Unified Matrix Theory of Lumped and Distributed
Directional Couplers M.A. Murray-Lasso 39

A Statistical Analysis of On-Off Patterns in Sixteen
Conversations P. T. Brady 73

Dielectric Loaded and Covered Rectangular Waveguide
Phased Arrays V. Galindo and C. P. Wu 93

Surface-Wave Effects on Dielectric Sheathed Phased
Arrays of Rectangular Waveguides C. P. Wu and V. Galindo 117

ADVISORY BOARD
P, A. GORMAN, President, Western Electric Company
J.B. FISK, President, Bell Telephone Laboratories

A. 8. ALSTON, Executive Vice President,
American Telephone and Telegraph Company

The usual design technique for waveguide band-rejection filters uses
narrow-band approximations and thus discrepancies generally exist be-
tween the designed and measured response, particularly in the fairly wide
passband. Nevertheless, this design technique has been used because of
its sumplicity and because the filter configurations obtained are relatively
sumple. Lately, a new design technique using transmission line synthesis
became available which, theoretically, would yield the desired response.
However, the physical realization results in a complicated configuration
which leads to certain practical problems. This paper presents a modified
technique which simplifies the structure without sacrificing performance.
With this modification the design procedure becomes very simple and many
of the practical problems can be avoided. This paper gives precise design
information and convenient design formulas. Furthermore, it shows that
excellent agreement between the designed and measured response can be
achieved. |

The microwave waveguide band-rejection filter (BRF) now used in
many radio systems has many undesirable features. The designer finds
that the actual bandwidth is consistently narrower than the designed
value, and that the filter has a unique passband VSWR behavior
which becomes worse as the frequency goes farther away from the

stopband. This becomes quite a severe problem in filters designed
for the high frequency or low frequency channels of the band. (For
example, in the 4 GHz band,* the return loss of a filter with the stop-
band at 3710 MHz becomes progressively poorer as the frequency
approaches 4190 MHz, and vice versa). In addition, the midband
frequency of the VSWR curve and of the corresponding delay curve
is found to differ from the midband frequency of the insertion loss
curve at which the filter is tuned. The offset of a typical 3-cavity
maximally flat BREF at the 4 GHz band with 3 dB points at +17 MHz
can be as much as 2 MHz. Such an uncontrolled shift causes extreme
difficulties in the delay equalization of a radio system.

The present BRF design is based on the lumped-element low-pass
prototype filter design technique: After making a proper frequency
transformation, the loaded Q of each cavity can be computed. The
cavities are then separated by waveguide lengths which give an ef-
fective spacing of an odd multiple of quarter wavelengths at the mid-
band frequency, which is a standard technique to realize ladder filters
in waveguide. It is desirable to keep the spacings as small as possible,
but, in order to avoid higher order mode coupling between cavities, it
has been determined that a three-quarter wavelength spacing is re-
quired.

In a recent investigation, it was determined that neglecting the
frequency-dependent phase shift of the connecting lines results in both
the narrower bandwidth and the problem of high VSWR in the pass-
band. A rigorous analysis was carried out for a three-cavity BRF.
The result shows that when the frequency dependence of the phase
shift of the connecting lines is accounted for, an exact control of band-
width is possible and the passband VSWR is improved; but the desired
characteristic of the passband VSWR cannot be obtained as long as
the present form of construction is retained.

Various techniques for the exact synthesis of transmission line filters
have been available for many years.? However, they often are found
to be not too practical for the design of waveguide filters because of
the large number of changes of characteristic impedances usually re-
quired. There is already a sufficiently large number of discontinuities
in a waveguide filter which are necessary to form the cavities. Addi-
tional discontinuities with their associated fringing susceptances, such
as would be needed to change the characteristic impedance, would
aggravate the practical realization difficulties. If these problems

could be overcome, the transmission line synthesis would offer the
advantage of being exact.

The possibility of constructing a waveguide BRF using transmis-
sion line synthesis was mentioned by Schiffman and Matthaei.? Un-
fortunately, the form they suggest involves three impedance changes
in each connecting line and therefore is not practical. This paper pre-
sents an improved form with only one impedance change in each
connecting line. The susceptance discontinuity present at each imped-
ance change is absorbed very naturally in the structure without caus-
ing any design difficulty. Design formulas for filters of up to five
cavities are given in a very convenient form.

The problem of the midband frequency shift between the VSWR
and the insertion loss was found to be the direct consequence of in-
accurate correction for the connecting lines. The precise design proce-
dure presented here guarantees the coincidence of the midband fre-
quency of the VSWR and insertion loss characteristic.

An experimental model was built and tested, and the measured
result agrees very well with the theoretical prediction. With this de-
sign technique, it is possible to achieve over 40 dB return loss across
the passband and to control the exact bandwidth and the midband
frequency. A pair of identical BRF were designed and in conjunction
with two hybrids, a constant resistance channel separation network
was built. Over 35 dB return loss across the band was observed; in
addition, the delay distortion of the dropped channel was symmetrical
with respect to the midband frequency. Channel separation networks
built with BREs designed with the old techniques always had un-
symmetrical delay distortion for the dropped channel.

II. BRF CONFIGURATIONS OBTAINED BY TRANSMISSION LINE FILTER
SYNTHESIS

The easiest way to synthesize a transmission line filter is to use the
known low-pass prototype filters as done by Schiffman and Mat-
thaei.? The reactive elements of the prototype circuit are replaced by
their equivalent transmission line stubs (all of the same length J and
of the same propagation constant @) using the frequency transforma-
tion:

where w is the normalized frequency of the prototype circuit and A is
a constant that determines the bandwidth of the filter. For a BRF, 1

can be chosen to be any odd multiple of a quarter wavelength of the
midband frequency. Schiffman and Matthaei chose 1 = dA,o/4. Then,
the shunt susceptances and series reactances of the prototype filter
transform as follows:

It is seen that the right side of (2) corresponds to the input suscep-
tance of an open-circuited stub of characteristic admittance

and that the right side of (3) corresponds to the input reactance of a
short-circuited stub of characteristic impedance

As shown in Fig. 1, a three-element low-pass prototype filter be-
comes a transmission line with three stubs using the transformation
(1). These stubs are connected to the main transmission line at the
same point, which would be impossible in a physical realization. To
solve this problem, connecting lines between stubs have to be inserted
by making use of Kuroda’s identity. This identity allows the inter-

Fig. 1— The 3 element low pass prototype filter and its corresponding BRF
with transmission line stubs.

change of a stub and a connecting line (both of the same length /) as
shown in Fig. 2. Adding a section of transmission line of appropriate
characteristic impedance on both ends of the filter (Fig. 1) as shown
in Fig. 3(a), the circuit retains its amplitude response. Applying
Kuroda’s identity to the two shunt open-circuited stubs, one gets the
desired form in Fig. 3(b). With a proper replacement of the three
series stubs by rejection cavities (only an approximation), Fig. 3(b)
forms three rejection cavities in cascade with quarter wavelength
lines separation between the cavities.

For the most commonly used aperture-coupled rejection cavity, a
quarter wavelength separation between the cavities would not be
enough because of strong coupling resulting from higher order modes
between the closely spaced discontinuities. It has been found, from
past experience, that a three-quarter wavelength separation is neces-
sary in most waveguide BRFs. For this reason, Schiffman and Mat-
thaei treated the case of three-quarter wavelength separation between
cavities. Following exactly the same technique as shown in Fig. 3,
one can add three pieces of quarter wavelength line on both sides of
the stubs, as seen in Fig. 4(a), without changing the amplitude re-
sponse of the filter. Then, the final form of Fig. 4b is obtained by ap-
plying Kuroda’s identity three times on each side. The characteristic
impedances of the connecting line sections are tabulated for two and
three cavity BRF,® but no test result has been given in the same
reference.

To realize the configuration of Fig. 4(b) in waveguide form, one
would, usually, change the height to control the characteristic imped-
ance such that the propagation constant @ remains unchanged. A
typical three cavity BRF would therefore assume the form shown in
Fig. 5. Theoretically, one may be able to compensate for the discon-
tinuity susceptances at each step by proper adjustment of the line
length of each section. This, however, complicates the design pro-
cedure unduly, and it is not sure how good the result may be. As
mentioned before, the higher order mode interaction between closely
spaced discontinuities may degrade the performance of the filter.
Since a better configuration, in every respect, is being proposed, no
attention was paid to the practical problems associated with this
suggested structure.

The only restriction which must be fulfilled in presently-available
transmission line synthesis is that all the line sections must have the
same length ! and the same propagation constant 8 so that the fre-
quency transformation (1) can be applied. To construct a BRF, one

Fig. 4— Suggested 3 stub transmission line BRI to be realized in waveguide
form.

may choose the length / to be any odd number of quarter wavelength
of the midband frequency. For the waveguide BRF, it is convenient
to choose 1 = 3A,g0/4. Then the shunt susceptance and series reactance
of the filter become

and the equivalence between prototype circuit and the transmission
line stubs shown in Fig. 1 still holds except that the constant A is

changed due to the change of stub length J. The value of A may be
computed from the given band edge attenuation specification; for ex-
ample, from the 3 dB point for maximally flat filters or from the
cutoff frequency for equal ripple filters. Let w; be the edge frequency
of the prototype filter and X,, be the corresponding waveguide wave-
length, then

After obtaining the characteristic impedances of all the stubs, one
can apply Kuroda’s identity to get the same form as Fig. 3(b), of
course with 1 = 3A,9/4. This is almost the desired form for waveguide
BRF except that the series stubs have to be replaced by cavities with
the proper Q values. There are many ways to find the equivalence be-
tween stubs and cavities, all of which are approximations. Among
them, one method is found to be particularly convenient. The input
impedance of a short-circuited stub of 3A,0/4 length with characteris-
tic impedance Zp 1s

By the well known partial fraction expansion of the tangent function,
one gets:

7 4(8d0\ < 1
Zin = 940 — (2) pS SS ge (8)
T g k=1 2 Ago
(2k — 1)° — (8h)
Ng
In the vicinity of the resonance frequency (A,0/A, ~ 1), it is clear
that the k = 2 term dominates (8). Then:

Taking this approximation, it is clear that the equivalent cavity must
possess a loaded Q of

This approximation is found to be sufficiently accurate for frequencies
close to the resonance frequency as is the case for most waveguide
filters.

Most of the advantages of this new configuration are obvious. As
seen in Fig. 6, a three-to-one reduction in the number of steps is

obtained, which makes the design procedure much simpler. In addi-
tion, the spacing between discontinuities has become large, hence, one
would expect less practical problems in realization (because of higher
order mode interaction). One rather subtle practical advantage of
this structure is that all the steps are located right under the center
of the coupling apertures. For all inductive coupling apertures (the
circular hole is one of them), a capacitive compensating stud is
always required at the location of the hole, which in this structure
happens to coincide with the location of the steps. Because of the
capacitive step susceptance, the size of the compensating studs just
has to be made slightly smaller. No additional compensation is
needed for all the step susceptances. |

A convenient table for filters with two to five cavities is presented
in this section which gives the @ values for each cavity and the char-
acteristic Impedance for each connecting line. The following symbols
are used:

n = number of cavities
R, = normalized generator impedance
FR, = normalized load impedance
Q; = loaded Q value of zth cavity
g; = normalized values of the low pass prototype filter
Z;.:., = normalized characteristic impedance of the connecting line
between the 7th and the (2 + 1)th cavity
A = bandwidth constant defined in (6)
n= 2

To construct connecting lines for a given characteristic impedance,
one may modify the heights 6; of waveguides according to

Information on cavity design has been available for a long time.
A set of curves was plotted of loaded Q vs aperture size for different

frequencies, of cavity length vs aperture size for different frequencies,
and of stud length vs aperture size for different frequencies. Such
frequencies were chosen that both the 4 GHz (8.7-4.2) and the 6 GHz
(5.925-6.425) bands are covered.

The equivalent circuit of a properly compensated cavity of Fig. 7(a)
is known (to the extent of the approximations used here) to be of
the form shown in Fig. 7(b) referred to the reference plane 7’. Aside

from the resonance circuit, an additional piece of transmission line
of length / must be added at each side, in order to make the connecting
line sections exactly 3\,,/4 and this line element must be taken into
account accurately. It was found that the unpredictable midband
frequency shift between VSWR and insertion loss response, which is
one of the major problems of the previously designed BRF, is the
direct consequence of an inaccurate correction for this line element.
A theoretical study showed that any inaccuracy in the length of the
connecting line may result in a shift in midband frequency of the
VSWR response with respect to the insertion loss response.* One
typical 3-cavity 4 GHz BRF with Afzsg,3 = +17 MHz was found to
have a 2 MHz shift with a 35 mil error in each connecting line.
Because of its importance, an attempt was made to measure ac-
curately the values of J for various sizes of coupling aperture. The first
set of data based on the symmetrical equivalent circuit (Fig. 7b) was
very disappointing since, for a simple measurement like this, the

* Dissipation must be taken into account since this phenomenon does not exist
in the lossless case.

widely spread measurement data was unexpected. It was found later
that the symmetry of the structure is destroyed by the tuning screw
on one side of the cavity (Fig. 7a); consequently, the symmetrical
equivalent circuit 7(b) is no longer valid.

To make the equivalent circuit unsymmetrical, the circuit in Fig. 7(c),
where the transmission lines on each side have different length, is
introduced. This does not solve the problem, however, because both
the length /, and 7, depend strongly on the amount of tuning added
to the cavity. A simple plot of J, and 7, does not make sense unless
one could specify a fixed amount of tuning. Fortunately, there is a
quantity, mainly 1, + 1,, which is very insensitive to the amount
of tuning. As measured in the 4 GHz band, this quantity is accurate
within a variation of 2 mils for a quite large tuning range. It is believed
that this would be true for the other frequency bands, too. Hence,
plots of (J, + 1.)/2 vs hole diameter are presented in Tigs. 8 and 9
for 4 and 6 GHz bands, respectively. For small tuning, it is true that
lL wel, & (1, + 1,)/2. Furthermore, it was found that 1, , the length
on the tuning screw side, is always larger than /, , and that the dif-
ference, 1; — 1,, may vary from zero to 20 mils depending on the
tuning range for the sizes of apertures used in this study.

To verify the theoretical work discussed above and the available
design data, a 3-cavity BRI was designed. Because of the approxima-

4 Fig. 8 — Measured information on waveguide rejection cavities in the 4 GHz
and,

Fig. 9— Measured information on waveguide rejection cavities in the 6 GHz
band.

tions involved in the equivalence between cavity and stub line, there
might still be deviations from the expected performance at frequencies
well removed from the center frequency. Therefore, the filter was
purposely designed to operate in the highest frequency channel (4190
MHz) of the 4 GHz band. In this case, the passband extends almost
500 MHz below the midband frequency of the filter. Any severe dis-
crepancy caused by the approximation should show up in this extreme
case. The filter was designed to have a maximally flat response with
the 3 dB points at +17 MHz, neglecting intrinsic losses. In order to
convey some idea of the ‘‘exactness” of the filter performance, the
measured results are compared with the computed theoretical response.
This computed curve is the theoretical response of a transmission
line filter of Fig. 3(b) (1 = 3),./4) with all line sections having an
attenuation constant:

As seen in Figs. 10 and 11, the filter shows lower VSWR value in
the vicinity of resonant frequency which is attributed to slightly
higher intrinsic loss than the theoretical estimated value. However,
over 40 dB return loss was obtained across the entire passband which
is in very good agreement with the computed results.

Measured data were also taken from a channel-separating net-
work constructed with two identical BRFs and two hybrid junctions.
The channel separating network has over 35 dB return loss across the
band. The delay distortion of the dropped channel was also measured,
and the result shows that the delay distortion vs Af curve is symmet-

Fig. 10—Computed and measured stop band performance of a typical 3
cavity maximally flat BRF.

Fig. 11— Computed and measured passband performance of a typical 3 cavity
maximally flat BRF

Fig. 12— Measured delay distortion of a channel separation network using
present design techniques as compared with that using previous techniques.

rical with respect to the midband frequency. The measured data is
shown in Fig. 12 where the delay distortion curve of the present exist-
ing channel separating network is plotted for comparison.

The author wishes to thank F. G. Joyal for carrying out all of the
measurements on the correction line lengths and on the filter per-
formance.

1, Young, L., Matthaei, G. L., and Jones, E. M. T., Microwave Bandstop Filters
with Narrow Stop Band, IRE Trans. on Microwave Theory and Tech-
niques, 10, November 1962, pp. 416-427.

2. For example, see A. I. Grayzel, A Synthesis Procedure for Transmission Line
ee IRE Trans. on Circuit Theory, CT-5, September 1958, pp.
172-181.

3. Schiffman, B. M. and Matthaei, G. L., Exact Design of Bandstop Microwave
Filters, IEEE Trans. on Microwave Theory and Techniques, 12, January
1964, pp. 6-15.

A new approach to speech synthesis by rule has been formulated and
evaluated. A discrete set of symbols (phonemes and stress marks) 1s con-
verted to a continuous acoustic waveform by a two-step transformation.
The first step involves conversion from phonemes to control signals capable
of driving a terminal analog speech synthesizer. The second step 1s con-
version from control signals to the acoustic waveform.

This paper presents a design for the terminal analog synthesizer and
discusses the new features of this device. It discusses in detail the method
of converting from phonemes to control signals. It places primary emphasis
on determining the formant frequency control signals and the fundamental
frequency contour, and presents models for determining these contours
from the input data. The paper includes an experimental evaluation of
the entire technique in terms of word intelligibility scores and consonant
confusion matrices.

Speech synthesis by rule is the method of converting from a discrete
representation of speech in linguistic units, that is, phonemes and
stress marks, to a continuous acoustic waveform. Fig. 1 shows the
technique for carrying out this transformation. The figure shows that
the discrete input is converted to continuous control signals by the
synthesis strategy. The synthesis strategy contains stored informa-
tion about the phonemes and stored rules about the mutual effects of
adjacent phonemes. The stored rules operate on the input sequence
to produce the control signals for the synthesizer. The speech synthe-
sizer converts the control signals to continuous speech. The synthesizer
may be a terminal analog, a dynamic analog of the vocal tract, or a
combination.

During the past ten years there have been many attempts at syn-
thesis by rule. The primary goal of these attempts has been to produce
natural sounding, intelligible speech. Of secondary importance has
been the preservation, in some natural way, of the dynamics of speech
production by embodying in the scheme the constraints imposed by
the human vocal tract.

Previous methods of synthesis by rule are generally classified as
either articulatory or acoustic domain approaches. An articulatory
approach uses physiological parameters such as tongue-tip position,
and lip opening as the control signals for the synthesizer. The stored
data of the synthesis strategy are of the form of vocal tract configura-
tions. An acoustic domain approach uses parameters such as formant
values and fundamental frequency as control signals. The stored data
include such information as target positions of formants and relative
amplitudes of phonemes.

Articulatory domain approaches to synthesis by rule’? have been
most successful in modelling the dynamics of the speech producing
mechanism. Acoustic domain methods, such as the one presented here,
can impose the natural constraints of the vocal tract only indirectly,
that is, by rules which often lack a firm physiological basis. How-
ever, acoustic domain approaches have enjoyed the most success in
producing intelligible, high quality speech,? * ** thus justifying and
motivating efforts along these lines. The technique for synthesis by
rule, described in this paper, is an acoustic domain approach.

The next section gives a general description of the synthesizer.
Terminal analog synthesizers of this type are common” *°® and we
discuss only the new features at any length.

A terminal analog synthesizer models the speech-producing mecha-
nism, which includes the vocal tract, excitation sources, and radia-
tion impedance. The transfer function of the vocal tract can be re-
duced to either a cascade of complex conjugate pole and zero pair
networks, or a parallel addition of complex pole pair networks. The
cascade representation was used because it reduced the complexity
of the synthesis strategy by reducing the number of synthesizer con-
trol parameters.

Fig. 2 is a block diagram of the synthesizer used in this work. The
synthesizer was simulated on a computer at 20 kHz sampling fre-
quency. There are two sources of excitation, a pitch impulse genera-
tor, and a frication (noise) generator. To produce voiced speech
(vowels, nasals, voiced stops, and voiced fricatives) the pitch im-
pulse generator output is gated by the switch to the upper arm of the
synthesizer. The nasal network is included in the upper arm only for
nasal consonants. To produce whispered or aspirated speech, the
frication generator is gated by the switch to the upper arm of the
synthesizer.

Fo | VA F, F Fs
(1-ANASAL)
PITCH HIGHER
IMPULSE SWITCH SHAPING FORMANT POLE ge & RADIATION
NETWORK NETWORK CORRECTION NETWORK
GENERATOR eislsate

FRICATIVE
FRICATION & oo) POLE AND SHAPING
GENERATOR ZERO NETWORK
NETWORK

To produce a voiceless fricative or the unvoiced component of
voiced fricatives, the frication generator excites the lower arm of the
synthesizer. For a voiceless fricative, the output of the voiced fricative
excitation network is constant. For a voiced fricative the output of
the voiced fricative excitation network modulates the frication gen-
erator output. The details of this network are explained in Section 2.1
because this is an original design.

The higher pole correction network in the upper arm of the synthe-
sizer compensates for the missing higher order poles.® *° A new design
for this network, based on the properties of sampled data systems, has
been formulated and is discussed in Gold and Rabiner’s paper.**

One last feature of the synthesizer is provision for generating a
voice bar. (A voice bar is the quasi-periodic low frequency energy
radiated from the region of the vocal cords during the closure inter-
val of voiced stop consonants. During this interval the vocal cords
are vibrating thus acting as the source of energy for the voice bar.)
To produce a voice bar, the middle arm of the synthesizer is used with
the switch gating the pitch impulse generator output to the shaping
network. The voice bar has energy only at low frequency similar to
voice bars of natural speech.

