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M. D. McILROY W. FIGHTNER B. W. KERNIGHAN
Technical Editor L. E. GALLAHER S. G. WASILEW

On the Use of Vector Quantization for Connected-Digit
Recognition
S. C. Glinski

Incorporation of Temporal Structure Into a Vector-
Quantization-Based Preprocessor for Speaker-Independent,
Isolated-Word Recognition

Sojourn Time Distribution in a Multiprogrammed Computer
System
K. M. Rege and B. Sengupta

Application of Decomposition Principle in M/G/1 Vacation
Model to Two Continuum Cyclic Queueing Models—Especially
Token-Ring LANs

A trellis code is a “sliding window” method for encoding a binary data
stream {a‘}, a' = 0, 1, as a sequence of signal points drawn from R”. The rule
for assigning signal points depends on the state of the encoder. In this paper
n = 4, and the signal points are 4-tuples of odd integers. We describe an
infinite family of eight-state trellis codes. For k = 3, 4, 5, --- we construct a
trellis encoder with a rate of k bits/four-dimensional signal. We propose that
the codes with rates k = 8 and 12 be considered for use in modems designed
to achieve data rates of 9.6 kb/s and 14.4 kb/s, respectively.

A trellis code is a “sliding window” method for encoding a binary
data stream {a'}, a' = 0, 1, as a sequence of signal points {x'} drawn
from R”. The set of possible signal points is finite, and this set is
called the signal constellation. The purpose of coding is to gain noise
immunity beyond that provided by standard uncoded transmission at
the same data rate. In this paper n = 4, and the signal points are
drawn from (2Z + 1)*, the lattice of 4-tuples of odd integers. We shall
regard transmission of a four-dimensional signal as one use of the
channel, and we measure the rate of the code in bits per channel use.
The four-dimensional signal space can be realized by using two space-
orthogonal electric field polarizations to communicate on the same
carrier frequency. It is also possible to regard each four-dimensional
symbol as two consecutive two-dimensional symbols.

Ungerboeck’ described a technique called set partitioning, which
assigns signal points to successive blocks of input data. The rule for
assigning signal points depends on the state of the encoder. Unger-
boeck constructed simple trellis codes providing the same noise im-
munity as is given by increasing the power of uncoded transmission
by factors ranging from 2 to 4 (coding gains ranging from 3 to 6 dB).
Calderbank and Mazo” have given a different algebraic description of
trellis codes. Trellis codes with a rate of 4 bits/two-dimensional symbol
have recently been proposed for use in modems designed to achieve
data rates of 9.6 kb/s on dial-up voice telephone lines. These codes
use the signal constellation shown in Fig. 1, which was originally
described by Campopiano and Glazer,® and gain 4 dB over uncoded
transmission at the same rate. In Section III of this paper we describe
the first code in our infinite family. This code has a rate of 8 bits/
four-dimensional symbol and promises a gain of 4.7 dB over uncoded
transmission. The signal constellation consists of 512 four-dimen-
sional signal points. Transmission of two consecutive two-dimensional
signals using one of the proposed trellis codes with a rate of 4 bits/
two-dimensional symbol requires 1024 = 32? four-dimensional signal
points. Furthermore, the restriction of the 512-point constellation to
the first two coordinates, or to the last two coordinates, is the 32-point
constellation shown in Fig. 1. The 0.7-dB improvement in performance
is derived from reducing the average transmitted power.

Fig. 1—Signal constellation for proposed trellis codes with rate 4 bits/two-dimen-
sional symbol.

coding gain of 4.9 dB over uncoded transmission. We propose using
this code in modems designed to achieve data rates of 14.4 kb/s.

We assume that binary data are being encoded at a rate of k bits/
signal point and that the data enter the encoder in k parallel sequences,
fai}, {a5}, --- , {ai}. We assume that the output x! of the trellis
encoder at time i depends not only on the present values a}, a},

- , a}, of the input sequences, but also on the previous »; > 0 bits of
the jth sequence. If v; = 0 for some j, then a}, a}, a}, --- is said to be
a sequence of uncoded bits. The constraint length v is given by
y= >*, v;. The output x’ of the encoder is a fixed vector-valued

encoder and there are 2” states. Figure 2 shows a state transition
diagram for a trellis code with k = 3, vy; = 0, ve = 1, vg = 2. The average
transmitted signal power P is given by

Basic to the trellis codes constructed below is a certain rate 3/4
binary convolutional code with total memory 3 and free distance 4.
The encoder is presented in Fig. 3, which is taken from Ref. 6 (Fig.
10.3, p. 292). The three parallel input sequences determine the output
sequence {v' = (vj, v5, U3, v4)} according to the following rules:

The triple a} ‘a5a5°? is the state of the encoder. The possible transi-
tions between states are shown in Fig. 2. The edge joining state
asa ak? to state abasay! is labeled with the outputs v’' =
(vi, vb, vs, vi) and v' = (0), 04, 05, 04) corresponding to this transi-
tion. Note that

Fig. 2—A state transition diagram for a trellis code with k = 3, »; = 0, vp = 1, v3 = 2.
(Every edge represents two possible transitions.)

We change from 0, 1 notation to +1 notation (0 @ +1 and 1< -—1).
An edge joining two states is now labeled with pairs of vectors +(w,,
We, W3, W4), Where w; = £1, i = 1, 2, 3, 4. This defines a trellis encoder
with a rate of 3 bits/four-dimensional symbol. The minimum squared
distance of this trellis code is simply four times the free distance of
the original binary convolutional code, namely, 16. This is because 0
opposite 1 contributes 1 to the free distance, whereas 1 opposite —1
contributes 4 to the squared minimum distance.

Transmission at the higher rates of 8 and 12 bits/four-dimensional
symbol requires more channel symbols. Indeed, to achieve any coding
gain, we have to use more symbols than are required by uncoded
transmission at the same rate.

Uncoded transmission at the rate 4 bits/two-dimensional symbol
uses the rectangular signal constellation shown in Fig. 4. To achieve
uncoded transmission of a four-dimensional symbol at a rate of 8 bits/
symbol, simply take two copies of this scheme. There are 256 possible
signals and the average power is 4(17 + 3°)/2 = 20. Since the minimum
squared distance between distinct signals is 4, we have

For coded transmission we shall use 2 X 256 = 512 signal points.
Representative signal points are listed in Table I. The remaining
points are obtained from these representatives by permuting the

Fig. 4—The rectangular constellation for uncoded transmission at 4 bits/two-dimen-
sional symbol.

Table I—The signal
constellation for coded
transmission at 8 bits/four-
dimensional symbol. All
permutations of coordinates
and all sign changes are

coordinates and changing signs in all possible ways. For example,
(3131), (1815), and (5111) are all signal points (where x denotes
—x). For every vector w = (Wj, Wo, W3, W4) With w; = +1, i = 1, 2, 3, 4,
let S(w) be the set of 32 signal points (x, x2, x3, x4) satisfying x; = w;
(mod 4), for 1 = 1, 2, 3, 4. The sets S(w) partition the signal
constellation into 16 equal parts. The set S(1111) is shown in Table
II and the other sets are obtained from S(1111) by changing signs. For
example, S(1111) is obtained from S(1111) by changing the signs of
the second and fourth entries. The distance d(A, B) between two sets
of vectors A and B is given by

d(A, B) = min {|x — yl}.
xE€A,yEB
The partition into sets S(w) satisfies the following metric properties:

In Section II we described a trellis code with a rate of 3 bits/four-
dimensional symbol and minimum squared distance d?,i, = 16. To
achieve the higher transmission rate of 8 bits/four-dimensional sym-
bol, we add 5 uncoded bits. There are now eight parallel input se-
quences {ai}, --- , {a}. The sequences {a3}, {a5} determine the state
a ‘a as? of the encoder as in Fig. 3. An edge joining two states that
was originally labeled by the pair of vectors +v is now labeled by the
64 vectors in S(v) U S(—v). This is because there are 64 parallel
transitions between states ab ‘ta}‘ay? and abaai* corresponding to
the 64 possible inputs aja), --- a}. We allow any fixed assignment of
channel symbols in S(v) U S(—v) to inputs aja) --- a}.

Consider the distance properties of the high-rate code. Properties
(M1) and (M2) guarantee that the squared distance of any error event
of length 1 is at least 16. Consider any error event in the eight-state
trellis of length greater than 1. If the squared distance for the low-rate
code is

l
»y Ive - v'I?,
then the squared distance for the high-rate code is at least
>» a*(S(v'), S(v')).
i=l

Property (M2) now implies that the minimum squared distance of the
high-rate code is at least 16.

The average signal power P of the 512-point signal constellation is
given by

P 51D 27.
Thus,
Grn) _ 16
P coded 7 27
and the coding gain (in decibels) is
(Ginin/P) coded 16/27
10 1 Ta aan ame (Goa = 4.
ee eae i lo8i0 4/20 : ut ob

Uncoded transmission at the rate of 6 bits/two-dimensional symbol
uses the 64-point rectangular constellation shown in Fig. 5. To achieve
uncoded transmission of a four-dimensional symbol at a rate of 12
bits/symbol, simply take two copies of this scheme. There are 64? =
2" possible signals, and the average signal power P is 4(17 + 3? +
5? + 7°)/4 = 84. Thus,

Fig. 5—A rectangular constellation for uncoded transmission at 6 bits/two-dimen-
sional symbol.

Table II[—The signal constellation for coded transmission at 12 bits/
four-dimensional symbol. All representatives are taken from $(1111).

For coded transmission we use 2 X 2” = 2} signal points. As in
Section III we partition the signal constellation into 16 sets S(w)
according to congruence of the entries modulo 4. Each set S(w)
contains 512 signal points. Representative signal points are listed in
Table III, where the representatives are all taken from S(1111).

To achieve the transmission rate of 12 bits/four-dimensional sym-
bol, we add 9 uncoded bits to the low-rate trellis code described in
Section II. There are now 1024 parallel transitions between states
ab aias? and ajajai' in the eight-state trellis. If the edge corre-
sponding to this transition was originally labeled +v, it is now labeled
with the 1024 vectors in S(v) U S(—v). The metric properties (M1)
and (M2) guarantee that the squared minimum distance of the high-
rate code is equal to the squared minimum distance of the low-rate
code, which is 16. An easy calculation shows that the average signal

To achieve coded transmission at the rate of k bits/four-dimensional
signal, we add k — 3 uncoded bits to the low-rate trellis code described
in Section II. There are 2" parallel transitions between states
ab tai as? and ajasai" in the eight-state trellis. Coded transmission
requires 2”*! signal points. The points of the lattice (2Z + 1)‘ lie in
shells around the origin consisting of 16 vectors of energy 4, 64 vectors
of energy 12, and so on (see Table III). The 2"*' signal points are
obtained by taking all points of energy 4, 12, 20, --- and just enough
points of a final shell to bring the total number up to 2**!. The signal
constellation is partitioned into 16 sets S(v) according to congruence
of the entries modulo 4. Each set contains 2’~° signal points. Edges in
the eight-state trellis originally labelled +w are now labeled with the
2*-2 vectors in S(v) U S(—v). The metric properties (M1) and (M2)
guarantee that the minimum squared distance of this trellis code is
16.

Consider the asymptotic performance of this family of codes. For
simplicity suppose that the signal constellation of each code in the
family is a complete union of energy shells. If x is a vector in the
lattice (2Z + 1)*, then ||x||? = 4 (mod 8), since ||x||* is the sum of four
odd squares. A classical result, due to Jacobi and to Legendre, is that
every positive integer of the form 8n + 4 is a sum of four odd squares
in o(2n + 1) ways, where o(m) is the sum of divisors of m. The
generating function

m odd
expresses the fact that there are 16 o(m) vectors of energy 4m in the
lattice (2Z + 1)*. The factor of 16 arises from the 16 possible sign
changes.

2:22
146 + o(m)= aD O(n log n).
out
For uncoded transmission we use just half this many points. The signal
constellation is the set of all 4-tuples x = (x1, xo, x3, x4), where x; =
+1, +3, --- , +(2a — 1). The number of points in this constellation is

1. G. Ungerboeck, “Channel Coding With Multilevel/Phase Signals,” IEEE Trans.
Inform. Theory, /7-28, No. 1 (January 1982), pp. 55-67.

2. A. R. Calderbank and J. E. Mazo, “A New Description of Trellis Codes,” IEEE
Trans. Inform. Theory, /T-30 (December 1984), pp. 784-91.

4, G. D. Forney, Jr., et al., “Efficient Modulation for Band-Limited Channels,” IEEE
J. Selected Areas Commun., SAC-2 (August 1984), pp. 632-47.

5. S. G. Wilson, H. A. Sleeper, and N. K. Smith, “Four-Dimensional Modulation and
Coding: An Alternative to Frequency-Reuse,” in Science, Systems and Services
for Communications, P. Dewilde and C. A. May, eds., New York and Amsterdam:
IEEE/Elsevier-North Holland, 1984, pp. 919-23.

6. S. Lin and D. J. Costello, Jr., Error Control Coding: Fundamentals and Applications,
Englewood Cliffs, N.J.: Prentice-Hall, 1983.

7. T.M. Apostol, Introduction to Analytic Number Theory, New York: Springer-Verlag,
1976.

nn?
= cy + O(n log n).
The estimates for partial sums are obtained using Euler’s summation
formula (see Ref. 7, p. 54). To prove

A. Robert Calderbank, B.Sc. (Mathematics), 1975, Warwick University,
England; M.Sc. (Mathematics), 1976, Oxford University; Ph.D. (Mathe-
matics), 1980, California Institute of Technology; AT&T Bell Laboratories,
1980—. Mr. Calderbank is presently employed in the Mathematics and Sta-
tistics Research Center. His research interests include applications of mathe-
matics to electrical engineering and to computer science, and algebraic meth-
ods in combinatorics.

N. J. A. Sloane, B.E.E., 1959, and B.A. (Hons.), 1960, University of Mel-
bourne, Australia; M.S., 1964, and Ph.D., 1967, Cornell University; AT&T
Bell Laboratories, 1969—. Mr. Sloane is the author of four books. He was the
editor in chief of the IEEE Transactions on Information Theory from 1978 to
1980. In 1979 Mr. Sloane was awarded the Chauvenet Prize by the Mathe-
matical Association of America. Fellow, IEEE.

Astigmatic launchers that would permit a single earth station antenna to
communicate with all the satellites along the geosynchronous arc have been
fabricated and measured at a frequency of 100 GHz. Good agreement between
measured data and calculated values has been obtained for astigmatic correc-
tions required by feeds displaced 18 and 29 degrees from the focus.

For high-capacity satellite communication systems, communication
satellites are placed at different locations along the geosynchronous
arc with the usual practice of using a separate earth station antenna
to communicate with each satellite in the system. If both the satellites
and earth stations are equipped with multiple-beam antennas, these
high-capacity communication systems could be achieved by using a
single earth station antenna and simultaneously communicating with
all the satellites in the system.’

Measurements and theory have indicated that the geometry of an
offset Cassegrainian antenna results in an ideal configuration?” for
both earth station and satellite antennas. Since the antenna aperture
has no blockage, this significantly reduces the sidelobe levels and, in
turn, reduces interference. However, since only one of the multiple

beams can be aimed along the axis of the antenna reflector, the
remaining beams must be displaced from the focus. The loss in

Fig. 1—Astigmatic correction can be obtained by a feed with two different phase
centers, F and F’, in the two principal planes of its beam.

efficiency that these displaced beams exhibit is a function of the
amount of astigmatism introduced as a result of the displacement
from the focus. By using a feed with different phase centers in the two
principal planes of its beam, shown in Fig. 1, one can eliminate the
astigmatic loss.® For efficient operation over a wide band of frequen-
cies, both the two phase centers (F, F’) and the beamwidths in the
two principal planes (0, 0’) must be frequency independent.

