of bits in error to the total number of bits combining network: network allowing the addition of two or more test signals produced by different sources (e.g. for connection to a receiver input) NOTE: Sources of test signals should be connected in such a way that the impedance presented to the receiver should be 5O Ω. The effects of any intermodulation products and noise produced in the signal generators should be negligible. correction factor: numerical factor by which the uncorrected result of a measurement is multiplied to compensate for an assumed systematic error confidence level: probability of the accumulated error of a measurement being within the stated range of uncertainty of measurement directivity: ratio of the maximum radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions (i.e. directivity = antenna gain + losses) duplex filter: duplex filter is a device fitted internally or externally to a transmitter/receiver combination to allow simultaneous transmission and reception with a single antenna connection error of measurement (absolute): result of a measurement minus the true value of the measurand error (relative): ratio of an error to the true value estimated standard deviation: from a sample of n results of a measurement the estimated standard deviation is given by the formula: 1 1 2 − − = ∑ = n ) x (x n i i σ xi being the ith result of measurement (i = 1, 2, 3, ..., n) and x the arithmetic mean of the n results considered. A practical form of this formula is: 1 2 − − = n n X Y σ Where X is the sum of the measured values and Y is the sum of the squares of the measured values. The term standard deviation has also been used in the present document to characterize a particular probability density. Under such conditions, the term standard deviation may relate to situations where there is only one result for a measurement. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 12 expansion factor: multiplicative factor used to change the confidence level associated with a particular value of a measurement uncertainty The mathematical definition of the expansion factor can be found in clause D.5.6.2.2. extreme test conditions: extreme test conditions are defined in terms of temperature and supply voltage NOTE: Tests should be made with the extremes of temperature and voltage applied simultaneously. The upper and lower temperature limits are specified in the relevant ETS. The test report should state the actual temperatures measured error (of a measuring instrument): indication of a measuring instrument minus the (conventional) true value free field: field (wave or potential) which has a constant ratio between the electric and magnetic field intensities free space: region free of obstructions and characterized by the constitutive parameters of a vacuum impedance: measure of the complex resistive and reactive attributes of a component in an alternating current circuit impedance (wave): complex factor relating the transverse component of the electric field to the transverse component of the magnetic field at every point in any specified plane, for a given mode influence quantity: quantity which is not the subject of the measurement but which influences the value of the quantity to be measured or the indications of the measuring instrument intermittent operation: manufacturer should state the maximum time that the equipment is intended to transmit and the necessary standby period before repeating a transmit period isotropic radiator: hypothetical, lossless antenna having equal radiation intensity in all directions limited frequency range: specified smaller frequency range within the full frequency range over which the measurement is made NOTE: The details of the calculation of the limited frequency range should be given in the relevant deliverable. maximum permissible frequency deviation: maximum value of frequency deviation stated for the relevant channel separation in the relevant deliverable measuring system: complete set of measuring instruments and other equipment assembled to carry out a specified measurement task measurement repeatability: Closeness of the agreement between the results of successive measurements of the same measurand carried out subject to all the following conditions: - the same method of measurement; - the same observer; - the same measuring instrument; - the same location; - the same conditions of use; - repetition over a short period of time measurement reproducibility: Closeness of agreement between the results of measurements of the same measurand, where the individual measurements are carried out changing conditions such as: - method of measurement; - observer; - measuring instrument; - location; ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 13 - conditions of use; - time. measurand: quantity subjected to measurement noise gradient of EUT: function characterizing the relationship between the RF input signal level and the performance of the EUT, e.g. the SINAD of the AF output signal nominal frequency: defined as one of the channel frequencies on which the equipment is designed to operate nominal mains voltage: declared voltage or any of the declared voltages for which the equipment was designed normal test conditions: defined in terms of temperature, humidity and supply voltage stated in the relevant deliverable normal deviation: frequency deviation for analogue signals which is equal to 12 % of the channel separation psophometric weighting network: Should be as described in ITU-T Recommendation O.41 polarization: for an electromagnetic wave, this is the figure traced as a function of time by the extremity of the electric vector at a fixed point in space quantity (measurable): attribute of a phenomenon or a body which may be distinguished qualitatively and determined quantitatively rated audio output power: maximum output power under normal test conditions, and at standard test modulations, as declared by the manufacturer rated radio frequency output power: maximum carrier power under normal test conditions, as declared by the manufacturer shielded enclosure: structure that protects its interior from the effects of an exterior electric or magnetic field, or conversely, protects the surrounding environment from the effect of an interior electric or magnetic field SINAD sensitivity: minimum standard modulated carrier-signal input required to produce a specified SINAD ratio at the receiver output stochastic (random) variable: variable whose value is not exactly known, but is characterized by a distribution or probability function, or a mean value and a standard deviation (e.g. a measurand and the related measurement uncertainty) test load: 50 Ω substantially non-reactive, non-radiating power attenuator which is capable of safely dissipating the power from the transmitter test modulation: test modulating signal is a baseband signal which modulates a carrier and is dependent upon the type of EUT and also the measurement to be performed trigger device: circuit or mechanism to trigger the oscilloscope timebase at the required instant NOTE: It may control the transmit function or inversely receive an appropriate command from the transmitter. uncertainty: parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to that measurement uncertainty (random): component of the uncertainty of measurement which, in the course of a number of measurements of the same measurand, varies in an unpredictable way (and has not being considered otherwise) uncertainty (systematic): component of the uncertainty of measurement which, in the course of a number of measurements of the same measurand remains constant or varies in a predictable way uncertainty (type A): uncertainties evaluated using the statistical analysis of a series of observations uncertainty (type B): uncertainties evaluated using other means than the statistical analysis of a series of observations ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 14 uncertainty (limits of uncertainty of a measuring instrument): extreme values of uncertainty permitted by specifications, regulations etc. for a given measuring instrument NOTE: This term is also known as "tolerance". uncertainty (standard): for each individual uncertainty component, an expression characterizing the uncertainty for that component NOTE: It is the standard deviation of the corresponding distribution. uncertainty (combined standard): uncertainty characterizing the complete measurement or part thereof NOTE: It is calculated by combining appropriately the standard uncertainties for each of the individual contributions identified in the measurement considered or in the part of it which has been considered. In the case of additive components (linearly combined components where all the corresponding coefficients are equal to one) and when all these contributions are independent of each other (stochastic), this combination is calculated by using the Root of the Sum of the Squares (the RSS method). A more complete methodology for the calculation of the combined standard uncertainty is given in annex D; see in particular, clause D.3.12 of TR 100 028-2 [8]. uncertainty (expanded): expanded uncertainty is the uncertainty value corresponding to a specific confidence level different from that inherent to the calculations made in order to find the combined standard uncertainty NOTE: The combined standard uncertainty is multiplied by a constant to obtain the expanded uncertainty limits (see clause 5.3 of the present document and also clause D.5 (and more specifically clause D.5.6.2 of TR 100 028-2 [8]). upper specified AF limit: upper specified audio frequency limit is the maximum audio frequency of the audio pass-band and is dependent on the channel separation wanted signal level: for conducted measurements the wanted signal level is defined as a level of +6 dB/µV emf referred to the receiver input under normal test conditions. Under extreme test conditions the value is +12 dB/µV emf NOTE: For analogue measurements the wanted signal level has been chosen to be equal to the limit value of the measured usable sensitivity. For bit stream and message measurements the wanted signal has been chosen to be +3 dB above the limit value of measured usable sensitivity. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 3.2 Symbols | For the purposes of the present document, the following symbols apply: β 2π/λ (radians/m) γ incidence angle with ground plane (°) λ wavelength (m) φH phase angle of reflection coefficient (°) η 120π Ω - the intrinsic impedance of free space (Ω) µ permeability (H/m) AFR antenna factor of the receive antenna (dB/m) AFT antenna factor of the transmit antenna (dB/m) AFTOT mutual coupling correction factor (dB) Ccross cross correlation coefficient D(θ,φ) directivity of the source d distance between dipoles (m) δ skin depth (m) d1 an antenna or EUT aperture size (m) d2 an antenna or EUT aperture size (m) ddir path length of the direct signal (m) drefl path length of the reflected signal (m) E electric field intensity (V/m) EDHmax calculated maximum electric field strength in the receiving antenna height scan from a half wavelength dipole with 1 pW of radiated power (for horizontal polarization) (µV/m) ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 15 EDVmax calculated maximum electric field strength in the receiving antenna height scan from a half wavelength dipole with 1 pW of radiated power (for vertical polarization) (µV/m) eff antenna efficiency factor φ angle (°) ∆f bandwidth (Hz) f frequency (Hz) G(θ,φ) gain of the source (which is the source directivity multiplied by the antenna efficiency factor) H magnetic field intensity (A/m) I0 the (assumed constant) current (A) Im the maximum current amplitude k 2π/λ k a factor from Student's distribution k Boltzmann's constant (1,38 x 10 - 23 J/°K) K relative dielectric constant l the length of the infinitesimal dipole (m) L the overall length of the dipole (m) l the point on the dipole being considered (m) λ wavelength (m) Pe (n) probability of error n Pp (n) probability of position n Pr antenna noise power (W) Prec power received (W) Pt power transmitted (W) θ angle (°) ρ reflection coefficient r the distance to the field point (m) ρg reflection coefficient of the generator part of a connection ρl reflection coefficient of the load part of the connection Rs equivalent surface resistance (Ω) σ conductivity (S/m) σ standard deviation SNRb* Signal to noise ratio at a specific BER SNRb Signal to noise ratio per bit TA antenna temperature (°K) U the expanded uncertainty corresponding to a confidence level of x %: U = k × uc uc the combined standard uncertainty ui general type A standard uncertainty ui01 random uncertainty uj general type B uncertainty uj01 reflectivity of absorbing material: EUT to the test antenna uj02 reflectivity of absorbing material: substitution or measuring antenna to the test antenna uj03 reflectivity of absorbing material: transmitting antenna to the receiving antenna uj04 mutual coupling: EUT to its images in the absorbing material uj05 mutual coupling: de-tuning effect of the absorbing material on the EUT uj06 mutual coupling: substitution, measuring or test antenna to its image in the absorbing material uj07 mutual coupling: transmitting or receiving antenna to its image in the absorbing material uj08 mutual coupling: amplitude effect of the test antenna on the EUT uj09 mutual coupling: de-tuning effect of the test antenna on the EUT uj10 mutual coupling: transmitting antenna to the receiving antenna uj11 mutual coupling: substitution or measuring antenna to the test antenna uj12 mutual coupling: interpolation of mutual coupling and mismatch loss correction factors uj13 mutual coupling: EUT to its image in the ground plane uj14 mutual coupling: substitution, measuring or test antenna to its image in the ground plane uj15 mutual coupling: transmitting or receiving antenna to its image in the ground plane ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 16 uj16 range length uj17 correction: off boresight angle in the elevation plane uj18 correction: measurement distance uj19 cable factor uj20 position of the phase centre: within the EUT volume uj21 positioning of the phase centre: within the EUT over the axis of rotation of the turntable uj22 position of the phase centre: measuring, substitution, receiving, transmitting or test antenna uj23 position of the phase centre: LPDA uj24 stripline: mutual coupling of the EUT to its images in the plates uj25 stripline: mutual coupling of the 3-axis probe to its image in the plates uj26 stripline: characteristic impedance uj27 stripline: non-planar nature of the field distribution uj28 stripline: field strength measurement as determined by the 3-axis probe uj29 stripline: Transform Factor uj30 stripline: interpolation of values for the Transform Factor uj31 stripline: antenna factor of the monopole uj32 stripline: correction factor for the size of the EUT uj33 stripline: influence of site effects uj34 ambient effect uj35 mismatch: direct attenuation measurement uj36 mismatch: transmitting part uj37 mismatch: receiving part uj38 signal generator: absolute output level uj39 signal generator: output level stability uj40 insertion loss: attenuator uj41 insertion loss: cable uj42 insertion loss: adapter uj43 insertion loss: antenna balun uj44 antenna: antenna factor of the transmitting, receiving or measuring antenna uj45 antenna: gain of the test or substitution antenna uj46 antenna: tuning uj47 receiving device: absolute level uj48 receiving device: linearity uj49 receiving device: power measuring receiver uj50 EUT: influence of the ambient temperature on the ERP of the carrier uj51 EUT: influence of the ambient temperature on the spurious emission level uj52 EUT: degradation measurement uj53 EUT: influence of setting the power supply on the ERP of the carrier uj54 EUT: influence of setting the power supply on the spurious emission level uj55 EUT: mutual coupling to the power leads uj56 frequency counter: absolute reading uj57 frequency counter: estimating the average reading uj58 Salty-man/Salty-lite: human simulation uj59 Salty-man/Salty-lite: field enhancement and de-tuning of the EUT uj60 Test Fixture: effect on the EUT uj61 Test Fixture: climatic facility effect on the EUT Vdirect received voltage for cables connected via an adapter (dBµV/m) Vsite received voltage for cables connected to the antennas (dBµV/m) W0 radiated power density (W/m2) Other symbols which are used only in annexes D or E of TR 100 028-2 [8] are defined in the corresponding annexes. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 17 |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 3.3 Abbreviations | For the purposes of the present document, the following abbreviations apply: AF Audio Frequency BER Bit Error Ratio BIPM International Bureau of Weights and Measures (Bureau International des Poids et Mesures) c calculated on the basis of given and measured data d derived from a measuring equipment specification emf electromotive force EUT Equipment Under Test FSK Frequency Shift Keying GMSK Gaussian Minimum Shift Keying GSM Global System for Mobile telecommunication (Pan European digital telecommunication system) m measured NSA Normalized Site Attenuation p power level value v voltage level value r indicates rectangular distribution RF Radio Frequency RSS Root-Sum-of-the-Squares u indicates U-distribution VSWR Voltage Standing Wave Ratio |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 4 Introduction to measurement uncertainty | This clause gives the general background to the subject of measurement uncertainty and is also the basis of TR 102 273 [3]. It covers methods of evaluating both individual components and overall system uncertainties and ends with a discussion of the generally accepted present day approach to the calculation of overall measurement uncertainty. For further details and for the basis of a theoretical approach, please see annex D of TR 100 028-2 [8]. An outline of the extensions and improvements recommended is also included in this clause. This clause should be viewed as introductory material for clauses 5 and 6, and to some extent, also for annex D of TR 100 028-2 [8]. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 4.1 Background to measurement uncertainty | |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 4.1.1 Commonly used terms | UNCERTAINTY is that part of the expression of the result of a measurement which states the range of values within which the true value is estimated to lie. ACCURACY is an estimate of the closeness of the measured value to the true value. An accurate measurement is one in which the uncertainties are small. This term is not to be confused with the terms PRECISION or REPEATABILITY which characterize the ability of a measuring system to give identical indications or responses for repeated applications of the same input quantity. Measuring exactly a quantity (referred to as the measurand) is an ideal which cannot be attained in practical measurements. In every measurement a difference exists between the TRUE VALUE and the MEASURED VALUE. This difference is termed "THE ABSOLUTE ERROR OF THE MEASUREMENT". This error is defined as follows: Absolute error = the measured value - the true value. Since the true value is never known exactly, it follows that the absolute error cannot be known exactly either. The above formula is the defining statement for the terms of ABSOLUTE ERROR and TRUE VALUE, but, as a result of neither ever being known, it is recommended that these terms are never used. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 18 In practice, many aspects of a measurement can be controlled (e.g. temperature, supply voltage, signal generator output level, etc.) and by analysing a particular measurement set-up, the overall uncertainty can be assessed, thereby providing upper and lower UNCERTAINTY BOUNDS within which the true value is believed to lie. The overall uncertainty of a measurement is an expression of the fact that the measured value is only one of an infinite number of possible values dispersed (spread) about the true value. This is further developed in clause D.5.6 of TR 100 028-2 [8]. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 4.1.2 Assessment of upper and lower uncertainty bounds | One method of providing upper and lower bounds is by straightforward arithmetic calculation in the worst case condition, using the individual uncertainty contributions. This method can be used to arrive at a value each side of the measured result within which, there is utmost confidence (100 %) that the true value lies (see also clause D.5.6.1 in TR 100 028-2 [8]). When estimating the measurement uncertainty in the worst case e.g. by simply adding the uncertainty bounds (in additive situations), (extremely) pessimistic uncertainty bounds are often found. This is because the case when all the individual uncertainty components act to their maximum effect in the same direction at the same time is, in practice, very unlikely to happen (it has to be noted, however, that the usage of expansion factors in order to increase the confidence levels (see also clause 5.3.1 and clauses D.5.6.2.2 and D.3.3.5.2 in TR 100 028-2 [8]) may have a balancing effect). To overcome this (very) pessimistic calculation of the lower and upper bounds, a more realistic approach to the calculation of overall uncertainty needs to be taken (i.e. a probabilistic approach). The method presented in the present document is based on the approach to expressing uncertainty in measurement as recommended by the Comité International des Poids et Mesures (CIPM) in 1981. This approach is founded on Recommendation INC-1 (1980) of the Working Group on the Statement of Uncertainties. This group was convened in 1980 by the Bureau International des Poids et Mesures (BIPM) as a consequence of a request by the Comité that the Bureau study the question of reaching an international consensus on expressing uncertainty in measurement. Recommendation INC-1 (1980) led to the development of the Guide to the Expression of Uncertainty in Measurement [1] (the Guide), which was prepared by the International Organization for Standardization Technical Advisory Group 4 (ISOTAG 4), Working Group 3. The Guide was the most complete reference on the general application of the BIPM approach to expressing measurement uncertainty. Further theoretical analysis has been introduced in the third edition of the present document (see, in particular, annexes D and E in TR 100 028-2 [8]). Although the Guide represented the current international view of how to express uncertainty it is a rather lengthy document that is not easily interpreted for radiated measurements. The guidance given in the present document is intended to be applicable to radio measurements but since the Guide itself is intended to be generally applicable to measurement results, it should be consulted for additional details, if needed. The method in both the present document and the Guide apply statistical/probabilistic analysis to estimate the overall uncertainties of a measurement and to provide associated confidence levels. They depend on knowing the magnitude and distribution of the individual uncertainty components. This approach is commonly known as the BIPM method. Basic to the BIPM method is the representation of each individual uncertainty component that contributes to the overall measurement uncertainty by an estimated standard deviation, termed standard uncertainty [1], with suggested symbol u. All individual uncertainties are categorized as either type A or type B. Type A uncertainties, symbol ui, are estimated by statistical methods applied to repeated measurements, whilst type B uncertainties, symbol uj, are estimated by means of available information and experience. The combined standard uncertainty [1], symbol uc, of a measurement is calculated by combining the standard uncertainties for each of the individual contributions identified. In the case where the underlying physical effects are additive, this is done by applying the "Root of the Sum of the Squares (the RSS)" method (see also clause D.3.3 in TR 100 028-2 [8]) under the assumption that all contributions are stochastic i.e. independent of each other. The table included in clause D.3.12 of TR 100 028-2 [8] provides the way in which should be handled contributions to the uncertainty which correspond to physical effects which are not additive. Clause D.5 of the same annex provides an overview of several general methods. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 19 The resulting combined standard uncertainty can then be multiplied by a constant kxx to give the uncertainty limits (bounds), termed expanded uncertainty [1], in order to provide a confidence level of xx %. This is further discussed in clause D.5.6.2 of TR 100 028-2 [8]. One of the main assumptions when calculating uncertainty using the basic BIPM method is that the combined standard uncertainty of a measurement has a Normal or Gaussian distribution (see also clause D.1.3.4 in TR 100 028-2 [8]) with an associated standard deviation (the present document often uses the term Normal). This may be true when there is an infinite number of contributions in the uncertainty, which is generally not the case in the examples discussed in the present document (an interesting example is provided in clause D.3.3.5.2.2 of TR 100 028-2 [8]). Should the combined standard uncertainty correspond to a Normal distribution, then the multiplication by the appropriate constant (expansion factor) will provide the sought confidence level. The case where the combined standard uncertainty corresponds to non-Gaussian distributions is also considered in clauses D.5.6.2.3 and D.5.6.2.4 of TR 100 028-2 [8]. The Guide defines the combined standard uncertainty for this distribution uc, as equal to the standard deviation of a corresponding Normal distribution. The mean value is assumed to be zero as the measured result is corrected for all known errors. Based on this assumption, the uncertainty bounds corresponding to any confidence level can be calculated as kxx × uc (see also clause D.5.6.2 of TR 100 028-2 [8]). To illustrate the true meaning of a typical final statement of measurement uncertainty using this method, if the combined standard uncertainty is associated with a Normal distribution, confidence levels can be assigned as follows: - 68,3 % confidence level that the true value is within bounds of 1 × uc; - 95 % confidence within ±1,96 × uc, etc. Care must be taken in the judgement of which unit is chosen for the calculation of the uncertainty bounds. In some types of measurements the correct unit is logarithmic (dB); in other measurements it is linear (i.e. V or %). The choice depends on the model and architecture of the test system. In any measurement there may be a combination of different types of unit. The present document breaks new ground by giving methods for conversion between units (e.g. dB into V %, power % into dB, etc.) thereby allowing all types of uncertainty to be combined. Details of the conversion schemes are given in clause 5, and theoretical support in annexes D and E of TR 100 028-2 [8]. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 4.1.3 Combination of rectangular distributions | The following example shows that the overall uncertainty, when all contributions of a measurement have the same rectangular distribution, approaches a Normal distribution. The case of a discrete approach to a rectangularly distributed function, (the outcome of throwing a die), is shown and how, with up to 6 individual events simultaneously, (6 dice thrown at the same time) the events combine together to produce an output increasingly approximating a Normal distribution. Initially with 1 die the output mean is 3,5 with a rectangularly distributed "error" of ±2,5. With 2 dice the output is 7 ± 5 and is triangularly distributed see figure 1. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 20 2 3 4 5 6 7 8 9 10 11 12 Probability of a number (2 dice) 1 2 3 4 5 6 Probability of a number (1 die) Figure 1: One and two die outcomes By increasing the number of dice further through 3, 4, 5 and 6 dice it can be seen from figures 2 and 3, that there is a central value (most probable outcome) respectively for 2, 3, 4, 5 and 6 dice of (7), (10,5), (14), (17,5) and (21) and an associated spread of the results that increasingly approximates a Normal distribution. It is possible to calculate the mean and standard deviation for these events. Probability of a number (3 dice) Probability of a number (4 dice) 3 4 5 6 7 8 9 101112131415 161718 4 6 8 10 12 14 16 18 20 22 24 Figure 2: Three and four die outcomes 6 8 10 12 14 16 18 20 22 24 26 28 30 Probability of a number (5 dice) 10 15 20 25 30 35 Probability of a number (6 dice) Figure 3: Five and six die outcomes ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 21 The practical interpretation of the standard deviation of a Normally distributed quantity is that 68,3 % of all its possible values will lie within ±1 standard deviation of the mean value, 95,45 % will lie within ±2 standard deviations. Another way to regard these standard deviations is "as confidence levels", e.g. a confidence level of 68,3 % attaches to one standard deviation, 95,45 % to two standard deviations. Using the mathematical definition of a Gaussian (see annex D in TR 100 028-2 [8]), it is possible to calculate the expanded measurement uncertainty for other confidence levels. This illustration shows that in the case of individual throws of a die (which corresponds to a set of identical rectangular distributions since any of the values 1 to 6 is equally likely) the overall probability curve approximates closer and closer that of a Normal distribution as more dice are used. The BIPM method extends this principle by combining the individual standard uncertainties to derive a combined standard uncertainty. The standard uncertainties (corresponding to the distributions of the individual uncertainties) are all that need to be known (or assumed) to apply this approach. From the assumption that the final combined standard uncertainty corresponds to a Normal distribution, it is possible to calculate the expanded uncertainty for a given confidence level. The confidence level should always be stated in any test report, in the case where the resulting distribution is Gaussian. In such case, it makes it possible for the user of the measured results to calculate expanded uncertainty figures corresponding to other confidence levels. For similar reasons, in the case where there is no evidence that the distribution corresponding to the combined uncertainty is Normal, the expansion factor, kxx, should be stated in the test report, instead. Usually, for the reasons stated above, kxx = 1,96 is used (see also clause D.5.6.2 of TR 100 028-2 [8]). An expansion factor, kxx = 2,00 could also be acceptable; it would provide a confidence level of 95,45 % should the corresponding distribution be Normal. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 4.1.4 Main contributors to uncertainty | The main contributors to the overall uncertainty of a measurement comprise: - systematic uncertainties: those uncertainties inherent in the test equipment used (instruments, attenuators, cables, amplifiers, etc.), and in the method employed. These uncertainties cannot always be eliminated (calculated out) although they may be constant values, however they can often be reduced; - uncertainties relating to influence quantities i.e. those uncertainties whose magnitudes are dependent on a particular parameter or function of the EUT. The magnitude of the uncertainty contribution can be calculated, for example, from the slope of "dB RF level" to "dB SINAD" curve for a receiver or from the slope of a power supply voltage effect on the variation of a carrier output power or frequency; - random uncertainties: those uncertainties due to chance events which, on average, are as likely to occur as not to occur and are generally outside the engineer's control. NOTE: When making a measurement care must be taken to ensure that the measured value is not affected by unwanted or unknown influences. Extraneous influences (e.g. ambient signals on an Open Area Test Site) should be eliminated or minimized by, for example, the use of screened cables. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 4.1.5 Other contributors | Other contributors to the overall uncertainty of a measurement can relate to the standard itself: - the type of measurement (direct field, substitution or conducted) and the test method have an effect on the uncertainty. These can be the most difficult uncertainty components to evaluate. As an illustration, if the same measurand is determined by the same method in different laboratories (as in a round robin) or alternatively by different methods either in the same laboratory or in different laboratories, the results of the testing will often be widely spread, thereby showing the potential uncertainties of the different measurement types and test methods; ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 22 - a direct field measurement involves only a single testing stage in which the required parameter (ERP, sensitivity, etc.) is indirectly determined as the received level on a receiving device, or as the output level of a signal generator, etc., and is subsequently converted to ERP, field strength, etc., by a calculation involving knowledge of antenna gain, measurement distance, etc. This method, whilst being of short time duration, offers no way of allowing for imperfections (reflections, mutual coupling effects, etc.) in the test site and can results in large overall uncertainty values; - the substitution technique, on the other hand, is a two stage measurement in which the unknown performance of an EUT (measured in one stage) is directly compared with the "known" performance of some standard (usually an antenna) in the other stage. This technique therefore subjects both the EUT and the known standard to (hopefully) the same external influences of reflections, mutual coupling, etc., whose effects on the different devices are regarded as identical. As a consequence, these site effects are deemed to cancel out (this has also been addressed in clause D.5.3.2 in TR 100 028-2 [8]). Some residual effects do remain however, (due to different elevation beamwidths, etc.) but these tend to be small compared to the uncertainties in the direct field method. All the test methods in the present document are substitution measurements; - for their part, test methods can contain imprecise and ambiguous instructions which could be open to different interpretations; - an inadequate description of the measurand can itself be a source of uncertainty in a measurement. In practice a measurand cannot be completely described without an infinite amount of information. Because this definition is incomplete it therefore introduces into the measurement result a component of uncertainty that may or may not be significant relative to the overall uncertainty required of the measurement. The definition of the measurand may, for example, be incomplete because: • it does not specify parameters that may have been assumed, unjustifiably, to have negligible effect (i.e. coupling to the ground plane, reflections from absorbers or that reference conditions remain constant); • it leaves many other matters in doubt that might conceivably affect the measurement (i.e. supply voltages, the layout of power, signal and antenna cables); • it may imply conditions that can never be fully met and whose imperfect realization is difficult to take into account (i.e. an infinite, perfectly conducting ground plane, a free space environment) etc. Maximum acceptable uncertainties and confidence levels (or expansion factors) are both defined in most ETSI standards. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 4.2 Evaluation of individual uncertainty components | As discussed in clause 4.1.4, uncertainty components can be categorized either as "random" or "systematic". Such categorization of components of uncertainty can be ambiguous if they are applied too rigorously. For example, a "random" component of uncertainty in one measurement may become a "systematic" component of uncertainty in another measurement e.g. where the result of a first measurement is used as a component of a second measurement. Categorizing the methods of evaluating the uncertainty components rather than the components themselves avoids this ambiguity. Instead of "systematic" and "random" uncertainty the types of uncertainty contribution are grouped into two categories: - type A: those which are evaluated by statistical methods; - type B: those which are evaluated by other means. The classification into type A and type B is not meant to indicate that there is any difference in the nature of the components, it is simply a division based on their means of evaluation. Both types will possess probability distributions (although they may be governed by different rules), and the uncertainty components resulting from either type may be quantified by standard deviations. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 23 |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 4.2.1 Evaluation of type A uncertainties | When we carry out a measurement more than once and find the results are different, the following questions arise: - What to do with the results? - How much variation is acceptable? - When do we suspect the measuring system is faulty? - Are the conditions repeatable? Variations in these repeated measurements are assumed to be due to influence and random quantities that affect the measurement result and cannot be held completely constant. Therefore none of the results is necessarily correct. In practice, repeated measurements of the same measurand can help us evaluate these type A uncertainties. By treating the results statistically, we can derive the mean (the best approximation to the "true value") and standard deviation values. The standard deviation can then be incorporated as a standard uncertainty into the calculation of combined standard uncertainty, when the corresponding component is part of some measurement system. Uncertainties determined from repeated measurements are often thought of as statistically rigorous and therefore absolutely correct. This implies, sometimes wrongly, that their evaluation does not require the application of some judgement. For example: - When carrying out a series of measurements do the results represent completely independent repetitions or are they in some way biased? - Are we trying to assess the randomness of the measurement system, or the randomness in an individual EUT, or the randomness in all of the EUT produced? - Are the means and standard deviations constant, or is there perhaps a drift in the value of an unmeasured influence quantity during the period of repeated measurements? - Are the results stable with ambient conditions? If all of the measurements are on a single EUT, whereas the requirement is for sampling, then the observations have not been independently repeated. An estimate of the standard uncertainty arising from possible differences among production EUT should, in this case, be incorporated into the combined standard uncertainty calculation along with the calculated standard uncertainty of the repeated observations made on the single equipment (e.g. for characterizing a set of pieces of equipment). If an instrument is calibrated against an internal reference as part of the measurement procedure, (such as the "cal out" reference on a spectrum analyser), then the calibration should be carried out as part of every repetition, even if it is known that the drift is small during the period in which observations are made. If the EUT is rotated during a radiated test on a test site and the azimuth angle read, it should be rotated and read for each repetition of the measurement, for there may be a variation both in received level and in azimuth reading, even if everything else is constant. If a number of measurements have been carried out on the same EUT/types of EUT, but in two groups spaced apart in time, the arithmetic means of the results of the first and second groups of measurements and their experimentally derived means and standard deviations may be calculated and compared. This will enable a judgement to be made as to whether any time varying effects are statistically significant. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 4.2.2 Evaluation of type B uncertainties | Some examples of type B uncertainties are: - mismatch; - losses in cables and components; - non linearities in instruments; - antenna factors. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 24 Type B uncertainties do not reveal themselves as fluctuations as do type A uncertainties; they can only be assessed by careful analysis of test and calibration data. For incorporation into an overall analysis, the magnitudes and distributions of type B uncertainties can be estimated based on: - manufacturers' information/specification about instruments and components in the test set-up; - data in calibration certificates (if the history of the instrument is known); - experience with the behaviour of the instruments. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 4.2.3 Uncertainties relating to influence quantities | Uncertainties relating to influence quantities are, as a result of the way they are treated in the present document, regarded as a subgroup of type B uncertainties. Some examples of influence quantities are: - power supply; - ambient temperature; - time/duty cycle. Their effect is evaluated using some relationship between the measured parameter e.g. output power and the influence quantity e.g. supply voltage. Dependency functions (e.g. the relationship between output power and the fluctuating quantity), as those given in the present document, should be used to calculate the properties corresponding to the effect considered. A theoretical approach to influence quantities and dependency functions can be found in TR 100 028-2 [8] (see clause D.4). |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 4.3 Methods of evaluation of overall measurement uncertainty | The uncertainty of the measurement is a combination of many components. Some of these components may be evaluated from the statistical distributions of the results of a series of measurements (type A uncertainty) whilst other components are evaluated from assumed probability distributions based on experience or other information (type B uncertainty). The exact error of a result of a measurement is, in general, unknown and unknowable. All that can be done is to estimate the values of all quantities likely to contribute to the combined standard uncertainty, including those uncertainties associated with corrections for recognized systematic offset effects. With knowledge of the magnitudes of their individual standard uncertainties, it is then possible to calculate the combined standard uncertainty of the measurement. At present the assessment of the number of uncertainty components for any particular test is very variable. Whilst some general agreement has been reached on the manner in which individual uncertainties should be combined (the BIPM method, see also the discussion of such methods in TR 100 028-2 [8], annex D, in particular, in clause D.5), no such agreement has been arrived at concerning the identity of those individual components. Consequently, it is left to the particular test house/engineer/etc. to decide the contributory uncertainties, and to assess which are independent and which are not. This can lead to considerable test house to test house variation for the same test and is heavily dependent, in general, on the experience of the test engineer. A model of the measurement can assist in the evaluation of combined standard uncertainty since it will enable all known individual components of uncertainty to be rigorously included in the analysis, and correctly combined (see annex D in TR 100 028-2 [8], and, in particular, the table in clause D.3.12). ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 25 |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 4.4 Summary | The measured result can be affected by many variables, some of which are shown in figure 4. Corrections Coupling Equipment under test Contributions from the test method Influence quantities Temperature, supply voltage etc. Random uncertainties Inadequate definition of the measurand Statistical fluctuations Measuring system Measured result Exisiting knowledge Systematic uncertainties Figure 4: The measurement model |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 4.5 Overview of the approach of the present document | The present document proposes an approach to the calculation of the combined standard uncertainty of a measurement which includes solutions to the present day imperfections. For example, in clause 5, a technique is put forward for converting linear standard deviations into logarithmic ones (and vice versa) so that all uncertainty contributions for a particular test can be combined in the same units (dB, Voltage % or power %), and as stated above, comprehensive lists of the individual uncertainty sources for the tests are attached. Instructions within the test methods have been made more detailed and thereby less ambiguous. A global approach for the analysis of the uncertainties corresponding to a complete measurement set up (i.e. "a complete system") is also proposed in clause D.5. This approach addresses, in particular, the concept of "sub-systems" and how to combine the uncertainties relating to each "sub-system". Such an approach could help in cases where different units are to be used (e.g. dBs in one sub-system, linear terms in another). A set of files (spread sheets) has been included in the present document, in order to support some of the examples given and to help the user in the implementation of his own methodology. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 5 Analysis of measurement uncertainty | This clause develops the approach to measurement uncertainty beyond the introduction given in clause 4. It details the improvements to the analysis which the present document is proposing and presents solutions for all the identified problems associated with the BIPM method for calculating measurement uncertainty in radiated measurements. Clause 6 presents numerous worked examples which illustrate the application of the proposed new techniques. In the beginning of this clause, a review is given of the BIPM method, along with an outline of where it is inadequate for radiated measurements. The means of evaluation of type A and type B uncertainties are also given. This is followed by a discussion of the units in which the uncertainties are derived and the technique for converting standard deviations from logarithmic to linear quantities (% voltage or % power and vice versa) is presented. The conversion technique allows all the individual uncertainty components in a particular test to be combined in the same units and overcomes a major current day problem of asymmetric uncertainty limits (e.g. x + 2, -3 dB, as found in edition 2 of ETR 028 [6]). The clause concludes with clauses on deriving the expanded uncertainties in the case of Normal distributions, how influence quantities are dealt with, calculating the standard deviation of random effects and an overall clause summary. Theoretical and mathematical support for this clause can be found in annex D of TR 100 028-2 [8]. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 26 |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 5.1 The BIPM method | Basic to the BIPM method is the representation of each individual uncertainty component that contributes to the overall measurement uncertainty by an estimated standard deviation, termed standard uncertainty [1], with suggested symbol u. All individual uncertainties are categorized as either type A or type B. Type A uncertainties, symbol ui, are estimated by statistical methods applied to repeated measurements, whilst type B uncertainties, symbol uj, are estimated by means of available information and experience. The combined standard uncertainty [1], symbol uc, of a measurement is calculated by combining the standard uncertainties for each of the individual contributions identified. In the case where the underlying physical effects are additive, this is done by applying the Root of the Sum of the Squares (the RSS) method under the assumption that all contributions are stochastic i.e. independent of each other. The table included in clause D.3.12 of TR 100 028-2 [8] provides the way in which should be handled contributions to the uncertainty which correspond to physical effects which are not additive. Clause D.5 of the same annex provides an overview of several more general methods. The resulting combined standard uncertainty can then be multiplied by a constant kxx to give uncertainty limits (bounds), termed expanded uncertainty [1]. When the combined standard uncertainty corresponds to a Normal distribution (see clause 4.1.3) the expanded uncertainty corresponds to a confidence level of xx %. This is the broad outline of the analysis technique employed in the present document, but there are numerous practical problems when applying the basic BIPM rules to measurements, such as: - how uncertainty contributions in different units (dB, % voltage, % power) can be combined; - whether individual uncertainties are functions of the true value (e.g. Bit error ratios); - how to deal with asymmetrically distributed individual uncertainties; - how to evaluate confidence limits for those standard uncertainties which are not Normal by nature (see also clause D.5.6.2 in TR 100 028-2 [8]). These problem areas are discussed below and have resulted in modifications and extensions to the BIPM method. For most cases, examples are given in clause 6. In order to help understanding some of these questions and to bring some more theoretical support, annexes D and E (found in TR 100 028-2 [8]) have been added to the third edition of the present document. Clause D.3 supports various combinations (e.g. additive, multiplicative, etc.), conversions (e.g. to and from dBs) and functions (see clauses D.3.9 and D.3.11). A complete approach, encompassing the "BIPM method" is included in clause D.5 of TR 100 028-2 [8]. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 5.1.1 Type A uncertainties and their evaluation | Type A uncertainties are evaluated by statistical methods, estimating their standard deviations (corresponding to "standard uncertainties"). These normally play a minor part in the combined standard uncertainty. Annex D (in TR 100 028-2 [8]) shows that, in most cases, it is only the standard uncertainty that needs to be known in order to find the combined uncertainty. In the BIPM approach, the shape of the individual distributions is relatively unimportant. However, annex D shows how to combine the various individual distributions, when needed, and that the result of a combination does not necessarily correspond to a Normal distribution. In such a case, the actual shape of the resulting distribution may be fully relevant (see, in particular, clauses D.5.6.2.3 and D.5.6.2.4). |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 5.1.2 Type B uncertainties and their evaluation | Type B uncertainties are estimated by various methods. Figure 5 illustrates a selection of uncertainty distributions which can often be identified in RF measurements. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 27 'U' Distribution +a -a 0 Rectangular Distribution +a -a 0 Normal (Gaussian) Distribution Figure 5: Types of uncertainty distribution Mismatch uncertainties have the "U" distribution, see annex G. The value of the uncertainty contribution is more likely to be near the limits than to be small or zero. If the limits are ±a, the standard uncertainty is: 2 a (see annex D) Systematic uncertainties (e.g. those associated with the loss in a cable) are, unless the actual distribution is known, assumed to have a rectangular distribution. The result of this assumption is that the uncertainty can take any value between the limits with equal probability. If the limits are ±a, the standard uncertainty is: 3 a (see annex D) If the distribution used to model the uncertainty is a Normal distribution, it is characterized by its standard deviation (standard uncertainty) (see annex D). In the present document the standard uncertainties are symbolized by uj xx or uj description. In all cases where the distribution of the uncertainty is unknown, the rectangular distribution should be taken as the default model. It will be noted that all the distributions illustrated in figure 5 are symmetrical about zero (clause D.1 addresses also distributions showing an offset and/or which are not symmetrical). An unexpected complication in combining standard uncertainty contributions may result from the use of different units, since a symmetrical standard uncertainty in % voltage is asymmetrical in dB (and vice versa). Similarly for % power. This "major" complication (for any particular test, the contributions may be in a variety of units) is the subject of clause 5.2. See also clause D.3 and in particular clause D.3.10.7 of TR 100 028-2 [8]. 