The outputs of the three arms of the synthesizer are added to
produce the speech. The synthesizer control signals are indicated in
Fig. 2 by arrows. These include four amplitude controls (avoicing,
anasal, avb, afric); a derived amplitude control (1l-anasal); 14 pole-
zero controls (both center frequency and bandwidth of F,, Pe, Fs,
npol, nzer, fpol, fzer); a switch control (va); and a fundamental
frequency control (Fo).

The network connecting the pitch impulse generator to the lower
arm of the synthesizer is used to provide the excitation for the un-
voiced component of voiced fricatives. Fig. 8 shows the relevant details
of this network. (For clarity, certain components of the synthesizer
have been omitted from Fig. 2.) |

The ouput of the pulse generator is shaped to produce a suitable
pitch pulse. A complex conjugate pole pair resonator was used, but
any suitably chosen network could have been used. The pitch pulses
excite a resonator tuned to the first formant of the fricative sound.
A single resonance is the first order approximation to the transfer
function of volume velocity (the signal of interest in Fig. 3) from the

NOISE FRICATIVE
GENERATOR NETWORK
Fig. 3 — Excitation network for voiced fricatives.

glottis through the point of constriction of the vocal tract. A threshold
level is subtracted from the output of the resonator and the result is
half-wave rectified. These operations model the physical situation
where turbulence is not produced until the volume velocity of the air-
flow exceeds a threshold value. The output of the half-wave rectifier
modulates the output of a noise generator, preducing a pitch syn-
chronous excitation for the unvoiced component of the fricative. The
final unvoiced component is produced by exciting the fricative net-
work by this excitation. The voiced component is produced in the
standard manner, that is, by exciting the formant network by the
pitch pulses.

Spectrograms of voiced fricatives produced by the above technique
are quite similar to spectrograms from natural speech. Experimental
evidence presented later shows that the synthetic fricatives are highly
intelligible.

Since formant contours are crucial to speech intelligibility (see
pages 220-234 of Ref. 9), the first step in transforming a discrete set
of input symbols to the synthesizer control signals is to generate the
formant contours. The method is explained in Sections 3.1, 3.2, and 3.3.

Once the formant contours are specified, the remaining control
parameter contours are determined by delimiting certain characteristic
times along the formant contours. At these times, motion of the other
parameters is initiated or terminated. The techniques for generating
these contours are discussed in Section 3.4.

Part IV treats the generation of a fundamental frequency con-
tour. Table I shows international phonetic alphabet symbols and the
letter equivalents used in the following sections.

In order to generate the formant contours from the phoneme input
sequence, certain information must be supplied. Corresponding to

each possible phoneme there must be data on formant target positions.
These data, along with other data, are included in a phoneme char-
acterization table. Certain durational data are necessary, such as for
stressed vowels. Finally a technique for generating formant transi-
tions must be supplied.

Each phoneme has a characterization independent of adjacent
phonemes. The characterization includes formant information, source
characteristics in the production of the phoneme, a description of
whether it is nasal or fricative, and a set of frequency regions sur-
rounding the formant positions.

The formant information is a set of target positions for both center
frequency and bandwidth of formants one, two and three. The source
characteristics describe the condition of the vocal cords during the
production of the phoneme. If the vocal cords are vibrating the sound
is voiced. The frequency regions of a phoneme represent the degree
to which certain acoustic parameters must approximate the target
values of these parameters in the context of connected speech. In an
articulatory analog, the corresponding concept would be the extent
to which a given vocal tract configuration must approximate the
target configuration for the phoneme.

The frequency regions represent a compromise between choosing
a single characterization for a phoneme and considering it inviolate,
and the realization that there are many acceptable characterizations
for a phoneme—especially in the context of connected speech.

Table II shows the phoneme characterizations we have used. The
first three columns list the formant target positions of the phonemes.
The second three columns show the frequency regions of the phonemes.
(The figures represent both + values.) The final three columns de-
scribe nasality, fricative, and voicing characteristics of the phonemes.
A + in any column indicates the presence of the feature and a —
indicates its absence. The voicing condition of the voiced fricatives z
and ZH is + indicating the two sources used to produce these sounds
on the synthesizer. The bandwidths of fF, and F; are held fixed at
100 Hz and 120 Hz for all phonemes. The bandwidth of F, is 60 Hz
except for nasals where it 1s 150 Hz. When a -+ appears in the nasality
or fricative columns, a table look-up procedure is used to specify
pole-zero locations.*

* All data referred to but not included in this paper are available in the au-
thor’s Ph.D. thesis which is available from the MIT library. (See Ref. 12.)

Vowel duration is specified only for stressed vowels. The durations
of unstressed vowels are determined by the methods illustrated in Sec-
tion 3.8. The duration of a stressed vowel is modified by its following
phoneme. The longest vowels are those followed by voiced fricative
consonants; the shortest are followed by voiceless stop consonants.”

For certain consonants maximum durations are specified. Conso-
nant duration (as measured from human speech) is not a fixed quan-
tity but is very dependent on context. (Four example, initial conso-
nants are much longer than medial consonants.) The synthesis
strategy generates consonants whose duration is variable within cer-
tain limits. Maximum durations are specified to prevent the consonant
from being unnaturally long, hence objectionable.

Maximum stop gap durations are specified for stops; aspiration
duration, as a function of the succeeding phoneme, is specified for
voiceless stop consonants. Values of amplitude control signals are
specified for all phonemes. Rates of change of control signals are
specified for various phoneme classes (that is, vowels, nasals, frica-
tives, and stops).

The technique for generating formant transitions (and hence for-
mant contours) is a new one. We present it in detail because the entire
synthesis strategy is built around it.

As we stated, the motion of formants is one of the most significant
factors contributing to the intelligibility of speech. Smooth, continuous
formant transitions are generally observed on spectrograms of real
speech. To match these characteristics, we used the solution to a
critically-damped second degree differential equation to describe the
transitions of formants. We chose a second degree equation because it
provided a good fit to data on formant transitions. We used a critically-
damped solution because it was completely specified from a single
time constant. Values of time constants were determined from examin-
ing formant transitions for real speech on spectrograms.

The input to the differential equation represents the formant target
position appropriate for the current phoneme. Since the current
phoneme changes its value discretely, the input to the differential
equation changes in a steplike manner. The formant motion, in re-
sponse to this step input, is smooth and continuous. Thus motion
from steady state value A1z to target value Af, beginning at time f =
0 is of the form:

In general, motion between target positions does not proceed from a
steady state condition; that is, there are initial conditions. Motion to
a target whose formant value is Af from an initial formant position
Av with an initial formant velocity Vi = dx/dt\o_ is of the form:

At times when the input to the differential equation is changed dis-
cretely, both the output value and slope are continuous. Thus the
concept of smooth, continuous formant transitions is realized in all
cases.

The time constants of the differential equation are functions of the
individual formant and the pair of phonemes between which the
transition is being made. Hence, for each possible pair of phonemes,
and each formant, a time constant is specified. Certain simplifying
approximations reduce (by an order of magnitude) the number of
time constants that have to be specified.

The inputs to the differential equations change discretely in time in
a steplike manner. However, provision is made for delaying, to any
formant, the steplike change in formant target position. Thus, in the
most general case, formants move independently of each other with
unequal time constants of motion. This delay feature was found
necessary for only a few cases.

The phonemic goals change discretely in time. The decision of when
to make the discrete changes, that is, when to initiate motion to new
sets of formant targets, is based on the criterion that the formants
must first be within the phoneme frequency regions of the targets, and
then satisfy durational requirements of the phoneme, if there are any.

Formants, in general, are in motion towards target values ap-
propriate for the phonemes to be generated. Their motion is charac-
terized by the solution to a differential equation. The time constant of
motion is a function of the phoneme from which motion began and
the phoneme which is being generated. Each formant moves with its
own time constant and there is provision for delay in time of initiation
of the motion of formants. When all formants are within the fre-
quency regions of the target, a decision is made. If a stressed vowel
is being generated, then a table look-up procedure determines the
correct vowel duration and motion continues for the specified time.
Once a vowel of proper duration has been generated, motion towards
target positions characteristic of the next phoneme is begun. If the
current phoneme is not a stressed vowel, motion towards the new
phoneme targets is initiated as soon as all the formants are within
their specified frequency regions.

The decision to start motion to new target values results in three
separate operations. First, new time constants for each formant are
inserted into the respective difference equations. Second, the forcing
functions (input) to the difference equations are changed in a step-

like manner indicating the changes in target positions. Finally, the
initial conditions of the difference equation are set to preserve con-
tinuity of formant values and formant velocities. If the motion of
any formant is to be delayed, the changes in the difference equation
for that formant are delayed appropriately.

Fig. 4 shows a typical cycle of events. Initially formants one and
two (we shall neglect formant three in this example) are at target
positions appropriate for phoneme 1. At time ¢, motion is initiated
to phoneme 2. Formants one and two begin motion simultaneously
(no delay is used here) with time constants 7}, and 77, respectively.
Ti. 18 much smaller than 77, so formant one moves more rapidly to
its target value than formant two. Periodically the formant values
are tested to see whether they are within the specified frequency regions
of the targets. (The frequency regions are indicated by Al, A2 in
Fig. 4.) If they are not, the formants continue their motion, thus
moving closer to target. For the example in Fig. 4, formant one enters
its frequency region prior to formant two. Until formant two enters
its frequency region at time ¢, , formant one moves closer to its target
position. At time ¢, both formants are within the specified frequency
regions and so a check is made on whether phoneme 2 is a stressed
vowel or not. In this example phoneme 2 is not a stressed vowel, so
motion to phoneme 3 is initiated at t,. However, we now have the

case when the time of initiation of motion of formant one is delayed.
Hence at ¢. the target value and time constant for formant two is
changed, but formant one’s target is unchanged. At time ¢{ the delay
is terminated and formant one begins its motion.

Phoneme 3 of Fig. 4 is a stressed vowel. So at time ¢;, when both
formants are within the frequency regions for phoneme 3, motion to
targets for phoneme 3 continues for the specified vowel duration.
At time ¢ , following the vowel duration, motion begins toward targets
for phoneme 4. New time constants and targets are again inserted
in the equations of motion. The process continues in this manner
until all the input phonemes have been generated.

The motion of the remaining synthesizer control parameters (that
is, nasal and fricative poles, and zeros and source amplitudes) is time-
locked to the formant motion. The source amplitudes (avoicing,
anasal, afric, avb) begin to switch approximately one time constant
after the discrete phoneme goal is changed. The amplitudes change
linearly at predetermined rates. The nasal and fricative poles and
zeros initiate motion at the time the phoneme goal is changed. The
motion is linear and the slopes are arranged so that the poles and
zeros just reach their targets at the time the source amplitudes are
switched. The target positions are specified in a table.

For nonnasal sounds, the target positions of the nasal zero and pole
are set to 1400 Hz. Thus the pole and zero will cancel each other in
these cases. Furthermore, for nasal sounds, the bandwidth of formant
one (nominally 60 Hz) is changed linearly to 150 Hz for the duration
of the nasal. The bandwidth begins to change 50 msec before the
amplitudes switch and is linearly changed back to its nominal value
in 50 msec after the nasal. For nonfricative sounds the fricative pole
and zero target positions are set to 1500 Hz.

Our model for generating fundamental frequency data is based on
the assumption that these data can be derived from data on laryngeal
tension (LT) and subglottal pressure (Ps). A description of an ut-
terance in terms of these variables is then used to produce the desired
fundamental frequency data.

The model is based on that of P. Lieberman, in which the breath-
group is defined as an underlying phonetic feature of American Eng-

lish.1# The unmarked breath-group is characteristic of a simple, de-
- Clarative sentence, whereas the marked breath-group characterizes a
simple interrogative sentence.

The feature breath-group is converted to a global description of an
utterance in terms of Ps and LT. Fig. 5 shows the archetypal Ps con-
tour, as suggested by Lieberman’s data. Ps increases over the first
300 msec of the utterance, and then remains constant until the last
300 msec of the breath-group, at which point it decreases rapidly to
zero. The LT contour for an unmarked breath-group is constant,
whereas for a marked breath-group it is characterized by a steady
increase over the last 175 msec of phonation. Fundamental frequency
is linearly proportional to both Ps and LT. Since the archetypal Ps
contour falls at the end of a marked breath-group, the increase in LT
must compensate for the decrease in Ps to give fundamental frequency
a rising terminal contour. A slope of 0.6 Hz/msec, for the last 175
msec of phonation, was assigned to the LT contour. This resulted in
a terminal rise of 60 Hz in fundamental frequency for a question.

The subglottal pressure contour is modified by both consonants
and vowels. Two levels of stressed vowels are adopted. One level is
referred to as emphasis and only one vowel in a breath-group is em-
phasized. The emphasized vowel provides the highest peak in the Ps
contour. All other stressed vowels are treated similarly. When a
vowel is stressed, there is an increase in subglottal pressure for a pe-

Fig. 5— Archetypal subglottal pressure contour showing effects of vowel stress.

riod of 500 msec, centered on the vowel. Fig. 5 shows an example of
this effect. The dashed curve shows the effects of placing stress on a
vowel at the beginning of the breath-group. There is a rise in Ps
early in the breath-group and the increase is centered at ts, the mid-
point of the vowel steady state. If the stressed vowel had been the
emphasized one, the only difference in Fig. 5 would be the amplitude
of the increased Ps. It would have been 2.5 em H.O as compared to
1.0 cm HO.

The effects of consonants on subglottal pressure have also been
included in our scheme. Tor a voiceless consonant, subglottal pressure
automatically increases whereas subglottal pressure automatically de-
creased for voiced consonants. Thus the consonants introduce local
perturbations to the Ps contour. The change in Ps for consonants
is 1.0 cm H,0, and this change occurs over a period of 150 msec
centered on the consonant.

All vowels are unstressed except those followed by the symbol
strss. The symbol strss1 signifies the emphasized vowel. Word boun-
daries are signified by the symbol space and pauses by the symbol
pause. A question is signified by the symbol ques. The sentence boun-
dary is indicated by the symbol end. The examples are:

(2) This is an olive.
THEI strss S$ space 1 Z space AEN space A strss1 L1v end.
(11) Why are you sad?
w A Strss1 IY space AR space ¥ 00 space § AB strss D end.
(272) We sang all day.
W IY space Ss AE Strss1 NG space OW L space DETY Strss end.

Whenever a word boundary (space in our code) occurs, consonants
on either side of the word boundary are affected. In this strategy an
initial consonant is lengthened by about 20 percent, whereas a final
consonant is shortened by a similar amount. A word boundary has no
effect on phonemes which do not lie on either side of the word bound-
ary.

Intelligibility tests were conducted to evaluate the scheme. To test
the rules in a limited environment, consonant intelligibility tests were
run. One test was intended to test perception of consonants in pre-

stressed position. The schwa vowel uu always preceded the consonant
and was used as a perceptual cue for stop consonants because it pro-
vided a basis for perceiving the stop gap. The second test was intended
to test perception of consonants in post-stressed position. The schwa
vowel uH always followed the final consonant—again providing a
basis for perceiving stop gap duration, bursts, and aspiration for
stops.

Sixteen consonants were used: B, D, G, P, T, K, M, N, F, TH, S, SH, V,
THE, Z, and ZH. Five vowels (besides schwa) were used: Iy, AE, A, OW,
and oo. For each test there were 80 possible stimuli. Twenty additional
stimuli were used, ten initiating the test and ten concluding it, giving
a total of 100 stimuli per test. Only the middle 80 were used for eval-
uation, and these were presented in random order.

Three subjects were tested. Their results are summarized in the
two confusion matrices shown in Table III. In prestressed position
(uH-c-v), 73 percent were correct; in post-stressed position (v-c-UH),
77 percent were correct. If r, rH and v, THE responses are pooled, as 1s
often done, then the correct percentages increase to 79 in prestressed
position and 81 in post-stressed position. Ten prestressed consonants
were identified correctly more than 75 percent of the time: B, D, P, T,
N, TH, 8, SH, Z, and zH. The post-stressed consonants identified cor-
rectly more than 75 percent of the time were B, P, T, K, TH, S, SH, 2,
and sH. The consonants which were identified incorrectly most often
were G, M, D, and kK.

An examination of the errors in the confusion matrices of Table III
shows:

(1) The voiced stop c was often confused with T and x; and D in
post-stressed position was often confused with «a.
(72) The unvoiced stop K was often confused with T in prestressed
position.
(22) The nasal Mm was often confused with B and v.
(1v) The fricative pairs v, THE and F, TH were often confused.

These errors were the major confusions in the tests. The stop con-
fusions primarily were caused by errors in frication burst positions.
The fricative pair errors were anticipated because of the acute acoustic
similarities between these particular fricatives. The cause for the con-
fusions between m and other phonemes is unknown. Further work re-
mains to be done in this area.

One test contained simple declarative, interrogative and imperative
sentences. A second test contained sentences chosen from a list of sen-
tences used often in intelligibility tests.15

The sentences were presented to listeners who wrote down what
they heard. They were told to guess whenever in doubt. The sentences
were played a second time and the listeners were allowed to make
changes. The tests were scored on the number of words which were
correctly identified (excluding only the and a as words). The results
of the tests are as follows. For the test using simple sentences, eight
listeners had an average of 92 percent of the words correct after one
try, and 95 percent after the second try. For the test using the longer

standard sentences four listeners had an average of 83 percent of the
words correct after one try; and 86 percent after the second try.

As shown above, the percent intelligibility scores for sentences were
significantly higher than for isolated syllables, primarily because of
the context of speech in a meaningful utterance. However, the longer
the utterance, the less intelligible 1t became. This is because rhythm
and timing are much more important for a long sentence than for a
short, simple one.

Fig. 6a shows wideband spectrograms of the utterance “Larry and
Bob are here.” The spectrogram in the upper half of the figure is the

Fig. 6a— Wideband spectrograms of synthetic (top) and natural versions of
“Larry and Bob are here.”

synthetic version. The lower spectrogram was made from the author’s
speech. (The synthetic utterance was in no way modelled after or
modified by the natural utterance.) Fig. 6b shows narrowband spec-
trograms of both the synthetic and natural versions of this utterance.

This is a high degree of similarity between the spectrograms of the
real and synthetic speech. The durations of both the synthetic and
natural utterances are comparable. Fig. 6a shows that the variation
of the formants for both versions is quite similar. Even the funda-
mental frequency contours for these utterances are quite similar. As
Fig. 6b shows, both contours are peaked during the stressed vowels
A in BoB and I in HERE. A careful examination of the narrowband
spectrograms shows the decrease of fundamental frequency, for both
utterances, during the initial and final B of Bos.

The stressed vowels of this utterance can easily be identified from
either the long steady state duration of Fig. 6a or the peak in the
fundamental frequency contour of Fig. 6b.

The results of the consonant intelligibility tests showed that most
consonants were reproduced accurately. The sentence intelligibility
tests also produced good intelligibility scores, indicating a high degree
of success for the major goal of this project.

Many of the listeners made informal comments concerning the ma-
chine-like quality of the speech, but no formal tests were run to meas-
ure the naturalness or quality of the synthetic speech. Current studies

Fig. 6b — Narrowband spectograms of synthetic (top) and natural versions of
“Larry and Bob are here.”

about the characteristics of the source of voiced speech are expected
to produce valuable information about the determinants of synthetic
speech quality.

Among the topics that will be considered for future work are the
effects of stress and rhythm on timing of an utterance, the inclusion
of more than one breath-group in an utterance, and studies of further
correlates of word boundaries.

An acoustic domain approach to speech synthesis by rule has been
formulated and programmed on a digital computer. Samples of speech
have been generated using our scheme and their intelligibility has
been measured. The problem of automatically generating a funda-
mental frequency contour by rule has also been investigated and one
possible solution has been found.

(iv) Natural fundamental frequency contours for both interroga-
tive and declarative sentences.

(v) A new synthesizer design having potential for high quality
voiced fricatives and provision for inclusion of a voice bar
for voiced stops.

(vi) A new method of handling formant transitions (a differential
equation approach) and transition durations (the matrices
of time constants).

(vii) Internal control of timing using system parameters with few
external constraints.

(viii) A new method of generating fundamental frequency data
based on a physiological theory involving simulated laryn-
geal tension and subglottal pressure information.

I wish to acknowledge the invaluable advice and guidance of Pro-
fessor Kenneth N. Stevens of the Massachusetts Institute of Tech-
nology and Doctor James L. Flanagan of the Bell Telephone Labora-
tories. I would also like to thank Doctors Harry Levitt and Peter
Brady for valuable comments and criticisms of this manuscript.

Henke, W., Dynamic Articulatory Model of Speech Production Using Com-
puter Simulation, Ph.D. thesis, Massachusetts Institute of Technology,
1966.

Rules, Speech Communication Seminar, Stockholm, August 29 to Septem-
ber 1, 1962, Paper F2

Gold, B. and Rabiner, L., Analysis of Digital and Analog Formant Synthe-
sizers. Paper presented at 1967 Conference on Speech Communication and
Processing, Massachusetts Institute of Technology, and accepted for pub-
lication in IEEE Trans. Audio and Elec., March 1968.

Rabiner, L., Speech Synthesis by Rule: An Acoustic Domain Approach, Ph.D.
thesis, Massachusetts Institute of Technology, 1967.

Lieberman, P., Intonation, Perception and Language, Research Monograph
38, MIT Press, Cambridge, Mass., 1967.