Earlier work by Dragone’ and Chu® shows that frequency-independ-
ent astigmatic corrections can be obtained by combining a small horn
with two cylindrical reflectors whose focal lengths are such that a
magnified image of the feed horn is produced over the main reflector
aperture.” However, this feed arrangement is not very suitable for an
earth station antenna supporting multiple beams, since the distance
between the two phase centers is fixed and cannot be varied after the
feed is constructed. If one were to vary this distance, the beamwidths
in the two principal planes would change, causing a reduction in
aperture efficiency. This is an important restriction, for it implies that
a given feed can only be used at certain locations in the vicinity of the
focus; at other locations corresponding to other beam displacements,
different feed parameters are required, necessitating the design of
different feeds for different displacements. In addition, a large feed
aperture is required, along with relatively large dimensions for one of
the two reflectors.

while maintaining constant beamwidth in the two principal planes.
Using the principles of Ref. 6, two launchers (one long and one short)
were designed and fabricated for operation at 100 GHz.

The electroformed feed horn used with the launchers is shown in
Fig. 2. To permit polarization rotation in the rectangular aperture of
the feed horn, the horn was fabricated in two sections—one section
tapering down to a square aperture and the second section tapering
down to rectangular waveguide. The complete feed horn can be seen
mounted together with the mixer on the short astigmatic launcher
shown on the right of Fig. 3.

The long astigmatic launcher, shown without feed horn on the left
of Fig. 3, has the top parallel plate removed to display the first reflector
that would be illuminated by the feed horn. Both the short and long
launchers shown here have identical pairs of reflectors; the only
difference is the length of the parallel plates.

The cylindrical wave radiated by the feed horn positioned at the
focus of the first reflector is guided to the first reflector by the parallel
plates. After being reflected, the wave is again guided by the parallel
plates in the direction of the second reflector. After some distance the
parallel plates are truncated and the aperture illuminated by the
reflected cylindrical wave is defined by this truncation. The width of

iter aaahisimatitbain itech dacniniieiboniataNnisibuahtiinlslnaiait ites inntKitioniash ialieitbMtdahitisiitaieniscwunititarrisatsiitataunniosiiincttisilistiatasnthitifsiuaserimiiteasssitat tn nubtinititbascuiuaidiisatatncha’isshuseshubivbausiiisiedniinitecs

Fig. 3—The short astigmatic launcher (right) and the long astigmatic launcher (left).

the aperture is defined by the spacing between the two parallel plates;
the wave radiated by this aperture illuminates the second cylindrical
reflector.

To produce an image of the feed horn aperture over the aperture of
the main reflector, the distances of the phase centers of the feed horn
and the truncated parallel plate aperture must satisfy the optical thin
lens equation.®

Using the newly constructed anechoic chamber at the radio range
facilities at Holmdel, New Jersey, measurements were made of the
radiation characteristics of the two 100-GHz astigmatic launchers
depicted in Fig. 3. The measured data,* both amplitude and phase, are
presented in Figs. 4 through 7 for both launchers and are shown by
the solid curves. The dashed curves are the calculated theoretical
values.

* These data were obtained at a distance equivalent to that of the main reflector of
an offset Cassegrainian Antenna, i.e, the data represent the actual illumination at the
aperture of the main reflector.

Fig. 4—(a) Measurements for an electric field parallel to the plates of the short
launcher, and the theoretical amplitude distribution. The phase center lies in front of
the aperture. (Cont.)

To avoid any difficulty in visualizing the polarization of the electric
field, the electric field will always be referred to the plane of the
parallel plates. Therefore, the electric field will be either parallel to or
orthogonal to the plates. Further, the insert in each of these figures
shows the position of the launcher with respect to the plane of
measurements. The location of the phase center is also shown on the
insert.

The amplitude measurements shown by the solid curve of Fig. 4a
were obtained with the electric field parallel to the plates. The agree-

Fig. 4—(b) The electric field remains parallel to the plates, but the launcher is rotated
90 degrees. The phase center lies behind the aperture.

ment with the theoretical calculations is very good. An examination
of the phase measurements shows a maximum phase variation of the
order 6 degrees. Over much of the aperture, the phase is essentially
constant.

For the amplitude data shown by the solid curve of Fig. 4b, the
electric field is still parallel to the plates. As shown by the insert on
this figure, the launcher is rotated 90 degrees. The dashed curve is
that for a uniformly illuminated aperture. The measured data, given
by the solid curve, are in good agreement. From the phase data shown
here, one sees that the phase change over the aperture is of the order

Fig. 5—(a) Measurements for an electric field orthogonal to the plates of the short
launcher, and the theoretical amplitude distribution. The phase center lies behind the
aperture. (Cont.)

8 degrees. However, over most of the aperture the phase is essentially
constant.

Measurements made with the electric field orthogonal to the plates
are shown in Figs. 5a and b. As shown here by the dashed curves, the
agreement between the calculated values and the measured data is
very good. From the phase measurements shown on these figures, one
can see that the phase is essentially constant across the aperture. The
two pairs of measured data shown by Figs. 4 and 5 are essentially
identical.

Fig. 5—(b) The electric field remains orthogonal to the plates, but the launcher is
rotated 90 degrees. The phase center lies in front of the aperture.

The feed horn and mixer assembly were then transferred to the long
astigmatic launcher shown at the left of Fig. 3. The amplitude and
phase measurements obtained with the long launcher are shown in
Figs. 6 and 7 by the solid curves. Again, the dashed curves are the
theoretical calculations.

The data shown in Figs. 6a and b were obtained with the electric
field parallel to the plates. As shown by the inserts on these figures,
the launcher was rotated 90 degrees to obtain the second set of data.
Here again, the measurements agree well with the calculated values.
The phase deviations across the aperture are very small.

Fig. 6—(a) Measurements for an electric field parallel to the plates of the long
launcher, and the theoretical amplitude distribution. The phase center lies in front of
the aperture. (Cont.).

The data for the long launcher were completed with the measure-
ments shown in Figs. 7a and b. For these data the electric field is
orthogonal to the plates. Again, the agreement between amplitude
measurements and theoretical calculations is very good and the phase
variations across the aperture are small.

A comparison of the data presented in Figs. 4 through 7 for both
the short and long launchers confirms the fact that the phase center
separation for this arrangement of launcher can indeed be varied and
still maintain constant beamwidth in the two principal planes. The

Fig. 6—(b) The electric field remains parallel to the plates, but the launcher is rotated
90 degrees. The phase center lies behind the aperture.

Fig. 7—(a) Measurements for an electric field orthogonal to the plates of the long
launcher, and the theoretical amplitude distribution. The phase center lies behind the
aperture. (Cont.)

frequency independence of these launchers was checked over a 20-
percent band with no discernible change in beamwidth.

Fig. 7—(b) The electric field remains orthogonal to the plates, but the launcher is
rotated 90 degrees. The phase center lies in front of the aperture.

It is with appreciation that I acknowledge G. F. Apgar, for the
successful results obtained are due to the attention he gave to the

fabrication of the feed horn and launchers. The design discussions
with C. Dragone are acknowledged.

1. L. C. Tillotson, “A Model of a Domestic Satellite Communication System,” B.S.T.J.,
47, No. 10 (December 1968), pp. 2111-37.

2. C. Dragone and D. C. Hogg, “The Radiation Pattern and Impedance of Offset and
Symmetrical Near-Field Cassegrainian and Gregorian Antennas,” IEEE Trans.
Ant. Propag., AP-22, No. 3 (May 1974), pp. 472-5.

3. M. J. Gans and R. A. Semplak, “Some Far-Field Studies of an Offset Launcher,”
B.S.T.J., 54, No. 7 (September 1975), pp. 1319-40.

4, Ta-Shing Chu and R. H. Turrin, “Depolarization Properties of Offset Reflector
Antennas,” IEEE Trans. Ant. Propag., AP-21, No. 3 (May 1973), pp. 339-45.

5. R. A. Semplak, “100-GHz Measurements on a Multiple-Beam Offset Antenna,”
B.S.T.J., 56, No. 3 (March 1977), pp. 385-98.

6. C. Dragone and R. A. Semplak, “An Antenna Feed Arrangement for Correcting for
Astigmatism,” U.S. Patent 4482898, issued November 1984.

7. C. Dragone, “An Improved Antenna for Microwave Radio Systems Consisting of
Two Cylindrical Reflectors and a Corrugated Horn,” B.S.T.J., 53, No. 7 (Septem-
ber 1974), pp. 1351-77.

9. C. Dragone and M. J. Gans, “Imaging Reflector Arrangements to Form a Scanning
Beam Using a Small Array,” B.S.T.J., 59, No. 3 (March 1980), pp. 449-61.

10. C. Dragone, “First Order Treatment of Aberrations in Cassegrainian and Gregorian

Ralph A. Semplak, B.S. (Physics), 1961, Monmouth College; AT&T Bell
Laboratories, 1955-1985. Mr. Semplak’s main research interest is in studies
of atmospheric effects on micro- and millimeter-wave propagation. He was a
member of the Radio Communications Research Department when he retired
in 1985. Member, Sigma Xi, Commission F of the International Union of
Radio Science (URSI/USNC).

Recent work at AT&T Bell Laboratories has demonstrated the efficacy of
vector quantization in greatly reducing both the computational and memory
requirements of isolated-word recognition systems. This efficiency is obtained
at the expense of a marginal decrease in performance, and thus is an attractive
approach. The purpose of this paper is to report on the results of a series of
experiments in the application of vector-quantization strategies to a small-
vocabulary, connected-word recognition task. Several strategies are investi-
gated, including the use of speaker-trained code books versus universal code
books, the use of binary and higher-order tree searches versus full searches of
these code books, and the quantization of both test and reference frames
versus reference frames only. For various strategies, the effect on error rate of
varying the code-book size is also reported. Results indicate that the vector
quantization approach is attractive for linear predictive coding-based con-
nected-digit recognition.

In the area of speech coding, the technique of Vector Quantization
(VQ) has recently been successfully applied.’ In the standard ap-
proach, a speech signal is framed and a feature vector is extracted
from each frame. Each element of the feature vector is then separately
quantized. In other words, the value of each element is replaced by its
closest match from a set of discrete values. The set of values is chosen
to minimize some error criterion while reducing the number of bits

required to identify the element value. This feature vector is typically
a set of Linear Predictive Coding (LPC) coefficients and perhaps an
energy term. In the VQ approach, a feature vector is extracted as
before. The entire vector of features is then quantized by replacing
the vector with its closest match from a set or “code book” of feature
vectors. Similarly, the entries in the code book are chosen to minimize
some distortion measure while reducing the number of bits required
to identify each frame of speech. In addition to reducing the number
of bits required to represent each feature vector, by using a suitably
compact code book in place of a large set of reference vectors, it is
possible to greatly reduce the number of comparisons made between
the unknown (test) feature vector and the stored-feature vectors. Since
this comparison (or distortion measure) is the current bottleneck in
many speech recognition systems, the VQ approach is quite effectively
used in them. In fact, VQ has been used heavily in several different
approaches to speech recognition, including Hidden Markov Modeling
(HMM)?* and Dynamic Time Warping (DTW),° among others.*>

The purpose of this paper is to report on the results of a series of
experiments in the application of VQ strategies to a small-vocabulary,
connected-word recognition task.’ These strategies include the use
of Speaker-Dependent (SD) versus Speaker-Independent (SI) code
books; the use of binary and higher-order tree searches versus full
searches of these code books, as suggested in Ref. 2; and the quanti-
zation of both test and reference frames versus reference frames only.®”
The effect on error rate of varying the code-book size is also reported.
In all experiments, the reference and test data sets are disjoint, and
code books are trained with reference data. That is, speaker-dependent
reference templates are quantized by vocabulary-dependent code books
that are either speaker dependent or speaker independent. All test
strings consist of deliberately spoken, connected words.

In Section II some useful terminology is presented. In Section HI
theory is developed. In Section IV experimental results are presented.

d(a;, V;) distortion between reference and training frames
p(l) perturbation vector

{Tu (i)} set of training vectors whose best match is code word 1
Cy (2) number of training vectors in {T'y(i)}.

The VQ procedure consists of two main parts: code-book generation
and the classification of test frames. In both cases, a code-book search
procedure must be employed to classify the training or test (unknown)
frames, respectively. The code-book generation process will be pre-
sented first and the search strategy second.

The basic generation procedure discussed in Refs. 1, 2, and 10 is
employed and is illustrated in Fig. 1. The flowchart is from Ref. 10,
but has been generalized to allow B-way splitting of centroids versus
the original two-way splitting (B is a power of 2 in this work). The
overall goal is to find a set of code words {a}, such that the mean
distortion D,, produced by replacing each of the J training frames by
its closest match from the code book {a}, is minimized. Succinctly
stated,

The distortion d can be calculated using the likelihood-ratio-dis-
tance metric as follows:

where row m, column n of matrix V; contains (1/N)R;(|m — n|). N
is the window length and R is the autocorrelation vector. Only the
spectral-shape information is used in the quantizer.

of size M = B (where B is typically 2), and then successively splitting
its code words B ways to obtain larger and larger code books until a
code book of the desired size is obtained (see outer loop of Fig. 1).
Each successive code book is corrected by iteratively classifying the
training set, and adjusting each code word to be the centroid of the
subset of training frames which best match that code word (inner
loop). The final iteration is determined as that for which the mean
distortion changes by less than a threshold ¢ in relation to the previous
mean distortion. Typically, « = 0.01.
Initialization of centroids is as follows:

where k = LogoB and 0 <1 Ss B — 1. This splits the LPC reflection
coefficient space on the first Log.B coordinate axes. For instance, for
B=4,

accomplishes a B-way split of each reflection coefficient vector in code
book k. The factor 6 is typically set to 0.01. LPC model stability is
ensured by requiring that -1 <r<1.

The new centroid (code-word) computation depicted in Fig. 1 is
done by averaging the normalized autocorrelations of those training
frames that best match a given code word:

This computes the centroid as a spectral average of training frames.”
The total-average-distortion calculation is

The only program step in Fig. 1 not yet discussed is the classification
of training vectors. Since this is common to both the code-book
training and ultimate quantization of test vectors, it will be discussed
in the next section.

In the original development of VQ, a full search of the code book
was used to classify training (or test) frames. The best matching code
word in a k-bit code book is

For classification of the test, k = Log,.M*. During the code-book
generation, k varies from a = Log,B to a = Log,M*, depending on
how many times the code words have been split.

However, Fig. 2a shows that a binary tree search is also possible. In
any given code book (level), only two code words are searched; those
that were split from the best matching code word in the previous code
book (level). The search is initialized by setting [(O) = 0, and then
executing

The search stops at code book k = Log2M* for test classification, or
at an intermediate k during code-book training.

The tree search may be carried out using a different branching
factor, B (which in this work is specified 1 to be an integer power of 2).
Initialize J(O) = 0, and then

The stopping criteria are the same as for eq. (9). If LogzM* ¥ af for
some positive integer 8, then the branching factor to the last code
book (level) is not B = 2°, but B = 2’, y < a, where

Figure 2b shows an example of the tree-search classification proce-
dure for a branching factor of 4 (B = 4).

Note that a full search of a k-bit code book requires M = 2" distortion
- computations, while a tree search of the same requires only B Logg M
distortion computations with a branching factor of B. Thus, a great
computational savings is incurred by using the tree-search strategy.
The two search strategies suggest two approaches to connected-word
recognition via DTW, as pictured in Fig. 3. In the following discussion
an LPC distortion will be referred to as a distance, which is more
common in the jargon of speech recognition. In Fig. 3a (full-search
strategy) a vector of distances representing the distances between the
current test frame and the entire code book is passed to the DTW
procedure.® The vector is sufficient to represent the distance between
any test and reference template frames. Since a full search was

employed, all these distances are available along with J(k), the
best matching code word. In the tree-search strategy (see Fig. 3b) this
is not the case. Only the index J(k) and corresponding distance
d(ar.), V;) are produced. Thus, it is necessary to precompute and store
a cross table of distances between all pairs of code words. This implies
that both reference and test frames are quantized to the code-book
entries, unlike in Fig. 3a, where solely the reference frames are quan-
tized. Thus, the tree-search strategy will require much less computing
power but probably do more poorly than the full-search strategy,
because of both the implicit quantization of the test signal and errors
introduced by the tree search.

In the HMM approach, however, for the procedure analogous to
time warping (Viterbi scoring, for example), the only information
necessary to pass from the quantizer is the index J(k) (no distance
vector). Thus, there is no need for the precomputed distance matrix
in the tree-search strategy, and the only error introduced results from
the tree search.