5.2 Combining individual standard uncertainties in different units The BIPM method for calculating the combined standard uncertainty of any test involves combining the individual standard uncertainties by the RSS method. If there are n individual standard uncertainty contributions to be combined, the combined standard uncertainty is: 2 2 )1 ( 2 3 2 2 2 1 2 2 )1 ( 2 3 2 2 2 1 .... in n i i i i jn n j j j j c u u .... u u u u u .... u u u u + + + + + + + + + + + + = − − (5.1) However, this is correct only if all the individual contributions, represented by their standard uncertainties: (1) combine by addition; and (2) are expressed in the same units. It does not matter whether the contributions are expressed in percent or logarithmic terms or any other terms as long as these two conditions are fulfilled… noting that the result of the corresponding combination will be expressed in the same way (see also conversions in clause D.3 and the discussion on the concept of sub-systems in clause D.5 of TR 100 028-2 [8]). ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 28 To use formula 5.1 for standard uncertainties of individual contributions which combine by addition, linear terms only i.e. voltage, percentage, etc., should be used. This is essential for the RSS combination to be valid. This is the case in many measuring instruments. To use formula 5.1 for standard uncertainties of individual contributions which combine by multiplication, logarithmic terms only i.e. dB should be used as they can then be combined by addition. This is essential for the RSS combination to be valid where uncertainty multiplication occurs. This is the case where gains and/or losses (i.e. attenuators, amplifiers, antennas, etc.) are involved as well as under mismatch conditions where modules (i.e. attenuators, cables, RF measuring instruments, etc.) are interconnected in RF measurements. If all parameters and their associated standard uncertainties in a measurement are in the same unit and combine by addition, the RSS method can be applied directly. The table in clause D.3.12 of TR 100 028-2 [8]) shows how to handle other cases. Clause D.5 (in TR 100 028-2 [8]) discusses general methods usable in most cases. For small (< 30 % or 2,5 dB) standard uncertainties however, both additive and multiplicative contributions can be incorporated into the same calculation (with negligible error) provided they are converted to the same units prior to calculating the combined standard uncertainty. The conversion factors are given in table 1. This is supported by the theoretical analysis provided in annex D, clause D.3 and annex E (TR 100 028-2 [8]). Annex E gives the justification for this statement by firstly mathematically converting the distribution of an individual uncertainty from logarithmic to linear (and vice versa) and secondly comparing the standard deviation of the two distributions before and after the conversion. One of the outcomes of annex E is that the conversion between linear and logarithmic standard uncertainties can, under some conditions, be approximated by the first order mathematical functions given in table 1. As can be seen from annex E there are, however, some problems involved in converting distributions. - It is not a linear procedure; the conversion factor is not only dependent on the magnitude of the standard uncertainty, but it is also dependent on the shape of the distribution. - The mean value of the converted uncertainty distribution is not necessarily zero, even if that was the case before the conversion. However if the standard uncertainties to be converted are less than 2,5 dB, 30 % (voltage), or 50 % (power) the errors arising may be considered as negligible. Table 1 shows the multiplicative factors to be used when converting standard uncertainties with a first order approximation. As an example, if the standard uncertainty is 1,5 dB then this, converted to voltage %, gives a corresponding standard uncertainty of 1,5 × 11,5 % = 17,3 %. Table 1: Standard uncertainty conversion factors Converting from standard uncertainties in …: Conversion factor multiply by: To standard uncertainties in …: dB 11,5 voltage % dB 23,0 power % power % 0,0435 dB power % 0,5 voltage % voltage % 2,0 power % voltage % 0,0870 dB It should be noted after any conversions that may be necessary before using equation 5.1, that the combined standard uncertainty, uc, that results from the application of equation 5.1, does not, by itself give the expanded uncertainty limits for a measurement. When uc corresponds to a Normal distribution, these can be calculated (see clause 5.3) from uc (assumed in this case to be in units of dB) as the 95 % confidence limits in dB of ±1,96 × uc (which is very asymmetric in linear terms). ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 29 Similarly, in voltage as ±1,96 × uc × 11,5 % (which is very asymmetric in dB terms).The major factor determining whether the combined standard uncertainty, uc, will have the symmetrical dB interval or the symmetrical % interval (or somewhere in-between) is whether the individual uncertainties combine by multiplication or by addition. In radiated measurements as well as most conducted measurements where the RF level is of importance, the overwhelming majority of the uncertainties combine by multiplication. It is, therefore, safe to assume that, in general, the resulting uncertainty limits are symmetrical in logarithmic terms (dB). This assumption has been confirmed by computer simulations on a large number of measurement models. This is also clear from the relations found in annex D. 5.3 Calculation of the expanded uncertainty values and Student's t-distribution This clause discusses two different problems, both relating to the handling of uncertainties, which have to be very clearly identified and handled separately. Unfortunately, in the previous editions of the present document, this clause had not been subdivided into two clauses. The two clauses address: - the situation where the statistical properties of a number of samples are to be evaluated; in this case, the Student's t-distribution is a powerful tool allowing the evaluation of the performance of those properties; it can be helpful in supporting the evaluation of properties of "type A uncertainties"; - the situation where only one measurement is performed, in conditions where the various sources of uncertainty have been evaluated; as a result, the combined standard uncertainty of that measurement may be evaluated (see clause D.5 of TR 100 028-2 [8]), and the knowledge of the shape of the distribution corresponding to that combined uncertainty allows for changes in the confidence level. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 5.3.1 Student's t-distribution | The Student's t-distribution gives coverage factors (i.e. multipliers) for measurements, whereby the confidence level of a series of measurements can be calculated from a limited number of samples, assuming those samples have been taken from a Normal distribution. The fewer the number of samples, the bigger the coverage factor for a given confidence level. For example: - if a type A standard deviation is calculated on only 3 samples and the required confidence level is 95 % the appropriate Student's t-factor is 3,18; - if the standard deviation had been based on 20 samples, the factor would have been 2,09; - for an infinite number of samples the multiplier would have been 1,96. When using such an approach, any measurement should be repeated a large number of times. In radio measurements, however, by using the approach recommended in the present document, only one measurement is usually performed. As a result, the Student's t-distribution is of no help. The Student's t-distribution can, however, be very useful for the statistical evaluation of the properties of individual uncertainty components (i.e. type A uncertainties which may happen to be part of some test set up). |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 5.3.2 Expanded uncertainties | When the combined standard uncertainty, uc, has been calculated from equation 5.1 (or by any other method) and it can be expected that the corresponding distribution is Normal, then, the uncertainty limits relate to a confidence level of 68,3 % (due to the properties of the Gaussian curve). By multiplying uc by "a coverage factor" (or "an expansion factor") other confidence levels may be obtained when the distribution corresponding to the combined standard uncertainty, uc, is Normal. Why? ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 30 When: - all the individual sources of uncertainty are identified for all the tests; - the distributions of the uncertainties of the individual sources are all known (or assumed); - the maximum, worst-case values of all of the individual uncertainties are known. Then, under these conditions, annex D of TR 100 028-2 [8] applies and the combined standard uncertainty can be calculated (see clause D.5). Assuming that the combined standard uncertainty corresponds to a Normal distribution then the magic factor of 1,96 applies: this is due to the shape of the Gaussian curve used to describe the distribution corresponding to the combined uncertainty (see the interpretation in clause D.5.6.2 of TR 100 028-2 [8]). As already indicated above (see clause 4.1.3), for a Gaussian shaped curve: - a surface of 68 % (2 x 34 % , the value which can be found in some tables) corresponds to one standard deviation (i.e. a combined standard deviation); - a surface of 95 % (2 x 47,5 % , the value which can be found in some tables) corresponds to two standard deviations (more precisely 1,96 standard deviations). and the surface referred to above can be interpreted as the probability of the true value being within the stated uncertainty bounds. The probability of remaining inside this surface is, by definition, the confidence level. It has to be made clear that, when the combination of the various components of the uncertainty corresponds to a distribution which is not Normal, then other expansion factors apply in order to convert from one confidence level to another. The values of these factors depend on the mathematical properties (i.e. the shape) of the corresponding distribution. It has to be made clear also that, as indicated in particular in annex D, when the number of components added (or combined linearly) in order to obtain the uncertainty can be considered as an infinity, and under some other conditions, then the distribution can be considered as Normal (based on the “Central Limit Theorem”). Under such conditions, the factor 1,96 is valid (for a 95% confidence level). This is why it has been used extensively in the examples given in the present document. The usage of a value of 2 for this expansion factor has also been suggested (this would provide a confidence level of 95,45 % in the case of Normal distributions). The tools given in annex D could allow for the calculation of the actual distribution corresponding to the combination of various components for the uncertainty. Under such conditions, the appropriate expansion factors could also be calculated, in the case where the distribution found would not have happened to be Normal. 5.4 Combining standard uncertainties of different parameters, where their influence on each other is dependant on the EUT (influence quantities) In many measurements, variations in the influence quantities, intermediate test results or test signals can affect the uncertainty of the measurand in ways that may be functions of the characteristics of the EUT and other instrumentation. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 31 It is not always possible to fully characterize test conditions, signals and measurands. Uncertainties are related to each of them. These uncertainties may be well known, but their influence on the combined standard uncertainty depends on the EUT. Uncertainties related to general test conditions are: - ambient temperature; - the effect of cooling and heating; - power supply voltage; - power supply impedance; - impedance of test equipment connectors (VSWR). Uncertainties related to applied test signals and measured values are: - level; - frequency; - modulation; - distortion; - noise. The effect of such uncertainties on the test results can vary from one EUT to another. Examples of the characteristics that can affect the calculation of the uncertainties are: - receiver noise dependency of RF input signal levels; - impedance of input and output connectors (VSWR); - receiver noise distribution; - performance dependency of changes of test conditions and test signals; - modulator limiting function e.g. maximum deviation limiting; - system random noise. If the appropriate value for each characteristic has not been determined for a particular case, then the values listed in TR 100 028-2 [8] table F.1 should be used. These values are based on measurements made with several pieces of equipment and are stated as mean values associated with a standard uncertainty reflecting the spread from one EUT to another. When the EUT dependent uncertainties add to the combined standard uncertainty, the RSS method of combining the standard uncertainties is used, but in many calculations the EUT dependency is a function that converts uncertainty from one part of the measurement configuration to another. In most cases the EUT dependency function can be assumed to be linear; therefore the conversion is carried out by multiplication, as shown in the theoretical analysis provided in clause D.4 of TR 100 028-2 [8]. The standard uncertainty to be converted is uj 1. The mean value of the influence quantity is A and its standard uncertainty is uj a. The resulting standard uncertainty uj converted of the conversion is: ) u + A ( u = u a j j converted j 2 2 2 1 (5.2) The standard uncertainty of this contribution is then looked upon as any other individual component and is combined accordingly (see annex D). A fully worked example of an influence quantity is given in clause 6.4.6. The conditions under which the expression 5.2 is valid can also be found in clause D.4 of TR 100 028-2 [8]. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 32 If the function is not linear another solution must be found: - the theoretical relation between the influence quantity and its effect has to be determined; - the expressions providing the conversion can then be found based on the table contained in clause D.3.12 of TR 100 028-2 [8]. When the theoretical relation between the influence quantity and its effect is not known, the usage of a simple mathematical model can be tried. In this case, an attempt can be made in order to determine the numerical values of the parameters of the model by some statistical method (see also clause D.5.4 in TR 100 028-2 [8]). In all cases, it is recommended to determine first the mathematical relation between the parameters, and only after try and find the appropriate numerical values. As a consequence, tables similar to table F.1 in annex F (TR 100 028-2 [8]) should also include the mathematical relation between the parameters for each entry (for further details, see clause D.4.2.1.2 in TR 100 028-2 [8]). |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 5.5 Uncertainties and randomness | The major difficulty behind this clause is to understand exactly what “randomness uncertainty” is supposed to cover in this context (i.e. what this clause or contribution is expected to cover): the BIPM method and the corresponding analysis is supposed to cover all components of the uncertainty, so it is fundamental to understand what is left over for the "uncertainty of randomness", in order to avoid taking into account the same effects twice, under different names (in a complex set up)… The standard uncertainty of randomness can be evaluated by repeating a measurement (e.g. of a particular component of the measurement uncertainty). The first step is to calculate the arithmetic mean or average of the results obtained. The spread in the measured results reflects the merit of the measurement process and depends on the apparatus used, the method, the sample and sometimes the person making the measurement. A more useful statistic, however, is the standard uncertainty σi of the sample. This is the root mean square of the differences between the measured values and the arithmetic mean of the samples. If there are n results for xm where m = 1, 2, ..., n and the sample mean isx, then the standard deviation σi is: ∑ = − n m _ m i x x n = 1 2 1 σ (5.3) This should not be confused with the standard deviation of the A uncertainty being investigated. It only covers n samples. If further measurements are made, then for each sample of results considered, different values for the arithmetic mean and standard deviation will be obtained. For large values of n these mean values approach a central limit value of a distribution of all possible values. This distribution can usually be assumed, for practical purposes, to be a Normal distribution. From the results of a relatively small number of measurements an estimate can be made of the standard deviation of the whole population of possible values, of which the measured values are a sample. Estimate of the standard deviation σi/: 2 1 / 1 1 − = ∑ _ m n m= i x x ) n- ( σ (5.4) ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 33 A practical form of this formula is: 1 2 / n - n X Y - i = σ (5.5) where X is the sum of the measured values and Y is the sum of the squares of the measured values. It will be noted that the only difference between σi/ and σi is in the factor 1/ (n-1) in place of 1/n, so that the difference becomes smaller as the number of measurements is increased. A similar way of calculating the standard deviation of a discrete distribution can be derived from this formula. In this case X is the sum of the individual values from the distribution times their probability, and Y is the sum of the square of the individual values times their probability. If the distribution has m values xi, each having the probability p (xi): ) p(x x X i m i i ∑ = = 1 (5.6) and ) p(x x Y i m i i ∑ = = 1 2 (5.7) The standard uncertainty is then: 2 X Y i − = σ (5.8) When measured results are obtained as the arithmetic mean of a series of n (independent) measurements the standard uncertainty is reduced by a factor √n thus: n = i / 1 σ σ (5.9) This is an efficient method of reducing measurement uncertainty when making noisy or fluctuating measurements, and it applies both for random uncertainties in the measurement configuration and the EUT. Having established the standard deviation, this is directly equated to the standard uncertainty: ui = σi As the uncertainty due to random uncertainty is highly dependent on the measurement configuration and the test method used it is not possible to estimate a general value. Each laboratory must by means of repetitive measurements estimate their own standard uncertainties characterizing the randomness involved in each measurement. Once having done this, the estimations may be used in future measurements and calculations. NOTE: See also the note found in clause 6.4.7 concerning the usage of this component. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 5.6 Summary of the recommended approach | The basic BIPM method, with specific modifications, remains the adopted approach used for the calculation of combined standard and expanded uncertainty in the examples given in this report. That is to say that once all the individual standard uncertainties in a particular measurement have been identified and given values, they are combined by the RSS method provided they combine by addition and are in the same units (otherwise, methods such as those detailed in annex D, e.g. in clause D.5, have to be used). ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 34 In order to ensure that this proviso is satisfied as often as possible, the present document supplies the factors necessary to convert standard uncertainties in linear units to standard uncertainties in logarithmic units (and vice versa). The present document also shows that small additive standard uncertainties (% V, % power) can be combined with multiplicative standard uncertainties (dB) in the RSS manner with, hopefully, negligible error. Having derived the combined standard uncertainty, an expanded uncertainty for 95 % confidence levels can then be derived, when the corresponding distribution is Normal, by multiplying the result by the expansion factor of 1,96. The multiplication by this factor (or simply by a factor equal to 2) is to be done in all cases, in order to obtain the expanded uncertainty. However, if the corresponding distribution is not Normal, then the resulting confidence level is not necessarily 95 % (see clause D.5.6.2 of TR 100 028-2 [8]). In all cases, however, the actual confidence level can be calculated, once the distribution corresponding to the combination of all uncertainty components has been calculated. Clause D.3, in TR 100 028-2 [8], provides the equations allowing for the calculation of this combined distribution. The practical implementation of this modified BIPM approach, adopted throughout the present document, is for each test method (including the verification procedures) to have appended to it a complete list of the individual uncertainty sources that contribute to each stage of the test. Magnitudes of the standard uncertainties can then be assigned to these individual contributions by consulting annex A (converting from linear units to dB, if necessary). All uncertainties are in dB units since the great majority of the individual contributions in radiated measurements are multiplicative i.e. they add in dB terms. In those cases in which annex A instructs that the values of the uncertainty contributions be taken from a manufacturer's data sheet, that data should be taken over as broad a frequency band as possible. This type of approach avoids the necessity of calculating the combined standard uncertainty every time the same test is performed for different EUT. 6 Examples of uncertainty calculations specific to radio equipment |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.1 Mismatch | In the following the Greek letter Γ means the complex reflection coefficient. ρx is the magnitude of the reflection coefficient: ρx = Γx. Where two parts or elements in a measurement configuration are connected, if the matching is not ideal, there will be an uncertainty in the level of the RF signal passing through the connection. The magnitude of the uncertainty depends on the VSWR at the junction of the two connectors. The uncertainty limits of the mismatch at the junction are calculated by means of the following formula: Mismatch limits = |Γgenerator| × |Γload| × |S21| × |S12| × 100 % Voltage (6.1) where: - |Γgenerator| is the modulus of the complex reflection coefficient of the signal generator; - |Γload| is the modulus of the complex reflection coefficient of the load (receiving device); - | S21| is the forward gain in the network between the two reflection coefficients of interest; - | S12| is the backward gain in the network between the two reflection coefficients of interest. NOTE: S21 and S12 are set to 1 if the two parts are connected directly. In linear networks S21 and S12 are identical. The distribution of the mismatch uncertainty is U-shaped, If the uncertainty limits are ± a, the standard uncertainty is: % 2 % 100 12 21 : Voltage S S u load generator individual j mismatch × × × Γ × Γ = (6.2) ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 35 This can be converted into equivalent dB by dividing by 11,5 (see clause 5.2): dB 5, 11 2 % 100 12 21 : × × × × Γ × Γ = S S u load generator individual j mismatch (6.3) If there are several connections in a test set-up, they will all interact and contribute to the combined mismatch uncertainty. The method of calculating the combined mismatch uncertainty is fully explained in annex G. In conducted measurements, when calculating the mismatch uncertainty at the antenna connector of the EUT, the reflection coefficient of the EUT is required. In this case, the laboratory should either measure it in advance or use the reflection coefficients given in TR 100 028-2 [8], table F.1. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.2 Attenuation measurement | In many measurements the absolute level of the RF signal is part of the measured result. The RF signal path attenuation must be known in order to apply a systematic correction to the result. The RF signal path can be characterized using the manufacturers' information about the components involved, but this method can result in unacceptably large uncertainties. Another method is to measure the attenuation directly by using, for example, a signal generator and a receiving device. To measure the attenuation, connect the signal generator to the receiving device and read the reference level (A), see figure 6, and then insert the unknown attenuation, repeat the measurement and read the new level (B), see figure 7. Receiving device Γ load Generator Γ generator Figure 6: Measurement of level (A) In figure 6, Γgenerator is the complex reflection coefficient of the signal generator and Γload is the complex reflection coefficient of the load (receiving device). Generator Attenuator Receiving device Γload Γoutput Γgenerator Γinput Figure 7: Measurement of level (B) In figure 7, Γgenerator is the complex reflection coefficient of the signal generator, Γload is the complex reflection coefficient of the load (receiving device), Γinput is the complex reflection coefficient of the attenuator input, Γoutput is the complex reflection coefficient of the attenuator output. The attenuation is calculated as A/B if the readings are linear values or A-B if the readings are in dB. Using this method, four uncertainty sources need to be considered. Two sources concern the receiving device, namely its absolute level (if the input attenuation range has been changed) and its linearity. The other two sources are the stability of the signal generator output level (which contributes to both stages of the measurement) and mismatch caused by reflections at both the terminals of the network under test and the instruments used. The absolute level, linearity and stability uncertainties can be obtained from the manufacturers data sheets, but the mismatch uncertainty must be estimated by calculation. For this example, we assume that an attenuator of nominally 20 dB is measured at a frequency of 500 MHz by means of a signal generator and a receiving device. The magnitude of the reflection coefficient of the generator |Γgenerator| is 0,2, the magnitude of the reflection coefficient of the receiving device |Γload| is 0,3 and the magnitude of the reflection coefficients of the attenuator |Γinput| and |Γoutput| are 0,05. Since the mismatch uncertainty of the attenuation measurement is different in figure 7 to that in figure 6, it therefore has to be calculated (for figure 6 and figure 7) and both values included in the combined mismatch uncertainty as shown below. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 36 Mismatch uncertainty: Reference measurement: The signal generator is adjusted to 0 dBm and the reference level A is measured on the receiving device. Using equation 6.1 with S21 = S12 = 1, and taking the standard uncertainty, uj mismatch: reference measurement: dB 0,37 5 11 24 4 % 24 ,4 % 2 100 3 0 2 0 : = , , , , u t measuremen reference j mismatch ≈ = × × = Attenuator measurement: The attenuator is inserted and a level (B) = -20,2 dB is measured after an input attenuation range change on the receiving device. NOTE: The measured attenuation is 20,2 dB, for which S21 = S12 = 0,098. The following three components comprise the uncertainty in this part of the measurement: - the standard uncertainty of the mismatch between the signal generator and the attenuator: % 0,71 % 2 100 05 0 2 0 : = × × = , , u attenuator to generator j mismatch - the standard uncertainty of the mismatch between the attenuator and the receiving device: % ,06 1 % 2 100 05 0 3 0 : = × × = , , u device receiving to attenuator j mismatch - the standard uncertainty of the mismatch between the signal generator and the receiving device: % ,041 0 % 2 100 098 ,0 2 0 3 0 2 : = × × × = , , u device receiving to generator j mismatch The combined standard uncertainty of the mismatch of the attenuation measurement uc mismatch: att. measurement, is calculated by RSS (see clause 5.2) of the individual contributions. dB 0,11 5 11 28 1 % ,28 1 041 0 06 1 71 0 2 2 2 . : = ≈ = + + = , , , , , u t measuremen att c mismatch A comparison of uj mismatch: reference measurement (0,37 dB) and uc mismatch: att. measurement (0,11 dB) shows clearly the impact of inserting an attenuator between two mismatches. Other components of uncertainty: Reference measurement: The stability of the signal generator provides the only other uncertainty in the present document. The receiving device contributes no uncertainty here since only a reference level is being set for comparison in the attenuation measurement stage. The output level stability of the signal generator is taken from the manufacturer's data sheet as 0,10 dB which is assumed (since no information is given) to be rectangularly distributed (see clause 5.1). Therefore the standard uncertainty, uj signal generator stability, is: dB 0,06 3 10 0 = , u stability generator j signal = Therefore, the combined standard uncertainty, uc reference measurement, for the reference measurement is: dB 0,37 06 0 37 0 2 2 2 2 : = + = + = , , u u u stability generator signal j t measuremen reference mismatch j t measuremen reference c Attenuation measurement: Here the output stability of the signal generator as well as absolute level uncertainty of the receiving device (the input attenuation range has changed) contribute to the uncertainty. However as a range change has occurred there is no linearity contribution as this is included in the absolute level uncertainty of the receiver. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 37 The signal generator stability, uj signal generator stability, has the same value as for the reference measurement, whilst the uncertainty for the receiving device is given in the manufacturer's data sheet as 1,0 dB absolute level accuracy. A rectangular distribution is assumed for the absolute level accuracy so the standard uncertainty, uj signal generator level, of its uncertainty contribution is: dB 0,58 3 00 1 = = , u level generator j signal The uncertainty contribution of the linearity of the receiving device uj linearity is zero. Therefore the combined standard uncertainty, uc att. measurement, for the attenuation measurement is: 2 2 2 2 . : . y j linearit level generator j signal stability generator j signal t measuremen att c mismatch t measuremen c att u u u u u + + + = dB 0,59 00 0 58 0 06 0 08 0 2 2 2 2 = + + + = , , , , So, for the complete measurement, the combined standard uncertainty, uc measurement, is given by: dB 0,70 59 0 37 0 2 2 2 . 2 = + = + = , , u u u t measuremen c att t measuremen e c referenc t measuremen c The expanded uncertainty is ±1,96 × 0,70 = ±1,37 dB at a 95 % confidence level. This is an exaggerated example. Smaller uncertainty is possible if a better receiving device is used. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.3 Calculation involving a dependency function | The specific dependency function is the relationship between the RF signal level at the EUT antenna connector (dB) to the uncertainty of the measurement of SINAD at the EUT's audio output i.e. how does SINAD measurement uncertainty relate to RF level uncertainty at the EUT antenna connector. The following example is based on a typical ETR 028 [6] type (conducted) RF measurement for clarity. The sensitivity of a receiving EUT is measured. The outline of the measurement is as follows. The RF level at the input of the receiver is continuously reduced until a SINAD measurement of 20 dB is obtained, see figure 8. The result of the measurement is the RF signal level causing 20 dB SINAD at the audio output of the receiver. SINAD meter EUT dependancy function mean value = 1 standard deviation = 0,3 EUT ± 1,0 dB ± 1,0 dB Signal generator | = 0,07 |S11 | | = 0,07 S22 ± 0,5 dB cable |ρgenerator|= 0,30 |ρ |= 0,4 EUT Figure 8: Typical measurement configuration The combined standard uncertainty is calculated as follows. For the mismatch uncertainty (annex G): Generator: Output reflection coefficient: |ρgenerator| = 0,30 Cable: Input and output reflection coefficients: |S11| and |S22| = 0,07 Attenuation: 1 dB = |S21| = |S12| = 0,891 EUT: Input reflection coefficient: |ρEUT| = 0,4 ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 38 All these contributions are U distributed. There are three contributions: - the standard uncertainty of the mismatch between the signal generator and the cable: dB 0,13 5 11 2 % 100 07 0 30 0 : = × × × = , , , u cable to generator signal j mismatch - the standard uncertainty of the mismatch between the cable and the EUT: dB 0,17 5 11 2 % 100 07 0 4 0 : = × × × = , , , u EUT to cable j mismatch - the standard uncertainty of the mismatch between the signal generator and the EUT: dB 0,59 5 11 2 % 100 891 0 4 0 3 0 2 : = × × × × = , , , , u EUT to generator signal j mismatch - the combined standard uncertainty of the mismatch: dB ,63 0 59 ,0 17 ,0 13 ,0 2 2 2 = + + = c mismatch u uc mismatch = 0,63 dB The uncertainty due to the absolute output level of the signal generator is taken as ±1,0 dB (from manufacturers data). As nothing is said about the distribution, a rectangular distribution in logs is assumed (see clause 5.1), and the standard uncertainty is: uj signal generator level = 0,58 dB The uncertainty due to the output level stability of the signal generator is taken as ±0,02 dB (from manufacturer's data). As nothing is said about the distribution, a rectangular distribution in logs is assumed (see clause 5.1), and the standard uncertainty is: uj signal generator stability = 0,01 dB The uncertainty due to the insertion loss of the cable is taken as ±0,5 dB (from calibration data). As nothing is said about the distribution, a rectangular distribution in logs is assumed, and the standard uncertainty is: uj cable loss = 0,29 dB Dependency function uncertainty calculation: The uncertainty due to the SINAD measurement corresponds to an RF signal level uncertainty at the input of the receiving EUT. The SINAD uncertainty from the manufacturer's data is ±1 dB which is converted to a standard uncertainty of 0,577 dB. The dependency function converting the SINAD uncertainty to RF level uncertainty is found from TR 100 028-2 [8], table F.1. It is given as a conversion factor of 1,0 % (level)/ % (SINAD) with an associated standard uncertainty of 0,3. The SINAD uncertainty is then converted to RF level uncertainty using formula 5.2: ( ) dB 0,60 = 3 0 0 1 577 0 2 2 2 ) ( , , , u converted level j RF + × = The RF level uncertainty caused by the SINAD uncertainty and the RF level uncertainty at the input of the receiver is then combined using the square root of the sum of the squares method to give the combined standard uncertainty. 