Unified Matrix Theory of Lumped and
Distributed Directional Couplers

This paper presents the theory of directional couplers using matrix
formulation for lumped, distributed, or any linear tume-invariant black-box.
The starting point is the matrix theory of untform multiple coupled trans-
mission lines in terms of which are brought in the concepts of characteristic
ampedance matrix, propagation matrix, reflection matrix, and scattering
matrix. Next, we use the results obtained with the theory of multiple coupled
transmission lines to derive expressions for the loading ampedance and
voltage ratios for distributed directional couplers. We do this using the
spectral properties of the matrices. Then we generalize the concepis of
characteristic impedance matrix and propagation matrix for a class of
black-boxes with the aid of the A, B, C, D transmission matrix. We give
conditions for the loading impedance and expressions for the voltage
ratios, using the spectral theory of the transmission matrix. We discuss
the physical significance of the directional coupler effect at all frequencies
in a vector-matrix framework and analyse in detail some lumped dtrec-
tional couplers. Finally we discuss hybrid (lumped and distributed)
directional couplers.

The directional coupler is an important device in many transmission
systems. The theory for the electrical design of certain types of dis-
tributed-parameter directional couplers is well established through the
contributions of a number of researchers.*~’

The purpose of this paper is to present the theory of transmission-line
symmetric directional couplers in matrix form and then extend that
theory to arbitrary lossless reciprocal circuits. There are many nota-
tional and conceptual advantages in using a matrix formulation which
give further insight for the synthesis of different types of directional
couplers.

The starting point is to examine the coupled-line directional coupler
as a particular application of multiple coupled transmission line theory
and then to use the concepts of characteristic impedance matrix and
propagation matrix to examine the whole problem, rather than one
line at a time or one mode at a time. By using a matrix approach the
fundamental properties of the modes become more evident and thus
physical intuition and mathematical reasoning blend to give a clearer
picture of the situation. For instance, when the lines are no longer
identical it is the eigenvectors of the matrices which provide the in-
formation of how to extend the mode concept.

Several researchers’ *’ have analyzed the behavior of a set of multiple
coupled transmission lines. For reference in subsequent sections of this
paper we present a brief account of this theory. For simplicity the
discussion is restricted to two identical lines operating in the TEM
mode. Fig. 1 shows schematically a differential section of two lines
and a ground plane. The circuit of which Fig. 1 is a differential section
obeys the following vector differential equations* in the steady state

By solving for I in (1) and substituting its value in (2), or solving
for V in (2) and substituting its value in (1) the following are obtained:
av

* The matrices V, I, Z, Y are all functions of frequency. For simplicity in the
notation this dependency is not explicitly indicated.

where the 2-vectors.v, and v_ are arbitrary constants (dependent on
frequency) which depend on the boundary conditions. Their inter-
pretation is very similar to the one of single line theory, v. may be
called the forward wave and v.. the reflected wave.

Equations (8), (8) and (9) have the same form as the single trans-
mission line equations. For this reason the matrices T, Y,, Z), are
called propagation matrix, characteristic admittance matrix, and char-
acteristic impedance matrix, respectively. Many properties of the
single transmission line hold for the multiple case. In particular, if
the set of lines is terminated in a network whose open circuit impedance
matrix is equal to the characteristic impedance matrix of the set of
lines, a vector of incident voltage waves traveling down the lines will
experience no reflection. The manner in which the voltages and currents

* In evaluating WZY to calculate F the convention is made to associate with I
eigenvalues whose real part are positive and with —I the ones with negative real
part.

in the lines interact is best understood by examining the matrix func-
tions e~** and e**. (See Appendix.)

2.2 feflection and Scattering Matrices
In single transmission line theory the voltage reflection coefficient
I', at a discontinuity is defined as
v. = pd. , (10)

where v, is the incident voltage and v_ is the reflected voltage. The
expression for Ip in terms of the characteristic admittance Y, of the
line and the input impedance Z, at the discontinuity is

a Vig Y — 1
~ Z,Y,+1
The concept of reflection coefficient can be generalized very simply

for the case of a multiple set of lines. Consider Fig. 2 depicting a pair
of coupled lines of length 7 and matrices IT, and Y, terminated at

x = Qin a device of open circuit impedance matrix Z,. According
to (8) and (9) for z = 0
V0) =v+Vv_, (12)
I(0) = Y,.(v. — v_). (13)
The box marked Z, obeys
V(O) = Z_I(0). (14)
Substitution of V(0) and I(0) from (12) and (13) into (14) gives
v, +tv_ = Z_Y,(vi — V-), (14b)

Fig. 2— Pair of coupled lines terminated in a circuit of impedance matrix Zz.

DIRECTIONAL COUPLERS 43
Thus, defining the reflection matrix Tp as a generalization of (10)
according to
v. = Fev,, (16)
it follows from (15) that
| Ty = (Z:Y, + I) '(ZzY, — D), (17)

which is the generalization to multiple lines of (11).
The scattering matrix § is a reflection matrix in which the incident
and reflected voltages are normalized.* It is defined by

Defining the normalized load impedance matrix Z, according to
Z, = YiZ1¥i , (22)
equation (21) may be rearranged as follows:
w= Z@. +N". — De... (23)
Comparison of (23) and (18) gives
S=@Z,+1"@, — 0. (24)

From either (17) or (24) it is clear that if the load has an open circuit
impedance matrix equal to Z, of the lines the reflected wave is zero
since both the reflection matrix Ip and the scattering matrix S vanish.

(The superscript -++ denotes transposed conjugate of a matrix.) For a
lossless device the incident power must be equal to the reflected power,

* In the literature on the scattering matrix the variables are normalized with
respect to a diagonal matrix (usually a real matrix). Here the matrix may be complex
and not necessarily diagonal. This derivation provides the physical interpretation
for a scattering matrix normalized with respect to a complex nondiagonal matrix.

Viewed from its terminal behavior the two coupled lines and ground
of Fig. 3 can be studied as a four port which may be characterized by,
among others, an impedance matrix ¢ or a transmission E matrix.
(The E matrix is the extension to 2N ports of the concept of the A, B,
C’, D parameter matrix of two ports. See Ref. 11.)

(The prime indicates the transpose matrix.) The E matrix, written in
partitioned form, is

— Bal _ | cosh T-l
C|D Y, sinh T-l
If the two lines in Fig. 3 are identical then I is symmetric and hence
I’ = I so that in (29), A = D. Furthermore, for this case all the
matrices Z, , Yo , Y, A, B, C, D and their analytic functions commute
and therefore may be treated unambiguously as ordinary numbers.
For instance (29) may be written

The analogy between the E matrix of (30) and the A, B, C, D parameter
matrix of a single transmission line considered as a two port is now
complete.

A symmetric matched directional coupler is a four port device whose
scattering matrix is of the form of S of (81) when the ports are ad-
equately numbered. 8 and 6 are, in general, frequency dependent.
It is well known in microwave circuits that, given any lossless reciprocal
4-port, if all ports are matched, then the device is a directional coupler.”
By properly numbering the ports it may be assumed that the directional
coupler has the following scattering matrix

For the case of two identical lossless coupled lines equally loaded
at the four ports all that is necessary to obtain a directional coupler
is to match one of the ports. This results from the great symmetry.

Fig. 4 shows two identical lines loaded at ports 2, 3, and 4 with
equal resistances, #. To calculate the driving point impedance at
port 1 the following procedure can be used. Consider Fig. 5. The
equation

Fig. 4 — Two-terminal circuit formed by a pair of coupled lines above a ground
plane loaded with resistances at three ports.

is the matrix version of the well known formula for calculating the
driving point impedance at one port of a two-port when the other
port is loaded. In (82) Z is the open circuit impedance matrix of the
two-port at the left in Fig. 5, A, B, C, D are the matrices given by (29).

Once the Z of (82) is known the resistor R may be connected to
port 2 as shown in Fig. 6 and the driving point impedance at port 1
calculated by a second application of (82) although now, instead of
matrices, A, B, C, D are scalars. The matrices in (82) are all of the form

Fig. 5—Intermediate circuit for calculating the driving point impedance at
port 1 of the circuit in Fig. 4.

Fig. 6.—Second intermediate circuit for calculating the driving point im-
pedance at port 1 of the circuit in Fig. 4.

which hold for matrices (and hence for scalars considering them as
1 X 1 matrices) may be used. The second application of (82) with
the aid of (88)-(41) gives

For port 1 to be matched z must equal R, hence, the condition for
directional coupler effect is

When the values given by (86) and (37) are substituted in (48), after
some algebra, the following equation results:

If a matched directional coupler at all frequencies is desired, (44)
must be satisfied at all frequencies. Some possible mathematical solu-
tions are the following:

Equation (45) or (46) and (48) give possible conditions for directional
coupler effect at all frequencies. If a narrow band directional coupler
is desired one may match the coupler at the discrete frequencies which
satisfy (47). Since (47) is a transcendental equation it 1s not unreason-
able to expect an infinity of roots. For instance, suppose (46) holds,
and J is made so that

Then the device will be a directional coupler at the frequencies that
are roots of (49).

Some explicit relationships for two identical lossless lines of inductance
per unit length Z,, , mutual inductance per unit length L,. , capacitance
per unit length of one line alone C, and capacitance between the two
lines per unit length Cy, , are:

which is the same as (55). Equation (54) implies that the lines have
negative mutual inductance, which is not achievable with parallel
lines. This condition can, however, be satisfield with counter-wound
lumped elements.

That is, for the frequencies given by (57), independent of the value
of the loads (as long as they are all equal) the lines will be matched
and will exhibit directional coupler effect.

Equations (54) and (55) or alternatively (56) and (55) are not fre-
quency-dependent. This means that the resulting circuit will be matched
for all frequencies. Therefore, the directional coupler effect will exist
for all frequencies, meaning that the coupling between uncoupled ports
is zero at all frequencies. However, the coupling between coupled
ports is frequency-dependent. This dependency is derived as follows.

Considering the coupled transmission lines as the load to four un-
coupled lines, each of characteristic impedance FR as shown in Fig. 7,
the scattering matrix of the load is calculated according to (24) and
(22) with

and Z, given by ¢ of (28). Applying the methods of the appendix the
matrix ¢ may be expressed as follows:

where R, , R; , R. , Ra are the spectral set of { and they are given by
(165)—(168) of the appendix and dz, , Ags , Age » Aza ATE the eigenvalues
of ¢ and are given by

Fig. 7— Two coupled lines above a ground plane of impedance matrix given
by equation (28) serve as load to four uncoupled lines of characteristic imped-
ance FR for calculation of scattering matrix of coupled lines.

If the values of Aya, Ara , Age » Ara aLe Substituted in (65)—(68) and those
are in turn substituted into (64) while the conditions indicated by
(46) and (48) are imposed, the following result is obtained after some

If, instead of the condition of (46), the one of (45) is imposed plus (48)
then the following result is obtained

where
1 7
Sis = Gshyl-+amhyl? cahyl+embyt? 9)
; i
Su = cosh y T+ sinhyl coshyl+sinhyl? = /
where

The matrices of (69) and (75) with the aid of (70)—(74) and (76)-—(80)
give the frequency dependency of the coupling between the ports for
the two types of directional couplers derived (matched at all frequencies)

Two sets of conditions have been derived for obtaining the directional
coupler effect at all frequencies for transmission lines. Fig. 8 represents
a pair of lossless lines c, d which are coupled from x = —/ to xz = 0.
The coupled lines are connected to four (uncoupled) lossless transmission
lines a, b, e, f, each of characteristic impedance & and each terminated
in &. The matrix Z, is the characteristic impedance matrix of the set
of coupled lines (c and d in Fig. 8) and I its propagation matrix.

Consider a case in which the eigenvalues of the matrix Z, of the
lines c, d satisfy

where I is the unit matrix. Assume that a pulse travelling down line
a occupies at time ¢, the position shown in Fig. 8. Lines a and b together
may be considered as a particular case of a set of multiple lines, in
which both (Z,),, and (I),,—the characteristic impedance and propaga-
tion matrices of lines a, b—are diagonal because lines a and b are
uncoupled. Thus it is convenient to think of the pulse traveling down
line a as a vector of pulses traveling down the two lines a, b; the second
component of the vector which corresponds to line b being zero. When
the vector of pulses arrives at the position « = —l, indicated by
in Fig. 8, the vector of pulses continue ‘‘seeing’”’ the same characteristic

Fig. 8 — Pair of coupled lossless lines c and d above a ground plane which has
uncoupled terminated lines a, b, e, f of characteristic impedances R. This shows
the progress of a pulse to explain the directional coupler effect in physical terms
for a coupler with diagonal characteristic impedance matrix.

impedance matrix RI as before and hence no reflection of the vector
is created at AZ.

In a vector formulation one speaks of reflections in a multidimen-
sional sense. The voltage on line a may give rise to a reflected voltage
on line a [self reflection] or to a reflected voltage on line b [mutual
reflection]. If all the lines are self matched and mutually matched
there will be no reflections whatever. Often a line might be self matched
but not mutually matched; then no self reflection will occur, but a
mutual reflection will.

At time ¢, the second component of the vector pulse is no longer
zero because the pulse on line ec which is coupled to line d induces a
pulse on line d as shown in Fig. 8. These component pulses will be

* Although for simplicity in the explanations, parallel lines are assumed, this

kind of coupler requires negative mutual inductances which are in general achieved
with counter-wound helices.

distorted because continually varying portions of them travel at
different speeds on the lines c and d.* When they reach x = 0 marked
by N in Fig. 8, the vector again continues to see the same characteristic
impedance matrix and hence no reflections are caused at N. Finally
the pulses travel out on lines e and f and are dissipated at the resist-
ances &.

From this it is clear that the two ports at Jf are uncoupled and also
the two ports at N are uncoupled. However a port at J/ is coupled to
both ports at N and vice versa. All the ports are self matched. This
explanation suggests a simple way of determining the conditions for a
nonsymmetric coupler realized with transmission lines. Assume lines
ec and d are no longer identical but that the characteristic impedance
matrix of the set is still diagonal

while the propagation matrix I is not. If the lines a and e have char-
acteristic impedance (Z,),, and lines b and f have characteristic imped-
ance (Zo)o. and a, b, e, f are properly terminated, the coupled lines
c, d will constitute a matched nonsymmetric directional coupler since
the discussion above holds for this case without modification.

Tig. 9 shows the same arrangement as Fig. 8. A pulse traveling
to the right on line a is shown at ¢ = ¢, . The second component of the
incident pulse corresponding to line b is zero since lines a and b are
uncoupled. As the vector of pulses reaches the position M, the vector
of voltage is reflected according to a reflection matrix because for
this case, the vector no longer sees the same characteristic impedance
matrix in the transition from lines a, 6 to lines c, d. However, because
line a matches line ¢ there is no self reflection; only a mutual reflection
appears on line b. Besides the reflected pulse, an identical transmitted
pulse appears on line d at time #, as indicated in Fig. 9. The appearance

* The propagation matrix should not have equal eigenvalues, otherwise it will be

diagonal which, together with a diagonal characteristic impedance, implies un-
coupled lines.

a \M € N e
|
t, ita t3)t3
w: VL. SULLLELLLLLA meV LEO ELEELELALLELAALAL LA ELL LLL LLL LET EY x 0) | 7 VALLE EELS Et
_> <—;— > <—|—> R
t to lt oF (-—)
1 2 ve t3, ts
fie ec ee es es a ee pe ee ee en etre ee A Ee
| d ' f

Fig. 9—Same structure as in Fig. 8 showing the progress of a pulse for a
coupler with scalar propagation matrix.

After ¢, , as the two forward pulses travel along lines c and d they do
so at the same speed, without distortion and do not interact with each
other since the propagation matrix is diagonal. As the two pulses
arrive at point N they encounter a reflection matrix of opposite sign
to the one they encountered at 17. This means that the pulse on line c
passes through N undisturbed but creates on lines d and f reflected
and,transmitted pulses identical to the ones created at M but of op-
posite sign. Likewise, the incident pulse on line d goes right through NV
(since each individual line is matched) creating transmitted and reflected
pulses on lines e and c, but being cancelled on line f by the transmitted
pulse created by the incident pulse on line c; thus, nothing comes out
of line f.

At time ¢; right after the reflection at N the situation is depicted
in Fig. 9. After t; the reflected pulses are traveling to the left at the
same speed undisturbed and undistorted on lines c and d. As they
arrive from the right at point M the pulse on line d goes out line b
undisturbed but creating transmitted and reflected pulses on lines a
and c. Likewise the pulse on line c goes out line a undisturbed creating
transmitted and reflected pulses on lines 6 and d, but being cancelled
on line a by the transmitted pulse created by the incident pulse on
line d. This eliminates any delayed reflections on line a to the original
incident pulse. The process continues in the same manner, the outgoing
pulses on lines a and f always being such that they cancel. This means
that the ports associated with lines a and f.are uncoupled, but the
ports of lines a, b and e are coupled.

It is clear that what is necessary for directional coupler effect on this
type of coupler is: all ports self matched, equal propagation velocities
without attenuation or distortion. Hence it should be possible to
realize a nonsymmetrical directional coupler of this type whose propaga-

tion matrix is a scalar matrix* by self matching its ports. This may
be useful for interconnecting lines a and b of different characteristic
impedances.

Because the reasoning above was made in the time domain with
pulses of arbitrary shape, the results hold for all frequencies. This
is an example of the gain in insight owing to the vector-matrix for-
mulation.

Let us generalize several concepts introduced in Section 2.1. Con-
sider a (2N + 1)-terminal network in which terminal 2N + 1 will
be grounded and ports from terminals 1 through 2N to ground will
be considered. Ports 1 to N will be considered input ports and Ports
N + 1 to 2N output ports. Suppose the 2N-port is characterized
by A, B, C, D N X N matrices. (Extensions to 2N-ports of the A, B,
C’, D parameters of a two-port). Assume the circuit is such that

where I is the N X N unit matrix and A, B, C are N X N symmetric
matrices which commute.

By analogy with a multiple transmission line the characteristic
impedance matrix Z,) and the propagation matrix I are defined so
that they satisfy the following equations: :

The N X N matrices © and Z, will also be symmetric and commute.
The matrix Z, is the open circuit impedance matrix of that network
which, when connected to the output ports N + 1 through 2N of

the circuit whose A, B, C matrices are those of (86) and (87), will
result in an open circuit impedance matrix of Z, when the circuit
is viewed at its ports 1 through N. This property is analogous to the
one of the characteristic Impedance matrices of a set of multiple
coupled transmission lines. The matrix TF, has virtually the same
properties of the matrix Tl of a set of coupled lines (although the
single quantity J looses its significance as a length in case the 2N-port
is a lumped circuit). For instance, if n identical 2N-ports are cascaded
the resulting 2N-port has a propagation matrix equal to nT.

The matrices I and Z, may be expressed in terms of the 2N XK 2N
impedance matrix Z of the 2N-port with the aid of (88) through (41)

the submatrices Z,, , Zi2 , Zo, , Zoo are N X N symmetric matrices
and commute with each other. The characteristic impedance matrix
may also be expressed in terms of the so-called open and short impedance
matrices. If the N-vector V, and I, denote the voltages and currents
at the N input ports and V, and I, denote the voltages and currents
at the output ports, then if I, = 0, that is, the terminals on the output
ports are open then

Equation (99) gives an experimental method of determining Z, if
it is known that the network satisfies equations (81) and (82), and
the symmetry and commutativity conditions.

It is often convenient to analyze some lumped or distributed (or
combinations of lumped and distributed) systems as though they were
multiple transmission lines using such concepts as reflection matrix
and incident voltage.

Consider the connection shown in Fig. 10. Each network is an (2N-+1)-
terminal network in which ports from each terminal to ground are
made. Ports 1 through N and 1’ through N’ are considered input ports.
Ports N + 1 through 2N and (N + 1)’ through (2N)’ are considered
output ports. The voltage vector at the junction B whose components
are the voltages of nodes N + 1, N + 2, --- , 2N to ground is denoted
by Vz.

The vectors v, , V_, i, , i_, called incident voltage, reflected voltage,
incident current, and reflected current at the junction B (assuming
the direction of propagation from left to right), are defined to satisfy

Fig. 10 — Connection of two 2N ports in cascade used to define the reflection
matrix.

from the ports NV + 1, N + 2, --+ , 2N with network 1’ disconnected.
The reflection matrix Vy,;,,- is defined according to

Vi = TV acarce-V— . (104)
The matrix T'y,;,,- satisfies the following relationship:
TD arare — (Z/Z* + | Od A Ae “= I), (105)

where Z’ is the N X N open-circuit impedance matrix of network
M’ as seen from ports 1’, 2’, --- , N’ with network M disconnected.
Tis the N X WN unit matrix. The indices MM’ on Yara, indicate the
direction of propagation from M to M’. If the indices are reversed
the roles of Z’ and Z are reversed, that is

Taking advantage of the derivations done for transmission line
directional couplers and the analogies introduced in Sections 5.1 and
5.2, it is possible to write without further work, the equations of a
directional coupler having the same mathematical symmetry of a
multiple transmission line directional coupler but which may have
lumped components or combinations of lumped and distributed com-
ponents. Suppose a four-port is characterized by its E matrix whose
2 X 2 submatrices A, B, C, D have the form of the matrix K of equa-
tion (1) of the Appendix and satisfy equations (81) and (82). Without
any further work it can be stated that if the load impedance z and the

the device will be a directional coupler for that frequency.
This result is deduced from equation (45). Likewise from equation
(46) it may be deduced that if the four port is such that

then the device will also be a directional coupler for those frequncies
for which (110) and (111) are satisfied.

It is convenient at this point to exemplify with a simple lumped
circuit.