In practice, during the classification of training frames, it may
happen that certain code words may not match any training frame. In
other words, the set {7y,(i)} may be empty for one or more i, and the
count Cy(i) = 0 for the same i. This is especially a problem as the
code-book size approaches the number of training frames. In this
work, the problem is addressed by simply ignoring the empty clusters
(which reduces the effective code-book size). During a full-search
strategy, the empty clusters are simply skipped. In a tree-search
strategy, however, it may happen that as the tree is traversed, a “dead
end” is reached before the last level is reached. That is, all branches
from a given node lead to code words corresponding to empty clusters.
In this case, the test frame is classified as the code word corresponding
to the last nonempty cluster encountered as the tree was traversed.

The advantage of this approach is that experiments may be run
where the number of training frames is fewer than the specified
number of code words. This is especially true for the speaker-depend-
ent code-book experiments in this work.

To evaluate the various VQ strategies, the connected-word recog-
nizer of Ref. 7 was employed. It can be shown that this recognition
algorithm performs identically to the level-building algorithm of Myers
and Rabiner! in the case where no constraint is imposed on the
number of words in the test. When this constraint is removed, the
level-building algorithm performance is marginally superior.’? The

distance measure used in the recognizer was the likelihood ratio of eq.
(2), clipped at 2.0. The test data consist of 40 random strings, spoken
by each of 18 different speakers, 9 male and 9 female. These are equal
numbers of strings of length 2, 3, 4, and 5 digits. The strings used in
this work are all deliberately spoken (about two words per second).

The reference data consist of 33 reference templates (about 960
reference frames) for each speaker. Three groups of 11 templates each
contain the digits 0 through 9, and one repetition of the digit with an
unreleased final consonant “t.” The first group includes standard
isolated-word reference templates; while the second contains em-
bedded, noncoarticulated, deliberately spoken references; and the third
contains embedded, coarticulated, normally spoken references. Em-
bedded templates are those extracted from between two other tem-
plates in running speech. The three groups are used to compensate for
differences in speaking rates and coarticulation effects.

Both the reference and test data were passed through an endpoint
detector. All the speech data are identical to those used in previous
work.!? Note that all experiments entail speaker-dependent, con-
nected-word recognition, and code books are trained with the reference
template data. Reference templates and test data sets are disjoint.

For each of the 18 speakers, a speaker-independent code book was
generated from the reference templates of the other 17 speakers. These
code books are “vocabulary dependent,” since the same vocabulary
was included in the code-book training data as in reference and test
data.

For speaker-dependent code books, the reference database for an
individual speaker was used to train the code book for that speaker.
This corresponds to an implementation in which reference templates
for a given speaker are created first, these templates are then used to
train a code book, and then the reference templates are quantized with
the same code book. Thus, in this case the code books are both speaker
dependent and vocabulary dependent.

¢ = 0.01 (distortion change threshold)
6 = 0.01 (perturbation factor).
The baseline string error rate for the recognizer is 5.4 percent. This

baseline error rate is for the case where no vector quantization is used.

In the first experiment, speaker-independent code books were used
to quantize reference frames only (one-sided quantization’). Code
books of rate 4, 6, 8, and 10 bits were investigated for both binary and

Fig. 4—Recognizer performance and code-book distortion for one-sided VQ and SI
code books.

full-search classification during code-book generation and reference
quantization. The mean distortion of the binary and full-search code
books is shown in Fig. 4. This is the mean distance D; between the
training vectors and their best matching code words. The recognizer
performance, as a function of code-book rate, is shown in Fig. 4 and
listed in Table I. Results show that: (1) Only the use of a 10-bit, full-
search code book results in performance approaching the baseline
performance. Other rates result in approximately a doubling (or worse)
in error rate. (2) About a 1-bit savings in code-book rate is obtained
by using a full-search strategy versus a tree-search strategy, while
recognizer performance is held constant.

The second experiment was identical to the first except that speaker-
dependent code books were used in place of speaker-independent code
books. The mean distortion of the binary- and full-search code books
is shown in Fig. 5. The recognizer performance as a function of code-
book rate is shown in Fig. 5, and listed in Table I. Results indicate the
following: (1) The recognizer shows little loss in performance at rates
of 8 bits and higher. This represents a compression on the reference

Fig. 5—Recognizer performance and code-book distortion for one-sided VQ and SD
code books.

database of about 4:1 for each speaker. (2) The binary tree-search
strategy rivals the full-search strategy with regard to recognizer per-
formance at rates of 6 bits and up. (3) For a given performance, the
SD code book incurred a bit-rate savings from 1 bit (at a low-bit rate)
to 3 bits (at a high-bit rate) over the SI code book.

The third experiment was identical to the second, except that both
reference and test frames were quantized (two-sided quantization’).
Recognizer performance is plotted in Fig. 6 and listed in Table I. The

Table II—Recognizer performance using tree-search strategies
Branching Factor

results show the following: (1) Two-sided quantization in the full-
search case rivals one sided for bit rates of 6 and up. (2) The binary-
search case does not compete as favorably with the full-search ap-
proach as in experiment 2.

In experiment 4, experiment 2 was repeated for the 8-bit code-book
rate, while the branching factor B in the tree-search approach was
varied between 2 and 16. Only reference frames were quantized (SD
code book). Performance data listed in Table II and plotted in Fig. 7
indicate that performance drops off only slightly with an increasing
branching factor.

Four experiments were run to determine the efficacy of VQ as
applied to a small-vocabulary, connected-word recognition task. Re-

Fig. 7—Recognizer performance for tree-search strategies using one-sided VQ and
SD code books at 8 bits per frame.

sults show that both full- and binary-search, one-sided VQ, at code-
book rates of 6 bits and higher, offers an attractive cost-performance
trade-off in an LPC-based connected-digit recognizer. Full-search two-
sided VQ is also a competitive approach. For tree-search strategies,
recognizer error rate is relatively insensitive to branching factor.

1. Y. Linde, A. Buzo, and R. Gray, “An Algorithm for Vector Quantizer Design,” IEEE
Trans. Commun., COM-28, No. 1 (January 1980), pp. 84-5.

2. A. Buzo et al., “Speech Coding Based Upon Vector Quantization,” IKEE Trans.
Acoust., Speech, and Signal Processing, ASSP-28, No. 5 (October 1980), pp. 562-

. L. R. Rabiner, S. E. Levinson, and M. M. Sondhi, “On the Application of Vector
Quantization and Hidden Markov Models to Speaker-Independent, Isolated Word
Recognition,” B.S.T.J., 62, No. 4 (April 1983), pp. 1075-105.

4. S. E. Levinson, L. R. Rabiner, and M. M. Sondhi, “An Introduction to the
Application of the Theory of Probabilistic Functions of a Markov Process in
Automatic Speech Recognition,” B.S.T.J., 62, No. 4 (April 1983), pp. 1035-74.

5. J. Shore and D. Burton, “Discrete Utterance Speech Recognition Without Time

6. T. K. Vintsyuk, “Element-wise Recognition of Continuous Speech Consisting of
Words of a Given Vocabulary,” Kibernetika (Cybernetics), 7, No. 2 (1971), pp.
361-72.

. J. S. Bridle, M. D. Brown, and R. M. Chamberlain, “An Algorithm for Connected
Word Recognition,” Proc. 1982 ICASSP, pp. 899-902.

. H. Sakoe, “Device for Recognizing and Input Pattern With Approximate Patterns
a for Reference Patterns on Mapping,” U.S. Patent 4,256,924, March 17,
1981.

9. K. Shikano, “Spoken Word Recognition Based Upon Vector Quantization of Input
Speech,” Trans. Committee on Speech Research (December 1982), pp. 473-80.

10. L. R. Rabiner, M. M. Sondhi, and S. E. Levinson, “Note on the Properties of a
Moa pomniieet for LPC Coefficients,” B.S.T.J., 62, No. 8 (October 1983), pp.
2603-16.

11. C. S. Myers and L. R. Rabiner, “A Level Building Dynamic Time Warp Algorithm
for Connected Word Recognition,” IEEE Trans. Acoust., Speech, and Signal
Processing, ASSP-29, No. 2 (April 1981), pp. 284-97.

12. S. Glinski, “A Comparison of Dynamic Time Warp Algorithms for Connected Word
Recognition,” paper presented at 106th meeting of Acoustical Society of America,
San Diego, Calif., November 1983.

13. L. R. Rabiner, A. Bergh, and J. G. Wilpon, “An Embedded Word Training Procedure
for Connected Digit Recognition,” Proc. 1982 ICASSP, pp. 1621-4.

Stephen C. Glinski, B.S. (summa cum laude, Electrical Engineering), 1977,
Lowell Technological Institute, Mass.; M.S., 1978, Ph.D., 1981 (Electrical
Engineering), University of Illinois at Urbana-Champaign; AT&T Bell Labo-
ratories, 1982—. While at the University of Illinois, Mr. Glinski was a research
assistant at the Computer Education Research Laboratory involved in the
area of speech synthesis. At AT&T Bell Laboratories, he has been with the
Speech Recognition Group engaged in the development of efficient algorithms
and architectures for connected-word speech recognition. Member, IEEE, Eta
Kappa Nu.

Incorporation of Temporal Structure Into a
Vector-Quantization-Based Preprocessor for
Speaker-Independent, Isolated-Word
Recognition

Recently a new structure for isolated-word recognition was proposed in
which a separate Vector Quantization (VQ) code book was designed for each
word in the vocabulary. The word-based VQs were used as a front-end
preprocessor to eliminate word candidates whose distortion scores were large;
a dynamic time-warping processor then resolved the choice among the re-
maining word candidates. The above scheme worked very well for small
vocabularies; however, the major flaw was the lack of temporal information in
the word-based VQ processor. As such, as the vocabulary grew in size and
complexity, the ability of the VQ processor to resolve among similar sounding
words decreased dramatically, and the effectiveness of the proposed recogni-
tion structure similarly decreased. To alleviate this difficulty a technique for
incorporating temporal structure into the preprocessor is proposed. In partic-
ular, the probability density function of the time of occurrence for each vector
in the code book is estimated from a training sequence. In the recognizer, the
spectral distance score of the VQ is combined with a temporal distance score,
for each frame in the word. An evaluation of the modified recognizer showed
slightly improved performance on the digits vocabulary and greatly improved
performance on a vocabulary of 129 airlines terms.

There has been a great deal of interest recently in isolated-word
recognition techniques that maintain high performance, but do so at

low computational cost.’° The reason for this renewed interest in
“low-cost” recognizers is the desire to implement such systems on
conventional microprocessors, where the computational power is no-
where near as great as needed for the “higher-cost” recognition sys-
tems.

One of the most promising of the low-cost recognizers is the Vector-
Quantization (VQ)-based recognizer, originally proposed by Shore and
Burton,” and modified by Burton et al.* and Pan et al.° The basic idea
in this recognition system is to design a separate VQ code book for
each word in the vocabulary, based on a training sequence of several
tokens of each word by one or more talkers. In the original Shore and
Burton implementation,” the recognizer chose the word in the vocab-
ulary whose average quantization distortion (according to its particular
code book) was minimum. In the implementation of Pan et al.,° the
word-based VQs were used as a front-end preprocessor to eliminate
word candidates whose distortion scores were large; a Dynamic Time
Warping (DTW) processor then resolved the choice among the re-
maining word candidates.

Both of the above implementations of the word-based VQ recognizer
worked very well for small vocabularies; however, as the vocabulary
size and/or complexity grew, the ability of the VQ processor to resolve
among similar sounding words decreased dramatically, and the effec-
tiveness of the recognizer similarly decreased.

The major problem with the word-based VQ processor, for large
vocabularies, was its inability to use temporal information, i.e., to
integrate information about the times of occurrence of the speech
sounds with the fact that the sounds occurred within the word. One
simple method for incorporating this type of temporal information
was proposed by Buzo et al.,° and developed by Burton et al.* In this
approach, gross temporal information was incorporated into the re-
cognizer by subdividing each input word into R nonoverlapping re-
gions, using a separate code book for each region. In this manner each
word was characterized by R code books, obtained from a training
procedure in which a similar subdivision of each training word was
made. Burton et al. reported good success with this method.*

An alternative procedure for incorporating temporal information
into the VQ-based preprocessor is proposed in this paper. In particular,
for each vector in each word-based code book, the probability density
function of the time of occurrence (on a normalized time scale) is
estimated from the same set of training sequences used to derive the
code-book vectors. In the recognizer, the spectral distance score of the
VQ preprocessor is combined with a (scaled) temporal distance score,
for each frame in the word. We use the structure of a preprocessor to
screen out unlikely word candidates (based on the combined spectral

and temporal distance), and resolve the fine word distinctions with a
DTW processor.

An evaluation of the modified recognizer structure, described above,
was performed using both a small vocabulary (the 10 digits), and a
moderate-size vocabulary (129 airline terms). Both vocabularies were
tested in a speaker-independent mode, i.e., code books and probability
histograms were generated from speaker-independent training sets.
Results showed recognition error performance on both vocabularies
was comparable to that of the best recognizers; however, computational
costs were comparable to those of a “low-cost” recognizer.

The organization of this paper is as follows. In Section II we discuss
the proposed recognition algorithm, which combines temporal infor-
mation along with the spectral information of the word-based VQ
preprocessor. In Section III we describe an experimental evaluation of
the new recognition structure. In Section IV we review the results of
the evaluation, and discuss potential ways of lowering the cost of the
recognizer even further. Finally, in Section V, we summarize our
findings.

A block diagram of the proposed recognizer is given in Fig. 1. The
input speech signal is digitized at a 6.67-kHz rate, the word endpoints
(beginning and ending frames) are detected, and a Linear Predictive
Coding (LPC) analysis is performed on all frames within the word.
The LPC analysis is an eighth-order analysis of 45-ms frames (300
samples), spaced every 15 ms (100 samples) along the word. Each
overlapping 45 ms section of speech is windowed using a Hamming
window, and an eighth-order autocorrelation analysis is performed
(giving nine autocorrelation values per frame). The results of the LPC
analysis are the set of frame log energies (suitably normalized to the
peak log energy of the word), H;, 1 < i < J, and the LPC vectors a;,
1 <1 <I, where J denotes the number of frames in the word.

Fig. 1—Block diagram of isolated-word recognizer that incorporates a word-based
vector quantization preprocessor and a dynamic time-warping processor.

The word-based LPC preprocessor uses the analysis results (i.e., the
frame log energies and the LPC vectors) to eliminate all unlikely
candidates from further analysis. Thus the output of the preprocessor
is a list of candidates for the unknown word. A DTW processor then
decides among the words in the candidate list using a conventional
dynamic time-warping alignment of the unknown test word against a
set of stored word reference patterns. A K Nearest Neighbor (KNN)
decision rule chooses the word whose average DTW distance of the K-
best word patterns is smallest. In cases where the list of candidates
from the preprocessor contains only a single choice, the DTW proc-
essor is bypassed and a final decision is made by the preprocessor.

A block diagram of the word-based VQ preprocessor is given in Fig.
2. Each word in the vocabulary is characterized, in the preprocessor,
by a code book, B, and by a temporal probability table, P. The code
book consists of a set of LPC vectors (supplemented by a log energy
scalar), by, 1 = k Ss L, which characterize the LPC vectors of a training
set of multiple occurrences of the word. The code-book vectors are
chosen by a VQ design algorithm, which minimizes the average dis-
tortion between the training vectors and the code-book vectors.”®
Typically, for word recognition applications, values of L (the total
number of vectors in each word code book) range from 4 to 32.

Fig. 2—Block diagram of the word-based vector quantization preprocessor, which
combines spectral and temporal distance scores.

The temporal probability table, P, is derived from both the code
book, B, and the word training data in the following way. The elements
of P are the values p;(t), defined as:

pDx(t) = probability that the code-book vector, k, occurs at normal-
ized time t = i/] within the word.

Thus the values p,(t) (where suitably quantized values of t are used
in practice) constitute a temporal probability table for the code-book
vectors. The way in which values of p;,(t) are obtained, from the
training set, is as follows:

1. Each training sequence is linearly warped to a fixed length, f =
40 frames. (‘Thus values of p;(t) are obtained for t = 1/40, 2/40, --- ,
40/40.)