2 ) ( 2 2 2 2 converted level j RF loss j cable stability generator j signal level generator j signal c mismatch ent c measurem u u u u u u + + + + = dB 1,08 = 60 0 29 0 01 0 58 0 63 0 2 2 2 2 2 , , , , , = + + + + The expanded uncertainty is ±1,96 × 1,08 = ±2,12 dB at a 95 % confidence level. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 39 |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.4 Measurement of carrier power | The example test is a conducted measurement. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.4.1 Measurement set-up | The EUT is connected to the power meter via a coaxial cable and two power attenuators, one of 10 dB and one of 20 dB (see figure 9). Transmitter under test Power meter Sensor 20 dB power attenuator 10 dB power attenuator Cable Figure 9: Measurement set-up The nominal carrier power is 25 W, as a result the power level at the input of the power sensor is (nominally) 25 mW. The carrier frequency is 460 MHz and the transmitter is designed for continuous use. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.4.2 Method of measurement | The transmitter is in an environmental chamber adjusted to +55°C. The attenuators and the power sensor are outside the chamber. Prior to the power measurement the total insertion loss of cable and attenuators is measured. The attenuation measurements are done using a generator and a measuring receiver and two 6 dB attenuators with small VSWR. Also the power sensor is calibrated using the built in power reference. The result of the measurement is the power found as the average value of 9 readings from the power meter, corrected for the measured insertion loss. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.4.3 Power meter and sensor module | The power meter uses a thermocouple power sensor module and contains a power reference. Power reference level: Power reference level uncertainty: ±1,2 % power. As nothing is stated about the distribution it is assumed to be rectangular and the standard uncertainty is converted from % power to dB by division with 23,0 (see clause 5.2). Standard uncertainty dB 0,030 0 23 3 2 1 = × = , , u level e j referenc Mismatch whilst measuring the reference: - Reference source VSWR: 1,05 (d): ρreference source = 0,024; - Power sensor VSWR: 1,15 (d): ρload = 0,07. Using formula 6.3 the standard uncertainty of the mismatch is: dB 0,010 5 11 2 % 100 07 0 024 0 : = × × × = , , , u reference j mismatch ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 40 Calibration factors: Calibration factor uncertainty = ±2,3 % power. As nothing is stated about the distribution it is assumed to be rectangular. The standard uncertainty is converted from % power to dB by division with 23,0. standard uncertainty dB 0,058 0 23 3 3 2 = × = , , u factor ion j calibrat Range to range change: Range to range uncertainty (one change) = ±0,5 % power. As nothing is stated about the distribution it is assumed to be rectangular. The standard uncertainty is converted from % power to dB by division with 23,0. standard uncertainty dB 0,006 0 23 3 25 0 = × = , , u change range j Noise and drift is negligible at this power level and can be ignored. Combined standard uncertainty of the power meter and sensor: Using formula 5.1: 2 2 2 : 2 change j range factor ion j calibrat reference j mismatch level e j referenc sensor and c meter u u u u u + + + = dB 0,066 006 0 058 0 010 0 03 0 2 2 2 2 = + + + = , , , , u sensor and c meter |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.4.4 Attenuator and cabling network | Standing wave ratios involved in the attenuation measurement (taken from manufacturers data): - Signal generator: VSWR ≤ 1,5 ρ = 0,200 - Power sensor: VSWR ≤ 1,15 ρ = 0,070 - 6 dB attenuators: VSWR ≤ 1,2 ρ = 0,091 - 10 dB power attenuator: VSWR ≤ 1,3 ρ = 0,130 - 20 dB attenuator: VSWR ≤ 1,25 ρ = 0,111 - Cable: VSWR ≤ 1,2 ρ = 0,091 Nominal attenuations converted to linear values: - 6 dB = S21=S12= 0,500; - 10 dB = S21=S12= 0,316; - 20 dB = S21=S12= 0,100; - 0,3 dB = S21=S12= 0,966 (assumed cable attenuation in the uncertainty calculations). The attenuation measurement is carried out using a signal generator and a measuring receiver. In order to have a low VSWR two 6 dB attenuators with low reflection coefficients are inserted. The measurement of the attenuation in the attenuator and cabling network is carried out by making a reference measurement (figure 10). The measurement receiver reading is "A" dBm. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 41 Then the cables and the attenuators are inserted. First the cable and the 10 dB power attenuator is inserted between the two 6 dB attenuators, and a new reading "B" dBm is recorded (see figure 11). Finally the 20 dB attenuator is inserted between the two 6 dB attenuators, and the reading "C" dBm is recorded (see figure 12). The total attenuation is then ("A"-"B") dB + ("A"-"C") dB. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.4.4.1 Reference measurement | Figure 10 details the components involved in this reference measurement. 6 dB (2) attenuator 6 dB (1) attenuator RF signal generator Power meter and sensor Figure 10: The reference measurement The individual mismatch uncertainties between the various components in figure 10 are calculated using formula 6.3: - the standard uncertainty of the mismatch between the signal generator and 6 dB attenuator (1): dB 0,112 5 11 2 % 100 091 0 2 0 . 6 : = × × × = , , , u att dB to generator j mismatch - the standard uncertainty of the mismatch between the 6 dB attenuator (1) and 6 dB attenuator (2): dB 0,051 5 11 2 % 100 091 0 091 0 2. 6 1. 6: = × × × = , , , u att dB to att dB j mismatch - the standard uncertainty of the mismatch between the 6 dB attenuator (2) and power sensor: dB 0,039 5 11 2 % 100 07 0 091 0 . 6: = × × × = , , , u sensor power to att dB j mismatch - the standard uncertainty of the mismatch between the signal generator and 6 dB attenuator (2): dB 0,028 5 11 2 % 100 5 0 091 0 2 0 2 2. 6 : = × × × × = , , , , u att dB to generator j mismatch - the standard uncertainty of the mismatch between the 6 dB attenuator (1) and power sensor: dB 0,010 5 11 2 % 100 5 0 07 0 091 0 2 1. 6: = × × × × = , , , , u sensor power to att dB j mismatch - the standard uncertainty of the mismatch between the signal generator and power sensor: dB 0,005 5 11 2 % 100 5 0 5 0 07 0 2 0 2 2 : = × × × × × = , , , , , u sensor power to generator j mismatch It can be seen that the mismatch uncertainty between the RF signal generator and the 6 dB attenuator (1) uj generator to 6 dB att 1, and the mismatch uncertainty between the 6 dB attenuator (2) and the power sensor uj 6 dB att. 2 to power sensor, add to both the reference measurement and the measurements with the unknown attenuators inserted. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 42 It is the result of the methodology adopted in annex D that these terms cancel and hence do not contribute to the combined standard uncertainty of the final result. The reference measurement mismatch uncertainty uj mismatch: reference (formula 5.1): 2 2 1. 6 2 2. 6 2 2. 6 1. 6 : sensor power to r j generato sensor power to att dB j att dB to r j generato att dB to att dB j reference j mismatch u u u u u + + + = dB 0,059 005 0 010 0 028 0 051 0 2 2 2 2 : = + + + = , , , , u reference j mismatch NOTE: If the two uncertainties of the generator and the power sensor did not cancel due to the methodology, the calculated reference measurement uncertainty would have been 0,131 dB. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.4.4.2 The cable and the 10 dB power attenuator | Figure 11 shows the clause of the reference set-up which concerns this part of the calculation. 6 dB (2) attenuator RF signal generator 6 dB (1) attenuator Cable 10 dB power attenuator Power meter and sensor Figure 11: The cable and the 10 dB power attenuator The individual uncertainties are calculated using formula 6.3: - the standard uncertainty of the mismatch between the signal generator and 6 dB attenuator (1): dB 0,112 5 11 2 % 100 091 0 2 0 . 6 : = × × × = , , , u att dB to generator j mismatch - the standard uncertainty of the mismatch between the 6 dB attenuator (1) and cable: dB 0,051 5 11 2 % 100 091 0 091 0 1. 6: = × × × = , , , u cable to att dB j mismatch - the standard uncertainty of the mismatch between the cable and 10 dB power attenuator: dB 0,073 5 11 2 % 100 130 0 091 0 . 10 : = × × × = , , , u att dB to cable j mismatch - the standard uncertainty of the mismatch between the 10 dB attenuator and the 6 dB attenuator (2): dB 0,073 5 11 2 % 100 091 0 130 0 2. 6 . 10 : = × × × = , , , u att dB to att dB j mismatch - the standard uncertainty of the mismatch between the 6 dB attenuator (2) and power sensor: dB 0,039 5 11 2 % 100 07 0 091 0 . 6: = × × × = , , , u sensor power to att dB j mismatch - the standard uncertainty of the mismatch between the signal generator and cable: dB 0,028 5 11 2 % 100 5 0 091 0 200 0 2 : = × × × × = , , , , u cable to generator j mismatch - the standard uncertainty of the mismatch between the 6 dB attenuator (1) and 10 dB power attenuator: dB 0,068 5 11 2 % 100 966 0 130 0 091 0 2 . 10 1. 6: = × × × × = , , , , u att dB to att dB j mismatch ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 43 - the standard uncertainty of the mismatch between the cable and 6 dB attenuator (2): dB 0,005 5 11 2 % 100 316 0 091 0 091 0 2 2. 6 : = × × × × = , , , , u att dB to cable j mismatch - the standard uncertainty of the mismatch between the 10 dB power attenuator and the power sensor: dB 0,014 5 11 2 % 100 500 0 070 0 130 0 2 . 10 : = × × × × = , , , , u sensor power to att dB j mismatch - the standard uncertainty of the mismatch between the signal generator and 10 dB power attenuator: dB 0,037 5 11 2 % 100 966 0 500 0 130 0 200 0 2 2 . 10 : = × × × × × = , , , , , u att dB to generator j mismatch - the standard uncertainty of the mismatch between the 6 dB attenuator (1) and 6 dB attenuator (2): dB 0,005 5 11 2 % 100 316 0 966 0 091 0 091 0 2 2 2. 6 1. 6: = × × × × × = , , , , , u att dB to att dB j mismatch - the standard uncertainty of the mismatch between the cable and power sensor: dB 0,001 5 11 2 % 100 500 0 316 0 070 0 091 0 2 2 : = × × × × × = , , , , , u sensor power to cable j mismatch - the standard uncertainty of the mismatch between the signal generator and 6 dB attenuator (2): dB 0,003 5 11 2 % 100 316 0 966 0 500 0 091 0 200 0 2 2 2 2. 6 : = × × × × × × = , , , , , , u att dB to generator j mismatch - the standard uncertainty of the mismatch between the 6 dB attenuator (1) and power sensor: dB 0,001 5 11 2 % 100 500 0 316 0 966 0 070 0 091 0 2 2 2 1. 6: = × × × × × × = , , , , , , u sensor power to att dB j mismatch - the standard uncertainty of the mismatch between the signal generator and power sensor: dB 0,000 5 11 2 % 100 500 0 316 0 966 0 500 0 070 0 200 0 2 2 2 2 : = × × × × × × × = , , , , , , , u sensor power to generator j mismatch The combined mismatch uncertainty when measuring the power level when the cable and the 10 dB power attenuator is inserted is the RSS of all these components except uj mismatch: generator to 6 dB attenuator and uj mismatch: 6 dB attenuator to power sensor: 2 : . 2 1. 6: 10 : ...... sensor power to generator mismatch j cable to att dB mismatch j cable and dB c mismatch u u u + + = dB 142 ,0 000 ,0 001 ,0 .... 073 ,0 051 ,0 2 2 2 2 10 : = + + + + = cable and dB c mismatch u The combined standard uncertainty of the mismatch when measuring the 10 dB attenuator and cable is: 2 : . 2 10 : 10 : reference mismatch c cable and att dB mismatch c t measuremen cable and dB c mismatch u u u + = dB 0,154 059 ,0 142 ,0 2 2 10 : = + = t measuremen cable and dB c mismatch u ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 44 The combined standard uncertainty of the mismatch uc mismatch: 10 dB and cable is 0,154 dB. NOTE: The result would have been the same if only the 6 dominant terms were taken into account. This illustrates that combinations of reflection coefficients separated by attenuations of 10 dB or more can normally be neglected. The exceptions may be in cases where one or both of the reflection coefficients involved are approaching 1,0 - which can be the case with filters or antennas outside their working frequencies. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.4.4.3 The 20 dB attenuator | Figure 12 shows the clause of the set-up which concerns this part of the calculation. Power meter and sensor 6 dB (1) attenuator RF signal generator 20 dB power attenuator 6 dB (2) attenuator Figure 12: The 20 dB attenuator In this part only terms separated by less than 10 dB are taken into account. The individual uncertainties are calculated using formula 6.3: - the standard uncertainty of the mismatch between the signal generator and 6 dB attenuator (1): dB 0,112 5 11 2 % 100 091 0 2 0 . 6 : = × × × = , , , u att dB to generator j mismatch - the standard uncertainty of the mismatch between the 6 dB attenuator (1) and 20 dB attenuator: dB 0,062 5 11 2 % 100 111 0 091 0 . 20 1. 6: = × × × = , , , u att dB to att dB j mismatch - the standard uncertainty of the mismatch between the 20 dB attenuator and 6 dB attenuator (2): dB 0,062 5 11 2 % 100 091 0 111 0 2. 6 . 20 : = × × × = , , , u att dB to att dB j mismatch - the standard uncertainty of the mismatch between the 6 dB attenuator (2) and power sensor: dB 0,039 5 11 2 % 100 07 0 091 0 . 6: = × × × = , , , u sensor power to att dB j mismatch - the standard uncertainty of the mismatch between the signal generator and 20 dB attenuator: dB 0,034 5 11 2 % 100 500 0 111 0 200 0 2 . 20 : = × × × × = , , , , u att dB to generaor j mismatch - the standard uncertainty of the mismatch between the 20 dB attenuator and power sensor: dB 0,012 5 11 2 % 100 500 0 070 0 111 0 2 . 20 : = × × × × = , , , , u sensor power to att dB j mismatch The rest of the combinations are not taken into account because the insertion losses between them are so high, that the values are negligible: - 6 dB attenuator (1) and 6 dB attenuator (2); - signal generator and 6 dB attenuator (2); ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 45 - 6 dB attenuator (1) and measuring receiver; - signal generator and measuring receiver. The combined standard uncertainty of the mismatch when measuring the attenuation of the 20 dB attenuator is the RSS of these 4 individual standard uncertainty values: dB 095 ,0 012 ,0 034 ,0 062 ,0 062 ,0 2 2 2 2 20 : = + + + = dB j mismatch u The combined standard uncertainty of the mismatch involved in the 20 dB attenuator measurement is: 2 : 2 20 : 20 : reference c mismatch dB c mismatch t measuremen dB c mismatch u u u + = dB 0,112 059 0 095 0 2 2 20 : = + = , , u t measuremen dB c mismatch NOTE: If the two 6 dB attenuators had not been inserted, the result would have been 0,265 dB. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.4.4.4 Instrumentation | Linearity of the measuring receiver is ±0,04 dB (from manufacturers data) as nothing is said about the distribution, a rectangular distribution in logs is assumed and the standard uncertainty is calculated: standard uncertainty dB 0,023 3 04 0 = = , u linearity j receiver |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.4.4.5 Power and temperature influences | Temperature influence: 0,0001 dB/degree (from manufacturers data), which is negligible, the power influence for the 10 dB attenuator is 0,0001 dB/dB × Watt (from manufacturers data) which gives 0,0001 × 25 × 10 = 0,025 dB as nothing is said about the distribution, a rectangular distribution in logs is assumed and the standard uncertainty is calculated: dB 0,014 3 025 0 10 = = , u dB fluence in j power The power influence for the 20 dB attenuator is 0,001 dB/dB × Watt (from manufacturers data) which gives 0,001 × 2,5 × 20 = 0,05 dB as nothing is said about the distribution, a rectangular distribution in logs is assumed and the standard uncertainty is calculated: dB 0,028 3 050 0 20 = = , u dB nfluence i j power |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.4.4.6 Collecting terms | 10 dB attenuator and cabling network uncertainty: 2 10 2 2 10 dB nfluence i j power linearity j receiver c mismatch cable and attenuator dB c u u u u + + = dB 0,156 014 ,0 023 ,0 154 ,0 2 2 2 10 = + + = cable and attenuator dB c u ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 46 20 dB attenuator and cabling network uncertainty: 2 20 2 2 20 dB nfluence i j power linearity j receiver c mismatch attenuator dB c u u u u + + = dB 0,122 028 ,0 04 ,0 112 ,0 2 2 2 20 = + + = attenuator dB c u The combined standard uncertainty of the attenuator and cabling network uncertainty: 2 20 2 10 attenuator dB c cable and attenuator dB c cabling and ion c attenuat u u u + = dB 0,201 122 ,0 160 ,0 2 2 = + = cabling and ion c attenuat u |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.4.5 Mismatch during measurement | Standing wave ratios involved in the power measurement: - EUT: ρ = 0,500; - Power sensor: VSWR ≤ 1,15 ρ = 0,070; - 10 dB power attenuator: VSWR ≤ 1,3 ρ = 0,130; - 20 dB attenuator: VSWR ≤ 1,25 ρ = 0,111; - Cable: VSWR ≤ 1,2 ρ = 0,091. The mismatch uncertainties are calculated using formula 6.3 for the individual mismatch uncertainties between: - the standard uncertainty of the mismatch between the EUT and cable: dB 0,112 5 11 2 % 100 091 0 200 0 : = × × × = , , , u cable to EUT j mismatch - the standard uncertainty of the mismatch between the cable and 10 dB power attenuator: dB 0,073 5 11 2 % 100 130 0 091 0 . 10 : = × × × = , , , u att dB to cable j mismatch - the standard uncertainty of the mismatch between the 10 dB power attenuator and 20 dB attenuator: dB 0,089 5 11 2 % 100 111 0 130 0 . 20 . 10 : = × × × = , , , u att dB to att dB j mismatch - the standard uncertainty of the mismatch between the 20 dB attenuator and power sensor: dB 0,048 5 11 2 % 100 070 0 111 0 . 