Consider the four-port lumped circuit shown in Fig. 11. The E
matrix of the circuit of Fig. 11 may be calculated by cascading 3 sec-
tions, the first and third containing only capacitors and the second
containing the inductors and mutuals. By proceeding carefully much
labor can be saved using the spectral sets given in the Appendix. The
results are

DIRECTIONAL COUPLERS 61
From (86) the hyperbolic sine and cosine of the propagation matrix is

If this condition is to be satisfied at all frequencies then
(Li — Dy)(C + 2Cu)” = in — Li2)C, (118)
21a, + LC + 2Cy) = 2h — Ly)C. (119)

Both (118) and (119) are satisfied for the following choice: L,, = —Ly.,
C = 0. Thus the circuit of Fig. 11 with C = 0 and Z,, = —JZy,,. (per-
fectly coupled counterwound inductors) is a directional coupler at all
frequencies, provided it is loaded at all ports with the impedance

The impedance z is frequency dependent. The voltage ratios will be
given by (76) and (77),* that is

It is often convenient to express the equations of a directional coupler
in terms of the A, B, C, D matrices directly instead of the Z,. and
matrices. Tor this purpose assume a lossless reciprocal four-port is
characterized in terms of its A, B, C, D matrices which are of the form
of the matrix K of equation (150) of the Appendix. Assume A = D.
The condition A” — BC = I is automatically satisfied if the circuit
is reciprocal. A, B, C, D commute, since they have the same eigen-
vectors. Because all matrices commute they may be treated without
ambiguity as scalars. The open circuit impedance matrix is

Suppose the ports are loaded with equal impedance z. The impedance
matrix ¢ normalized with respect to the matrix zI is

where A”, A’, C*, C” are the eigenvalues of A and C associated with
the sum and difference modes. (See Appendix.) The reflection matrix

* Because for lumped elements F corresponds to rl in using the formulas derived
for distributed elements for circuits with lumped elements one should take 1 = 1.

where R,, R,, R., Rz are the members of the spectral set of %, and
which are given by equations (165) to (168) of the appendix.

To find the condition for self match at port 1 (which because of the
symmetry gives the condition of self match at any port) the eigenvalues
given by equations (125) to (128) are substituted in (1380) and the
upper left corner of 'z is equated to zero. After some algebra this yields

(Apes) CA) sae yes 1
ei CAP ea") = A eee a0, 6 IB)
Equation (131) 1s a quartic in z which may be rewritten
z2(C*C’) + 2(C*A7 + C7 A*)
—2(A*B +B°A)-BB =0 (132)

The solutions of Equation (118) give the values of the impedances which
will match the four ports in terms of the eigenvalues of the matrices
A, B, C. Although a quartic algebraic equation can be solved in terms
of the coefficients, the solution is extremely cumbersome algebraically
and it would be very difficult to see the effect of varying the quantities
A*, A’, B*, B’, C’, C’. A sounder approach is probably to look at
particular simple cases. For instance if

Eq. (182) then reduces to
—2(B*A’) —-B°B’ = 0,
whose solution is
z= —-— > (136)
Obviously there are many possibilities. Some of the solutions are not

is a root of equation (132). This fact can only be seen after a good deal
of algebra, for this reason it is convenient to express (132) in different
ways so that different possibilities may be “seen.’”? With this in mind
equation (129) may be written

Using the spectral set of %, the following alternative expression for
the condition for the self-match of all ports is obtained

It is simpler (although not trivial) to verify that the conclusions asso-
elated with equations (137) and (138) are true from (140) than from
(132).

The reflected voltages caused by an incident voltage at port 1 may
be obtained from equation (130).

Vig CA EO a7. Oa Se
Equations (123) through (148) all are good for any four-port, whether
lumped, distributed, or made with combinations of lumped and dis-
tributed elements.
5.5 A Degenerate Situation:

Consider the circuit of Fig. 12. Notice the structure is not physically
symmetrical. The A, B, C, D matrices of the circuit are

Although in equation (144) A and D are apparently not equal, it turns
out that Y and Z are orthogonal and therefore YZ = 0. Thus (144) reads

PS ps :
E 4 ~ LY¥{I | a
The matrices A, B, C, D commute and satisfy equations (81) and
(82). Thus, although the structure is not physically symmetrical, it
is electrically symmetrical. When one attempts to use equation (87)

to determine Z, one finds that the matrix C~* does not exist because
C = Y is singular. Since Z, does not exist, equations (48), (70), and

Fig. 12— Lumped directional coupler which, when connected to a resistive load,
exhibits directional coupler effect of all frequencies.

The load impedances are frequency invariant, which indicates that
the circuit may be matched with a constant at all frequencies and
should exhibit directional coupler effect at all frequencies when loaded
with a positive resistance of value ~WL,,/C,,. Using equations (141)-
(143) the voltage ratios are found to be

Equation (149) corroborates that the coupler exhibits directional coupler
effect at all frequencies. This example illustrates the use of the direc-
tional coupler equations in terms of the A, B, C, D matrices.

The formulation that has been developed allows the handling of
circuits with both lumped and distributed elements without any changes
because the formulas are good for “black boxes.” For example, for
the circuit shown in Fig. 13, the total E matrix is found by multiplying
the individual E matrices of the sections. The E matrix of section P
or 7 is given by equation (145) while that of Q is given by equation
(29). Once the total E matrix is known, it is partitioned into A, B, C, D
matrices and equation (132) applied to determine the proper z for
terminating the coupler. When the coupler is thus terminated, equa-
tions (141)—(1438) yield the voltage ratios.

Fig. 13— Directional coupler configuration containing both lumped and dis-
tributed elements.

In general, the algebra will get quite unmanageable and one will
have to resort to numerical calculations on a digital computer at
discrete frequencies..° The z may be found by numerically solving
the quartic equation (132) at a set of discrete frequencies and then
realizing it as a driving point impedance through successive approx-
imations, or some similar procedure. The processes of normalizing the
impedance z and of making frequency transformations can be used
very effectively in the realization of directional couplers of this sort.

Although strictly speaking all physical devices are distributed in
space and thus, in general, have transcendental transfer functions for
certain frequency regions, it might be possible to model the devices
accurately enough with conventional ideal lumped elements, or more
generally with elements having given frequency curves, which may
be given analytically or numerically. In this paper we give a matrix
theory for lumped and distributed circuits, keeping this fact in mind.

By using matrix formulation and treating the circuits as black boxes,
it is possible to extend the classic theory of stripline directional couplers
to more general circuits while still keeping many of the concepts (such
as even and odd characteristic impedance) that have been found useful.

The paper makes evident the fact that the concepts of odd and even
mode arise because of the special symmetry of the matrices and that
they correspond to their eigenvectors and eigenvalues. We indicate
in the appendix that when such special symmetry is lost, the odd and
even modes are also lost and it might be necessary to introduce one
set of modes for the currents and another for the voltages. This fact
is not simple to see without the matrix formulation.

effect and gain considerable insight which is useful for realizing direc-
tional couplers lacking double symmetry.

We have given general equations for analyzing and designing black-
box directional couplers in terms of the characteristic impedance and
propagation matrices, and in terms of the transmission A, B, C, D
matrices. The latter may be necessary to analyze circuits whose char-
acteristic impedance matrix (hence even or odd characteristic imped-
ances) does not exist but which have a scattering matrix.

The topic of the actual design of directional couplers with lumped
or with lumped and distributed elements, and more specifically the
design of multisection directional couplers using computer aids, is not
treated because it is the subject of a forthcoming paper.

Here are some spectral properties of the principal matrices in the
paper for ease of reference. *

The quantities associated with the vector U, are called the “even mode”’
or “sum mode.” The quantities associated with the vector U, are
called the ‘‘odd mode”’ or ‘‘difference mode.”

In our main paper, the eigenvalues of the matrices Z, and I which
are in the form of equation (1), are denoted by Z4 and y” for the sum
mode and Z;, and y for the difference mode.

This can be verified by a matrix multiplication similar to that of equa-
tion (154).

Concerning the so-called “‘modes of propagation” of a set of two
coupled lines, when the I matrix of the two lines has the double sym-
metry exhibited by the matrix K of equation (1), the eigenvectors
of the matrix are those given by equation (4). Since the matrix is
symmetrical, the eigenvectors of the transposed matrix I’ are the
same; therefore, one may speak of the ‘‘sum mode voltages and cur-
rents” and ‘‘difference mode voltages and currents.’”’ However, it might
happen that I does not have the form of K in equation (1). Then the
concept of sum and difference modes disappear because the eigenvectors
that will result will not be quite so simple. If he wishes, one may then
speak of ‘“‘first mode” and ‘‘second mode,” associating each mode with
each eigenvector.

coincide. However, if I is not symmetrical then the eigenvectors of
Ir’ will not be the same as those of Fr. It will be necessary to speak of
“first voltage mode,” “second voltage mode,” “first current mode,”
and “second current mode” because the voltage modes will differ from
the current modes if I is not symmetrical. The eigenvalues of a matrix
and its transpose are always the same, hence no distinction is necessary
for the propagation constants of the voltage and current modes.

Firestone, W. L., Analysis of Transmission Line Directional Couplers, Proc.
IRE, 42, October 1954, pp. 1529-15388.

. Fel’dshtein, A. L., Synthesis of Stepped Directional Couplers, Radiotekhnika

i Elektronika, 6, No. 2, February 1961, pp. 234-240; English translation,
Pergamon Press, New York, 1961, pp. 75-85.

and Transmission-Line Stepped-Impedance Filters, Proc. IEEE, 110, Feb-
ruary 1963, pp. 275-281.

Pipes, L. A., Matrix Theory of Multiconductor Transmission Lines, Phil.
Mag., Series 7, 24, July 1937, pp. 97-113.

Huelsman, L. P., Circuits, Matrices and Linear Vector Spaces, McGraw-Hill
Book Company, Inc., New York, 1963.

Montgomery, C. G., Dicke, R. H., and Purcell, E. M., Principles of Micro-
wave Circuits, McGraw-Hill Book Company, Inc., New York, 1948.

Murray-Lasso, M. A., A Digital Computer Simulation of a Class of Lumped
and/or Distributed Four-Ports, Proc. SHARE-A.C.M. Design Automation
Workshop, Los Angeles, Calif., June 19-22, 1967. .

Frazer, R. A., Duncan, W. J., and A. R. Collar, Elementary Matrices, Cam-
bridge University Press, Cambridge, 1963.

This ts a summary of data from an extensive analysts of on-off speech
patterns in 16 experimental telephone conversations. The on-off patterns
are determined by a fixed threshold speech detector having certain rules
for rejecting notse and for filling in short gaps (for example, from stop
consonants). Distributions are obtained for ten events, including talk-
spurts, pauses, double talking (stmultaneous speech from both parties),
mutual silence, etc. Particular emphasis ts placed on events surrounding
interruptions. The entire analysis 1s performed for three speech detector
thresholds, since most of the data are strongly influenced by choice of
threshold. Observations are made about the influence of threshold on the
data, properties of speech invariant with choice of threshold, and differences
between male and female speech patterns.

A statistical analysis of the on-off speech patterns of 16 recorded
conversations has been obtained by a computer program written by
Mrs. N. W. Shrimpton in 1968, and recently modified by the author.
The data can serve the following purposes.

(i) They can illustrate the effect of variation of threshold setting
on the resulting speech data. This problem has been plaguing virtually
all researchers who have attempted to arrive at the “basic” talkspurt-
pause patterns, that is, patterns which represent the subjective on-off
behavior, either as intended by the speaker or as perceived by the
listener. (There is, of course, no certainty that such on-off classifica-
tion actually occurs during normal talking and listening.)

(7) They can guide the design of voice operated devices, such as
conventional echo suppressors? or an adaptive transversal filter echo
canceller,? both of which have critical timing problems in the inter-
vals surrounding interruptions.

(12) They can provide material for building stochastic models of
speech patterns in conversations. Several studies have already used
basic models (such as Markov processes) to approximate talkspurt
and pause durations.*: * These models could be useful in predicting
conversational behavior over special circuits, such as those containing
transmission delay.

Applicability of the data to the above-mentioned purposes is influ-
enced by the source and nature of the specch material and by the
speech detector used to obtain on-off patterns. Section IT is a descrip-
tion of the speech material, and Section III contains a description
of the speech detector. This detector tries to yield patterns as close
as possible to the original waveform, while making certain correc-
tions to make the pattern representative of perceived speech patterns.
These corrections, requiring two arbitrary parameters, include rejec-
tion of impulse noise operation and bridging of gaps caused by stop
consonants. The third parameter, threshold, has such an effect on the
data that the analysis is performed for a range of thresholds. In other
respects however, the detector preserves fine details of timing of
events; for example, the attack and release times are less than 5 msec.

Characterization of speech for speech detectors in the telephone
system is a different problem from characterization of speech for
modeling conversational speech patterns. The data of the present
study are not intended to provide a basis for characterizing speech
for telephone system speech detectors. Within the constraints of the
corrections described in the preceding paragraph, however, the data
can be extended to predict the behavior of certain speech detectors
as explained in Section 5.2.

Of the 16 conversations, eight, obtained from four male pairs and
four female pairs, lasted about 7 minutes each and were documented
in a previous paper.> The remaining eight, also four male and four
female pairs, lasted about 10 minutes each. The subjects talked over
a 4-wire circuit such as illustrated in Fig. 1. The losses were typical
of a long distance call, and there were no degrading factors such as
noise, echo, or delay. The voices were recorded at the zero transmis-
sion level points (0 TLP), determined to be 6 dB “away from” the
transmitters. The 0 TLP is an arbitrary reference level used to es-
tablish relative levels in a telephone circuit.)

The members of each pair were close friends; we have found that
conversation between strangers can be restrained and halting. Their
instructions were as follows.

“Your task in this experiment will be to converse with each other
for approximately 10 minutes. You may talk about anything you
wish, but keep in mind that you will be recorded. The recording will
be kept private and will be used for computer analysis of speech.
We ask that you both talk frequently; if only one person talks the
conversation will be of almost no value to us.”

This method seemed to produce natural conversational speech which
was not restrained by the subjects’ knowledge that they were being
recorded.

(1) The experimental calls are recorded, and can be studied for
contextual material, etc.

(w) The subjects are indeed conversing, rather than momentarily
setting the phone down, or even switching off to other persons. In
short, in the experimental calls, the subjects and tasks are known.

of a call with active interchange, as our experimental calls generally
had. This is especially true in echo suppressor? and speakerphone‘
studies.

The technique of obtaining on-off speech patterns, although already
documented,” is summarized as follows. A flip-flop is set any time
speech (full-wave rectified and unfiltered) from speaker A crosses a
threshold. This flip-flop is examined and cleared every 5 milliseconds,
with the output being a 1 if the threshold was crossed, 0 otherwise.
The resulting string of 1s (spurts) and Os (gaps) is examined for short
spurts; all spurts <15 msec* are erased. After this is done, all gaps
<200 msec are filled in to account for momentary interruptions, such
as those due to stop consonants. The resulting on-off pattern consists,
by definition used here, of talkspurts and pauses. An identical procedure
is used for speaker B.

Three thresholds have been chosen: —45 dBm0Ot (most sensitive),
—40, and —35. These values seemed to bracket the range between
excessive noise operation and insufficient speech operation. The average
peak level (apl)f for all 32 speakers was —18.9 dBm re 0 TLP, 26.1 dB
above the most sensitive threshold. If one prefers VUs, a previous
study’ showed that VUs obtained by Miss K. L. McAdoo (an ex-
perienced VU meter reader) are roughly 6 dB below the apls, hence,
the average VU for that observer would have been near —25 dBm.

Although all means reported here are exact (in that they equal the
total time in an event divided by the number of event occurrences)
the medians are not exact because the measuring intervals are arbitrarily
categorized. For example, the median talkspurt at the —45 dBm
threshold is somewhere between 750 and 800 msec, and is reported
at 775 msec, the interval midpoint. The measuring intervals are roughly
proportional to the lengths of events; the intervals are as short as
10 msec for events $200 msec and as long as 1 second for events
> 6 seconds.

* This was originally 10 msec, but 15 msec seems to be required for good im-
pulse noise rejection.

< Average peak level is a measure of speech level based on the average log
rectified speech voltage.’

(z) Percent of time each person talked, averaged over 32 persons,
obtained for each person by dividing his total speech time by the
length of his conversation.

Table IL shows two measures made on the entire sample of 137.4
minutes of conversation: the percent of time in double talking, and
the percent of time in mutual silence. Notice that mutual silence is
the complement of the event that one or both speakers are talking.

Ten events were defined and measured. Figs. 2 through 11 are cumu-
lative distribution plots of the events. The arrows show which event
is being measured. For example, in Fig. 2, which shows the talkspurt
cumulative distribution, there are three events illustrated and indi-
cated by the arrows.

(v) Alternation silence—the period of mutual silence between the
end of one speaker’s talkspurt and the beginning of the other’s. Event
5 is a subset of 4. If a speaker alternation results from an interrup-
tion so that there 1s no mutual silence period, then an alternation
silence has not occurred. (There are no negative alternation silences.)

(vz) Pause in isolation—a pause in which the other speaker is silent
throughout the pause. Event 6 is a subset of both 2 and 4.

(viz) Solitary talkspurt—a talkspurt which occurs entirely within
the other speaker’s silence. Event 7 is a subset of 1.

(vii) Interruption—if A interrupts B, the time at which A’s talk-
spurt begins determines the start of an interruption. The interruption
terminates at the end of A’s talkspurt, unless B stops and then inter-
rupts A, in which case A’s interruption terminates upon B’s counter
interruption.

10 20 40 60 100 200 400 600 | 2 4 6 6810 20 40 60
——— MILLISECONDS SECONDS
SPEAKER
A DOOM EXQy SON MOM wW OOOO
SPEAKER [eB 70 A ATO Bke--->| B To Ales
B

Fig. 6— Alternation silences for 32 subjects. A to B and B to A have been
combined.

Fig. 7 — Pauses in isolation for 32 subjects. Events from A and B have been
combined.

Fig. 8— Solitary talkspurts for 32 speakers. Events from A and B have been
combined.

SPEECH ROSES QQC.®Meuuw MSG RSP Mowm A837
B KB kB KB>  k--B-=ft TA kB
SPEECH QO MON DOH REX ™|QAQQ4yg Ry

SPEECH MS S| SBS SX “ngn SSS SSn1w _ ESXV““oyr
4 AA =| KA KAA as SA Kye
SPEECH Sy SQ99y Ss“ MEXSSAA’WN SXs9ou

of 6’s talkspurt is entered here, unless A pauses and then again inter-
rupts the same B talkspurt. The first “speech after interruption”
would terminate upon A’s reinterruption, and a second speech after
interruption would begin.

(x) Speech before interruption—if A interrupts B, B’s speech in-
terval up to the interruption is entered here. If A then pauses at time
t; and reinterrupts at time ts, (assuming B continues talking), a new
B speech before interruption (f2-¢,) is entered. If A continues talking
and B pauses and then counter interrupts, the length of B’s pause is
entered as A’s speech before interruption.

Table III lists the mean, median, and number of events for talk-
spurts, pauses, and doubletalks of the entire 137.4-minute conversa-
tion sample. Notice that the talkspurts and pauses represent 274.8
minutes of speech, since the A and B speech samples can be sepa-
rated and placed end to end.

Table IV lists the means of the averages of the categorized events
per person (or in some cases per conversation). For example, the

mean talkspurt length of 1.866 second for a —45 dBm threshold is
the average of 32 numbers, each in turn being the average talkspurt
length for a particular speaker. The o reported* is the standard devia-
tion of the 32 (or 16) averages among speakers (or conversations).

} Values of n were obtained by dividing the total number of events for —40 dBm
threshold by k. These numbers thus give a rough idea of the frequency of these events
per person (or conversation). These same values also apply to Table V.

Table V lists the means of the medians of the categorized events
per person (or conversation). For example, the “average of median’’
talkspurt length of 0.788 second for a —45 dBm threshold is the
average of 32 talkspurt medians, with 0.229 second as the standard
deviation of the 32 medians among speakers.

Table VI lists the means of the averages of the events per person
or conversation, for the men and women separately. (Data on aver-
ages of the medians are available from the author.)

As we mentioned, some researchers have postulated first-order
Markov processes to model speech patterns. A conversation, at any
instant, can exist in one of four states depending on who is talking:

neither, A, B, or both. If the conversation is in state 21 (1 = 1,2,3,4) at
some time ¢#, it may be of interest to know the probability of being
in state 7 (7 = 1,2,3,4) at ¢ + At, possibly to establish a crude Markov-
ian model for distributions of times in each state. Notice, however,
that the simplest Markovian model will predict that each state will
have an exponential distribution, which is a hypothesis not generally
supported by the data. (For example, mutual silence is the event rep-
resenting the first state, and a glance at Fig. 5 shows that this dis-
tribution is strongly colored by the 200 msec fill-in time.)

This paper is primarily a collection of data, and is not intended
to pursue the problem of modeling conversational behavior. We shall
therefore simply list the transition matrix for the —40 dBm threshold*

* Transition probabilities for the other thresholds may be obtained from the
author.

Threshold
Event (dBm)
—45
Talkspurt —A0
—35
—45
Pause —40
—35
—45
Double talk —40
—35
Mutual —45
silence —40
—35
Alternation —45
silence —40
—35
Pause in —45
isolation —40
—35
Solitary —45
talkspurt —40
—35
—45
Interruption — 40
—35
Speech after —45
interruption —40
—35
Speech before —45
interruption —40
—35

* Compares events at a common threshold.
+ Compares males at —40 dBm with females at —45 dBm threshold; similar for
last column. All significance levels from #-test.

in Table VII. The table indicates transition probabilities for 5 msec
time steps. For example, if both are talking, the probability 1s 0.98095
that they will still both be talking 5 msec later. The conversation will,
therefore, leave the state with p = 1.0 — 0.98095 = 0.01905. If a
Poisson termination process* is assumed for terminating the event,
the conversation would leave the state in 1 msec with p = 0.01905/5.

Events of one subject that do not involve interaction with talk-
spurts of his partner include pauses in isolation and solitary talk-
spurts. Data on behavior during double talking should be contrasted

with data on these “isolated” events rather than, for example, the
distribution of all talkspurts, since this distribution includes events
during double talking.

The data are notably influenced by threshold changes. The author
does not believe it is possible, from results reported here, to establish
a single “correct” threshold. It is possible, however, to draw certain
conclusions which are threshold independent (see (vz) and (vit) below,
and Section 5.2.)