2. Each vector of each linearly warped training sequence is vector
quantized, using code book B.

3. At each time ¢, all code-book vectors whose spectral distortion
distance score is within a fixed threshold, A, of the minimum distortion
score for the frame are considered to have occurred.

code-book vector k occurred at time t (as defined in step 3 above), and
the number of times any code-book vector occurred at time t, over the
entire training set for the word. In this manner Y‘4_, p,(t) = 1 for
all t.
To illustrate the results of the above procedure, Fig. 3 shows the
resulting p,(t) temporal probability tables for an L = 8 vector code
book for the word six with a training set of 150 tokens of the word
derived from 150 different talkers (75 male, 75 female). A value of
A = 0.25 was used in computing p;(t). Experimentation with A showed
the resulting temporal probability tables were insensitive to A over a
broad range; this was because with L = 8 (or 16) vectors, generally
there was only a small fraction of the code-book vectors whose distor-
tion scores were low. Given a large enough training set, the exact value
of A (as long as it was relatively small) is almost irrelevant.

The temporal probability tables of Fig. 3, for the word six, show
that a smooth probability density was obtained for all vectors. Further,
we see that for some vectors a unimodal distribution resulted; for
other vectors distinct multimodal distributions are found. In this
example, the code-book vectors whose sounds represent the vowel /I/
have a unimodal distribution, since this sound occurs only at a single
place in the word six. Code-book vectors whose sounds represent the
fricative /s/ have a distinct two-mode distribution, since /s/ occurs at
both the beginning and end of the word six. Finally, code-book vectors
whose sounds represent silence have three modes, since silence can be

Fig. 3—Estimates of temporal probability density functions for the eight code-book
vectors of the digit six. (Forty-frame normalization of the word duration is assumed.)

found at the beginning, end, and in the stop gap of the word six. All
three types of distributions are clearly seen in the data of Fig. 3.

For convenience, and to reduce computation, the temporal proba-
bility tables were stored as

iLe., as negative log probabilities, so they could be combined readily
with the LPC distances. The multiplier, y, was chosen so that, aver-
aged over the entire training set, the average value of p,(t) was the
same as the average LPC distance. Typically, the value of ~ was about
0.45 for L = 8 vector code books, and about 0.22 for L = 16 vector
code books. Also, values: of p,(t) were clipped at a level of 10~*; hence
no temporal probability score was 0.

After a great deal of investigation into ways of combining LPC
distance and temporal probability scores, the resulting distance score
that was used was .

where dsp was the spectral (LPC combined with energy) distance and
dyp was the temporal probability distance. The scaling value a was
chosen by optimization and determined the mix of spectral and tem-
poral “distances.” A value of a = 0 represents pure spectral distance;
similarly, a value of a = 1.0 represents pure “temporal distance.”

The spectral distance, which combined the LPC distance with the
energy distance, had the form

with V,, being the autocorrelation matrix of the input frame, E; being
the normalized log energy of the input frame, and E;, being the
normalized log energy of the kth code-book vector. We then have

where c, Exo, Ey, and Eor were suitably chosen constants. (We used
c=0.1 Eto =6 dB, and Fur = 20 dB.)
The temporal distance of eq. (2) was of the form

The following sequence of steps was required to generate a combined
distance score in the preprocessor:

1. Vector quantize the input frame (by each word-based code book),
at time ¢ = 1/I, consisting of LPC vector a; and normalized energy E;,
and determine the minimum spectral distance, dsp, and the index of
the best code-book vector, k;.

2. Access the temporal distance as p,,(t), where ¢ is quantized to the
nearest 1/40 (since tables with 40 entries were used).

The above procedure is performed at each frame for each word in the
vocabulary, and the resulting distance scores are accumulated for each
word, as shown in Fig. 2.

2. If only a single word candidate exists, then the recognition is
over—i.e., no DTW processing is required.

3. If more than one word candidate exists, then use the DTW
processor to make the final recognition decision among the word
candidates.

Two databases were used to evaluate the performance of the recog-
nizer. All recordings were made over a standard, local, dialed-up
telephone line. The first database was a digits set consisting of four
sets of 1000 digits each (100 talkers-10 digits/talker). We call the
digits sets DIG1, DIG2, DIG3, and DIG4. Their characteristics are as
follows:

DIG1—100 talkers (50 male, 50 female), 1 replication of each digit
by each talker.’° These recordings have been used as a training set in
a wide variety of evaluations of isolated-word recognizers.

DIG2—Same 100 talkers and recording conditions as DIGI; record-
ings made several weeks later than those of DIGI.

DIG38—100 new talkers (50 male, 50 female), 1 averaged occurrence
of each digit by each talker obtained from averaging a pair of robust
tokens of the digit.’’!” The transmission conditions (i.e., analog front
end, filter cutoff frequencies, etc.) differed slightly from those used in
recording the DIG1 and DIG2 databases.

DIG4—A second group of 100 new talkers (50 male, 50 female), 20
recordings of each digit by each talker.!? A random sampling of 1 of
the recordings of each digit by each talker was used. The transmission
conditions differed substantially from those used in recording the
other databases.

The templates (12 per word, speaker independent) for the DTW
processing were created from the data of set DIG1. The training data
for the word-based VQ preprocessor (to get the code books, B’, and
the temporal probability tables, P”) were derived from a randomly
chosen set of 150 tokens of each word from sets DIG1, DIG3, and
DIG4. (Of course these same training data could have been used to
create the speaker-independent reference templates for DTW process-

ing; however, a conveniently available template set was used.) For
testing the recognizer, all four digit sets were used.

The second database was a vocabulary of 129 words used in an
airlines information and reservation system.’ Two sets of data, called
AIR1 and AIR2, were used. Their characteristics were:

AIR1—100 talkers (50 male, 50 female), 1 averaged occurrence of
each word by each talker obtained from averaging a pair of robust
tokens of the word.”

AIR2—20 new talkers (10 male, 10 female), 1 replication of each
word by each talker. The data of set AIR1 were used to create both
the word reference templates (speaker independent, 12 per word), and
to give the word code books and word temporal probability tables. The
data of set AIR2 were used to test the recognizer.

For each of the digit test sets, a preliminary test run was performed
in which the preprocessor was used by itself to make the final recog-
nition decision based on the word with the lowest combined spectral
plus temporal distance score. (Equivalently, 6, in the decision logic,
was set to 0.) The distance combining parameter, a, in eq. (2) was
then varied from 0 to 1 (in steps of 0.1) and a curve of the preprocessor
recognition accuracy versus a was computed. A typical such curve for
the test set DIG1 is given in Fig. 4. The behavior of the recognition
rate, shown in this figure, is typical for all the digit test sets. It can be
seen that for a = 0 (only spectral distance) and for a = 1.0 (only
temporal distance), the recognition rate of the preprocessor (91.4
percent for a = 0, 91.2 percent for a = 1.0) is significantly lower than
its value at the peak of the curve (97.5 percent for a = 0.7). This result
strongly points out the value of combining spectral and temporal

a
Fig. 4—Curve of average digit recognition rate versus the combining multiplier, a,

for the data of test set DIG1. (Note that a = 0 corresponds to pure spectral distance
and a = 1.0 corresponds to pure temporal distance.)

Table I—Average error rates for digits vocabulary
Average Digit Error Rate (%)

distances in the preprocessor. It also can be seen that in the vicinity
of the peak (near a = 0.7), the recognition rate is fairly constant (its
value at a = 0.5 is 97.1 percent); hence, a fairly broad region of choices
for a is possible. Across the four digit test sets, the optimum value of
a varied from 0.4 to 0.7. If we used the value a = 0.5 for all digit sets,
the preprocessor recognition rate changed less than 0.2 percent, on
average.

A complete set of performance results on the digits test sets is given
in Table I. Table Ia gives average digit error rates for the preprocessor
working without the DTW processor, for the four test sets (and an
overall average), for code books with 8 and 16 vectors per word. The
average digit error rate is 3.3 percent for 8 vector code books, and 2.8
percent for 16 vector code books. Table Ib gives average digit error
rates for the complete recognizer, as a function of code-book size. The
threshold, 5, in the preprocessor was set so that, on average, about 83
percent of the time no DTW was required (i.e., the preprocessor made
the final decision), and about 17 percent of the time, the average
number of word candidates passed on to the DTW processor was 2.25.
No quantization of the reference templates in the DTW processor was
used; previous experience with this data set indicates that no degra-
dation need occur if the reference template quantization is done
correctly.°

From Table Ib it can be seen that the entire recognizer achieved an
average digit error rate of 2.6 percent for L = 8 vector code books, and
2.2 percent for L = 16 vector code books. These results represent
improvements of 0.6 to 0.7 percent in word accuracy; for 4000 test
digits, such a result is statistically significant.

For the airline vocabulary, a curve of preprocessor average perform-
ance versus the combining multiplier a was again run, and the results
are given in Fig. 5. Although the form of the curve is similar to that
of the digits case (Fig. 4), performance improves significantly when

Qa
Fig. 5—Curve of average word recognition rate versus the combining multiplier, a,
for the airline test data.

using both spectral and temporal distance, as opposed to either spectral
or temporal distance alone. We see from Fig. 5 that for a = 0 (spectral
distance only), the preprocessor achieves a 65.4-percent accuracy; for
a = 1.0 (temporal distance only), the accuracy is 73.2 percent (it is
better than the result for a = 0). However, for a = 0.5, the combined
distance yields a performance of 88.1-percent word accuracy, an im-
provement in accuracy of from 15.5 to 22.7 percent over the individual
distances.

The overall recognizer performance on the airline vocabulary is
given in Table II. The 8-vector-per-word system has a preprocessor
error rate of 14.8 percent, whereas the 16-vector-per-word system has

a preprocessor error rate of 11.9 percent. By setting the preprocessor
decision threshold so that a unique decision was made by the prepro-
cessor on 76 percent of the trials, and on 24 percent of the trials, an
average of 2.5 candidates (out of 129 possible) were passed on to the
DTW processor, the overall word error rates fell to 11.7 percent for
the 8-vector code books, and to 8.9 percent for the 16-vector code
books.

To illustrate how the addition of temporal information aids the
preprocessor, Fig. 6 shows a recognition case in which a word (the

Fig. 6—The enhanced recognition performance obtained by combining temporal and
spectral distances in the preprocessor: (a) the test word (zero) log energy contour; (b)
spectral and (c) temporal distances on a frame-by-frame basis (solid curve is for the
word three, dashed curve is for the correct word zero); (d) accumulated spectral and (e)
accumulated combined distance scores.

digit zero) would have been misrecognized (as the digit three) based
on VQ spectral distances alone, but is correctly recognized based on
the combination of spectral and temporal distance. Shown in this
figure are the log energy contour of the test word zero (Fig. 6a), the
VQ spectral distance (frame by frame) for both the word zero (dashed
line) and the word three (solid line, Fig. 6b), the log probability
distances for both words (Fig. 6c), the accumulated spectral distance
scores for both words (Fig. 6d), and the combined, accumulated total
distance scores for both words (Fig. 6e). On the basis of VQ spectral
matches, the preprocessor would have made a hard error since the
distance for zero was not close enough to the distance for three;
however, using the combined distance the correct word zero was
uniquely recognized. The reason that the temporal distance helped so
much, in this case, was the large temporal distance during the /O/
vowel in zero for the word three. Thus, although there is a code-book
vector that matches the /O/ spectrum well in three, the probability of
it occurring at the end of the word is very small. Although this example
is an extreme case, it does illustrate well why the addition of temporal
information to the preprocessor can help the performance to improve.

The results presented in the previous section clearly show that the
addition of temporal information to a word-based VQ preprocessor
increases the accuracy of the recognizer and makes it more robust to
vocabulary size and complexity.

To gain perspective on how the current system performance com-
pares with previous recognizers, Table III gives digit recognition error
rates for the current system, for the best DTW recognizer,’ and for a
previous recognizer using a word-based VQ preprocessor.’ Similarly,
Table IV gives word recognition error rates, for the airline vocabulary,
for the current system, for the best DTW recognizer, and for a previous
recognizer using a word-based VQ preprocessor.”

For the digits, the DTW system performs slightly better, on average,
than the current system. However, the best performance is on test
sets DIG1 and DIG2, from which the word reference templates were
derived. On the test sets DIG3 and DIG4, the current system performed

Table I!|—Average digit error rates for three recognition systems
Average Digit Error Rate (%)

Recognizer DIG1 DIG2 DIG3 DIG4 Overall
Current system 1.3 2.3 2.2 2.8 2.2
DTW alone 0.0 0.6 2.7 3.9 1.8
Previous recognizer — 2.0 — — —

Table !V—Average word error
rates for three recognition
systems for the airline vocabulary

Average
Word
Error Rate
Recognizer (%)
Current system ' 8.9
DTW alone 10.2
Previous recognizer 12.6

slightly better than the DTW recognizer. On the test set DIG2 (which
was the only common one between the current system and the previous
recognizer with the preprocessor), the system performances were es-
sentially the same.

For the airline vocabulary we see that the error rate of the current
system is 1.3 percent lower than that of the DTW recognizer alone,
and 3.7 percent lower than the previous recognizer based on the VQ
preprocessor. For this vocabulary a real performance improvement
has been achieved.

It remains for us to show that this increase in system performance
is achieved at essentially no increase in system cost (i.e., computational
complexity). To do this we define the following system variables:

= Average fraction of words passed on to DTW processor, when
more than a single word candidate exists.

The computation of the preprocessor can be expressed as:
Cpre = V-I-L-(p +1) *, +
and the computation of the DTW postprocessor is
Cpost = (1 — y)8Cprw,

The ratio between the full DTW computation (without a preprocessor)
and the current recognizer computation is then

Thus, a computational reduction (over a standard DTW recognizer)
of from 10 to 20 times is achieved by the proposed recognizer.

Although the performance of the proposed recognizer is impressive,
it is possible to reduce its computational complexity even further. If
we analyze the computation above, the major computation is in the
preprocessor, where a total of V-L dot product distances need to be
computed for each test frame. In the case where V is large (e.g., the
129-word airline vocabulary), the total number of code-book vectors
becomes large. In such a case it would be less expensive to use a
universal code book (word and talker independent) of say 1024 vectors,
and to choose the word-based code books from the universal code
book. In this manner the number of distance computations per frame
is fixed, and does not grow with the vocabulary size V. Of course, it
must be shown that performance will not degrade, but it seems
reasonable that for a sufficiently large code book, this will indeed be
the case.

In this paper we have shown how the addition of temporal infor-
mation into the preprocessor of an isolated-word recognizer can im-
prove the system performance and make the overall recognizer more
robust to vocabulary size and complexity. The way in which the
temporal information was added was straightforward; namely, we
defined and measured from a training set a probability density function

on the time of occurrence of the code-book vectors in the word-based
VQ preprocessor. A temporal distance was defined as the scaled,
negative log probability of the probability of occurrence of the vector
chosen by the vector quantizer. A combined measure in which the
spectral distance (from the VQ) was added to the temporal distance
was used in the recognizer and shown to improve performance for
both a digits and moderate-size airline vocabulary. Finally, it was
shown that, on average, the computational complexity of the resulting
recognizer was less than that required for a conventional dynamic
time-warping implementation by at least a factor of ten, whereas the
recognition performances of the two systems were comparable.

1. K. Shikano, “Spoken Word Recognition Based Upon Vector Quantization of Input
Speech,” Trans. Comm. Speech Res. (December 1982), pp. 473-80.

2. J. E. Shore and D. K. Burton, “Discrete Utterance Speech Recognition Without
Time Alignment,” IEEE Trans. Inform. Theory, [T-29, No. 4 (July 1983), pp.
473-91.

3. L. R. Rabiner, S. E. Levinson, and M. M. Sondhi, “On the Application of Vector
Quantization and Hidden Markov Models to Speaker-Independent, Isolated Word
Recognition,” B.S.T.J., 62, No. 4 (April 1983), pp. 1075-105.

4. D. K. Burton, J. T. Buck, and J. E. Shore, “Parameter Selection for Isolated Word
Recognition Using Vector Quantization,” Proc. ICASSP 84, San Diego, Calif.
(March 1984), pp. 9.4.1-4.