20 : = × × × = , , , u sensor power to att dB j mismatch - the standard uncertainty of the mismatch between the EUT and 10 dB power attenuator: dB 0,149 5 11 2 % 100 966 0 130 0 200 0 2 . 10 : = × × × × = , , , , u att dB to EUT j mismatch - the standard uncertainty of the mismatch between the cable and 20 dB attenuator: dB 0,058 5 11 2 % 100 316 0 966 0 111 0 091 0 2 2 . 20 : = × × × × × = , , , , , u att dB to cable j mismatch ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 47 - the standard uncertainty of the mismatch between the EUT and 20 dB attenuator: dB 0,013 5 11 2 % 100 316 0 966 0 111 0 200 0 2 2 . 20 : = × × × × × = , , , , , u att dB to EUT j mismatch The rest of the combinations: - 10 dB attenuator to power sensor; - cable to power sensor; - EUT to power sensor; are neglected. The combined standard uncertainty of the mismatch during the measurement is the RSS of the individual components: dB 0,232 013 0 058 ,0 149 0 048 0 089 0 073 0 112 0 2 2 2 2 2 2 2 = + + + + + + = , , , , , , uc mismatch In the case where all contributions are considered as independent. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.4.6 Influence quantities | The two influence quantities involved in the measurement are ambient temperature and supply voltage. Temperature uncertainty: ±1,0°C. Supply voltage uncertainty: ±0,1 V. Uncertainty caused by the temperature uncertainty: Dependency function (from TR 100 028-2 [8], table F.1): Mean value 4 %/°C and standard deviation: 1,2 %/°C. Standard uncertainty of the power uncertainty caused by ambient temperature uncertainty (formula 5.2; see also clause D.4.2.1 of TR 100 028-2 [8]). ( ) dB 0,105 2 1 0 4 3 0 1 0 23 1 2 2 2 / = + = , , , , u e temperatur j power Uncertainty caused by supply voltage uncertainty: Dependency function (from TR 100 028-2 [8], table F.1): Mean: 10 %/V and standard deviation: 3 %/V power, Standard uncertainty of the power uncertainty caused by power supply voltage uncertainty (formula 5.2; see also clause D.4.2.1 in TR 100 028-2 [8]). ( ) dB 0,026 3 10 3 1 0 0 23 1 2 2 2 / = + = , , u voltage j power dB 0,108 026 0 105 0 2 2 2 / 2 / = + = + = , , u u u voltage power j e temperatur power j nfluence ic |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.4.7 Random | The measurement was repeated 9 times The following results were obtained (before correcting for cabling and attenuator network insertion loss): - 21,8 mW, 22,8 mW, 23,0 mW, 22,5 mW, 22,1 mW, 22,7 mW, 21,7 mW, 22,3 mW, 22,7 mW. The two sums X and Y are calculated: - X = the sum of the measured values = 201,6 mW; - Y = the sum of the squares of the measured values = 4 517,5 mW2; ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 48 - = − − = − − = 1 9 9 6 201 5 4517 1 2 2 , , n n X Y uc random 0,456 mW (formula 5.5); - Mean value = 22,4 mW. As the result is obtained as the mean value of 9 measurements the standard uncertainty (converted to dB by division with 23,0) of the random uncertainty is: dB 0,089 0 23 100 4 22 456 0 = × = , , , uc random NOTE: It is important to try and identify whether this value corresponds to the effect of other uncertainties, already taken into account in the calculations (e.g. uncertainties due to the instrumentation), or whether this corresponds to a genuine contribution (in which case it has to be combined with all the other contributions)… obviously, there are uncertainties in the measurements, so it has to be expected that performing the same measurement a number of times may provide a set of different results. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.4.8 Expanded uncertainty | The combined standard uncertainty for the carrier power measurement is the RSS of all the calculated part standard uncertainties: 2 2 2 2 2 c random fluence c in c mismatch cabling and ion c attenuat sensor and c meter power carrier c u u u u u u + + + + = dB 0,344 089 0 108 0 232 0 201 0 066 0 2 2 2 2 2 = + + + + = , , , , , u power carrier c The expanded uncertainty is ±1,96 × 0,344 dB = ±0,67 dB at a 95 % confidence level, should the distribution corresponding to the combined uncertainty be Normal (this is further discussed in clause D.5.6 in TR 100 028-2 [8]). The dominant part of this expanded uncertainty is mismatch uncertainty. In the calculations all the mismatch uncertainties were based on manufacturers data, which are normally very conservative. The relevant reflection coefficients could be measured by means of a network analyser or reflection bridge. This would probably give lower reflection coefficients thereby reducing the overall uncertainty. NOTE: In the case where these coefficient are measured a number of times, under conditions where it can be considered that the measurements are independent, then the comments found in clauses 5.3.1 and 6.4.7 may be relevant. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 49 6.5 Uncertainty calculation for measurement of a receiver (Third order intermodulation) Before starting we need to know the architecture and the corresponding noise behaviour of the receiver. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.5.1 Noise behaviour in different receiver configurations | The effect of noise on radio receivers is very dependant on the actual design. A radio receiver has (generally) a front end and demodulation stages according to one of the possibilities presented in figure 13. This simplified diagram (for AM and FM/PM systems) illustrates several possible routes from the front end to the "usable output". Front end IF Demodulator Demodulator FM / PM Sub carrier FM / PM Sub carrier FM / PM Sub carrier AM modulated data Speech Data Speech Data Speech Data Sub carrier mod Direct mod FM / PM AM Figure 13: Possible receiver configurations The Amplitude Modulation route involves a 1:1 conversion after the front end and the amplitude demodulation information is available immediately (analogue) or undergoes data demodulation. The frequency modulation/phase modulation route introduces an enhancement to the noise behaviour in non-linear (e.g. FM/PM) systems compared to linear (e.g. AM) systems, see figure 14, until a certain threshold or lower limit (referred to as the knee-point) is reached. Below this knee-point the demodulator output signal to noise ratio degrades more rapidly for non-linear systems than the linear system for an equivalent degradation of the carrier to noise ratio, this gives rise to two values for the slope: one value for C/N ratios above the knee and one value for C/N ratios below the knee. A similar difference will occur in data reception between systems which utilize AM and FM/PM data. Therefore "Noise Gradient" corresponds to several entries in TR 100 028-2 [8], table F.1. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 50 Knee point Linear system (AM) Non-linear system (FM/PM) Better > Better > S/N C/N Figure 14: Noise behaviour in receivers |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.5.2 Sensitivity measurement | The sensitivity of a receiver is usually measured as the input RF signal level which produces a specific output performance which is a function of the base band signal-to-noise ratio in the receiver. This is done by adjusting the RF level of the input signal at the RF input of the receiver. What is actually done is that the RF signal-to-noise ratio at the input of the receiver is adjusted to produce a specified signal-to-noise ratio dependant behaviour at the output of the receiver, i.e. SINAD, BER, or message acceptance. An error in the measurement of the output performance will cause a mis adjustment of the RF level and thereby the result. In other words any uncertainty in the output performance is converted to signal-to-noise ratio uncertainty at the input of the receiver. As the noise does not change it causes an uncertainty in the adjusted level. For an analogue receiver, the dependency function to transform the SINAD uncertainty to the RF input level uncertainty is the slope of the noise function described above in clause 6.5.1 and depends on the type of carrier modulation. The dependency function involved when measuring the sensitivity of an FM/PM receiver is the noise behaviour usually below the knee-point for a non-linear system, in particular in the case of data equipment. This function also affects the uncertainty when measuring sensitivity of an FM/PM based data equipment. This dependency function has been empirically derived at 0,375 dB RF i/p level / dB SINAD associated with a standard uncertainty of 0,075 dB RF i/p level / dB SINAD and is one of the values stated in TR 100 028-2 [8], table F.1. If the receiver is for data the output performance is a specified BER. BER measurements are covered by clause 6.6. In some standards the sensitivity is measured as the output performance at a specified input level. In this case the dependency functions converting input level uncertainty to output performance uncertainty are the inverse of the functions previously described. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 51 |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.5.3 Interference immunity measurements | Interference immunity (i.e. co-channel rejection, adjacent channel rejection) is measured by adjusting the RF level of the wanted signal to a specified value. Then the RF level of the interfering signal is adjusted to produce a specified performance at the output of the receiver. The interfering signal is normally modulated. Therefore for measurement uncertainty purposes it can be regarded as white noise in the receiving channel. The uncertainty analysis is therefore covered by clause 6.5.2. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.5.4 Blocking and spurious response measurements | These measurements are similar to interference immunity measurements except that the unwanted signal is without modulation. Even though the unwanted signal (or the derived signal in the receive channel caused by the unwanted signal) can not in every case be regarded as white noise, the present document does not distinguish. The same dependency functions are used. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.5.5 Third order intermodulation | When two unwanted signals X and Y occur at frequency distance d(X) and 2d(Y) from the receiving channel a disturbing signal Z is generated in the receiving channel due to non linearities in filters, amplifiers and mixers. The physical mechanism behind the intermodulation is the third order component of the non-linearity of the receiver: K × X3 When two signals - X and Y - are subject to that function, the resulting function will be: - K(X + Y)3 = K(X3 + Y3 + 3X2Y + 3XY2), where the component Z = 3X2Y is the disturbing intermodulation product in the receiving channel. If X is a signal Ix sin(2π(fo + d)t) and Y is a signal Iy sin(2π(fo + 2d)t), the component: - Z = K × 3X2Y will generate a signal having the frequency fo and the amplitude K × 3Ix2Iy. (A similar signal Z' = 3XY2 is generated on the other side of the two signals X and Y, as shown in figure 15). The predominant function is a third order function: Iz = Ic + 2Ix + Iy (6.4) where lz is the level of the intermodulation product Z, Ic is a constant, Ix and Iy are the levels of X and Y. All terms are logarithmic. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.5.5.1 Measurement of third order intermodulation | The measurement is normally carried out as follows. Three signal generators are connected to the input of the EUT. Generator 1 is adjusted to a specified level at the receiving frequency fo (the wanted signal W). Generator 2 is adjusted to frequency fo+ δ (unwanted signal X) and generator 3 is adjusted to frequency fo+ 2δ (unwanted signal Y). The level of X and Y (Ix and Iy) are maintained equal during the measurement. Ix and Iy are increased to level A which causes a specified degradation of AF output signal (SINAD) or a specific bit error ratio (BER) or a specific acceptance ratio for messages. Both the SINAD, BER and message acceptance ratio are a function of the signal-to-noise ratio in the receiving channel. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 52 The level of the wanted signal W is Aw (see figure 15). The measured result is the difference between the level of the wanted signal Aw and the level of the two unwanted signals A. This is the ideal measurement. A A z Z X Y Z' Level A w W fo fo+δ fo+2δ Figure 15: Third order intermodulation components When looked upon in logarithmic terms a level change δIx dB in X will cause a level change of 2 × δIx dB in Z, and a level change δIy dB in Y will cause the same level change δIz dB in Z. If the levels of both X and Y are changed by δI dB, the resulting level change of Z is 3 × δI dB. Since X is subject to a second order function, any modulation on X will be transferred with double uncertainty to Z (see also annex D, clauses D.3.2, D.3.4 and D.5 in TR 100 028-2 [8]), whereas the deviation of any modulation on Y will be transferred unchanged to Z. Therefore, as Y is modulated in the measurement, the resulting modulation of Z will be the same as with Y. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.5.5.2 Uncertainties involved in the measurement | The predominant uncertainty sources related to the measurement are the uncertainty of the levels of the applied RF signals and uncertainty of the degradation (the SINAD, BER, or message acceptance measurement). The problems about the degradation uncertainty are exactly the same as those involved in the co-channel rejection measurement if the intermodulation product Z in the receiving channel is looked upon as the unwanted signal in this measurement. Therefore the noise dependency is the same, but due to the third order function the influence on the total uncertainty is reduced by a factor 3 (see clauses D.3.2 and D.5 in TR 100 028-2 [8]). It is in the following assumed that the distance to the receiver noise floor is so big that the inherent receiver noise can be disregarded. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 53 |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.5.5.2.1 Signal level uncertainty of the two unwanted signals | A is the assumed level of the two unwanted signals (the indication of the two unwanted signal generators corrected for matching network attenuations): - Ax is the true level of X and Ay is the true level of Y. (Ax is A + δx and Ay is A + δy) see figure 16; - Az is the level of Z (the same as in the ideal measurement). A A z Ax Ay Z X Y Z' Level δX δY A w W fo fo+δ fo+2δ Figure 16: Level uncertainty of two unwanted signals If Ax and Ay were known the correct measuring result would be obtained by adjusting the two unwanted signals to the level At (true value) which still caused the level Az of Z. If there is an error δx of the level of signal X, the error of the level of the intermodulation product will be 2 × δx (see also clauses D.3 and D.5 in TR 100 028-2 [8]); to obtain the wanted signal-to-noise ratio the two unwanted levels must be reduced by 2 × δx/3. In other words the dependency function of generator X is 2/3. In the same way if there is an error δy of the level of signal Y, the error of the level of the intermodulation product will be δy; to obtain the wanted signal-to-noise ratio the two unwanted signals must be reduced by δy/3. In other words the dependency function of generator Y is 1/3. When looking at the problem in linear terms, the dependency functions are valid for small values of δx and δy due to the fact that the higher order components of the third order function can be neglected. δx and δy are the relative RF level uncertainties at the input of the EUT. They are combinations of signal generator level uncertainty, matching network attenuation uncertainty and mismatch uncertainties at the inputs and the output of the matching network. The standard uncertainties of the levels of X and Y are uj x and uj y. The standard uncertainty uj unwanted signals related to the uncertainty caused by level uncertainty of the two unwanted signals is thus (see also clause D.3.2.3 of TR 100 028-2 [8]): j y j x signals j unwanted u + u = u 3 1 3 2 2 2 (6.5) ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 54 |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.5.5.2.2 Signal level uncertainty of the wanted signal | Under the assumption that equal change of both the level of the wanted signal and the intermodulation product will cause no change of the SINAD, (or the BER, or the message acceptance) the error contribution from the uncertainty of the level of the wanted signal can be calculated. If there is an error δw on the wanted signal, the two unwanted signal levels must be adjusted by 1/3 × δw to obtain the wanted signal-to-noise ratio. The dependency function of generator W is therefore 1/3 and assuming the same types of uncertainties as previously the standard uncertainty, uj wanted signal, is (see clause D.3.2.3 of TR 100 028-2 [8]): signals j unwanted signal j wanted u = u 3 1 (6.6) |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.5.5.3 Analogue speech (SINAD) measurement uncertainty | Sensitivity is normally stated as an RF input level in conducted measurements. For analogue systems this is stated as at a specified SINAD value. For an analogue receiver, the dependency function to transform the SINAD uncertainty to the RF input level uncertainty is the slope of the noise function described above in clause 6.5.1 and depends on the type of carrier modulation. The dependency function involved when measuring the sensitivity of an FM/PM receiver is the noise behaviour usually below the knee-point for a non-linear system, in particular in the case of data equipment. This function also affects the uncertainty when measuring sensitivity of an FM/PM based data equipment. This dependency function has been empirically derived at 0,375 dB RF i/p level / dB SINAD associated with a standard uncertainty of 0,075 dB RF i/p level / dB SINAD and is one of the values stated in TR 100 028-2 [8], table F.1. The SINAD measurement uncertainty also contributes to the total measurement uncertainty. If the receiver is working beyond the demodulator knee point any SINAD uncertainty corresponds to an equal uncertainty (in dB) of the signal-to-noise ratio. If the receiver is working below the knee point the corresponding uncertainty of the signal-to-noise ratio will be in the order of 1/3 times the SINAD uncertainty (according to TR 100 028-2 [8], table F.1). Any signal-to-noise ratio uncertainty causes 1/3 times that uncertainty in the combined uncertainty: the unwanted signal levels must be adjusted by 1/3 of the signal-to-noise ratio error to obtain the correct value. Therefore if the receiver is working above the knee point the SINAD dependency function is 1/3, and if the receiver is working below the knee point the dependency function is in the order of 1/9. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.5.5.4 BER and message acceptance measurement uncertainty | Any BER (or message acceptance) uncertainty will influence the total uncertainty by the inverse of the slope of the appropriate BER function at the actual signal-to-noise ratio. As the BER function is very steep, the resulting dependency function is small, and it is sufficient to use the differential coefficient as an approximation. If the signalling is on a sub carrier, the relation between the signal-to-noise ratio of the sub carrier must be dealt with in the same way as with other receiver measurements. See clause 6.6.3. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.5.5.5 Other methods of measuring third order intermodulation | Some test specifications specify other methods of measuring the intermodulation rejection. The measured result is the SINAD, BER, or message acceptance at fixed test signal levels. This is the case with some digital communication equipment like DECT and GSM. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 55 In these measurements the uncertainty must be calculated in 3 steps: 1) the uncertainty of the resulting signal-to-noise ratio is calculated; 2) this uncertainty is then applied to the appropriate SINAD, BER, or message acceptance function; 3) and then combined with the measurement uncertainty of the SINAD, BER, or message acceptance measurement. The uncertainty of the signal-to-noise ratio due to uncertainty of the level of the test signals is: ( ) 2 2 2 2 j w j y j x j SNR u u u = u + + This uncertainty is then transformed to the measured parameter. If the measured value is a SINAD value and the receiver is working beyond the knee point the SINAD uncertainty is identical, but if the receiver is working below the knee point the dependency function is in the order of 3,0. If the measurand is a BER or a message acceptance, the dependency function is too non linear to be regarded as a first order function. The total uncertainty must then be calculated as described in clause 6.6.4.3. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.6 Uncertainty in measuring continuous bit streams | |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.6.1 General | If an EUT is equipped with data facilities, the characteristic used to assess its performance is the Bit Error Ratio (BER). The BER is the ratio of the number of bits in error to the total number of bits in a received signal and is a good measure of receiver performance in digital radio systems just as SINAD is a good measure of receiver performance in analogue radios. BER measurements, therefore, are used in a very similar way to SINAD measurements, particularly in sensitivity and immunity measurements. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.6.2 Statistics involved in the measurement | Data transmissions depend upon a received bit actually being that which was transmitted. As the level of the received signal approaches the noise floor (and therefore the signal to noise ratio decreases), the probability of bit errors (and the BER) increases. The first assumption for this statistical analysis of BER measurements is that each bit received (with or without error) is independent of all other bits received. This is a reasonable assumption for measurements on radio equipment, using binary modulation, when measurements are carried out in steady state conditions. If, for instance, fading is introduced, it is not a reasonable assumption. The measurement of BER is normally carried out by comparing the received data with that which was actually transmitted. The statistics involved in this measurement can be studied using the following population of stones: one black and (1/BER)-1 white stones. If a stone is taken randomly from this population, its colour recorded and the stone replaced N times, the black stone ratio can be defined as the number of occurrences of black stones divided by N. This is equivalent to measuring BER. The statistical distribution for this measurement is the binomial distribution. This is valid for discrete events and gives the probability that x samples out of the N stones sampled are black stones (or x bits out of N received bits are in error) given the BER: ( ) ( ) ( ) x N x x BER BER ! x N x! N! P − − × − = 1 (6.7) ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 56 The mean value of this distribution is BER × N and the standard deviation is: ( ) N BER BER × − × 1 (6.8) and for large values of N the shape of the distribution approximates a Gaussian distribution. Normalizing the mean value and standard uncertainty (by dividing by N) gives: Mean value = BER (6.9) ( ) N BER BER u BER j − = 1 (6.10) From these two formulas it is easy to see that the larger number of bits, the smaller the random uncertainty, and the relation between number of bits and uncertainty is the same as for random uncertainty in general. By means of formula 6.11 it is possible to calculate the number of bits needed to be within a specific uncertainty. For example: A BER in the region of 0,01 is to be measured. a) If the standard uncertainty, due to the random behaviour discussed above, is to be 0,001, then the number of bits to be compared, N, in order to fulfil this demand is calculated from the rearranged formula (6.11). ( ) 900 9 001 0 99 0 01 0 1 2 2 = × = − = , , , u BER BER N jBER b) If the number of bits compared, N, is defined, e.g. 2 500 then the standard uncertainty is given directly by formula (6.11). ( ) 002 0 2500 01 0 1 01 0 , , , u BER j = − = As stated earlier the binomial distribution can be approximated by a Normal distribution. This is not true when the BER is so small that only a few bit errors (< 10) are detected within a number of bits. In this case the binomial distribution is skewed as the p (BER < 0) = 0. Another problem that occurs when only few bit errors are detected, and the statistical uncertainty is the dominant uncertainty (which does not happen in PMR measurements, but it does, due to the method, occur in DECT and GSM tests) is that the distribution of the true value about the measured value can be significantly different from an assumed Normal distribution. 6.6.3 Calculation of uncertainty limits when the distribution characterizing the combined standard uncertainty cannot be assumed to be a Normal distribution In the calculations of uncertainty there is usually no distinction between the distribution of a measured value about the true value, and the distribution of the true value about a measured value. The assumption is that they are identical. This is true in the cases where the standard uncertainty for the distribution of the measured value about the true value is independent of the true value - which usually is the case. But if the standard uncertainty is a function of the true value of the measurand (not the measured value), the resulting distribution of the measurement uncertainty will not be a Normal distribution even if the measured value about the true value is. This is illustrated by the following (exaggerated) example. A DC voltage is to be measured. We assume that there is only one uncertainty contribution which comes from the voltmeter used for the measurement. In the manufacturers data sheet for the voltmeter it is stated that the measured value is within ±25 % of the true value. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 57 If the true value is 1,00 V then the measured value lies between 0,75 V and 1,25 V. However, if the measured value is 0,75 V and the true value is still 1,00 V corresponding to 1,3333 times the measured value. Similarly, If the measured value is 1,25 V and the true value is still 1,0 V this corresponds to 0,8 times the measured value. Therefore the limits are asymmetric for the true value about the measured value (-20 % and +33,33 %). When looking at the standard deviations, the error introduced is small. In the previous example the standard deviation of the measured value about the true value is 14,43 %. The standard deviation of the related true value about the measured value is 15,36 %. As the difference is small, and the distribution of the measured value about the true value is based on an assumption anyway, the present document suggests that it can be used directly. NOTE: The average value, however, is no longer zero, but in this case is approximately 4,4 %. Alternatively, also in this example, xt is the true value and xm is the measured value. Any parameter printed in square brackets, e.g. [xm], is considered to be constant. The distribution of the measured value xm about the true value xt is given by the function p (xm, [xt]). Based on this function the distribution p1 (xt, [xm]) of the true value xt about the measured value xm can be derived. The intermediate function is p (xt, [xm]) which is the same as the previous; the only difference being that xt is the variable and xm is held constant. This function is not a probability distribution as the integral from -∞ to +∞ is not unity. To be converted to the probability function p1 (xt, [xm]) it must be normalized. Therefore: [ ] ( ) [ ] ( ) [ ] ( ) ∫ ∞ ∞ − = dx x x, p x , x p x , x p m m t m t 1 (6.11) As this distribution is not Normal, the uncertainty limits must be found by other means than by multiplication with a coverage factor from Student's t-distribution. How the actual limits are calculated in practise depends on the actual distribution. An example: If the true BER of a radio is 5 × 10-6 and the BER is measured over 106 bits, the probability of detecting 0 bits is 0,674 %. On the other hand if the BER in a measurement is measured as 5 × 10-6 the true value cannot be 0. If the uncertainty calculations are based on the assumption of a Gaussian distribution, the lower uncertainty limit becomes negative (which of course does not reflect reality, and provides the evidence that not all distributions are Normal!). The standard uncertainty based on the measured value 3,0 × 10-6: ( ) 6 6 6 6 10 1,73 10 10 0 3 1 10 0 3 − − − × = × − × = , , u j The expanded uncertainty is ±1,96 × 1,73 × 10-6 = ±3,39 × 10-6 at a 95 % confidence level. The correct distribution p1 (xt) is the continuous function in figure 17. NOTE: The true value is not BER, but number of bit errors, where BER= (bit errors/number of bits tested)). The binomial function p (xm) based on the true value = 3 bit errors (corresponding to BER = 3 × 10-6) is the discrete function shown. The distribution p (xt) (based on the binomial distribution with 3 bit errors and 106 bits tested): ( ) ( ) % x x k x p t t t 3 10 6 3 6 6 10 1 10 100 − − × × × = where ( ) 17 6 6 10 67 1 3 10 3 10 × = − × = , ! ! ! k ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 58 The integral from -∞ to +∞ of p (xt) is very close to 1. Therefore p (xt) is a good approximation to the correct distribution p1 (xt). By means of numerical methods the 95 % error limits are found to be +5,73 and -1,91 corresponding to +5,73 × 10-6 and -1,91 × 10-6. Figure 17 shows the discrete distribution giving the probabilities of measuring from 0 to 14 bit errors when the true value is 3 bit errors corresponding to BER = 3 × 10-6, and the continuous distribution giving the probability function for the true value when the measured value is 3 bit errors corresponding to BER = 3 × 10-6. 0 5 10 15 20 25 % bit errors 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 p (x ) 1 t Figure 17: BER uncertainty |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.6.4 BER dependency functions | As in SINAD measurements, the BER of a receiver is a function of the signal to noise ratio of the RF signal at the input of the receiver. Several modulation and demodulation techniques are used in data communication and the dependency functions are related to these techniques. This clause covers the following types of modulation: - coherent modulation/demodulation of the RF signal; - non coherent modulation/demodulation of the RF signal; - FM modulation. The following assumes throughout that the data modulation uncertainty combines linearly to the carrier to noise ratio uncertainty. The uncertainty calculations are based on ideal receivers and demodulators where correctly matched filters are utilized. ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 59 The characteristics of practical implementations may differ from the theoretical models thereby having BER dependency functions which are different from the theoretical ones. The actual dependency functions can, of course, be estimated individually for each implementation. This, however, would mean additional measurements. Instead the theoretically deduced dependency functions may be used in uncertainty calculations. |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.6.4.1 Coherent data communications | Coherent demodulation techniques are techniques which use absolute phase as part of the information. Therefore the receiver must be able to retrieve the absolute phase from the received signal. This involves very stable oscillators and sophisticated demodulation circuitry, but there is a gain in performance under noise conditions compared to non coherent data communication. Coherent demodulation is used, for example, in the GSM system with Gaussian Minimum Shift Keying (GMSK). |
9fd31b6289d69846b992d5f7b2e5698e | 100 028-1 | 6.6.4.2 Coherent data communications (direct modulation) | The BER as a function of SNRb, the signal to noise ratio per bit for coherent binary systems is: BER (SNRb) = 0,5 × erfc (√SNRb) (6.12) where erfc (x) is defined as: dt e erfc(x)= -t x 2 2 ∫ ∞ π (6.13) It is not possible to calculate the integral part of (6.11) analytically, but the BER as a function of the signal to noise ratio is shown in figure 18 together with the function for non coherent binary data communication. There are different types of coherent modulation and the noise dependency of each varies, but the shape of the function remains the same. The slope, however, is easily calculated and, although it is negative, the sign has no meaning for the following uncertainty calculations: ( ) ( ) b SNR b b e SNR SNR d BER d − × × = π 2 1 (6.14) For the purpose of calculating the measurement uncertainty, this can be approximated: ( ) ( ) BER , SNR d BER d b × ≈2 1 (6.15) If the aim is to transform BER uncertainty to level uncertainty - which is the most likely case in PMR measurements, the inverse dependency function must be used (the result is in percentage power terms as it is normalized by division with SNRb*): ( ) ( ) % SNR BER , u % SNR SNR d BER d u u b jBER b b jBER nty uncertai BER to due j level 100 2 1 100 * * × × × ≈ × = (6.16) The SNRb* is a theoretical signal to noise ratio read from figure 19. It may not be the signal to noise ratio at the input of the receiver but the slope of the function is assumed to be correct for the BER measured. For example: The sensitivity of a receiver is measured. The RF input level to the receiver is adjusted to obtain a BER of 10-2. The measured result is the RF level giving this BER. The BER is measured over a series of 25 000 bits. The resulting BER uncertainty is then calculated using formula (6.11): ( ) 4 10 29 6 25000 01 0 1 01 0 − × = − = , , , u j BER ETSI ETSI TR 100 028-1 V1.4.1 (2001-12) 60 The uncertainty of the RF signal at the input is 0,7 dB (uj). The signal to noise ratio giving this BER is then read from figure 18: SNRb*(0,01) = 2,7 and the dependency function at this level is: ( ) ( ) ( ) 2 2 10 2 1 10 1 2 1 2 1 7 2 − − × = × × = × = , , BER , SNR d , BER d b The BER uncertainty is then transformed to level uncertainty using formula (6.16): power % 1,95 100 7,2 10 2 1 10 29 6 2 4 = × × × × = − − % , , u j level dB 0,085 dB 0 23 95 1 = ≈ , , .... 085 0 7 0 2 2 + + = , , u level j RF There is an additional uncertainty component due to resolution of the readout of the measured BER. If the RF input level has been adjusted to give a reading of 0,01 and the resolution of the BER meter is 0,001 the correct lies between 0,0095 and 0,0105 with equal probability. The standard deviation is therefore 4 3 10 89 2 3 10 5 0 − − × = × = , , u lution j BER reso This standard deviation is then by means of formula 6.16 converted to level uncertainty: % SNR ) d(SNR d(BER) u u b b lution j BER reso esolution e to BER r j level du 100 × × = dB , dB , , % , % , , , , u esolution e to BER r j level du 004 0 0 23 089 0 089 0 100 7 2 01 0 2 1 10 289 0 4 = ≈ = × × × × = − The total uncertainty of the sensitivity level is then: dB 71 ,0 7,0 004 ,0 085 ,0 2 2 2 2 2 2 = + + = + = j resolution BER to due level j level j level c RF u u u u As can be seen the BER statistical uncertainty and the BER resolution only plays a minor role. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.