5.1 The Data
We know from the data that:
(2) As the threshold 1s raised (speech detector made less sensitive),

events which measure periods of talking tend to decrease in length,
since the longer events tend to be broken up into short ones. These

* A good discussion of Poisson and Markovian processes may be found in Ref-
erence 8.

events include talkspurts, double talks, solitary talkspurts, interrup-
tions, and speech before and after interruption.

(it) As the threshold is raised, events which measure periods of
silence tend to increase in length. These events include pauses, mu-
tual silences, alternation silences, and pauses in isolation.

There are some individual speaker exceptions to these observations.
For example, male No. 14 talkspurt averages are 1.683, 1.620, and
1.759 seconds for —45, —40, and —35 dBm thresholds, respectively.
Two other male speakers exhibit such a reversal for talkspurts. In
general, however, conclusions drawn from the gross data are true of
most speakers or conversations.

(2) The distribution functions of events resulting from periods of
talking seem in general more strongly affected by threshold shifts
than those resulting from silences. Compare, for example, talkspurts
vs pauses, or solitary talkspurts vs pauses in isolation. The mutual
silence distribution, however, seems strongly influenced by threshold
changes.

(iv) For all events, the number of times they occur (7) is notably
influenced by the threshold. This is particularly true of pauses in
isolation, whose distribution remains virtually unaffected while n
changes from 1890 to 3322 for a 10 dB threshold shift.

(v) As the threshold is raised, the number of talkspurts tends to
increase. This trend will obviously be reversed if the threshold be-
comes so high that only a few spurts of energy clear it. But for low
thresholds, as threshold is raised, long talkspurts are apparently being
broken up into shorter segments at a faster rate than that of low
level talkspurts being left below the threshold.

(vt) For any particular threshold, the cumulative distributions of
speech before and after interruption are practically identical, as seen
from a comparison of Figs. 10 and 11.

(vit) Interruptions tend to be much shorter than solitary talk-
spurts, as would be expected because the interrupter might merely
be trying to get attention rather than make a statement. Also, some
interruptions are really not deliberate interruptions but rather ac-
knowledgments, such as “uh huh” and “um.” This effect may be seen,
for example, at the —40 dBm threshold for which 17 percent of the
interruptions are less than 100 msec long, while only 9.5 percent of
solitary talkspurts are less than 100 msec.

(vu) Many speech detectors operate with a hangover, rather than
a fill-in, to bridge short gaps. By shifting the talkspurt distribution
200 msec to the right and the pause distribution 200 msec left, one

can determine the distributions for these events which would have
resulted if a 200-msec hangover were used instead of fill-in. However,
the “interaction event” (double talking, etc.) distributions will be
changed in a manner which cannot be determined from our present
data.

Table VI shows that when male and female speech is compared at
the same threshold, four events show a statistically significant dif-
ference:* talkspurt, pause in isolation, solitary talkspurt, and speech
before interruption. With the exception of pause in isolation, which
is significant only at —45 dBm threshold, these are events resulting
from talking rather than silence.

Some of the apparent difference in male and female speech may
result from a difference in average levels. The average speech level
for the females was 5.94 dB below the average male speech level
(measured in apl). When male speech at —40 dBm threshold is com-
pared with female speech at —45 dBm, and when male speech at —35
dBm is compared with female speech at —40 dBm, thus roughly
compensating for the average 6 dB level difference, the significant
differences previously observed tend to disappear. New events—pause
in isolation, and possibly mutual silence and speech before interrup-
tion—become significant. It thus appears not possible to completely
eradicate differences in male and female speech with a simple level
adjustment, although a level difference does account for differences
observed in certain events.

These conclusions are of particular interest in view of a recent
study by Krauss and Bricker,?> who made measurements of verbal
interaction (measured from transcripts of the conversations) when
pairs of men and pairs of women talked over a circuit containing
voice-operated, fixed threshold devices. The verbal behavior of the
two sexes was significantly different in certain tasks. One wonders
if the devices operated differently on the male and female speech,
as they did in the present study. This could be a contributing factor
in bringing about the behavioral difference reported by Krauss and
Bricker.

We hope that the publication of these data will encourage other
researchers to make further observations leading toward a general

model of the speech patterns occurring in conversations. We also hope
that by emphasizing the events surrounding double talking and other
speaker interaction, it may be possible to draw conclusions regarding
difficulties in conversing on certain circuits that have voice-operated
devices.

I am especially grateful to Mrs. Lynn Evans, who cheerfully spent
considerable time making hand tallies and desk-calculator analyses
of the computer printouts, and to C. J. Gspann, who set up the inter-
face between the speech detector and computer.

1. Brady, P. T. and Helder, G. K., Echo Suppressor Design in Telephone Com-
munications, B.S.TWJ., 12, November 1963, pp. 2893-2917.

3. Jafie, J.. Cassotta, L., and Feldstein, S., Markovian Model of Time Patterns
of Speech, Science, 1d, May 15, 1964, pp. 884-886.

4. Brady, P. T., Queueing and Interference among Messages in a Communica-
tion System with Transmission Delay, Ph.D. Thesis, NYU Department of
Electrical Engineering, June 1966.

5. Brady, P. T., A Technique for Investigating On-Off Patterns of Speech,
BS.T.J., 44, January 1965, pp. 1-22. |

6. Clemency, W. F. and Goodale, W. D., Jr., Functional Design of a Voice-
Switched Speakerphone, B.S.T.J., 40, May 1961, pp. 649-668.

7. Brady, P. T., A Statistical Basis for Objective Measurement of Speech
ree. "BST J. , 44, September 1965, pp. 1453-1486.

Delay on the Efficiency of Verbal Communication, J. Acoust. Soc. Amer.,
41 No. 2, September 1966, pp. 286-292.

This study examines the effects of loading or covering a phased array
with dielectric materials. It studies in detail the effect of dielectric geometry,
dielectric constant, and sheath thickness on the wide angle array performance
of an array of rectangular waveguides in the H and quasi-E plane modes
of scan. We obtain numerical solutions of the integral equations describing
an array covered with thick dielectric material. The resulis show that we
can obtain a match over a wide scan angle for the array by appropriate
use of dielectric geometries, and we discuss the advantages and disadvantages
of several geometries.

The advent of swift aircraft, missile warfare, and the need for
modern radar to accomplish multifunction detection has given im-
petus for a considerable amount of research into phased-array anten-
nas. Such arrays consist of a large group of small radiators in a grid,
frequently a rectangular grid, and, most important, correlated in phase
and amplitude. The radiated beam can be steered by an electronically-
variable linear taper of the phase correlation among elements. (See
Fig. 1).

Considerable knowledge of the behavior and problems of such ar-
rays has been obtained in recent years by experimental and theoretical
study of phased linear and parallel plate arrays. For example, it is
well known that the coupling coefficients between any single excited
element in the array and any terminated inactive element is uniquely
determined by the inverse Fourier series transform of the reflection
coefficient as a function of scan angle determined when all elements
are excited.1 Hence, by studying the array behavior for all possible
linear tapers of phase, we can determine the behavior of the array
for any phase or amplitude distribution among the elements.

Fig. 1—JInfinite array geometry. Top: H-plane scanning in X-Z plane,
Bottom: Quasi-E plane scanning in Y-Z plane.

Of fundamental importance in designing such arrays is a knowledge,
and control, of the mutual coupling (the coupling coefficients) be-
tween elements in an array. For arrays which scan over wide angles,
this coupling very seriously affects the array, so substantial effort has
been made to understand mutual coupling. Because the arrays of in-
terest are very large and consist of very many elements, theoretical
studies have generally assumed the arrays to be infinite in extent.
The usefullness of this approximation for elements located near the
center of a large array has been verified, and in fact, the approxima-
tion is frequently valid to within several elements from the edge.

Because the arrays are generally very large, the coupling between
greatly separated elements, the asymptotic coupling, is also of interest
and has been theoretically studied.” ? In general, planar phased ar-
rays with terminated elements behave like lossy surfaces and have
an asymptotic 1/r? decay of coupling between elements separated by
the distance r along the array surface.

Among other fundamental and interesting early developments 1s
that the transmission coefficient of a phased array with all elements
excited, as a function of scan angle, is directly related and proportional
to the radiation pattern of a single-excited element in the array as a
function of far-field observation angle.t As we mentioned, the two
physical situations (all the elements excited with a linear phase taper
and only one element excited) are uniquely related. Hence, an ana-
lytical formulation for only one or the other situation is necessary.
Although most work has concentrated on the linear phase taper case,
with the consequent application of periodicity conditions and Floquet’s
theorem,! *® some work has proceeded by directly attacking the
case with only a single element excited.”

One of the most advantageous approaches to phased array problems
has been through the use of high-speed computers and numerical solu-
tions of the appropriate integral equations.’ We use this approach
in this study, which attempts to discover some problems and solu-
tions associated with covering phased arrays with radomes. The
principal problem is, of course, to maintain a good impedance match
to the array over a wide scan angle when the phased array radome is
included in the design. Now many antennas have radomes covering
their moving mechanical parts and their interior electrical components.
A planar phased array can be so protected by covering the array with
a dielectric sheath or by loading it with a dielectric material. Hence
our study concentrates on this type of cover. |

Whereas ordinary radomes usually are designed to have the least
effect on the antennas they cover, phased array covers often can be
made to very substantially improve the wide angle scan performance
of the array. In fact, Magill and Wheeler recently have shown that a
dielectric sheath cover can greatly improve the wide angle match of
the array.2 However, their analysis was a transmission line analysis
in the sense that it did not take into account the interaction of the
evanescent modes, generated at the array interface, with the dielectric
sheath.

More recently, Lee® has made an analysis restricted to an array of
thin-walled parallel plates, wherein the interaction of a limited num-
ber of evanescent modes with the dielectric sheath is taken into ac-
count. His results bear out the possibility of improving the array
match with a dielectric sheath.

By using a somewhat different and more powerful analytical ap-
proach, wherein the integral equations describing the array with a

dielectric covering are solved by an accurate numerical technique
(basically Galerkin’s method*’), we may obtain a solution with few
restrictions. The interaction of virtually all the modes at the array
interface with one or more dielectric sheaths is accounted for with
very little more difficulty than that entailed in the solution for the
uncovered array. We made an extensive study of the rectangular ar-
ray shown in Fig. 1 by this method for two modes of scan, the quasi-ld
and H planes of scan.’ We use the term ‘‘quasi-E”’ because the ad-
jacent columns of elements in the bottom part of Fig. 1 are out of
phase by 180°.* We assume that the waveguides in each case are
excited in the dominant mode and we compute the parameter of par-
ticular interest, the reflection coefficient R of this mode, from the
aperture field determined by the integral equations.

We divided the complete study into two parts. The first, considered
in this paper, analyzes the effects of loading the waveguide with di-
electric or covering the array with a very thick sheath where only one
grating lobe, at most, is present in the sheath. Generally speaking, a
thick sheath is used with only lower dielectric constant materials be-
cause with higher dielectric constants in a thick sheath a great and
very frequency-sensitive mismatch arises. Furthermore, the presence
of two grating lobes in the sheath gives rise to surface wave phe-
nomenon. This is the subject of the second part of our study, which
we have relegated to another part.‘ In that paper we plan to deal
with thinner sheaths, multiple sheaths, and some anomolous surface
wave effects that we have observed in arrays with dielectric covers.

Fig. 2 shows the three dielectric geometries that we analyzed. In
each case we assume a moderate fixed waveguide wall thickness’ to
exist in the plane of scan only. The top figure illustrates the “loaded”
array with a symmetrical iris in the aperture (quasi-/ scan only, Fig. 1).
Fig. 2 also shows two other thick sheath covers. By ‘“‘thick’’ we mean
that there is very little interaction between the evanescent modes
generated at the aperture plane (¢ = 0) and the second dielectric
boundary removed from the array interface (¢ = -+d,). We may test
the validity of this assumption by estimating the relative amplitudes
of the evanescent to propagating modes at z = +d, when the second
boundary is not present. We made such a validity check with most

* The waveguides are excited in this manner to reduce to a more easily
numerically tractable one-dimensional integral equation the two-dimensional
integral equations which result in the usual E-plane scan.

Fig. 2— Dielectric sheath geometry. Top: Dielectric loading. Middle: Sheath
inside guides. Bottom: Sheath outside guides.

of the thick sheath results. Notice that with this assumption the input
impedance or reflection coefficient is a periodic function of the distance
ae
Actually the integral equations that are solved require only a slight
modification to go from a “thick”? approximation to a “total” account-
ing of the interaction of higher order modes with the dielectric interface
removed from the aperture by d, . The actual equations solved for the
dielectric loaded case take the forms:”

2¥oe,.(y) = / - p> Vcr Deng 2. ARON CON ACD A
= a 1
for the unknown tangential electric field in the aperture in the quasi-E
plane scan case, and

for the unknown magnetic field over an entire cell (6 X d, see Fig. 1).
The ¢,(x) and e,,(y) are appropriate interior orthonormal waveguide

The wn(%) or un(y) are appropriate exterior orthonormal modes
pertinent to the periodic structure and are obtained from an applica-
tion of the Floquet theorem:

By the laws of transmission of a plane wave through a plane dielectric
boundary (Snell’s law), the quantity (kb sin @) or (kd sin 6) is unchanged
by the presence of the dielectric. Hence the interior and exterior modes
are independent of the dielectric constant. By using the h S c limits
in (1) we also allow for the presence of a thin metallic iris directly at
the aperture plane (see Fig. 1).

The incident electric field in (1) is given by e,,(y) exp (— 7852) and
the incident magnetic field in (2) by ¢,(x) exp (—j8%z), where the
interior modal propagation constants are given by

Now the coefficients of the interior and exterior dyads in (1) and
(2), the Y,, YZ, Z,, and Z/, take a form that is dependent on the
dielectric sheath geometry. For the dielectric loaded case they become
simply the modal admittances and impedances:

For the dielectric sheath cases, with one or more sheaths, the Y,,,
Y/,, Z,, and Z/% become the modal admittances or impedances ap-
propriately referred to the aperture plane (2 = 0). These modal admit-
tances or impedances are obtainable by the usual transmission line equa-
tions. For example, suppose a single dielectric sheath inside the guides
is considered (middle of Fig. 2) in the quasi-E plane scan case. Define

, = 6. in the empty portion of the guide. (9)
Y,, = admittance in the dielectric region.
Y, = admittance in the air region.
Then the coefficients Y,, become

while the exterior Y/ coefficients remain unchanged (unless an exterior
sheath is simultaneously included). The free term on the left becomes,
if we postulate the same incident field as earlier,

Solving equations (1) and (2) by the Galerkin’® (or Ritz) method
means that (1) and (2) are approximated in an NV-dimensional subspace*
of the complete Hilbert space.’ One way of testing the accuracy of this
approach is to choose two very dissimilar subspaces for approximation

* Approximation in an N-dimensional subspace leads to a set of N linear
equations to be solved by well-known matrix inversion methods.

and then compare the attained results. One subspace choice was that
spanned by a set of N equally-spaced pulses’ (that is, we sample the
field at N points along the z or y axis). In this case, R is determined by
averaging the coefficients of the pulses. Other bases used were the first
N modes, e,,(y) and y,,(z). In the case of e,, , R is determined directly
from only the coefficient of the e,,(y) term.

We made a number of additional checks on the solutions. We observed
the convergence of the solutions with increasing NV, and we verified the
conservation of energy between incident, transmitted, and reflected
waves. For certain angles of incidence we compared the results with
those obtained previously by Marcuvitz and Lewin.’*"*” We also checked
some of the results against values for R obtained experimentally.
(See Ref. 5 for example.) We checked the thin dielectric sheath results
(using the exact formula (10) for example) against thick dielectric
sheath results (using the approximate formula (8) and subsequent ap-
plication of the transmission line equations to the dominant mode).

In the thick sheath numerical results which follow, we restrict the
results to include only the cases wherein at most a single propagating
mode exists in any region where relative ¢ is greater than one.

We first consider the dielectric-loaded array (top of Fig. 2). In
reality this may be viewed as an infinitely thick sheath with only one
dielectric boundary interacting with the array interface (2 = 0). The
phase of #, the reflection coefficient, and the amplitude of R are
plotted as a function of scan angle (kb sin @ for the H-plane, and kd
sin 6 for the quasi-E plane), with e as a parameter. A moderate, but
fixed, guide wall thickness in the plane of scan is assumed in all the
data.

Fig. 3 gives some typical results for the H-plane scan direction
with the waveguides loaded with e = 0.9 to « = 3.0. The change in R
with « between curves is smooth. Between « = 0.9 and e« = 1.1, how-
ever, the change in | R | is great. This may be attributed to the fact
that cutoff of the dominant waveguide mode occurs at « = 0.872.

We notice that for e ~ 1.3, the angular response is nearly flat, both
in amplitude and phase. In fact, for all wavelengths examined there
appears to be at least one value of « for which a nearly flat angular
response for #& is obtained. It should be noted that even if the magni-

tude of & were large, the flatness of the response in both amplitude
and phase permits matching the array for all angles in the region of
flat response, at least at one frequency.

The discontinuity in slope of these curves at kb sin 6 = 2r(1—)A/b)
coincides with the onset of a grating lobe at that angle. Notice that
the singularity in the derivative of the modulus of R( | R | ) lies on the
right side of grating lobe incipience, but on the left side for the phase
of R. This is the same as that found for thin walls’ * and it is plausible
that this will lead to the same asymptotic coupling.

The table in Fig. 3 shows the self-reflection coefficient Co and the
coupling to the adjacent element C; when a single waveguide element
is excited and the others merely terminated with a perfect match.
The adjacent element coupling is found to be an order of magnitude
smaller for the H-plane than for the quasi-E plane. For e = 3.0, some
higher-order coupling coefficients for the H-plane case of Fig. 3 are:

Fig. 3 also shows the transmission phase and amplitude curves for
« = 3.0. These curves are in fact the far field patterns when a single
waveguide element is excited? (except that the maximum value for 6
is somewhat less than 90° since 6/A > 4). The very flat phase curve
is an indication that the phase center for the singly excited element
lies in the aperture plane.

For a wavelength further removed from the waveguide cutoff length,
the variation of R between curves of constant e is considerably re-
duced. Fig. 4 gives some typical results for b/A = 0.400. (The dielec-
tric loading here permits an element spacing of less than 2/2.)

Fig. 5 gives typical quasi-E plane results, where R(6), | Co |, and
| C1 | are shown as a function of « from e = 0.8 toe = 1.6. This range
of e« generally depicted all the important characteristics observed.
Furthermore, a slightly greater value of « than e = 1.6 would cause the
waveguides to multimode (more than one mode propagates). Notice

Fig. 4— Dielectric loading inside waveguides for H-plane scanning (element
spacing < A/2).a = 0.937b = 0.3748), b = 0.400X.

Fig. 5— Dielectric loading inside waveguides, for quasi-E plane scanning.
a=b=d= 0.5714), h = c = 0.937d = 0.5354).

in the table of coupling coefficients that the adjacent element coupling
magnitudes are an order of magnitude greater than those for the H-
plane scan case, independent of dielectric constant. This is also true
for higher coupling coefficients for the case depicted in Fig. 5, as shown
below:

This behavior is attributable to both element spacing and polarization.
(By making b < }/2, and with an appropriate dielectric loading,
H-plane results very similar to these quasi-E plane results are obtainable.
For example; see Fig. 4. The \ we refer to here is that which is appro-
priate at the aperture for z > 0.)

The curves of | & | in Fig. 5 show that total reflection occurs beyond
a critical angle (~100°). This occurs because the element spacing d
is less than \/[4 — (d/b)?]?. Again, the infinite slopes for | R | and for
the phase of R, which occur at this critical angle, exhibit the same
behavior found in the thin wall analysis’’” when e = 1. Hence the same
asymptotic behavior of coupling, exp (—jkr)/r?, may be expected for
thick walled dielectric loaded arrays. (Here r is the distance between
the excited and coupled element.)

A point of special interest in connection with these curves is the ap-
pearance of a resonance that occurs near the critical angle. A sharp
dip in | & | occurs precisely at the same angle for which the slope of
the phase of R curves has a maximum. Although the sharpness of the
resonance increases gradually with e¢, it is interesting that there is no
resonance for e < 1.0.

Fig. 6 illustrates the transmission phase and amplitude, or equiva-
lently, the far field pattern of a singly excited element.

In Fig. 7 we illustrate the effect of a capacitive iris loading (that
results when h < c) together with dielectric loading. In this case we

Tig. 6 — Transmission coefficient for dielectric loading inside waveguides, with
quasi-E plane scanning. a = b = d = 0.57144, h = c = 0.937d = 0.53542.

Fig. 7—Dielectric loaded waveguides and iris loaded apertures, for E-plane
scans, a=b= ,d = 0.400A, h = 0.6d = 0.240 and h = 0.937d = 0.3748),
= 0.937d (h = 0. 937d => fully open aperture; h < 0.937d => iris at aperture).

have let a = 6 > o so that a true E-plane scan in a parallel plate ar-
ray is considered. The effect of the iris tends to flatten the phase and
amplitude responses, particularly the phase response, at the expense
of a somewhat greater average | Ff |. The larger average | R | can, how-
ever, be uniformly reduced for all scan angles in a region of flat R(@)
response.