5. K. C. Pan, F. K. Soong, and L. R. Rabiner, “A Vector Quantization Based Prepro-
cessor for Speaker Independent Isolated Word Recognition,” IEEE Trans.
Acoust., Speech, and Signal Processing, ASSP-33, No. 3 (June 1985).

. Y. Linde, A. Buzo, and R. M. Gray, “An Algorithm for Vector Quantization,” IEEE
Trans. Commun., COM-28, No. 1 (January 1980), pp. 84-95.

. B. Juang, D. Wong, and A. H. Gray, Jr., “Distortion Performance of Vector
Quantization for LPC Voice Coding,” IEEE Trans. Acoust., Speech, and Signal
Processing, ASSP-30, No. 2 (April 1982), pp. 294-303.

9. L. R. Rabiner, M. M. Sondhi, and S. E. Levinson, “A Vector Quantizer Combining
Energy and LPC Parameters and Its Application to Isolated Word Recognition,”
AT&T Bell Lab. Tech. J., 63, No. 5 (May-June 1984), pp. 721-35.

10. L. R. Rabiner, S. E. Levinson, A. E. Rosenberg, and J. G. Wilpon, “Speaker
Independent Recognition of Isolated Words Using Clustering Techniques,” IEEE
Trans. Acoust., Speech, and Signal Processing, ASSP-27, No. 4 (August 1979),
pp. 336-49.

11. L. R. Rabiner and J. G. Wilpon, “A Simplified Robust Training Procedure for
Speaker Trained, Isolated Word Recognition Systems,” J. Acoust. Soc. Amer.,
68, No. 5 (November 1980), pp. 1271-5.

12. J. G. Wilpon, L. R. Rabiner, and A. Bergh, “Speaker Independent Isolated Word
Recognition Using a 129-Word Airline Vocabulary,” J. Acoust. Soc. Amer., 72,
No. 2 (August 1982), pp. 390-6.

13. A. E. Rosenberg, K. L. Shipley, and D. E. Bock, “A Speech Data Base Facility Using
a Computer Controlled Cassette Tape Deck,” J. Acoust. Soc. Amer., Suppl. 1, 72,
(Fall 1982), p. 580.

14. S. E. Levinson, and K. L. Shipley, “A Conversational Mode Airline Information
and Reservation System Using Speech Input and Output,” B.S.T.J., 59, No. 1
(January 1980), pp. 119-37.

15. J. G. Wilpon and L. R. Rabiner, “A Modified K-Means Clustering Algorithm for
Use in Speaker Independent Isolated Word Recognition,” IEEE Trans. Acoust.,
Speech, and Signal Processing, ASSP-33, No. 3 (June 1985).

Arthur F. Bergh, currently enrolled at Swarthmore College; AT&T Bell
Laboratories, 1980—. Mr. Bergh has been engaged in speech-recognition
research in the Acoustics Research Department.

Lawrence R. Rabiner, S.B. and S.M., 1964, Ph.D., 1967 (Electrical Engi-
neering), The Massachusetts Institute of Technology; AT&T Bell Laborato-
ries, 1962—. From 1962 through 1964 Mr. Rabiner participated in the coop-
erative plan in electrical engineering at AT&T Bell Laboratories, in Whippany
and Murray Hill, New Jersey. He worked on digital circuitry, military com-
munications problems, and problems in binaural hearing. Presently, Mr.
Rabiner is engaged in research on speech communications and digital signal
processing techniques. He is coauthor of Theory and Application of Digital
Signal Processing (Prentice-Hall, 1975), Digital Processing of Speech Signals
(Prentice-Hall, 1978), and Multirate Digital Signal Processing (Prentice-Hall,
1983). Member, National Academy of Engineering, Eta Kappa Nu, Sigma Xi,
Tau Beta Pi. Fellow, Acoustical Society of America, IEEE.

Frank K. Soong, B.S., 1973, National Taiwan University, M.S., 1977,
University of Rhode Island, Ph.D., 1983, Stanford University, all in Electrical
Engineering; AT&T Bell Laboratories, 1982—. From 1973 to 1975 Mr. Soong
served as a teacher at the Chinese Naval Engineering School at Tsoying,
Taiwan. In 1982 he joined the technical staff at AT&T Bell Laboratories,
where he engaged in research in speech coding and recognition. Member,
IEEE.

A new light pen system using a real-time algorithm for computing the
centroid of the intensity pattern seen by a photosensor is described. This new
light pen achieves an accuracy of better than one-quarter of a pixel on a
cathode-ray-tube screen with 1K xX 1K resolution. This corresponds to a
capability of resolving 0.004 inch on a 15-inch screen, which is an improvement
by a factor of at least 50 over the conventional light pens available today. The
transistor-transistor logic hardware implementation of the algorithm is de-
scribed in detail. The central part of the hardware is a real-time, moment-
generating circuit that implements an efficient moment calculation algorithm
reported earlier. Issues such as complexity of the hardware compared with the
conventional techniques, and the possibility of implementing the algorithm in
a single complementary metal-oxide semiconductor chip are addressed. Some
line and curve drawing results sketched by the new light pen are presented
and compared to similar drawings obtained by a conventional light pen.

The light pen, an input device for graphics displays and work
stations, has been around for quite some time. Among the many input
devices for graphics systems, the light pen is probably the most natural
one to use and, other than the touch-sensitive screen, the only one
that directly interacts with the screen. In the world of interactive
computer graphics, the light pen has been used mostly as a selection
device for pointing to objects and characters on a Cathode-Ray-Tube

* AT&T Bell Laboratories. t AT&T Bell Laboratories; present affiliation Bell Com-
munications Research, Inc.

Copyright © 1985 AT&T. Photo reproduction for noncommercial use is permitted with-
out payment of royalty provided that each reproduction is done without alteration and
that the Journal reference and copyright notice are included on the first page. The title
and abstract, but no other portions, of this paper may be copied or distributed royalty
free by computer-based and other information-service systems without further permis-
sion. Permission to reproduce or republish any other portion of this paper must be
obtained from the Editor.

(CRT) screen. Although very simple and inexpensive, the light pen
has not been very popular among the users of graphics systems. Some
have called it a clumsy, fragile device and predicted that it will become
less and less popular as raster graphics grows.’ One of the main reasons
for the light pen’s poor popularity is its limited resolution, which
makes it unsuitable for accurate pointing, or writing and sketching on
a CRT screen. Provided with improved resolution capability, the light
pen is useful in a variety of applications, such as telewriting, accurate
pointing and selecting, facsimile, brushing, and graphics generation.
An example of the light pen used in a telewriting system is described
in Ref. 2.

The poor resolution of the light pen is a combined result of its
relatively large field of view, the signal-to-noise ratio performance of
the pen’s photosensor, limited image sharpness,” and, most important,
the simple thresholding technique used to detect the position of the
pen. We recently proposed a new approach to estimating the position
of the pen by processing the analog signal generated by the light pen’s
photosensor and computing the centroid of the two-dimensional image
received by the sensor. Using this new technique we can achieve
subpixel accuracies on a 1K X 1K screen. The algorithm is extensively
treated in Ref. 4.

Conventional light pens used in today’s computer graphics systems
employ a very simple circuit for detecting the position of the pen on
the display screen. This is illustrated in Fig. 1. A photosensor is placed
in the tip of a penlike housing. Whenever the scanning electron beam
falls in the photosensor’s field of view, it generates an analog signal
whose amplitude is a function of the light intensity received by the
sensor. This analog signal is compared with a predefined threshold,
and a pulse is generated indicating a hit. This pulse latches the values
of two counters (the X-Y counters in Fig. 1) that track the horizontal
and vertical positions of the beam. The two counters are reset at the
beginning of each scan, namely, when the beam is at the top-most left
corner of the screen. The X counter tracks the position of the beam
on each line and is reset at the end of the line; the Y counter indicates
the number of the line currently being scanned by the beam.

Obviously, when the hit pulse is generated, the values of the X and
Y counters indicate the coordinates of the hit point. These coordinates

are latched into two registers to be used by a host processor (e.g., the
graphics station’s display processor). Using this simple thresholding
technique, the accuracy in estimating the position of the light pen is
a function of the photosensor’s signal-to-noise ratio and the aperture
of the photosensor (size of the field of view). The latter is more
important. Many attempts have been made to improve the accuracy
of the light pen by controlling the field of view of the pen and using
photosensors with high sensitivity.

Designers have employed various mechanical and optical arrange-
ments in the tip of the light pen to improve the accuracy. All of these
attempts, although successful in improving the performance, have not
achieved the ultimate desired accuracy. To the best of our knowledge,
the best of today’s available light pens do not have a repeatable
accuracy of better than +5 pixels (in both directions) on a screen with
a resolution of 500 X 500 pixels. Besides, even this poor accuracy is
only achievable when the pen is held perpendicular to the screen and
not at an angle.

Such poor accuracy in estimating the position of the light pen is
mainly due to the simple thresholding technique used in detecting the
hit point. In our approach, a different technique is used. Rather than
thresholding the analog signal generated by the photosensor, we take
advantage of the information contained in that signal to achieve an
accurate and highly robust estimate of the position of the light pen on
the display screen. As we will see in the following sections, using the
right kind of processing can convert an ordinary light pen into one
with an accuracy of one-quarter of a pixel on a display with 1024 x
1024 resolution.

Il. MOMENT COMPUTATION APPROACH
If the analog signal generated by the light pen’s photosensor is
displayed on a monitor, one sees a cometlike pattern similar to what

is shown in Fig. 2. The tail of the pattern is caused by the persistence
of the phosphor used in the CRT display at which the light pen is
pointed. A long persistence phosphor will cause a longer tail. In our
new light pen, the centroid of this intensity pattern is used as the
estimate of the position of the light pen. As shown in Ref. 4, this
approach produces a highly accurate and robust estimate of the posi-
tion.

The x and y coordinates of the centroid are calculated in real time
(60 fields/s) by computing various moments of the intensity pattern.
The scheme is illustrated in Fig. 3. After proper amplification, the

photosensor’s output is digitized and fed to a two-dimensional digital
filter, which can compute the intensity moments of the two-dimen-
sional signal represented by the photosensor’s output. The moments
m”? about the point (N, M) are defined as

N M
mt = YY xi, ji NYG MY, (1)
1=0 j=
where x(i, j) represent the digitized samples of the sensor’s output,
and N and M are the horizontal and vertical resolution of the display
system. We recently proposed a very efficient algorithm for real-time
computation of (1). This algorithm is described and analyzed in Ref.
5. It uses a set of identical single-pole digital filters and has a highly
regular and expandable structure. In Ref. 5 we also present a Very
Large-Scale-Integrated (VLSI) design for single-chip implementation
of this algorithm in Complementary Metal-Oxide Semiconductor
(CMOS) technology. The chip is capable of simultaneously computing
16 moments m?? (p = 0, 1, 2, 3; g = 0, 1, 2, 3) of a 512 * 512, 8
b/pixel image in real time (i.e., at conventional video rate).

In this application for estimating the position of the light pen, only
the three moments m°°, m®', and m?’ are used to compute the centroid
coordinates as
1,

The subpixel accuracy in our estimation stems from the fact that the
above division operations can be carried out in floating point mode.
This generates results accurate to almost one digit after the decimal
point, as we will see in Section V.

The light pen algorithm described above has been implemented as
part of a larger graphics test bed described in Ref. 6. Figure 4 shows
the light pen system used in the test bed. Except for the moment
generator circuit, the rest of the blocks in this figure are part of the
graphics test bed. As described earlier, the signal generated by the
light pen’s photosensor is digitized and fed to the moment generator
circuit, which generates a set of raw moments at the rate of 60 times
a second (video field rate). These moments are read by the test bed’s
control processor (a 68000 microprocessor), which calculates the cen-
troid coordinates and feeds them to the display processor for proper
action (e.g., writing, erasing, and brushing).

Fig. 4—Light pen system using the moment approach implemented on a graphics

Fig. 5—Digital filter representation of the moment computation algorithm.

The moment generator circuit is basically a Transistor-Transistor
Logic (TTL) implementation of the algorithm we described in Ref. 4.
It consists of a number of single-pole digital filters, with transfer
function 1/(Z — 1), interconnected as shown in Fig. 5. This is only
part of a larger filter network described in Ref. 5, mainly because
higher-order moments are not required in this application. Each filter
block is simply implemented by an accumulator, which is basically an
adder with an output register and a feedback path. This is shown in
Fig. 6 for the two blocks in the first row of Fig. 5, referred to as a row
filter in Ref. 5. These two blocks operate at the pixel clock rate. Figure

Fig. 6—Circuit and timing diagram of the row filter section of the moment generator.

6 illustrates the operation of this simple arrangement of adders and
registers. The rest of the blocks in Fig. 5, referred to as column filters,
are implemented in exactly the same way, except that they run at the
display’s scan-line clock rate rather than the much faster pixel clock
rate. This feature relaxes the speed requirement on the adders in the
column filter to the point where they can be implemented serially. .
This results in considerable savings in space and power consumption.
This feature is extremely important in the VLSI implementation of
the moment generator circuit, as discussed in Ref. 5.

Results obtained from the hardware implementation of our new
light pen aglorithm indicate that the moment computation approach
can indeed achieve subpixel accuracy in estimating the position of the
light pen. All the line and curve drawings presented in this section

were drawn by the light pen on a 1000-line raster scan black and white
monitor that uses a short persistence phosphor (2 to 3 us). For a
display monitor with a long persistence phosphor, the output of the
photosensor’s amplifier should be differentiated and rectified before
processing. See Ref. 4 for more details.

Figure 7 shows a number of curves and lines drawn by the light pen
using the moment computation approach. Figure 8 shows similar
drawings using the conventional method in determining the position

of the light pen (same photosensor was used). Except for Figs. 7a and
8a, a linear interpolation has been used for connecting the points. A
comparison of the drawings in Figs. 7 and 8 clearly shows the degree
of improvement achieved by the moment computation approach. Fig-
ure 7b is of special interest. It shows that, even with a simple linear
interpolation, light pens using this new algorithm can generate very
high-quality writing on CRT displays.

To obtain some quantitative measure of the accuracy, the following
experiment was performed. The light pen was attached to a micropo-
sitioning device facing the display screen. The micropositioner was
moved horizontally in steps of 0.001 inch, and the floating point x and
y coordinates generated by the moment processor were recorded. The
results are plotted in Fig. 9. From these results we can observe an
accuracy of one-quarter of a pixel on a 1K X 1K resolution monitor.
The pen can resolve 0.004 of an inch on a 15-inch screen. It should be
noted that, although the resolution of our display system is 1K x 1K,
the signal fed to the moment generator circuit is subsampled by a
factor of 2, resulting in an effective input resolution of about 500 x
500. It should also be noted that part of the error in the recorded data
for the x and y coordinates is due to the jitter in the mechanical setup
in the above experiment.

Fig. 9—Position computed by the moment algorithm in pixels as a function of the
movement of the pen by a micropositioner.

To be of practical use in today’s graphics systems, the cost of this
new light pen must be drastically reduced. We are currently studying
the possibility of implementing the algorithm in a single-hybrid CMOS
chip. This would include the Analog-to-Digital (A/D), moment gen-
erator circuit, and postprocessing. A VLSI design for the moment
generator part of this chip has already been prepared.° In applications
where an accuracy of one pixel is sufficient, the requirement on the
size of the A/D and the adder circuits in the moment generator can
be considerably relaxed. Preliminary investigations indicate that the
moments of the area of the pattern in Fig. 2, rather than the intensity
moments, are sufficient for one-pixel accuracy. Using the area mo-
ment, if possible, reduces the A/D to a simple comparator (1 b/pixel),
and the moment generator requires much less silicon area. All these
options are being investigated and will be reported in the future.

A new light pen system based on computing the centroid of an
intensity pattern generated by a photosensor was presented. The

hardware implementation of this system was described in detail. The
central part of the hardware is a TTL implementation of an efficient,
real-time, moment-generating algorithm reported earlier. The mo-
ments are used to compute the x and y coordinates of the centroid as
an estimate of the position of the light pen on the screen. It was shown
that this new technique can achieve an accuracy of at least one-quarter
of a pixel on a display screen with a resolution of 1K X 1K. The pen
can thus resolve 0.004 inch on a 15-inch screen. This is an improve-
ment by a factor of at least 50 over the conventional techniques used
in today’s light pens. Such an improvement is gained at the cost of
increasing the complexity of the hardware by a considerable factor.
However, due to the simplicity of our moment-generating algorithm
and the regularity of its structure, the required hardware can be
implemented in a single CMOS chip. Such a chip can have many
potential applications in the field of computer graphics. Work on this
single-chip implementation of the algorithm is in progress.

1. J. D. Foley and A. Van Dan, Fundamentals of Interactive Computer Graphics,
Reading, Mass.: Addison-Wesley, 1982.

2. P. Zorkoczy, Information Technology—An Introduction, Knowledge Industry Pub-
lications, 1983.

3. J. E. Scott, Introduction to Interactive Computer Graphics, New York: Wiley, 1982.

5. Z. L. Budrikis and M. Hatamian, “Moment Calculation by Digital Filters,” AT&T
Bell Lab. Tech. J., 63, No. 2 (February 1984), pp. 217-29.

6. E. F. Brown et al., “An Interactive Graphics Conferencing Test-Bed,” ICC-84 Conf.,
Amsterdam.

Earl F. Brown, 1955, RCA Institute; AT&T Bell Laboratories, 1955-1984.
Present affiliation Bell Communications Research, Inc. At AT&T Bell Labo-
ratories Mr. Brown was engaged in image processing bandwidth compression,
image quality studies, video teleconferencing, and audio teleconferencing. At
Bell Communications Research he is a member of the Wideband Data and
Video Services Division.

Mehdi Hatamian, B.S.E.E., 1977, Ary-Mehr University of Technology, Teh-
ran; M.S. and Ph.D. (Electrical Engineering), University of Michigan, Ann
Arbor, in 1978 and 1982, respectively; AT&T Bell Laboratories, 1982—. From
1979 to 1982 Mr. Hatamian was a Research Assistant on a NASA project
working on the design and development of hardware and software for one of
the space shuttle experiments. His research interests are in real-time signal
processing, circuit design, image processing, and VLSI design. Member, IEEE,
Sigma Xi.

We present a method for calculating the moments and the distribution of
sojourn time in a multiprogrammed computer system. We assume that the
CPU and I/O subsystem can be represented by a general state-dependent
server who works according to the processor sharing discipline. Further, at
most m jobs may be simultaneously receiving service. Thus, m is the multi-
programming level of the system. The arrival of jobs occurs according to a
Poisson process, and the arrivals must wait in a waiting area if m jobs are
already receiving service. The method presented may be useful in designing
the multiprogramming level needed to meet certain objectives on the charac-
teristics of the sojourn time.

This paper is concerned with finding the moments and the distri-
bution of sojourn time in a multiprogrammed computer system. We
assume that jobs arrive into a computer system according to a Poisson
process at a rate of \. The computer system is divided into two areas,
a waiting area and a service area. The service area can hold at most
m jobs, where m is the multiprogramming level of the system. The
jobs in the service area receive service from a server whose service rate
is assumed to be state dependent. The server operates at a rate py;
{using the Processor Sharing (PS) discipline] whenever there are j
jobs in the service area with y,, = u. We assume that each job’s service

time is exponentially distributed, so that on every service completion,
each customer in the service area is equally likely to leave the system.
An arrival into the system goes directly into the service area if it is
not full. An arriving job goes to the waiting area of unlimited size if
there are already m jobs in the service area. The customers are drawn
from the waiting area into the service area according to the First-In
First-Out (FIFO) discipline. Figure 1 depicts this situation schemati-
cally. The model may be useful in determining the multiprogramming
level needed to meet a certain response-time characteristic.

The model is an obvious generalization of the M/M/1 First-Come
First-Served (FCFS) queue (m = 1) and of the M/M/1 PS queue
(uj; = w and m = ©), A little thought should convince the reader that
the M/M/m FCFS queue is also a special case of this model with p; =
io for 1 = 1, --- , m, where o is the service rate of each server. Avi-
Itzhak and Heyman had proposed the use of a state-dependent server
to approximate the CPU and I/O subsystem.’ We depict a multiple
CPU and disk subsystem in Fig. 2, which is approximated by a state-
dependent server in our model. The service rate y»; of our model is
obtained by solving for the throughput in the closed queueing network
of Fig. 2 with a population size of 1. Reiser and Lavenberg describe a

s CPUs | 1 : SERVICE RATE= @
DISK 1 OC III z SERVICE RATE = B
e . e
: ° :
DISK d O {||| Pa SERVICE RATE = 8B

NOTES: 1. POPULATION SIZE =
2. FCFS DISCIPLINE
3. THROUGHPUT = fh;,/=1,...,m
Fig. 2—Model of a multiple CPU and disk subsystem.

method of solving the throughput of such a closed queueing network.”
Fredericks obtains approximations for mean delays in a multipro-
grammed computer system and discusses the question of accuracy of
the state-dependent server model.? Konheim and Reiser examine a
computer system with one disk and one CPU, subject to a bound on
the number of jobs present.* Mitra obtains the waiting time distribu-
tion for a computer system fed by jobs from a finite number of
terminals.° Salza and Lavenberg investigate hierarchical decomposi-
tion methods for approximating response-time distributions in certain
closed queueing network models of computer performance.® Coffman,
Muntz, and Trotter obtain the sojourn time distribution of the
M/M/1 queue with processor sharing discipline.’

In Section II of this paper, we discuss some preliminary results.
Sections III and IV are concerned with finding the moments of sojourn
time. In Section V, we provide a method of obtaining the distribution
of sojourn time. In Section VI, we present some numerical examples.

H. DEFINITIONS AND PRELIMINARIES
Let us define the following quantities:

N = Number in system seen by an arrival, excluding itself.
P,= P(N =n).
I = Number in system seen by a customer entering the service
area, including itself.
qi= P(I =i).
U = Time spent in the waiting area by an arrival.
F(u) = P(U Su).
2n = E(U").
T = Time spent in the service area by a tagged customer.
B;(s) = Laplace-Stieltjes transform of the conditional distribution
of T given I = 1, 1.e.,

By solving the equation of a birth and death process, one obtains the
following results directly.
Let

n=1 Lp
where
p = A/um = d/u
and
epee if n=1,---,m
‘i pe "Pm if n=m+i.,.
Let L be the mean number in the system. Then
at P.,, mP

In this section, we will derive the distribution and moments of U,
and the distribution of J, and we will characterize the moments of V.
It is clear that

This can be shown to be equal to
P,,n'
(1 = p)\(u — a)"
This result could also have been obtained as follows:
Zn = EU" = E(U"|NZ=m)P(N=™m) + E(U"|N< m)P(N < m).

like an M/M/1 FCFS queue. Thus, the wait time corresponds to the
sojourn time of an M/M/1 queue and is exponential with parameter
(u = d). So,
_ n! Pz

This can be explained by the fact that an arrival to and a departure
from the waiting area see the same distribution of the number in the
waiting area and that the number in the service area is constant, given
that N > m. For I < m, the number in the system including the arrival
is one more than the number seen by the arrival. So

For I = m, it is possible for an arrival into the service area to see m
customers including itself in one of two possible ways. First, there
were m — 1 customers in the system and an arrival occurred. Second,
the arrival saw at least m customers and no one arrived during its
wait. So,

We now characterize the moments of the sojourn time in terms of
Xin. In the next section, we will show how to calculate x;n.
2 n! ' ;
EV"? =) Gi at E(T’U"”)
joj inj)

The last step follows from the fact that T is independent of U given
I. We now consider three cases:
1. Forj =n,

In this section, we show how to calculate x;,. The transforms B;(s)
satisfy the following equations:

These equations are obtained by assuming that the system is in state
i + 1 in steady state and conditioning on the time of the next event.
An event is defined to be a service completion or an arrival, whichever
occurs first. By rearrangement,

Equation (3) is a second-order partial difference equation with variable
coefficients. However, we note that for i = m, the coefficients do not
vary with 1. So we will first show a method of solving an equivalent
system with coefficients that do not vary with i and then present a
procedure for the solution of eq. (3). The interested reader is referred
to Boole® or Jagerman’ for details of the techniques used to solve this
difference equation. Consider the equation

Vi+2,n — “+ Yi+1,n = Yin = + Yin (6)
for positive integer valued i and n, where c = c,,. For the homogeneous
version of this equation, the solution is

It can be shown that o, > 1 > oo. The constants A, and B, will be
evaluated later. To find a particular solution to (6), we define the
translation operator FE and the difference operator A such that

The last step is obtained from the shift formula (see footnote on page
73 of Boole®):
f(E)(a'u;) = a'f (aE)u.

for positive integer valued i and n. It is easy to verify that this satisfies

We first state that y;, has a probabilistic interpretation. We replace
the original system by a new one in which the service rate of the server
is n, regardless of the number of customers in the service area. Further,
consider a tagged customer whose probability of leaving the system is
1/m at each service completion, whenever there are at least two
customers in the service area. Then eq. (6) describes the behavior of
the conditional moments of the time spent in the service area by the
tagged customer in this new system for 1 = 2. Clearly, as i tends to
infinity, the conditional moments of the original and new systems are
identical. Further, as i tends to infinity, the server is going to operate
at rate » for a long time in the original system. If we now tag a
customer in the service area, it will leave the system with probability
1/m and, therefore, T'is exponential with rate u/m as i tends to infinity
in the original system. So

Yin = a ( »» OT Yin-1 + >; oF sin] +. B,o%. (8)
A(o1 — 2) \j=it1 j=l
At this point, it is possible to verify that this result indeed yields the
familiar answer for the M/M/1 FCFS queue. To do this, one has to
set m = 1. This yields c = 0 and oz = 0. After some algebra, one obtains
Vin = n!/p" for 1 = 1 as expected.

We have so far been able to find y,, up to a constant, and a
comparison of (3) and (6) should convince the reader that y,,, obtained
from (8) satisfies (3) for 1 = m. Thus, after finding x;, for 1 = m, we
can recursively compute Xm—in; °** » Xin by using (8). It is clear that
each of these variables is a linear function of the unknown constant
B,,. Finally, we will find B,, to satisfy (4). This is done by choosing two
trial values of B,, evaluating two sets of x;,, and using linear interpo-
lation to satisfy (4). The formal procedure to do this is as follows:

3. For each B® (k = 1, 2), use (8) to obtain y‘, for i= 1 and k = 1,
2. In this step, the terms ¥,,,-: on the right-hand side of (8) should not
be indexed by Rk.

5. Use (3) recursively to compute x*,1,, +--+, xf, for k = 1, 2. In
this step, the terms x;4;,,-1 on the right-hand side of (3) should not be
indexed by k.

6. From x*, (k = 1, 2) evaluate the left-hand side of (4). Let L* be
the left-hand side corresponding to B® for k = 1, 2.
7. Since x;, is a linear function of B,, for all 1, we have

In this section, we provide a direct method of obtaining the distri-
bution of sojourn time. First, we observe that U and T are independent
random variables conditional on J. Thus,

qi
from which we obtain, after some routine algebra (for t = 0),
1 if 1<m-1
P(U st|I=1)= L= Pe / dn if t=m

* (wu)™ pe
o (i-—m)!
If we let V(s) and U;(s) be the Laplace-Stieltjes transforms of
P(V st) and P(U s t|J = 1), respectively, then from (9) we have

It is possible to use the method of Section IV to solve eq. (1) for B;(s)
directly. In fact, for 1 = m, B;(s) has the form

After finding B;(s) for 1 = m, we can recursively compute B,-1(s),
--- , By (s) by using (1). It is clear that all of these quantities depend
on the unknown function B(s). Finally, one can find B(s) to satisfy
(2). This would have to be done by choosing two trial values of B(s),
evaluating two sets of B;(s), and using linear interpolation to satisfy
(2). It is possible to shown that V(s) simplifies [by using eqs. (10),
(11), and (12)] to

In the remainder of this section, we will show a method of calculat-
ing P(T <s t|J = 1) directly. Let us assume that at all times, events
occur in this system according to a Poisson process at a rate of \ +
Max( 1, 2, °** , Lm). Let a be the maximum value of the arguments.
Whenever the system is in state 1, arrivals occur with probability \/(A
+ a), departures occur with probability u;/(\ + a), and with probability
(a — wi)/(A + a) the system does not change its state. We assume that
Ho = 0. Let v;, be the probability that a tagged customer departs on
the kth event given that J = i. Then, v;, can be recursively computed
from

Cipi a — pi ;
Rao. beige Vi-1,k-1 + vie for 12=1,k=2
Vik aa ee Joanne 1,k-1 A icary k-1
1 — ¢;)p; é
if os ES 5 ihe
(A + a)
and

Since events are occurring at a Poisson rate of \ + a, departure of the
tagged customer at the kth event means that its time in the service
area is a Gamma random variable of order k and parameter \ + a.
Thus,

second for the M/M/5 FCFS queue. We approximate the M/M/1 PS
queue in the third example by choosing a high value (50) for m. The
fourth example is likely to be typical for a multiprogrammed computer
system where y; is increasing in i but at a diminishing rate. We get a
good correspondence between W (obtained from Little’s formula) and
EV in all cases. It is possible to verify that the higher moments in the
first two examples are very accurate.

The authors are grateful to Dave Jagerman for his help in solving
the difference equation of Section IV.

1. B. Avi-Itzhak and D. P. Heyman, “Approximate Queueing Models for Multipro-
gramming Computer Systems,” Oper. Res., 21, No. 6 (November 1973), pp. 1212-
0

2. M. Reiser and S. S. Lavenberg, “Mean Value Analysis of Closed Multichain
Queueing Networks,” IBM Research Report RC7023, 1978.

3. A. A. Fredericks, “Approximations for Customer Viewed Delays in Multipro-
grammed, Transaction Oriented Computer Systems,” B.S.T.J., 59, No. 9 (Novem-
ber 1980), pp. 1559-74.

4. A. G. Konheim and M. Reiser, “Finite Capacity Queueing Systems With Applica-
ao in Computer Modeling,” SIAM J. Computing, 7, No. 2 (May 1978), pp. 210-
29.

5. D. Mitra, Waiting Time Distributions From Closed Queueing Network Models of

Shared Processor Systems, Performance 81, F. J. Kylstra (Ed.), New York: North
Holland, 1981, pp. 113-31.

6. S. Salza and S. S. Lavenberg, Approximating Response Time Distributions in Closed
Queueing Network Models of Computer Performance, Performance 81, F. J.
Kylstra (Ed.), New York: North Holland, 1981, pp. 133-45.

7. E. G. Coffman, R. R. Muntz, and H. Trotter, “Waiting Time Distributions for
Processor Sharing Systems,” JACM, 17, No. 1 (January 1970), pp. 123-30.

8. G. Boole, A Treatise on the Calculus of Finite Differences, New York: G. E. Stechert,
1931.

Kiran M. Rege, B. Tech. (Electrical Engineering), 1977, I.I.T., Bombay;
Ph.D. (Electrical Engineering), 1981, University of Hawaii; AT&T Bell Lab-
oratories, 1982—. Mr. Rege spent 1984 on leave of absence, teaching at I.I.T.
Bombay in the Department of Electrical Engineering. His technical work at
AT&T Bell Laboratories includes modeling and analysis of switching, com-
puter, and communication systems. His research interests include communi-
cation theory, queueing theory, and performance analysis of computer and
communication systems.

Bhaskar Sengupta, B. Tech. (Electrical Engineering), 1965, I.I.T. Kharag-
pur, Eng. Sc.D. (Operations Research), 1976, Columbia University; AT&T
Bell Laboratories, 1981—. Mr. Sengupta has worked at IBM and the Service
Bureau Company and was an Assistant Professor at the State University of
New York at Stony Brook. He was also a consultant to Turner Construction
Company in New York. At AT&T Bell Laboratories he works on performance
problems for communication, computer, and manufacturing systems.

Application of Decomposition Principle in M/G/1
Vacation Model to Two Continuum Cyclic
Queueing Models—Especially Token-Ring LANs

Cyclic queueing models, in which a single server switches back and
forth among a (large) number n of queues, have been studied by many
authors. These studies were motivated largely by the need to describe
the performance of electronic telephone-switching systems. Recent
technological developments in computer-communication networks (lo-
cal area networks, or LANs) have generated renewed interest in these
models. In the present paper we consider two continuum cyclic
queueing models, i.e., models where n — © while the total arrival rate
remains fixed. Model 1 describes the behavior of certain token-ring
LANs, while Model 2 has been used to describe mail pickup and
delivery systems.