The solutions of the integral equations (1) and (2) are actually
complete solutions of the boundary value problem. The fields as well
as the scattering matrix are determined. The variation of E, in the

aperture as a function of scan angle (kd sin @) is sharper when the
waveguides are loaded. Particularly interesting is the field change near
the critical angle kd sin 6 = d/[4 — (/b) *}?, which has the value 99.57°
for the results shown in Fig. 8. Since the relevant eigenvalue equation
(See p. 157 of Ref. 4.) for [#,(6) + H,(—6)] has a Hermitian kernel*
for |kd sin @| > 99.57°, the phase of [#,(6) + E,(—6)] should be
constant in this region. By observing the phase of the approximate field

*Since no power is radiated for | kd sin 6 | > X / [4 — (A/b)2]4 when the
phasing is directed in the +6 and —é directions simultaneously, the phased
array behaves lke a closed system, a cavity, and it is easy to show that a
Hermitian kernel results. It is well known, then, that the eigenfunctions (the
field solution here) of such a kernel have no varying phase.

solution at angles greater than the critical angle, we may obtain some
evaluation of the errors in this solution. For most angles the errors are
small. However, it is evident that errors in phase do occur for @ near
the critical angle and also for y near the singularity obtained for | E, |.
Nevertheless, since the computation of R is an averaged quantity over
the range of y, these errors do not greatly affect the values obtained
for R.

When we add another dielectric boundary inside the waveguides,
that is, when we place a dielectric sheath inside the guides, the results
obtained for R are substantially different. This is true despite the fact
that we will consider only thick sheaths, in the sense described earlier.
The large change in R(@) behavior occurs because, even when a thick
sheath is assumed, the second dielectric boundary is accounted for by a
bilinear transmission line transformation which changes the input varia-
tion of R with 6. (A linear transformation would leave R(6) functionally
unchanged.)

In the following we will keep the dielectric constant fixed and plot
R(@) versus kb sin @ with d,, the sheath thickness, as a parameter.
The phase of # will be referred to the aperture plane, z = 0, although
R is the reflection coefficient for, or into, the region zg < —d,. The
choice of ¢ in any given figure was made so that the illustrated results
were typical of a wider range of «. With the thick sheath approximation,
the results will repeat every half guide wavelength, so that d, is varied
over only one half wavelength. The minimum d, for which the thick
approximation is valid is determined by the relative decay of the first
evanescent mode in the distance d, . This decay factor, df, is presented
with each curve. It is found, generally, that df ‘<= 0.1 is sufficient for
the thick sheath approximation to be valid. This result is usually
satisfied for some d, = \,/2. Of course, by adding a sufficient number
of multiples of a half guide wavelength to d, , the results must become
valid to any accuracy desired.

In Fig. 9 we have illustrated some typical results with e = 2.0. For
any given e we have found that there exists a thickness, d,, for which
both the amplitude and phase of the reflection coefficient is flat over
the generally useful region of scan angle (region in which only one
lobe radiates). However, as the dielectric constant 1s increased, the

frequency sensitivity of the angular response is increased. This is
manifested, basically, by an increased spread between the curves
shown. Furthermore, since the nearly flat curve also is found to have
the maximum amplitude of R (d, = 0.618b in Fig. 9), the necessity of
matching out a larger | R |, when « is greater, further aggravates the
frequency sensitivity problem that occurs with increasing e«. The re-
sults in Fig. 10, when compared with those in Fig. 9, show the effect
of increasing e.

Qualitatively similar results are obtained in the quasi-E plane scan
case as illustrated in Figs. 11 and 12. The increasing dielectric con-
stant, illustrated by comparing the results in Figs. 11 and 12 (e = 1.2
and « = 1.6, respectively), causes a greater spread between curves and,
consequently, a greater frequency sensitivity.

In addition, we again notice a sharp resonance here (increasingly
sharper with greater «), just as we noticed when only a single dielectric
boundary was present. The primary difference between these results
and those for a single dielectric boundary is that we may keep the
dielectric constant fixed here and vary the thickness, d,, to obtain a
flat response. This is done, however, with the cost of increased fre-
quency sensitivity.

We notice that very similar results are obtained in the quasi-E
plane scan independent of whether the second dielectric boundary is
placed inside or outside the waveguides; that is, independent of
whether there is a dielectric sheath inside or outside the waveguides.
Fig. 13 illustrates a typical result. In this figure the value of « is
held fixed while the sheath thickness (see the bottom figure of Fig. 2)
is varied from curve to curve.

boundary, at 2 = +d,, and the array face is accounted for, with all
the evanescent modes, then the results shown by the circles and tri-
angles are obtained. The results agree very well with those results
in which higher order mode interaction with the second dielectric
boundary is neglected, that is, the sheath satisfies the earlier specified
thickness criteria.

Fig. 14 shows a useful way of estimating what value of d, will be
properly “thick”. Notice first that the rate of decay of higher order
modes away from the array face is a function of scan angle in the
exterior sheath case. Hence, a single value cannot be used as a decay
factor for all 6. In Fig. 14, however, the actual ratios of the first and
second evanescent mode amplitudes to the propagating modes is

Fig. 11— Dielectric sheath inside waveguides, for quasi-E plane scanning.
e-—12,¢,=—10,@a=-b=d = 0.5714, ¢c = h = 0.987d = 0.5354).

Fig. 12—Dielectric sheath inside waveguides, for quasi-E plane scanning.
e=16,¢ = 10,a = b= d = 0.5714), c = h = 0.987d = 0.5354).

plotted versus kd sin 6. Again we see that if RD1 and RD2 (see Fig.
14 for definitions) are less than about 0.1, then the parientar d, is
thick.

A “thick” d, means that, for a given scan angle, the results for R (6)
will repeat periodically so that

for alln > 0. The question remains whether \,, = 27/82" (see equation
(7)) varies rapidly with kd sin 6. This may be answered by examining the
grating lobe structures in the dielectric sheath as compared with that
in free space, as shown in the inset in Fig. 13. This structure is obtained

Fig. 13— Dielectric sheath outside waveguides, for quasi-E plane scanning.
e— 1818,a =b=d = 0.5596), c = h = 0.9387d = 0.52432.

(Actually (12) defines the intersections shown in the Fig. 13 inset.
The total grating lobe structure requires setting the two-dimensional
z-directed propagation constants to zero.) Now the z-directed wave-
length in the sheath is given by

which is a slowly-varying function of 6 in that part of the grating lobe
structure bounded by the dashed lines in the Fig. 13 inset. Only this
region is useful, because the propagating wave in the dielectric is
totally reflected at (kd sin 0)” = k* — (x/b)’(y,> 90.6°). Hence the
pole in \,, doesn’t affect the variation of \,,, and A,. varies very little
with 6, from y, = 0 to ¥? = (k* — (x/b)’), providing that

SE = 3, (13)
ve- &)
2b
We should mention that the quasi-E plane results for the sheath
inside and outside the waveguides are similar, primarily because of
the element spacing that causes total reflection to occur at the defined
critical angle. When the element spacing is changed, very markedly
different results can be obtained. The results depicted in Fig. 15 for
the H-plane are typical of this.
In examining the grating lobe diagram in the inset of Fig. 15 we
notice that, in the shaded region, two waves propagate in the dielectric

sheath whereas only one wave propagates in free space. This is a
potentially very useful operating region for the phased array because

RD = (AMPLITUDE OF Ist EVANESCENT MODE AT Z=ds
ne (: AMPLITUDE OF PROPAGATING MODE

a dg—=0224d
0.1
0.447 d
0.671 d
O beeas 1.006 d
0.3
Ae AMPLITUDE OF 2ND EVANESCENT MODE AT zat)
Gi. =( AMPLITUDE OF PROPAGATING MODE
ra)
aa
0.!
0

Fig. 14— Validity check of transmission line approximations, for quasi-E
plane scanning with a dielectric sheath outside waveguides. e = 1.818, a = b =
d = 0.5596, c = h = 0.937d = 0.5243.

”)
uJ
o:
©) GRATING
wl STRUCTURE
IN FREE SPACE
= GRATING |
ac STRUCTURE
IN SHEATH
Ww
oO
Ww
Y)
<
L
ao.

only one beam will be radiated. However, we find that interference
between the two propagating waves in the dielectric causes some very
interesting anomolous results associated with what might be described
as unattenuated surface waves. This effect 1s markedly different from
that which may occur for a sheath inside the guides (although as a
function of frequency or sheath thickness, as opposed to scan angle,
similar results may occur). Further discussion on this subject is
deferred to another paper.

The results in Fig. 15 for the | & | and the phase of R are shown in
the scan angle region in which only one wave, at most, propagates in
the sheath. In this angular region the results are also very different
than those for the interior sheath in that the response is comparatively
flat for a wide range of sheath thicknesses.

In summary we may state that dielectric loading or covering has a
very substantial effect on the array performance to the extent that an

array and dielectric cover should be designed as an integral unit in-
stead of being designed for minimal effect or match correction. Never-
theless, the additional parameters available in the design of an array
with a dielectric cover can be used to match the array over a wide
angular and frequency region.

No particular method of loading or covering appears to be uni-
versally superior except that when only one dielectric boundary is
present (Fig. 2, top ), the wide angle match appears to be less fre-
quency sensitive than when a sheath or two boundaries are present.
Furthermore, with thick sheaths, the maximum permissible value of «
is small before serious matching problems occur (a large | R | together
with a flat R[6]). Finally, by placing the sheath inside the wave-
guides instead of outside, certain anomolous reflection phenomena,
associated with the exterior sheath grating lobe structure and sur-
face waves, can be avoided."

Although we did not give analytic proof, the amenonl results do
indicate that the asymptotic behavior of the coupling coefficients? is
not altered by the presence of the dielectric materials.

1. Wu, C. P. and Galindo, V., Properties of a Phased Array of Rectangular
Waveguides with Thin Walls, IEEE Trans. on Antennas and Propagation,
AP-14, No. 2, 1966, pp. 163-172.

2. Galindo, V. and Wu, C. P., Asymptotic Behavior of the Coupling Coef-
ficients for an Infinite Array of Thin-Walled Rectangular Waveguides,
IEEE Trans. on Antennas and Propagation, AP-14, No. 2, March 1966,
pp. 248-9.

3. Galindo, V. and Wu, C. P., On the Asymptotic Decay of Coupling for Infinite
Phased Arrays, paper presented at International Union of Radio Science,
Washington, D. C., April 1966. (To be published.)

4, Galindo, V. and Wu, C. P., The Relation Between the Far-Zone Pattern
of the Singly Excited Element and the Transmission Coefficient of the
Principal Lobe in an Infinite Array, IEEE Trans. on Antennas and
Propagation, AP-14, No. 3, March 1966, pp. 397-398.

5. Galindo, V. and Wu, C. P., Numerical Solutions for an Infinite Phased
Array of Rectangular Waveguides with Thick Walls, IEEE Trans. on
Antennas and Propagation, AP-14, 1966, pp. 149-158.

6. Galindo, V. and Wu, C. P., Integral Equations and Variational Expressions
for Arbitrary Scanning of Regular Infinite Arrays, IKEE Trans. on An-
tennas and Propagation, AP-14, No. 3, May 1966, pp. 392-394.

7. Galindo, V. and Wu, C. P., A Variation Expression for the Dominant Mode
Coupling Coefficients Between The Elements in an Infinite Array, IEEE
Trans. on Antennas and Propagation, AP-14, No. 5, September 1966, pp.
637-639.

8. Magill, E. G. and Wheeler, H. A., Wide Angle Impedance Matching of a
Planar Array Antenna by a Dielectric Sheet, IEEE Trans. on Antennas
and Propagation, AP-14, 1966, pp. 49-53.

9. Lee, S. W., Impedance Matching of an Infinite Phased Array by Dielectric
Sheets, IEEE Elec. Letters, 2, No. 10, October 1966, pp. 366-368.

10. Kantorovich, L. V. and Krylov, .V. I., Approximate Methods of Higher
Analysts, Interscience Publishers, New York, 1958.

11. Wu, C. P. and Galindo, V., Surface-Wave Effects on Dielectric Sheathed
Phased Arrays of Rectangular Wave-Guides, B.S.T.J. 47, No. 1, January
1968, pp. 117-142 (next article in this issue).

12. Waveguide Handbook, N. Marcuvitz, ed., MIT Radiation Lab. Series, 10,
McGraw-Hill Book Company, Inc., New York, 1951.

Surface-Wave Effects on Dielectric Sheathed
Phased Arrays of Rectangular Waveguides

A further study of the effects of dielectric slabs on the radiation char-
acteristics of an infinite array of rectangular waveguides has been carried
out. It is found that, in addition to causing substantial and sometimes
beneficial changes in the array performance, the presence of dielectric slabs
can give rise to sharp resonant peaks in the reflection coefficient at certain
scan angles. T’'he occurrence of such resonant peaks at which total reflection
occurs 18 contingent upon the presence of space harmonics which have
surface-wavelike field distribution. Extensive data for both the H and E
planes of scan, when the array 1s covered with a single slab, have been obtained
and are presented here. This paper discusses the influence of the dielectric
constant, slab thickness, and waveguide wall thickness on the resonant
peak location, and points out the relationship between the resonance phe-
nomenon and the surface wave propagation over a corrugated surface. It
also presents some further results for the thin sheath and their extension
to multiple sheaths.

We presented the radiation properties of a dielectric-loaded rectan-
gular waveguide array in some detail in a previous paper? for two
planes of scanning: the H plane and a quasi-E plane. (See Fig. 1.)
We discussed the effects of a thick dielectric sheath placed inside or
outside the array aperture. We concluded that the presence of a
dielectric material can cause a substantial change in the array per-
formance, so that dielectric material should be considered an integral
part of the array when it is designed. Moreover, we demonstrated
that the effects of a dielectric may be used to improve the match
performance of an array, perhaps at the cost of a larger frequency
sensitivity, by a judicious choice of the added physical parameters.

on an extensive further analysis. We completely removed the restric-
tions placed on the parameters in the previous work in order to analyze
the full effects of a dielectric medium. In particular, since dielectric.
slabs can support surface waves, we direct special attention toward
investigating the possibility of anomalous array behavior when dielec-
tric slabs are used to cover the array. Indeed, a resonance phenomenon
is found to exist due to the presence of trapped- (or surface-) wave
type space harmonics at the air-dielectric interface. One important
effect of such space harmonics is to cause the appearance of sharp:
resonant peaks in the reflection coefficient at certain scan angles. The
peaks attain, for all practical purposes, values of unity. We present
results for a dielectric slab of arbitrary thickness placed over an array,
thereby, removing the previous restriction to thick slabs. We also dis-
cuss radiation through a stratified medium.

The approach we use in this work, as in previous ones,?? is based
on an integral equation having either the aperture electric or the
aperture magnetic field as the unknown function. One of the advan-
tages of such an approach is that the integral equation may be easily
and rigorously derived without any limitations from the physical
parameters of the problems, and the approach is readily adaptable
to a more general class of problems such as, for example, an array
covered by a stratified dielectric medium. Although the basic proce-
dures for formulating the integral equation have been discussed else-

where,” 74 in order to facilitate a further discussion of the formula-
tion and to make this paper more or less self-contained, we present
a brief derivation here and point out the modifications necessary
for an extension to a more general situation of stratified media.

The procedure in the integral equation method is first to expand
the fields into the appropriate normal modes in the various regions
and then to match the boundary conditions across the interfaces.
Consider, for example, an infinite array of parallel plates, covered
with a single dielectric slab and scanned in the H plane as shown
in Fig. 2. It is convenient in this case to divide the space into three
regions, the region inside the waveguides, the region inside the dielec-
tric slab, and the free space region. However, as will become evident
shortly, it turns out that the geometry of the problem is such that
it suffices to partition the space into two regions, inside and outside
the waveguides.

The orthonormal modal functions and the modal impedances per-
tinent to the waveguide region are the usual ones given by

are the z-directed propagation constants for a waveguide filled with
a dielectric of dielectric constant e;.

By using the Floquet theorem, it can be easily shown that the
dielectric slab and free space regions have identical modal functions,
owing to the requirement that the tangential fields must be continuous
at all points across the dielectric-free space interface. The normalized
modal functions take the usual form:

Ym(z) = Jie penny, m = 0, £1,
where JT, is the phase shift per unit length. The modal impedances

for these two regions are different, however. They are given respec-
tively by

When the fields in the various regions are expanded into the nor-
mal modes under the situation in which the waveguides are excited
in the fundamental mode of unit amplitude, we have

A time convention of exp (—jwt) is assumed and suppressed for brevity
throughout this paper. The J’s are the unknown modal coefficients.
Waves of all modes travelling in both the positive and negative z
directions are included in the fields for 0 S z S d,, because this re-
gion is situated between two interfaces. Likewise, the fields for z S$ 0
contain waves travelling in the negative z direction due to the scat-
tering at the array aperture.
We find, by applying the boundary conditions at z = 0, that

Hence, on account of the orthonormality among the modal functions,
it follows that

By observing that the right and left sides of (4) are the field expan-
sions with respect to the same set of modal functions, one may 1m-
mediately write

[ieitmts + Toe" 784 = I’
Ae aaa aes ad Ee
Equations (5) show that the air-dielectric interface at 2 = d, is a
simple one in that the m™ order mode in the dielectric region couples

only into the same order mode in the free space region and vice versa.
The implication of this result is that the fields at the air-dielectric
interface are completely determined when the fields at the array aper-
ture are obtained. Therefore, it is necessary only to solve for the
aperture field alone. With this in mind, we may then make use of
(2), (8) and (5) to derive an integral equation having only the aper-
ture magnetic field as the unknown function. Thus,

It is clear from expression (7) that the equivalent impedances Z,/
for the m“ order modes are the familiar input impedance of a trans-
mission line, which has a characteristic impedance Z? , propagation
constant 6,,, and length d,, and is terminated in a load impedance
Z,. Notice also that only quantities pertaining to the m‘ space
harmonic appear in (7). All these facts suggest that the space exterior
to the waveguides may be treated as a single region from the onset,
as observed earlier, provided that the effects of the dielectric slab are
taken into account through the use of appropriate modal impedances.
Moreover, the integral equation given by (6) is readily extended to a
more general situation in which the array is covered or loaded with
stratified dielectric media (or both covered and loaded). Only the
modal impedances need to be modified for this purpose.

As an example, suppose we wish to study the properties of an array
covered by a stratified medium as Fig. 3 shows. Equation (6) still is a
valid integral equation to use. In this case, the Z, are the usual wave-
guide modal impedances, whereas the Z,’ are the modal impedances
as seen at the array aperture of the N layer stratified dielectric medium,
which may be calculated by standard techniques.”

The integral equation appropriate to the quasi-E plane of scan may
be derived in a similar manner. In a quasi-E plane of scan, the scanning
takes place in the plane of the electric field while there is a sinusoidal
field variation in the direction normal to the plane of scanning.*

*Such a mode of scanning results from a planar array of rectangular wave-
guides, which is scanned in the E plane direction with a fixed scan angle of 180°

applied in the H plane direction. This special scan case is considered for the sake
of simplifying the problem. See Ref. 4.

The exterior modal functions for the quasi-E plane scan are the same
as those for H plane scan, according to the Floquet theorem, but the
wave modes which the waveguides support are different. The orthonor-
mal modal functions for this case are

Here we have used c for the internal waveguide width and d for the
element spacing. A sinusoidal variation of sin [ (7/b)x], which applies
to all tangential field components, is omitted for brevity.

The integral equation with the aperture electric field as the un-
known is given by

are the interior modal admittances and the Y/ are the exterior modal
admittances with the presence of dielectric slab(s) appropriately taken
into account.

The method used in solving the integral equations is basically that
of Galerkin,® * the method of moments. Briefly, in this method, the

first step is to expand the unknown function as a linear combination
of NV linearly independent functions. Substitution of the representa-
tion into the original integral equation leads to an approximate equa-
tion. The difference between the left and right sides of the approxi-
mate equation is then required to be orthogonal to the set of functions
individually, thus yielding a set of N equations in N unknowns. The
resulting set of equations may then be inverted with the help of an
electronic computer.

There are different sets of functions one may choose to use in this
approach. Although the choice, aside from consideration of computer
time and convenience, seems to be largely a matter of personal pref-
erence, it is desirable to incorporate as much prior knowledge about
the problem as possible. We have used the set of first N modal func-
tions to obtain most of the results reported here. This step is tanta-
mount to assuming the higher order modal coefficients to be zero. For
example, to solve (6), we set

The expression (9) is then required to be orthogonal to the set of func-
tions y¥(z), 1 = 0, +1, --- , 4M, individually, thus yielding

AL ro)
De , LC anC#, + Zh, 5h = 27,C%, 1=0,41,---,4M.
m=-—AL n=1
(10)
The set of equations (10) may be solved by a standard technique. The
reflection coefficient R is then obtainable from

Since it appears that there is no convenient scheme for estimating
the error in the type of problem being considered here, the following
procedures have been used to ascertain the accuracy of the results.
They include:

(t) Observing the “convergence” of the solution by increasing JN,
the number of functions used to represent the unknown aper-
ture field.

(iz) Using different sets of functions, such as the set of piecewise
constant or pulse functions for the approximation.

(271) Comparing the results with known solutions obtained by dif-
ferent methods where applicable.* * &

(iv) Applying the variational principle to check the adequacy of
the algorithm used in the numerical procedure.

In the previous report,! we placed the emphasis on the results for
the situations where the waveguides are either completely filled with
a dielectric material or loaded with a dielectric slab. We also dis-
cussed some preliminary results for covering the array with a di-
electric sheath. We obtained that data for a range of parameters such
that, at most, one propagating mode could exist inside the dielectric
region, and only relatively thick slabs were considered. Such a choice
of the parameters was necessary in order that an approximation based
on the transmission line theory could be applied.