* AT&T Bell Laboratories. t Florida Atlantic University, Boca Raton, Florida.

Copyright © 1985 AT&T. Photo reproduction for noncommercial use is permitted with-
out payment of royalty provided that each reproduction is done without alteration and
that the Journal reference and copyright notice are included on the first page. The title
and abstract, but no other portions, of this paper may be copied or distributed royalty
free by computer-based and other information-service systems without further permis-
sion. Permission to reproduce or republish any other portion of this paper must be
obtained from the Editor.

Exact analytic models for finite n tend to be very complicated (see,
for example, Refs. 1 through 8). One of the earliest techniques for the
analysis of these complicated multiqueue models was to define the
ordinary single-server vacation model, in which the server periodically
leaves the queue and takes a “vacation”; this vacation model is then
“connected” to the model of n queues served in cyclic order by
interpreting the vacation as the time interval from when the server
leaves a particular queue until its return to that queue after cycling
through the other n — 1 queues.”

The basis for the analysis in the present paper is to relate the server
vacations to the cyclic queueing model in an entirely different manner,
and then to apply a new stochastic decomposition result of Fuhrmann
and Cooper? for the M/G/1 vacation model. (As noted, in the present
paper we are interested in models where n is infinite. Another paper’®
uses a related method to analyze certain cyclic queueing models where
n is finite.)

In both of our continuum models, the server scans (or polls) at a
constant velocity along a closed path. Customers arrive according to a
Poisson process (with rate \) in time, and are uniformly and inde-
pendently distributed in space along the scanning path. Service times
have distribution function H(-), with mean 7 and variance o”, and are
independent of the arrival process and each other. In Model 1, the
server stops scanning and serves customers as they are encountered
along the scanning path. In Model 2, the server collects customers
(there is no time expended in collecting a customer) as they are
encountered along the scanning path; when the server reaches a unique
point on the path called the origin, the server stops and serves all the
customers it has collected since last leaving the origin. Model 2 is
closely related to a model studied by Nahmias and Rothkopf,"' which
we shall refer to as Model 3, in which customers are served at the
origin and are then randomly (uniformly) redistributed (delivered)
over the scanning path on the server’s next cycle. This is in contrast
to Models 1 and 2, where each customer departs from the system as
soon as his service is completed.

Model 1 provides a good description of a large, symmetric, token-
ring LAN. In such a network, a number n of terminals (devices, work
stations) are interconnected in either a physical or logical ring struc-
ture. The terminals’ access to the transmission medium is controlled
by a “token” (a signal) that circulates around the ring. A terminal
gains access to the medium by seizing the circulating token as it goes
by. It retains the token while it is transmitting, thereby preventing
other terminals from simultaneously accessing the medium; it then
releases the token to circulate around the ring, enabling another
terminal to gain access to the transmission medium. (For a general

description of LANs and token-passing protocols, see, for example,
Refs. 12 through 14.)

One identifies the scan time c as the time required for the server to
poll all the terminals once (equivalently, the time required for the
token to cycle once around the LAN ring when no terminals are
waiting to transmit). If the number n of terminals is large, and the
terminals submit statistically identical loads (i.e., each terminal is
characterized by the same arrival rate and distribution of service
times), then there is a good correspondence between Model 1 and the
LAN.

Model 3 has been studied by Nahmias and Rothkopf,"' who used it
to describe a delivery system in which a clerk (the server) traverses
(scans) at a constant velocity a route along which letters (customers)
are generated (arrive) randomly in space and time. As the clerk travels
along the route, he picks up the letters that have been generated since
his last traversal, and he delivers the letters (to locations distributed
uniformly along the route) that were previously picked up and sorted.
When the clerk reaches the end of the route he sorts (serves) the
letters he has just picked up; then he again traverses the route,
delivering the letters that have just been sorted, and picking up the
new letters that have been generated since his last traversal of the
route. This process is repeated indefinitely.

For each model, the equilibrium cycle time T;, defined as the time
(during equilibrium) between successive visits in Model j by the server
to any given point along the scanning path, has the same mean value
T = E(T;), given by

where c is the (constant) length of time the server spends scanning
during each cycle (which is the time to complete a scan cycle when
there is no work to be done), and p(=Ar) is the server utilization.
[Equation (1) is easily derived by the following argument, given by
Kuehn,’ for a very general model of n queues served in cyclic order
by a single server: The mean cycle time T is the sum of the (constant)
time c spent scanning and the mean time s spent serving per cycle;
that is, T = c +s. Clearly, s = pT, and (1) follows. Note that T does
not depend on the form of the service-time distribution function,
but only on its mean value; also, the parameter n does not appear
explicitly. ]

Our main results are these: Let W;(j = 1, 2) be the equilibrium
waiting time (time from request for service until start of service) in
Model j, and let Wo be the equilibrium waiting time in the correspond-
ing M/G/1 queue (Model 0). Then,

where T is given by (1), and E( Wo) is given by the celebrated
Pollaczek-Khintchine formula [see, for example, Ref. 15, p. 217, eq.
(8.39) ]:

The simplicity of (2) and (3) and their similarity are quite remarkable.
We also define S;(j = 0, 1, 2) to be the equilibrium sojourn time
(waiting time plus service time) in Model j. Since

For Model 3, we define the equilibrium delivery time D3 as the
elapsed time between the generation of a letter and its delivery to its
destination. We will show that E'(D3) is given by the following formula,
again remarkable in its simplicity:

The special case of (7) when o” = 0 (i.e., when the time required to
sort a letter is constant) was found (in a different form, by a more
complicated argument) by Nahmias and Rothkopf."! The general result
(7) was found also by Shanthikumar,”*® using level-crossing analysis.

The well-known textbook by Tanenbaum” discusses a model of a
token-ring LAN with an arbitrary number n of terminals that, for n
infinite, coincides with our Model 1. Tanenbaum gives eq. (1) and
then states (p. 310) that the mean waiting time is “about half”
the mean cycle time. It is interesting to note that, for the case
of n infinite, eq. (2) shows that Tanenbaum’s approximation (i.e.,
E(W,) = T/2) underestimates the correct value by exactly E( Wo), an
amount that can be considerable, being essentially proportional to o”
and inversely proportional to 1 — p. [The reason that 7/2 underesti-

mates the correct value is a manifestation of the phenomenon of
length biasing, i.e., the cycle to which an arbitrary customer (the test
customer) arrives is stochastically longer than an arbitrary cycle. In
particular, if Tf is the length of the cycle during which the test
customer arrives, then F(T?) = E(T,) + V(T,)/E(T;), where V(7T1)
is the variance of the cycle times in Model 1 (see, for example, Ref.
15, pp. 200-6). Since E( W,) = E(T?)/2, it follows from this observa-
tion and eq. (2) that V(T,) = 2TE(W,). A practical implication of
this observation is that the mean waiting time E( W;) can be estimated
using measurements of cycle times only.]

Several authors (see Refs. 4, 17, 18, and 19), using arguments more
complicated than ours, have obtained results for related models with
different queue disciplines (e.g., exhaustive service or gated service)
and a finite number n of terminals; our result (2) can be obtained from
their results when n — «. Coffman and Gilbert” have analyzed Model
1 for the case of constant service times and, for this special case,
derived a number of explicit distributional results, such as the distri-
bution of waiting times.

In Section II we state the M/G/1 decomposition result alluded
to earlier. In Section III we apply this decomposition result to obtain
eqs. (2), (3), (5), and (6). For completeness, in Section IV we quickly
derive eq. (7) by directly comparing the mean delays in Models 2
and 3.

At all times the server is either scanning or is serving customers.
The basis for the analysis of this paper is to interpret the time intervals
when the server is scanning to be vacations, and then to invoke
Proposition 3 of Fuhrmann and Cooper.’ We define

y;(-) = the p.g.f. (probability generating function) of the equilibrium
distribution of the number of the customers present in Model
J(j = 1, 2) at an arbitrary point in time;

x;(-) = the p.g.f. of the equilibrium distribution of the number of
customers present in Model j(j = 1, 2) at an arbitrary point
in time, given that the server is scanning (on vacation); and

a(-) = the p.g.f. of the equilibrium distribution of the number of
customers present in the corresponding M/G/1 queue at an
arbitrary point in time (or, equivalently, just after a service
completion epoch).

Thus, z(-) is given by a well-known formula [see, for example, eq.
(8.12), p. 210, Ref. 15]. It follows directly from Proposition 3 of
Fuhrmann and Cooper? that

In Section III we show that it is a simple matter to find x/(1) for j =
1, 2. Since (by Little’s theorem) 7’(1) = XE(So) and y¥/(1) = AE(S;),
eq. (9) yields eqs. (5) and (6) or, equivalently, (2) and (3).

In this section we derive formulas (2) and (8) for the mean waiting
times for Models 1 and 2. To do this, first note that (for either model)
during each customer’s service time, k new customers arrive to the
system with probability

We now define two auxiliary models, Auxiliary Model 1 and Auxil-
iary Model 2. Auxiliary Model 1 is defined in exactly in the same way
as Model 1 except for the following aspect: Now, whenever the server
encounters a customer, the customer is served in zero time and departs
from the system; coincidental with his departure, however, a batch of
k new customers joins the system (distributed along the scanning path
in a uniform and independent manner) with probability p,, given by
(10). Thus, while the lengths of all service times have been collapsed
to zero, the number of customers in the system just after a service
completion epoch is stochastically the same for both Model 1 and
Auxiliary Model 1. (This is true in a distributional sense. Or, if we go
to the trouble to define Model 1 and Auxiliary Model 1 on the same
sample space, we can make this statement true on every sample path.)
This leads to the following conclusion: If we define A, to be the mean
number of customers present in Auxiliary Model 1, then

To calculate Aj, let Af and Sf be the arrival rate and sojourn time
in Auxiliary Model 1. Then, by Little’s theorem,

himself) generated by a Poisson arrival in Auxiliary Model 1; then
Ai = AXE(N). Now observe that the average number of new customers

generated when a customer is served is Ar = p; and each of these
customers will generate, on average, E(N) additional customers.
Hence, E(N) = 1+ pE(N); that is, E(N) = (1 — p)7', and therefore
we. (14)
1—p
[Note that E(N) is precisely the mean number of customers served
during an M/G/1 busy period. Observe also that (14) follows imme-
diately from the requirement that the mean number of arrivals per
cycle be the same in Auxiliary Model 1 as in the original Model 1:
Nic = AT.] Equations /11) through (14) yield y{(1) = Ac/2(1 — p); in
light of (1), we have
\T
5
This completes the calculation of (9) for Model 1, from which the
main result (2) follows.

We now define Auxiliary Model 2 in a completely analogous manner,
that is, in exactly the same way as Model 2, except that now when a
customer is served (at the origin), he is served in zero time and is
instantaneously replaced by a batch of k customers with probability

Pr, given by (10). We define Az to be the mean number of customers
present in Auxiliary Model 2. By the same argument used earlier,

and the same argument that was used to derive eq. (15) for Model 1
applies. Hence, combining the equations that are analogous to (12)
and (14), we have
r
Az = —— E(S9). (17)
Lp

But, in contrast with (13), the mean sojourn time of a customer in
Auxiliary Model 2 is exactly the cycle time c,

For completeness, we now derive eq. (7). This is accomplished by
directly comparing the mean delays in Models 2 and 3. Recall that in

these models, all customers are served at the origin. For either model,
consider an arbitrary customer (the test customer) and define

y = the mean number of customers in the test customer’s batch;
yp = the mean number of customers in the test customer’s batch that
are served before the test customer; and
Yq = the mean number of customers in the test customer’s batch that
are served after the test customer.
Then
Y=M+tyat1 (20)
and, by symmetry,
Yo = Ya: (21)

Now let L. denote the number of customers present in Model 2,
excluding the test customer, when the test customer enters service.
Then

The term \c/2 equals the mean number of customers who arrived
during the last scan, but behind the server. (These customers will be
collected on the server’s next cycle.) The term y,(= 57) equals the
mean number of customers who arrived during the service times of
the customers (in the test customer’s batch) who were served before
the test customer. Finally, the term y, equals the mean number of
customers that have not yet been served.

Now observe, on the other hand, that Lz has the same distribution
as the number of customers present in Model 2 at an arbitrary point
in time, excluding the customer being served (if any). [This is true
because departures see the same distribution of customers that arrivals
see (see Ref. 15, p. 187) and the arrivals see time averages (see Ref.
21).] Hence, by Little’s theorem,

1. R. B. Cooper and G. Murray, “Queues Served in Cyclic Order,” B.S.T.J., 48, No. 3
(March 1969), pp. 675-89.
2. R. B. Cooper, “Queues Served in Cyclic Order: Waiting Times,” B.S.T.J., 49, No. 3
(March 1970), pp. 399-413.
3. M. Eisenberg, “Queues With Periodic Service and Changeover Times,” Oper. Res.,
20, No. 2 (March-April 1972), pp. 440-51.
. O. Hashida, “Analysis of Multiqueue,” Review of the Electrical Communication
Laboratories, N. T. T. Public Corp., 20, Nos. 3-4 (March-April 1972), pp. 189-

5. A. G. Konheim and B. Meister, “Waiting Lines and Times in a System With
Polling,” J. Ass. Comput. Mach., 27, No. 3 (July 1974), pp. 470-90.

6. S. Halfin, “An Approximate Method for Calculating Delays for a Family of Cyclic
Type Queues,” B.S.T.J., 54, No. 10 (December 1975), pp. 1733-54.

7. P. J. Kuehn, “Multiqueue Systems With Nonexhaustive Cyclic Service,” B.S.T.WJ.,
58, No. 3 (March 1979), pp. 671-98.

. S. W. Fuhrmann and R. B. Cooper, “Stochastic Decompositions in the M/G/1
Queue With Generalized Vacations,” Oper. Res., 33, 1985.

. S. Nahmias and M. H. Rothkopf, “Stochastic Models of Internal Mail Delivery
Systems,” Manage. Sci., 30, No. 9 (September 1984), pp. 1113-20.

13. W. Bux, “Performance Issues in Local-Area Networks,” Research Report RZ 1268
(45715), IBM Zurich Research Laboratory, November 1983.

14. H. Takagi and L. Kleinrock, “Analysis of Polling Systems,” Technical Report
TR87-0002, IBM Japan Science Institute, Chiyoda-ku, Tokyo, 1985.

15. R. B. Cooper, Introduction to Queueing Theory, second edition, New York: North-
Holland (Elsevier), 1981.

17. W. Bux, “Local-Area Subnetworks: A Performance Comparison,” IEEE Trans.
Commun., COM-29, No. 10 (October 1981), pp. 1465-73.

18. L. F. M. de Moraes and I. Rubin, “Analysis and Comparison of Message Queueing
Delays in Token-Rings and Token-Buses Local Area Networks,” Proc. IEEE Int.
Conf. Commun., Amsterdam, May 14-17, 1984, pp. 130-5.

19. H. Takagi, “Mean Message Waiting Time in N Symmetric Queues With Nonex-
haustive Cyclic Service Discipline,” IBM Japan Science Institute, Chiyoda-ku,
Tokyo, 1984.

Robert B. Cooper, B.S., 1961, Stevens Institute of Technology; M.S. (Sys-
tems Engineering and Operations Research), 1962, and Ph.D. (Electrical
Engineering), 1968, University of Pennsylvania; AT&T Bell Laboratories,
1961-1969; Georgia Institute of Technology, 1969-1976; University of Michi-
gan, 1975; New Mexico Institute of Mining and Technology, 1976-1978;
Florida Atlantic University, 1978—. Mr. Cooper’s primary interest is queueing
theory and its application in telecommunications and computer science.

Steve W. Fuhrmann, B.A. (Mathematics), 1970, University of Montana;
Ph.D. (Statistics), 1975, Purdue University; Rutgers University, 1975-1977;
AT&T Bell Laboratories, 1977—. Mr. Fuhrmann’s interests and work focus
on stochastic processes and stochastic models for the performance evaluation
of computer-communication systems.