The results presented here are concerned largely with the effects of
dielectric sheaths covering the array. In particular, we concentrate
on the situation in which more than one wave can propagate inside
the dielectric region so that there will be trapped- (or surface-) wave-
like space harmonics at the air-dielectric interface. Although in prin-
ciple it is still possible in such cases to apply a generalized transmis-
sion line approximation, this approach might not be very convenient
in practice. The modification required for generalization depends on
the number of modes which can propagate inside the dielectric, and
this number is, in turn, dependent on the dielectric constant being
used. Moreover, the minimum slab thickness necessary for a valid
approximate calculation might sometimes become so large that it
would exclude a useful range of practical interest. Therefore, it is
desirable to proceed with the solution of the appropriate integral
equations without introducing any intermediate steps. Thus, the ef-
fects of the air-dielectric interface at 2 = d, on all the modes gen-
erated at the array aperture may be fully taken into account. By
doing so, we are also able to obtain data for comparison with those
calculated by using the transmission line theory, and thus gain a

general feeling for the accuracy of this type of approximation. Some
results of such a comparison are presented in Ref. 1.

The effect of an ordinary dielectric (« > 1) on wave propagation is
to slow down the phase velocity, or equivalently to shorten the wave-
length. Hence, when a dielectric slab 1s placed over a phased array,
two apparent element spacings (in terms of the wavelength) have to
be considered: one inside the dielectric medium and the other in the
free space; the former always greater than the latter. A difference in
the element spacings as seen in the two regions results in different scan
angles for the appearance or disappearnce of grating lobes in the re-
spective regions. Consequently, when the fields are expanded into normal
modes according to the Floquet theorem, there will be a range of scan
angles over which the number of propagating modes inside the dielectric
is larger than that in free space. A mode is said to be propagating when
the corresponding z-directed propagation constant is real. In a linear
array, this propagation constant is given by

with an appropriate e« for each region. It is easy to show that for m <
b/d. < (m + 1/2), where m is an integer and X, is the wavelength of a
plane wave in a medium with dielectric constant ¢«, the number of
propagating modes will change from (2m-+1) forO S$ T,b < 2r(b/d,.—m)
to (2m) for 2r(b/A. — m) S T,b S +. On the other hand, when (m + 1/2)
< b/d. < (m + 1), the number of propagating modes will increase by
one from (2m + 1) to (2m + 2), when the scan angle passes from
0< T,b < 2x(m + 1 — DS.) to 2r(m +1 — B/D.) < T,b < o.

The wave modes which are propagating inside the dielectric and
are evanescent in free space have the same field distribution as that
of a surface wave. Such wave modes have profound effects on the
radiation characteristics of a phased array as we will see in the ex-
amples. It is important to emphasize that a single wave mode of this
type alone is not sufficient to satisfy the boundary conditions. In
other words, all the modes are required to constitute a correct solu-

3.2 H-Plane Scan Results
Tigs. 4 and 5 give the reflection coefficients as a function of scan
for an infinite array of rectangular waveguides covered with a single

Fig. 4— Reflection coefficient R vs scan angle T.b for H plane scan with
a/\ = b/d = 0.5714, and e = 3.0625.

dielectric sheath and scanned in the H plane. The results are obtained
for the following parameters: b/\ = a/\ = 0.5714, e = 3.0625 with the
thickness of the dielectric slab d, varied over one i, at an increment of
\./8, where \, is the wavelength of a plane wave in the dielectric medium.
Notice that the element spacing is measured in terms of the free space
wavelength \, for b/d is the quantity which determines the number
of radiated beams at a certain scan angle. With the given arrangement,
the element spacing is such that the array radiates one beam for 0 S$
T,b5 S 2r(1 — b/d) = 154°, and two beams when 27(1 — b/d) S T,b S

180°. The dielectric constant of the slab makes the apparent element
spacing in the medium to be b/A, = 1. Thus, there are always two
propagating modes inside the dielectric slab for all scan angles. As a
result, we have in 0 S T,b S$ 2r(1 — b/X) a mode which exhibits a
surface-wavelike behavior by being propagated inside the dielectric
and evanescent in free space. The effect of such a mode is to cause
the appearance of sharp resonant peaks in the reflection coefficient at
certain scan angles as is evident from the graphs.

Based on the results presented here and some further calculations
at different wavelengths, we may make the following general observa-
tions:

Fig. 5— Reflection coefficient R vs scan angle 7.b for H plane scan with
a/\ = b/A = 0.5714, and e = 3.0625.

(1) When the thickness of the dielectric slab is relatively small, no
resonant peak occurs.

(72) When the thickness is increased beyond a critical value,
usually in the neighborhood of 3A,/16, a resonant peak starts
to appear at a scan angle close to the value 27(1—b/A), and
the peak is usually preceded by a dip.

(17) Increasing the slab thickness causes the peak to move toward
the broadside direction and the peak becomes sharper.

(iv) A further increase in the slab thickness makes more than one
peak appear.

The dielectric constant used in obtaining the results for Figs. 4 and
5 was chosen to have a value e = (A/b)*. This creates a situation where
one space harmonic possesses a surface-wavelike field distribution in
0 < T,b S$ 27r(1 — D/)). It is possible for a resonant peak to appear
at any scan angle within this range. If a smaller dielectric constant
had been used, the range of scanning over which a resonant peak might
appear would be reduced accordingly. On the other hand, a larger value
of dielectric constant would give rise to more than one surface-wavelike
space harmonic, and it is possible for resonant peaks to appear with
dielectric slab of thickness even smaller than 3),/16.

Recall that when we use a dielectric slab of small dielectric constant
and large slab thickness, the transmission line approximation may be
effectively applied to yield useful results.t The reflection coefficients
calculated under such conditions are periodic functions of the slab
thickness; hence, the calculations only need be performed over a
period, that is, a half guided wavelength. If the dielectric constant is
large so as to permit more than one propagating mode inside the di-
electric, however, the periodic property 1s no longer present, and
separate calculations have to be carried out for different slab thick-
nesses,

Fig. 6 is typical of the transmission coefficients calculated for b/A
= a/r = 0.5714, « = 3.0625 and d; = ,/2. These transmission coef-
ficients are referred to the air-dielectric interface z = d,. The figure
shows both the transmission coefficients for the zero order mode
(corresponding to the main beam) and the first space harmonic (cor-
responding to a grating lobe). These coefficients are defined so that
the sum of the reflection coefficient squared and the square of the
transmission coefficient(s) equals one. Thus, if 7’9 and 7’; denote the

Fig. 6 — Transmission coefficients 7’ vs scan angle T.b for H plane scan with
a/\ = b/d’ = 0.5714, e = 3.0625, d, = 0.5r..

transmission coefficients of the main beam and the grating lobe re-
spectively, then

Clearly, 7’, has significance as a transmission coefficient only in the
scan range 27(1 — D/A) S$ T,b So.

The graph of | 7 | shows a sharp dip at the scan angle of 7, = 70°,
at which the reflection coefficient attains its peak value of one. Notice
that the phase of 7’) exhibits a discontinuity of 180° at this scan angle.
This is so because both the real and imaginary parts of 7) go through
zero and then change sign as the beam is scanned past this scan angle.
This appears to indicate that | 7) | does go to zero rather than approach
zero, or equivalently, that | R | actually attains the value one. The
significance of the difference between | R | approaching one and | R |
actually attaining one lies in whether the match of an array can be
improved by network compensation. Moreover, there is an intimate
connection between the fact that | R | = 1 and the surface wave propaga-
tion along a plane corrugated structure. We discuss this in detail in
Section IV).

Another point which might be of interest is that 7 is the radiation
pattern in the angular range 0 S @ S sin™*(\/2b), when a single ele-
ment is excited with the rest of the elements terminated in the char-
acteristic impedances.” The radiation pattern for the remaining angular
range sin-'(\/2b) < 6 < 2/2 may be obtained from the curve of 7;
in the scan range 27(1 — b/d) S T,b S 7, reflected with respect to
the T.b = a7 axis.

The incident wave in the E plane scan is a TEM wave with the
electric field polarized in the direction normal to the waveguide walls.
Figs. 7 and 8 give the results for both the amplitudes and phases of
the reflection coefficients as a function of scan. The set of parameters
used for obtaining these results is: d/A = 0.5714, c/d = 0.85. The ar-
ray is covered with a single layer of dielectric material flush with the
array aperture. A dielectric constant of e = 3.0625 is chosen for the
same reason as for the H plane, namely to have one surface wavelike
space harmonic present over as wide a scan angle as possible. Rela-
tively thick waveguide walls are used in this case in order that the
waveguides support only the dominant ITEM mode, because the ele-
ment spacing is larger than 4/2.

From the numerical data we may observe that, in the absence of
a dielectric material, the reflection coefficient is relatively flat over a
large range of scan angles, except in the vicinity of the grating lobe
formation angle (in this case T,d = 154°), at which we find a sharp

Fig. 7— Reflection coefficient R vs scan angle Tyd for E plane scan with
d/x = 0.5714, c/d = 0.85 and e = 3.0625.

Fig. 8— Reflection coefficient R vs scan angle T,d for E plane scan with
d/\ = 0.5714, c/d = 0.85 and e = 3.0625

peak. This is in marked contrast with the H plane scan case (see
Fig. 4). The peak value does not reach one, however. When a dielec-
tric sheath of thickness A./8 is used to cover the array, the effect is
to raise the level of reflection almost uniformly over all the scan
angles. In particular, the peak which occurs at the grating lobe forma-
tion angle is seen to reach a value of unity. Although not presented
here, the results for thinner dielectric sheaths show similar tendencies
with the exception that the peak values are not necessarily unity.
While the position for the appearance of the resonant peaks remains
practically unchanged when the dielectric slab is relatively thin, it
does start to shift toward the broadside direction as the thickness is
increased beyond a critical value of about \,/4. A further thickening
of the dielectric sheath eventually leads to the emergence of multiple
resonant peaks.

A quasi-E plane scan is a scan in which the fields have a sinusoidal
variation in the direction perpendicular to the plane of scanning.‘
The reason for considering such a scan condition is that it enables us
to simplify a vector three dimensional problem for a planar array to a
scalar two dimensional one.

The z-directed propagation constants of the exterior modes for this
case are given by

From this expression, it is easy to show that when m < dVe — (d/2b)"/d
< (m + 1/2), there will be (2m + 1) modes propagating for

the number of propagating modes will increase from (2m + 1) to
(2m + 2) as the scan angle is steered through the angle of transition
Td = 2r[m +1 — dWVe — (d/20)’/XI.

With the parameters of the array chosen to be a/\ = b/d = 0.5714,
c/d = 0.937, the array radiates one beaminO S 7T,d S T,,d = (2rd/))-
[1 — (\/2a)]?, and no beam in 7,,d S T,d S wz. Fig. 9 shows some

Fig. 9 — Reflection coefficient R vs scan angle T,d for quasi-E plane scan with
b/\ = d/x = 0.5714, c/d = 0.937, « = 3.825.

results of the reflection coefficient for this array when it 1s covered by a
dielectric sheath with dielectric constant 3.8385. The choice of this
relatively large value of dielectric constant is again dictated by the
interest in making as large a range of scan angle as possible for a mode
in the region outside the waveguides to have surface wavelike behavior.

From the results for both the H plane and £ plane scans as we already
discussed, it might be anticipated that a resonant peak would be
encountered with a relatively thin dielectric slab. The actual calculation,
however, shows that this is not so. In fact, the reflection coefficient
starts to show sharp peaks only when the slab thickness exceeds 3),/4,
where

Moreover, even when a resonant peak is present, the reflection coeffi-
cient as a function of scan is usually quite flat except near the scan
angle at which the resonant peak appears.

So far, we have presented results for the situation where the element
spacing and the dielectric constant were kept constant while the thick-
ness of the dielectric sheath was varied. We studied the effects of the
sheath thickness on the location of the resonant peak in some detail
under these conditions.

Now let us look at some data showing the effect of the waveguide
wall thickness. Fig. 10 shows both the amplitude and phase of the
reflection coefficient as functions of the scan angle for an array which
is scanned in the E plane and is covered by a single dielectric sheath.
We obtained the results with fixed values for the element spacing
b/X = 0.5714, a dielectric constant of the sheath e = 3.0625, and its
thickness d,; = 0.5 A,. However, the waveguide wall thickness (or
equivalently, the size of the radiating aperture) is varied over a
wide range. It is evident from the graphs in Fig. 10 that the size of
the radiating aperture has substantial effects on the reflection coef-
ficient. The scan angle for the appearance of the resonant peak is
shifted toward the broadside direction as the aperture decreases.
Similar shift in the scan angle for the resonant peak has also been
observed in the case of the H plane scan. Moreover, loading the wave-
guides with a dielectric material also causes similar effects. It is ap-
parent that the resonance phenomenon is strongly dependent on the
aperture impedance which is a function of the array parameters.

The results presented earlier indicate that when a dielectric slab
thicker than ),/4 is used to cover an array, an important effect is
manifested by the appearance of sharp resonant peaks in the reflec-
tion coefficient. Such resonances may be avoided if thin dielectric

sheaths are used. A thin dielectric sheath covered array for H plane
scanning has these dimensions: a/A = b/d = 0.5714, and the dielectric
slab has a fixed thickness ds = A/16. We obtained the results for
dielectric constants ranging from 1.25 to 3. As the graphs in Fig. 11
show, even though it does not cause ‘resonance,’ a thin dielectric
slab does have a considerable effect on the array match characteristics.
This is true even when the dielectric constant is close to one. In fact,
in the example shown here for e = 1.25, the reflection coefficient seems
to have a larger variation than that of an uncovered array. How-
ever, by varying the dielectric constant, and perhaps also the thick-
ness of the slab (as long as the slab is thin), it 1s possible to obtain a
suitable combination of these two parameters and thus achieve a flat
response in both amplitude and phase of the reflection coefficient at a
single frequency. Such a possibility has been demonstrated previously.*
The significance of this possibility is that a thin dielectric slab may be
used as an effective means for the wide angle match of an array over
a narrow frequency band.

We have shown the effects of a dielectric slab on the radiation char-
acteristics of a phased array. One outstanding feature of the substan-
tial changes which the dielectric slab brings to array performance is
the appearance of resonance peaks in the reflection coefficient of the
array at certain scan angles. This phenomenon is caused by surface-
wavelike space harmonics at the interface between the dielectric and
free space.

When the fields in the free space region are expanded into a gen-
eralized Fourier series according to the Floquet theorem, each space
harmonic may be viewed as homogeneous or inhomogeneous plane
waves, depending on whether it 1s propagating or evanescent. In the
absence of a secondary boundary at z = d, introduced by the dielectric
slab, the plane waves generated at the array aperture either propagate
away from it if they are homogeneous, or decay away if inhomo-
geneous.

When an air-dielectric interface is introduced at 2 = d,, it causes
each space harmonic generated at the aperture to be reflected and
refracted upon encountering the interface as it travels away from
the aperture. Naturally, only wave modes of the propagating type
are significantly affected, because those of the evanescent type are
usually rapidly attenuated away from. the aperture. Furthermore,

Fig. 11— Reflection coefficient R vs scan angle T.b for an array covered with
a thin dielectric slab (H plane scan, a/\ = b/A = 0.5714, d, = 2/16).

modes which exhibit a surface-wavelike field distribution suffer total
reflection at the interface. The energy reflected from the interface is
returned to the aperture as a wave incident to the array from the ex-
terior region, and is scattered there.

In the presence of a dielectric sheath, there usually exists a range
of scan angles over which there are two or more propagating modes
inside the dielectric while only one propagating mode is present in
the free space region. Under such circumstances, the two propagating

modes inside the dielectric interact as they are multiply-scattered at
the array aperture. The degree of interaction depends on the various
array parameters as well as the scan angle. It 1s then possible that,
given a suitable set of parameters, the multiple scatterings between
the two interfaces might lead to a situation in which a large reflection
is generated at certain scan angles. The numerical examples in Sec-
tion IIT indicate that the reflections can be so high as to reach unity
for all practical purposes. In fact, it has been inferred from the
numerical results for the transmission coefficient that the reflection
can indeed reach the value one exactly.

The fact that the modulus of the reflection coefficient can attain
exactly the value of unity assumes some special significance. For
when this happens, inside the waveguides at a distance away from
the array aperture such that all the evanescent modes are sufficiently
attenuated, the incident and reflected waves combine to form a pure
standing wave. This implies that the tangential electric and mag-
netic fields alternatively go through zero at one half guide wavelength
intervals. Specifically, by writing R = e/, the resultant tangential
electric field under these conditions may be expressed as

Hence, for 2 = —L, such that (a,;L,+9/2) = na, or equivalently Ln
= (nr — ¢/2)/a,, the tangential electric field vanishes. As a conse-
quence, electric conductors may be introduced at these positions with-
out disturbing the field distributions of the entire system. When this
is done, however, the array is then transformed into a corrugated
structure which is completely isolated from the source region. The
solution for the aperture fields obtained under these situations may
then be regarded as the solution of a surface wave which propagates
along a corrugated structure.

There are two aspects of this conclusion which deserve further
comments. A corrugated surface has long been known as a structure
which is capable of supporting surface waves. The characteristics of
such a system are that it supports only the TM type surface wave,
relative to the direction of propagation, in the absence of a dielectric
material and that the period of corrugation be less than a half free
space wavelength in order for the wave to propagate. Moreover, there
is some requirement in regard to the depth of the corrugation for the
surface to be properly reactive.

The introduction of a dielectric sheath above the array aperture or
corrugated surface produces two significant effects. One is to enable
a surface wave to propagate in the TE mode, corresponding to the
case of H plane scan, as well as in the TM mode. The other is that the
surface wave propagation is possible even though the period of cor-
rugation is larger than a half free-space wavelength. It 1s important
to observe that when the corrugation period is larger than a half wave-
length, the zero® space harmonic of the fields outside the corrugation
is a propagating mode. Therefore, it is necessary for the modal coef-
ficient of this mode to be identically zero in order for the fields to
conform to that of a surface wave. The results presented in this calcu-
lation indicate that indeed this is the case.

The scan angles at which resonant peaks appear may be determined
accurately by the transverse resonant method when the radiating
aperture is small. This situation applies readily to the case of the E
plan scan. In this method, the sum of impedances looking towards the
positive and negative z directions at some reference plane is set equal
to zero, thus yielding a characteristic equation. It is convenient to use
the array aperture as the reference plane. When the radiation aperture
is small in comparison to the size of a periodic cell, the “average” im-
pedance looking toward the array side is almost zero. The impedance
looking away from the aperture may be approximated by that of the
lowest order mode, in this case the (—1)st space harmonics. The im-
pedance of this mode is given by

By setting this expression to zero and using the appropriate modal
impedances, we find

When this equation is used to determine the scan angle for the ap-
pearance of the resonant peak, it is found that excellent agreement
with actual calculation is obtainable for aperture size up to 10 per-
cent of the size of the periodic cell. For larger aperture sizes, the
agreement gradually becomes poorer. This is because the impedance
looking toward the waveguide side no longer remains negligibly small.
It appears at this time that an accurate determination of the scan
angle for resonant peaks under such conditions is best carried out by

solving the boundary value problem directly. Fortunately, this is a
relatively easy task nowadays with the help of high speed electronic
computers.

The numerical results obtained so far have revealed that the occur-
rence of sharp resonant peaks is associated with rather thick dielec-
tric sheaths, and such resonance may be avoided by using thin sheaths.
Thus, dielectric covering of an array is still a useful tool for the pro-
tection of the array from its environment. More importantly, the scan
(or incident angle) dependent reflectivity of dielectric slabs may be
utilized to advantage in improving the match performance of an array.
The feasibility of this desirable feature has been demonstrated by
extensive data presented herein and in [1]. It has also been suggested
by other workers.*: 7° Although, based on the calculated results, the
improvement in array match by a thin dielectric sheath is obtainable
at the expense of a higher frequency sensitivity, it seems possible to
achieve a broadband compensation by using multiple thin sheaths.
Further work is being carried out along this line, and the results will
be reported at a later date.

1. Galindo, V. and Wu, C. P., Dielectric Loaded and Covered Rectangular
Waveguide Phased Arrays, presented at International Symposium on An-
tennas and Propagation, Palo Alto, California, December 1966.

2. Galindo, V. and Wu, C. P., Numerical Solutions for an Infinite Phased Array
of Rectangular Waveguides with Thick Walls, IEEE Trans. Antennas and
Propagation, AP-14, March 1966, pp. 149-158.

. Wu, C. P. and Galindo, V., Properties of a Phased Array of Rectangular
Waveguides with Thin Walls, IEEE Trans. Antennas and Propagation,
AP-14, March 1966, pp. 163-172.

. Hildebrand, F. B., Methods of Applied Mathematics, Prentice-Hall, Inc.,
New York, 1954, pp. 451-452.

. Kantorovitch, L. V. and Krylov, V. I., Approximate Methods of Higher
Analysis, Interscience Publishers, New York, 1958.

. Waveguide Handbook, N. Marcuvitz, ed., Radiation Laboratory, Series 10,
McGraw-Hill Book Co., New York, 1950.

. Lee, S. W., Impedance Matching of an Infinite Phased Array by Dielectric
Sheets, Electronics Letters, 2, October 1966, pp. 366-368.

. Galindo, V. and Wu, C. P., The Relation Between the Far-Zone Pattern
of the Singly Excited Element and the Transmission Coefficient of the
Principal Lobe in an Infinite Array, IEEE Trans. Antennas and Propaga-
tion, AP-14, May 1966, pp. 397-398.

10. Magill, E. G. and Wheeler, H. A:, Wide Angle Impedance Matching of a
Planar Array Antenna by a Dielectric Sheet, THEE Trans. Antennas and
Propagation, AP-14, January 1966, pp. 49-53.

11. Collin, R. E., Field Theory of Guided Waves, McGraw-Hill Book Co., New
York, 1960, Chapter 8.

A systematic approach to the analysts and construction of channel
codes for digital baseband transmission is presented. The structure of the
codes ts dominated by the set of requirements imposed by channel charac-
teristics and system operation. These requirements may be translated into
symbol sequence properties which, in turn, specify a set of permissible
sequence states. State-dependent coding of both fixed and variable length
1s a direct result. Properties of such codes are discussed and two examples
are presented,

Binary information to be transmitted over a baseband digital system
must typically be encoded into a sequence of symbols suitable for
transmission through the channel. The structure of such codes is
dominated by the set of requirements imposed by considerations
such as channel characteristics and system operation. Several codes
designed for baseband transmission have been discussed in the litera-
ture, 23 but the analysis of such coding has received little attention.
This paper presents a systematic approach to the description and
construction of codes for such application. The symbol sequence to
be generated by the code is viewed as the output of a sequential ma-
chine with a set of permissible states derived from the imposed re-
quirements. State-dependent codes, both of fixed and of variable
length, are a direct result. Various properties of such codes are dis-
cussed, and two examples are presented.

An advantage of using a sequence-state point of view is the ease
with which it may be adapted to computer analyses, which are feasi-
ble due to the typically short word length of such codes (usually less

than eight symbols), and useful because normal design procedure
often entails the construction of a number of codes, derived from
different sequence requirements, in order to compare properties.

A block diagram of a baseband digital transmission system is shown
in Fig. 1. Binary signals are encoded into a sequence of pulses (sym-
bols) which are transmitted over a channel with regularly spaced re-
constructive repeaters.* Typically, a number of factors strongly
influence the choice of a channel code. Examples are:

(2) The symbol sequence is often required to furnish timing infor-
mation to the repeaters.

(72) The channel may impose restrictions on the spectral shape of
the sequence. Most such systems, for example, have a transmission
null at de. This implies that long strings of SyvOr of one polarity
cannot be tolerated.

(122) Provisions for monitoring the channel error rate during normal
operation may be necessary.

(tv) Pulse sequence requirements must be satisfied independently
of the source statistics.

Requirements such as the above may usually be translated into
limitations on the length of strings of like symbols, specifications of
sets of allowable transitions between symbols (such as when a + 2
pulse must be followed by a — 2 pulse to avoid timing jitter due to
asymmetries), and bounds on the variation of the running digital
sum. The latter is defined as

T
£(T) = D Qi y
t=T'o
where {a;} are the weights assigned to the channel symbols and Ty
is an arbitrary but fixed time.

The objective of the code designer is to specify a code which meets
the imposed conditions while obtaining maximum information ca-
pacity (average number of bits per channel symbol) or alternatively
to optimize sequence properties.

PULSE STREAM
BINARY SUITABLE FOR BINARY
walks CopER LLRANSMISSION) REPEATER REPEATER DECODER

A code may be defined as a mapping from the binary input infor-
mation to the ensemble of symbol sequences which satisfy the im-
posed criteria. The allowable symbol sequences can, in turn, be speci-
fied by a set of permissible states {s;} for each point, describing such
quantities as the present value of the running digital sum, the pre-
ceding number of symbols of a given type, etc. In addition to the {s;},
it is convenient to define a set of allowable words {w,;}, which take the
sequence through a succession of allowable states, one for each sym-
bol in the word. A necessary condition for w; to be permissible is that
there exists at least one state s, such that the states resulting from the
use of w,; at s, are all allowable. In addition, the words to be used in
the code may be restricted to a subset of {w;} which satisfies require-
ments not specified by a particular choice of states, such as various
error-correcting properties. The latter restriction will not be treated
in this paper.

It is assumed in this paper that each code word carries an integral
number of information bits, and that the ratio of bits per symbol re-
mains constant over each word. The latter assumption is a sufficient
condition for synchronous transmission.

The simplest codes of the above type are those of fixed length,
which may be defined as mappings from the set of input binary words
of length logel to the set of allowable words {w,} of length N. These
mappings are generally state-dependent. That is, the choice of a code
word may be a function of the state occupied by the sequence.

Defimtion: W(s;) is the set of words (alphabet) of length N which
take the symbol sequence from state s; through a succession of allow-
able states.

The fact that each word w; must carry a = logeL information bits
implies that words used in a code must terminate in states for which
W (s) contains a minimum of 22 words.

Defintion: The principal states {S,} of the channel sequence are
those states for which W{S,,} contains at least 2¢ words each of which
takes the sequence to another principal state.

ficient condition for the existence of a code. It is possible to implement
computer search routines for finding such states. The appendix con-
tains a description of a search method which was used as an aid on
constructing the two codes in Section 5.2.

A given code actually occupies a subset {c,} of the principal states
at the end of code words. Let W*(c,) C W(c,) be the words actually
used for coding when the sequence occupies the state o, .

The above relation partitions the set of terminal states into alphabet
classes Rz, g = 1,2,...Q. A set of words W*(R,) is associated with
each such class. Each W*(R,) contain 2“ words which are used when
the sequence occupies one of the terminal states in the class Ry (1e.,
each word in W*(R,), g = 1, 2,... Q is associated with one binary
input word b,). It is advantageous to minimize the number of alphabet
classes in order to simplify the coding and decoding circuits.

The coder tracks the state of the symbol sequence, and at the end
of each word, codes the next binary word 6, into a word chosen from
the alphabet corresponding to the state occupied by the output se-
quence. For example, the Paired Selected Ternary code book con-
sists of two alphabets, which contain words of zero or positive weight
and zero or negative weight, respectively. The coder switches to a
different alphabet after use of a word with non-zero weight.

The class of codes defined in Section 2.1 may be uniquely decoded
provided the state of the symbol sequence is known. Frequently, how-
ever, it is not feasible for the decoder to distinguish which member
of a given subset of states the sequence occupies at the end of a code
word. For example, the states may be functions of the instantaneous
symbol sequence sum. Here a single error in detection may result in
an unbounded string of decoding errors. In addition, implementation
of state-dependent decoding requires circuits to track the sequence
state. Thus, state-independent decoding may often be necessary.

Decoding independently of the state is possible if and only if one
binary word 6, corresponds to each code word w; « {w,;}. That is, the
state-dependent mapping of {b.} into {w;} must have a unique in-
verse. If decoding is to be independent of the state within a given
subset of the terminal states, the above mapping must have a unique
Inverse within this subset.

Given a group of terminal states {o,}, q = 1, 2,...Q, and their
associated alphabets W*(o,) [each containing 2“ words], it is desirable
to determine whether it is possible to code so that decoding is in-
dependent of g. The following are necessary and sufficient conditions
for the above:

Condition 1: A sufficient, but not necessary, condition for the existence
of an assignment of binary words to the members of W*(c,) q = 1, 2, °°: Q
so that decoding is independent of t is that for any integers u, v such that
Ilsu<vsQ

Proof: Suppose that a binary word b, has been assigned to w, in the
alphabet W*(c,). Then it is possible to give w, the same binary assign-
ment in W*(o,41) «++ W*(o,). The lack of necessity of the above condi-
tion may be demonstrated by a renumbering of the alphabets.

Condition 2: A necessary and sufficient condition for the possibility of
assigning binary words to the members of W*(o9), 6 = 1, 2 ++: © so
that decoding 1s independent of 61s that for! SuSv Sx S Oand for each

be equivalent (in the sense that they are assigned the same binary word).
The necessity of the condition follows from the fact that if

then some other word w, « W*(c,) must be assigned the binary word
assigned to w, in the uth alphabet. But if w,,w,eW*(c,),1 Sms Tf,
state-independent decoding is precluded.

It is obvious from the above conditions that state-independent
decoding is always possible when there are only two alphabets (as in
Paired Selected Ternary’). An example of a group of alphabets which
do not admit state-independent decoding is shown in Table I.

The received symbol sequence must be correctly partitioned into
blocks of words length N before decoding. This may be done by ob-

serving whether the received words coincide with the words utilized
in a code, by tracking the sequence state to determine whether ends
of the received words coincide with sequence terminal states, or by
looking for the presence of nonallowable sequences of permissible
words.

Attempts to increase fixed length state-dependent code efficiency
with constraints on pulse sequence properties result in increased
word length, and thus in rapidly mounting coder and decoder com-
plexity, reframe times and sensitivity to detection errors. Hence,
variable length codes, which may combine the advantages of short
and long word lengths, are frequently advantageous.

Consider a sequence of symbols generated by a fixed length code
of word length N. The sequence by definition occupies a principal
state at the end of each word. Often, however, a principal state will
be entered in the middle of a word. This offers the possibility of using
the word fragment to convey information, and of starting any of at
least 2* words from this point (by definition of a principal state). The
result, subject to a number of restrictions outlined below, is a variable
length code.

compared to the 256 required for a single alphabet of a fixed length
code of equal efficiency and sequence properties.

The structure of variable length codes required to satisfy sets of
sequence requirements is quite similar to that of fixed length codes.
A number of special features, however, arise from the presence of
words of different lengths.

The requirement of synchronous transmission, coupled with the
asumption that each word carries an integer number of information
bits, implies that the word lengths in a given code are integer multiples
of a basic word length N, where N is the smallest integer for which
the bit per symbol ratio a/N is that of two integers (i.e., if the bit per
symbol ratio is 1.5, the basic word length is 2). The proof of this state-
ment follows from the fact that the word length K,;N must be such
that the number of bits per word Kja is an integer [this paper covers
the transmission of binary data]. Suppose that K; = n + b, where n
is an integer and 0 < b < 1. Then ba and DN must also be integers,
which violates the definition of N.

The set of allowable words may be written as {wy},7 = 1, 2,...,
M, where wi; is the 7th word of length 1.

Defintion: W;(s;) is the set of words of length 2:N,1 = 1,2,...,M,
which may be used in the permissible state s; without violating the
sequence requirements.

A set of principal states may be defined for a symbol sequence as-
sociated with a variable length code in a manner analogous to that
of Section IT.

Defimtion: The principal states {S,} are those allowable states for
which W(S,) contains sufficient words terminating in principal states
to maintain an information rate of a bits per N symbols.

The number of words of length 1V,27 = 1,2... M required to main-
tain a bit rate of a bits per N symbols is given by

where L,[W(s,)] is the number of words in W,,(s;). Equation (1) clearly
indicates the advantage of using as many short words as possible.

The terminal states {c,,} are as defined in Section II. Similarly,
W*(o,,) and W*(c,,) are those members of W;(c,,) and W(c,,), respec-
tively, which are actually used in the code; two terminal states o,,
and o, are in the same alphabet class if and only if W*(c,,) = W*(o,).

for all w,;; , wi, © W*(c.) and w;; ~ wi, . P;(w;,) denotes the first «NV
symbols of the word w,, . The above condition follows from the fact
that w;; « W(S.) implies that w;; , when used in the state S, , takes
the sequence to another principal state.

State-independent decoding of a variable-length code is desirable
for the same reasons as for one of fixed length. The criterion for such
decoding is the same as for a fixed length code — the state-dependent
mapping from the binary words {b,} to the words to be transmitted in
the channel must have a unique inverse. Two considerations which
affect the complexity of this mapping are the presence of words of
length 7V,2 = 1, 2,..., M and the existence of words which consist
of concatenations of two or more shorter words. Such word combina-
tions will be called concatenated words. It follows from Section 3.2
that a concatenated word and its prefix cannot occupy the same
alphabet. From Section 3.2, it can be concluded that a concatenated
word of length 2N is a legitimate word in a given alphabet if no
terminal state is reached with the KNth symbol for0 < K<1A
requirement for state-independent decoding is that the binary words
assigned to a concatenated word be concatenations of the binary
words assigned to the words of which the concatenated word is com-
posed. Section 5.2 contains an example of a variable-length code
which satisfies these criteria.

The process of framing (at the decoder) a symbol sequence gen-
erated by a variable-length code may be divided into two parts:
partitioning the sequence into blocks of length N (the basic word
length), and properly grouping these blocks into words of length iN,
a=1,2, ...,M. Framing may be performed in much the same man-
ner as for a fixed-length code. A misframe condition may be detected
by noting the presence of nonpermissible words, word sequences which

do not occur in the code, and/or the occupation of a nonterminal state
at the end of a word.

It is often possible to substantially simplify the framing process by
ensuring that secondary misframing (the condition that results from
improper grouping of blocks of length N into words) is self-terminat-
ing. This restricts the need for framing to blocks of length N (the
basic word length), and frequently results in superior framing per-
formance compared to a fixed length code of equal efficiency and
sequence properties.

Consider a state-independent decoder which stores all word prefixes
of length iN,2 = 1,2,. .. , M—1, and which operates in the following
manner: If the first fN symbols that are received form a prefix of a
legitimate word, the decoder considers the first (f+1) N symbols.
Otherwise the first fN symbols are decoded into an arbitrary binary
word of length fa. The concatenated words are treated as strings of the
shorter words of which they are composed.

A sufficient condition for secondary misframing to be self-termi-
nating in the above decoder is that (excluding concatenated words)

for all r, k, 7, m, e such that e > r andi > r. E,[wz;] denotes the last
rN symbols of the word w,;. This ensures that the secondary mis-
framing condition terminates with the ending of the word wx.

It is often necessary to monitor the level of channel performance
without interrupting service. Codes with bounded sequence sums
readily lend themselves to this objective, especially in the low noise
environment typical of telephone communications systems. This is
because an error is quite likely to eventually cause the sequence sum
to exceed the imposed bounds. Alternatively, the behavior of the
digital sequence at the bound can be observed. For example, not more
than one consecutive zero (null pulse) might be permitted at extreme
sequence sum values, while strings of length N > 1 may occur at
intermediate sums. Here, the detection of a string of zeros at an ex-
treme sequence sum value indicates that an error has occurred in
detection. An advantage of these methods' is that error monitoring
can be performed without framing.

Standard methods of error monitoring (or error detection) can also
be incorporated in a code of this form. A possibility is the require-

ment that the words have even digital sums, or more generally that
the terminal states {o;} be separated by a minimum “distance.”

(zv) Terminal states and code words are chosen for (if necessary)
state-independent decoding, to optimize spectral properties, error
monitoring and framing statistics. This usually requires much tedious
computation.

It may occur that a given set of sequence requirements are such
that a set of principal states does not exist for codes of word length
N, but that a length of MN yields an acceptable code. In this case, a
variable length code of basic word length N with maximums length
MN might be a possibility. It is advantageous to use as many words
of length N as possible, and to utilize the longer lengths to achieve a
sufficient number of words inn each terminal state.

The following two ternary codes, one of fixed length and one of
variable length, are examples of results of the above procedures.
These codes were designed for possible use in a future high-speed
baseband digital communication system.’ The symbols to be trans-
mitted on the line consist of positive, negative and null pulses. The
main objectives in the design of these codes was to restrict the varia-
tions in the running digital sum of the pulse stream (with positive,
negative, and null pulses assigned values of +1, —1, and O respec-
tively) to avoid de buildup in the channel, and to have as many transi-
tions between symbols as possible in order to provide energy for
self-timing in the repeaters.*

The M848 selected ternary code (Table II) is an example of a
fixed length, three alphabet code of word length three. The digital
Sequence sum varies over six levels, which are arbitrarily labeled
zero to five. These six levels are the allowable states. Terminal states
occur only at values of one to four inclusive. There are three alphabet
classes. R, and Rg correspond to terminal states with sequence sums
of one and four respectively, while R. corresponds to a sum of either
two or three. The choice of M843 words is such that strings of zeros
and symbols of like sign are limited to lengths of four and five re-
spectively.

Coding is performed as a function of the terminal state. For ex-
ample, if 00110100 is to be coded when the value of the sequence sum
is +1, the first four bits are coded as O—+ and the second as 0+-+.
It can easily be seen from Table II that decoding is independent of
the state.

The VL43 code [Table III] is an example of a three alphabet
selected ternary code of variable length. The basic word length is
three and the maximum length six. The bit per symbol ratio is 4/8,
as in MS48. Secondary misframing is self-terminating and decoding is
state-independent. The variable length feature permits a decrease
in the variation of the running digital sum from six to five levels, and
a significant increase in density of transitions. There are three
terminal states (01, 02, 03) each with its own alphabet. W*(c2) con-
tains only words of length three. W*(c,) and W*(o3) each contain 16
words of length six (each carrying eight bits) and 15 words of length
three. Both W*(o,) and W*(o3) contain concatenated words.

Coding is performed in much the same manner as for MS43. For
example, if 110000010000 is to be coded when the state is o2, the first
four bits are coded as ——+, after which the state is o,. The next ian
bits are coded into a six symbol word, -+—+—+.

This paper has presented a description of a number of properties of
multilevel coding for synchronous baseband transmission. The domi-
nant feature of such coding is the set of pulse sequence requirements
imposed by channel characteristics and system operation. It was
shown that multialphabet and variable length codes are a natural
consequence of attempting to attain high efficiency with codes of this

The author is indebted to Miss N. K. Shellenberger for program-
ming support, and to J. M. Sipress for many valuable discussions.

(«) The maximum number of different levels which can be assumed
by the running digital sum (with positive, zero and negative pulses
assigned values of +1, 0, and —1, respectively) is 2Q + 1.

The first step is a search for the set {w} of words of length N which
satisfy the above conditions.
+++ Qy; , with a;; = —1, 0 or +1. It is then required

Next, a set of allowable states s(t,u,v) is defined, where ¢ denotes
the value of the running digital sum of the pulse sequence, w the
preceding number of zeroes and v > 0 (v < OQ) the number of preced-
ing positive (negative) pulses. It is required that

The words {w}, when used in a particular state s(t,u,v) take the
sequence through N states. A word wy; is permissible in s(é,u,v) if
and only if the N resulting states are allowable. Each word in {w}
is tested to determine the states in which it is permissible. The re-
sult is that a set of words {w}:4,» is associated with each state s(t,u,v).

When the above procedure is completed, the program searches for
the principal states. States which contain less than 2* words are dis-
carded. Then, for each remaining state s(t,u,v), those words in
{whtu,o are eliminated which terminate in a state which has been
discarded. If {w}z.4,, contains fewer than 2¢ words after this opera-
tion, s(t,u,v) is eliminated. This procedure is continued until either
no states remain or the routine runs through a complete cycle of
states without eliminating any words or states. In the latter case, the
remaining states are the principal states, {o,}, and the remaining
words are {W (aq) }.

1. Sipress, J. M., A New Class of Selected Ternary Pulse Transmission Plans
for Digital Transmission Lines, IEEE Trans. Commun. Tech., COM-13,
No. 3, September, 1965, pp. 366-372.

2. Davis, C. G., An Experimental Pulse Code Modulation System for Short-
Haul Trunks, B.8.T.J., 41, January, 1962, pp. 1-24.

4, Mayo, J. 8., A Bipolar Repeater for Pulse Code Modulation Signals, B.S.T_J.,
41, January, 1962, pp. 25-97.

5. Dorros, I., Sipress, J. M., and Waldhauer, F. D., An Experimental 224 Mb/s
Digital Repeatered Line, B.S.TJ., 45, No. 7, September, 1966.

Paut T. Brapy, B.E.E., 1958, Rensselaer Polytechnic Institute;
M.S.E.E., 1960, Massachusetts Institute of Technology; Ph.D., 1966,
New York University; Bell Telephone Laboratories, 1961—. He has
worked in statistical modeling of on-off speech patterns and of
speech level distributions, and in studying two-way transmission
of speech and data on circuits containing transmission delay.

P. A. Franaszek, Se.B., 1962, Brown University; M.A., 1964, Ph.D.,
1965, Princeton University. Bell Telephone Laboratories, 1965—. Mr.
Franaszek has been concerned with the analysis of digital data trans-
mission problems. Member, Sigma Xi, Tau Beta Pi, AAAS.

Victor GALINnbo, B.S. (E.E.), 1954, New York University; M.S.
(E.E.), 1962, and Ph.D. (E.E.), 1964, University of California,
Berkeley; Hughes Aircraft Company, 1954-1957, 1958-1960; M.I.T.
Lincoln Laboratory, 1957-1958; Bell Telephone Laboratories, 1964—.
Mr. Galindo has been engaged in applying electromagnetic theory
to studies of microwave transmission devices, antennas, and phased
arrays. Member, Eta Kappa Nu, Tau Beta Pi, IEEE.

Marco A. Murray-Lasso, M.S., E.E., 1962, and Sc.D., E.E., 1965,
both from Massachusetts Institute of Technology; Bell Telephone
Laboratories, 1965—. Dr. Murray-Lasso has done theoretical research
in communication system components, computer studies involving
predicting the behavior of directional couplers, and design work on
control and data reduction systems. Member, IEEE, Sigma Xi, As-
sociation of University Professors of Mexico, and Association of
Mechanical Electrical Engineers of Mexico.

LAWRENCE R. Rasiner, §.B. and §.M., 1964, and Ph.D., in elec-
trical engineering, 1967, all from the Massachusetts Institute of Tech-
nology. From 1962 through 1964 he participated in the cooperative
plan in electrical engineering at Bell Telephone Laboratories, Whip-
pany and Murray Hill, New Jersey. He worked on digital circuitry,
military communications problems, and problems in binaural hearing.
He joined the staff of Bell Laboratories in 1967 and has been engaged

in speech communications research. Member, Eta Kappa Nu, Sigma
Xi, Tau Beta Pi, Acoustical Society of America, and IEEE.

Han-cuiu Wane, B.8.H.E., 1955, Cheng-kung University, Taiwan,
China; M.S.E.E., 1960, University of Notre Dame; Ph.D., 1965, Poly-
technic Institute of Brooklyn; Chinese Government Radio Adm.,
1956-1958; Polytechnic Institute of Brooklyn, 1964-1965; Bell Tele-
phone Laboratories, 1965—. He is currently engaged in developing
microwave components for radio relay systems.

C. P. Wu, B.S., 1956, National Taiwan University; M.S., 1959,
and Ph.D., 1962, Ohio State University; Bell Telephone Laboratories,
1962—. Mr. Wu was an assistant instructor at the National Taiwan
University, 1956-57. He has done research in electromagnetic radia-
tion in anisotropic media. His present work includes phased array
antennas and electromagnetic scattering. Member, IEEE, Sigma Xi.

magazine :: Bell System Technical Journal :: BSTJ V47N01 196801