Phase Locked Loop Analysis and Circuit Emulation (PLACE) is an inter-
active program to assist in the design of Phase Locked Loop (PLL) systems.
Written in C language, PLACE computes the loop filter components (up to
active third-order filters); performs a stability analysis (finding the phase
margin and damping ratio); and calculates the lockup time, hold range, and
capture range. PLACE computes the PLL output jitter response due to
incoming reference signal jitter, output jitter due to reference leakage through
the phase detector, and output jitter response due to phase noise of the PLL
components. The open and closed loop gain and phase, as well as jitter
response, is plotted. Additional features found in PLACE 2.0 are frequency
and magnitude of jitter peaking; a sensitivity analysis, which computes changes
in loop performance as a function of component variation; and an interactive
routine to help the designer optimize PLL performance. Currently residing on
Digital Equipment Corporation’s VAX-11/780, AT&T 3B20, and IBM Sys-
tem/370 processors, PLACE is presently available at most AT&T Bell Labo-
ratories locations. This paper illustrates the capabilities of PLACE, shows
several examples, and discusses the required calculations.

Phase Locked Loops (PLLs) are used extensively in communication
systems. Yet there has been no generally available computer-aided
design program specifically targeted to assist the PLL designer in
evaluating the performance of his system before building it in the

laboratory. Phase Locked Loop Analysis and Circuit Emulation
(PLACE) is an interactive program that allows the designer to opti-
mize his PLL, trading off one parameter (e.g., lockup time) for another
(e.g., jitter response). PLACE performs the following functions:

1. Determines if the PLL is stable by computing the phase margin,
damping ratio, and undamped natural frequency. PLACE accounts for
any parasitic Voltage Controlled Oscillator (VCO) or operational
amplifier poles, and phase shift due to the feedback counter.

2. Finds the appropriate loop filter components for a given damping
ratio or 3-dB frequency. PLACE notifies the user if desired loop
parameters result in unrealistic component values.

3. Computes the PLL system bandwidth, noise bandwidth, and loop
filter bandwidth.

4. Approximates the hold range, capture range, and lockup time.
(The hold range is normalized for the active loop filter case.)

5. Determines the output jitter response due to the internal phase
noise of the PLL components and the output jitter response due to
jitter on the incoming reference signal.

6. Determines the sensitivity of PLL performance to component
variation.

Section II of this paper discusses PLACE input/output, Section III
shows several examples, and Section IV explains the calculations that
PLACE performs.

PLACE consists of a nongraphics module followed by a graphics
module. The two modules are independent, and thus a user can invoke
PLACE on a nongraphics terminal. PLACE recognizes the PLL shown
in Fig. 1, where we have defined the following constants:*

The program will ask the user for the values of these constants.
Typical values for K, are 1.4 V/rad for an exclusive-or gate, 0.4 V/rad
for the RCA 4046 phase comparator II, 0.11 V/rad for the Motorola
4044 phase detector, and 0.16 V/rad for the Motorola 12040.

For best results, the VCO gain constant should be measured because
the PLL stability is highly dependent on Ky. The value of Ky will

most likely vary over the VCO’s frequency range; the sensitivity
analysis will compute the effect of this variation. If the results are
unacceptable, the user will need to build a linearizer in order to
maintain a constant Ky. (See Ref. 1 for an excellent example, or Ref.
2.) If a passive attenuator is used ahead of the VCO, the entered value
of Ky must be reduced by the attenuation factor. For frequency
synthesizer applications, the feedback divisor Nyg is often a variable,
and running PLACE twice with the maximum and minimum values
will show the change in loop performance.

Next, PLACE asks the user if it is desired to account for any
parasitic VCO pole; if a Voltage Controlled Crystal Oscillator (VCXO)
is used, it is strongly recommended that this pole be entered since it
is often the dominant pole of the PLL. (See Section 4.1.4 on how to
measure the VCO pole.) For the active loop filter case, if it is desired
to account for the op amp pole, the designer may lump this pole into
the VCO pole and enter the value at this time.

Next, PLACE asks for the reference frequency, i.e., the frequency
entering the phase detector. The reference frequency is the frequency
at which the phase comparison is performed. This information is
required for the jitter analysis and implicitly defines the output fre-
quency of the PLL.

PLACE will then ask the user for the type of loop filter desired.
Five types of loop filters are recognized (see Fig. 2):

g. 2—Loop filter topologies: (a) No loop filter, (b) RC loop filter, (c) lag-lead loop
rived (d) second-order active filter, and (e) third-order active filter.

We now list several guidelines for selecting a loop filter. The choice
between an active or passive loop filter must be made first. If the

designer chooses a passive loop filter, the lag lead is preferred over the
RC loop filter for the following reasons:

1. For the lag-lead loop filter, the designer can specify the loop
natural frequency w, and damping ¢ independently of each other, and
thus it is possible to have a narrowband loop with a substantial
damping. However, for the RC loop filter, a narrowband PLL (low w,,)
requires a high value of 7 (loop filter time constant), but a high 7
results in low damping and possible instability (see Section IV).

2. The VCO parasitic pole is less likely to render a lag-lead PLL
unstable compared with the RC loop filter PLL. (This is because the
open loop phase for the RC loop filter PLL approaches —180 degrees
at high frequencies, whereas the phase approaches —90 degrees for the
lag-lead case.)

3. The lag-lead PLL exhibits improved transient behavior. For a
given PLL damping value, the phase error transient of the lag-lead
PLL reaches its steady-state value much sooner than that of the RC
loop filter.

An active loop filter is required when a higher loop gain is required
(to increase the hold range, for example). The advantages of the third-
order active filter over the second-order active filter are improved
response to phase-step changes, zero steady-state phase error due to
frequency ramp inputs, and better reduction of VCO noise.

After selecting a loop filter type, the user has two options: the
desired damping and natural frequency may be specified, or the loop
filter time constants may be specified. The first option is usually used
when designing a new PLL, while the second option is normally used
when analyzing an existing PLL.

The user can request a stability analysis, a loop filter analysis, a
tracking analysis, a jitter analysis, a sensitivity analysis, and an
interactive optimization routine.

The stability analysis yields (1) second-order undamped natural
frequency w, in hertz, (2) phase margin in degrees, (3) phase margin
degradation due to the VCO (and/or op amp) pole, (4) phase margin
degradation due to any phase delay of the feedback counter, (5)
damping, and (6) PLL system bandwidth in hertz.

The loop filter analysis yields (1) the loop filter time constants in
seconds, (2) values for the loop filter resistors and capacitors, and (3)
the loop filter 3-dB frequency.

The tracking analysis yields (1) hold range in hertz (for active loop
filters, the hold range is normalized, i.e., assuming F(0) = 1); (2)

approximate capture range in hertz; (3) approximate pull-in range in
hertz; and (4) approximate lockup time in seconds.

The jitter analysis yields (1) output jitter due to incoming reference
signal jitter: jitter bandwidth in hertz, frequency of jitter peaking in
hertz, and noise bandwidth in hertz; (2) output jitter due to reference
leakage through the phase detector: frequency of first sideband in
hertz, magnitude of first sideband relative to the carrier in decibels,
peak phase jitter in degrees, peak frequency deviation in hertz, and
loop filter’s attenuation of reference frequency in decibels; and (3)
output jitter due to VCO phase noise: VCO phase noise reduction 3-
dB frequency.

The sensitivity analysis computes upper and lower bounds on the
hold range, capture range, pull-in range, and lockup time. The user
input is the tolerance of the gain constants and loop filter components.

The optimization routine asks if the user wants to design for any
one of the following: (1) larger hold range, (2) larger capture and pull-
in range, (3) faster lockup time, or (4) less output jitter.

PLACE then automatically changes the PLL parameters to achieve
the desired goal; then, the user can rerun the loop filter analysis to
observe the new component values and use the tracking analysis to
observe the new lockup time, etc.

The graphics portion of PLACE plots (on a TEK 4014 or similar
device) the open and closed loop gains and phase, and the PLL
response to VCO phase noise. For the graphics portion, the S package®
must be installed on the system.

This example demonstrates the lag-lead loop filter case; it also
demonstrates that the parasitic VCO pole may add sufficient phase
shift to cause possible instability. _

The user input to PLACE is as follows: The PLL for this example
phase locks a 3.088-MHz crystal oscillator to an incoming 1.544 MHz
signal. The VCXO has a measured gain constant of 800 Hz/V around
the center frequency of 3.088 MHz; it also has a parasitic pole at 10
Hz. The phase detector is a Complementary Metal-Oxide Semicon-
ductor (CMOS) exclusive-or gate measured at 1.4 V/rad; this was
derived by measuring the logic high/low levels: (4.6 — 0.2)/m = 1.4.
The phase comparison is done at 4 kHz; thus, the values for
the frequency dividers are Nrr = 386 = 1.544 MHz/4 kHz and Npgg =
772 = 3.088 MHz/4 kHz. The loop filter resistors and capacitors were
measured accurately, yielding time constants of 7; = 57.4513 ms and
T2 = 4.00336 ms.

The results from the stability, loop filter, tracking, and jitter analysis

Parameter Calculated Measured
Hold range +/—568 Hz +496, —588 Hz
Pull-in range +/—568 Hz +493, —585 Hz
Lockup time 324 ms 400 ms
Frequency of jitter peaking 1.6 Hz 1.5 Hz
Magnitude of jitter peaking 1.3 dB 3 dB
Jitter bandwidth 3 Hz 4 Hz
Frequency of first sideband 8 kHz 8 kHz
Magnitude of first sideband —52 dBc ~—50 dBc
Peak base jitter 0.279 degrees 0.343 degrees
Peak frequency deviation 20 Hz 25 Hz
Reference frequency attenuation 23 dB 23 dB

appear in Appendix B. Table I summarizes the calculated results and
also shows the measured values. The stability analysis indicates a
damping of 0.7 and natural frequency of 2 Hz. The phase margin is 60
degrees: the VCO pole reduced the phase margin by 7.4 degrees, while
the feedback counter caused negligible phase margin degradation of
0.02 degree. The PLL bandwidth is 1.2 Hz; this low PLL bandwidth
is due to the very low Ky of 800 Hz/V. The previous results can also
be found from the PLACE graphic display of open loop gain, shown
in Fig. 3. Unity gain occurs at 1.2 Hz, where the phase is —120 degrees.

Fig. 3—PLACE graphic display of open loop gain for the lag-lead loop filter example.
The magnitude is indicated by the solid line, and the phase is indicated by the dotted
line (right scale).

Note that the parasitic VCO pole has brought the total phase shift to
—180 degrees (instead of 0 degrees) for high frequencies. The tracking
analysis indicates a hold range of +/—568 Hz. The measured value is
+496 Hz, —588 Hz. The hold range is asymmetrical due to the larger
than 50-percent duty cycle phase detector output during the lock
condition. The pull-in range is +493, —582 Hz; it is common for low
gain loops to have pull-in ranges close to the hold range.

The calculated lockup time is 324 ms, while the measured time is
400 ms; this may be found from Fig. 4, a display of tuning voltage
during the lock-in process.

The jitter analysis indicates 1.3 dB of jitter peaking at 1.6 Hz, and
a jitter bandwidth of 3 Hz. The closed loop gain Bode plot is shown in
Fig. 5. The peaking is difficult to observe given the scale of the plot.
The measured values may be found from Fig. 6, a spectrum analyzer
display of closed loop gain versus jitter frequency from an HP 8557A.
There is 3 dB of jitter peaking at 1.5 Hz (picture only displays up to
50 Hz). (The measured jitter peaking is higher than calculated due to
additional poles in the VCXO.) The gain is down 3 dB (jitter band-
width) at 4 Hz. The test equipment configuration for measuring the
closed loop gain transfer function of Fig. 6 is shown in Fig. 7.

The calculated magnitude of the 8-kHz first sideband off the PLL
output carrier is —52 dB; the measured level is —50 dB. This is shown
in Fig. 8, a spectrum display from an HP 8557A. The peak phase jitter
is 0.279 degree; the measured value from an HP 8901A modulation
analyzer is 0.348 degree.

Fig. 4—Tuning voltage versus time during acquisition measured on a Paratronics
5000 logic analyzer.

Fig. 5—PLACE graphic display of closed loop gain. The magnitude is indicated by
the solid line, and the phase is indicated by the dotted line (right scale).

Fig. 6—Measured closed loop gain versus frequency showing 3 dB of jitter peaking.
Vertical scale is 10 dB/division, horizontal scale is 5 Hz/division, and resolution
bandwidth is 1 Hz.

The sensitivity analysis is also found in Appendix B. The upper and
lower bounds found there reflect a 5-percent tolerance on the entered
VCO gain constant, and 2-percent tolerance on the loop filter resistors
and capacitors.

The remainder of this paper explains the calculations that PLACE
performs.

IV. ANALYSIS
4.1 Stability analysis
The stability analysis requires computation of the PLL transfer

Fig. 7—Test equipment configuration used to obtain the closed loop gain transfer
function shown in Fig. 6.

Fig.8—Spectrum of VCO output carrier showing 8-kHz sidebands down 50 dB.
Vertical scale is 10 dB/division, horizontal scale is 5 kHz/division, and resolution
bandwidth is 1 kHz.
function. The components of a PLL are shown in Fig. 1; constants are
defined in Section 2.1.

For the following calculations, we define Ky, = 2rKy = VCO gain
constant in rad/s/V. We also define the lumped gain constant:

K,Kvr
. Nrp ©
The equivalent frequency domain model of the PLL is also shown in
Fig. 1, where F(s) and V;(s) are the Laplace transforms of the loop

filter transfer function and loop filter output voltage, respectively. The
VCO model of Ky,/s is derived as follows:

The transfer function for the PLL may be easily derived if we define
the feed-forward gain as

Then, following conventional control theory analysis, the open loop
gain is the product of the feed-forward and feedback gains:

sNep Ss
The closed loop gain is
— douwls) Gs)
As) ="j.(¢) 1+ BAGG)
= K,Ky,F(s)/s : K,Kyv,F(s) _ NrpK (4)
1+ K, Ky,F(s)/sNyp St+ K,Ky,-F(s)/Nre Ss re K

The loop type is specified by the number of poles at the origin of
B(s)G(s). The loop order is specified by the total number of poles in
G(s)G(s). We have ignored the phase delay through the feedback
counter in the above analysis; this phase delay is treated in Section
4,1.5.

From the above equations we see that the PLL response is highly
dependent on the form of loop filter F(s). By substituting a particular
loop filter function F(s) into eq. (4), we can (except for the no loop
filter case) get the denominator into the classic second-order control
system form of s? + 2¢w,s + w2, where w, is the undamped natural
frequency and ¢ is the damping. This is shown in Appendix C.

PLACE calculates the phase of 8(s)G(s) at unity open loop gain;
the difference from —180 degrees is the phase margin. This definition
is identical to the stability analysis of feedback amplifiers.

A parasitic pole causes the open loop gain to fall off more quickly
and thus affect the phase margin. The tuning element of a VCO
(typically a varactor) cannot respond to rapidly changing tuning
voltages. We represent this by assigning a pole to the VCO. It has
been the author’s experience that this VCO pole may be the dominant
pole of the PLL (especially if VCXOs are used). PLACE determines
the phase margin degradation due to the parasitic VCO pole by
multiplying the open loop gain by 1/(1 + 7,s), where 7, = 1/(2zfv),
where fy is the VCO pole. The VCO pole may be found by impressing
an ac modulating signal on the dc voltage to the varactor, and deter-
mining the highest modulating frequency for which the VCO output
will follow the input. (A typical value for a VCXO is fy = 10 Hz.)

If the user wants to account for the operational amplifier pole for
the active loop filter case, he may lump this pole into the VCO parasitic
pole. The designer should try to keep the operational amplifier band-
width much larger than the PLL system bandwidth (defined in Section
4,1.6).

The effect of the feedback counter is to add a phase delay of f/frer to
the open loop gain and thus degrade the phase margin by f,,/frer rad,
where f,, is the frequency for unity open loop gain and fier is the
frequency entering the phase detector.*” This is derived as follows.

Since any change in the VCO frequency can be observed by the
phase detector only after the feedback counter overflows T seconds
later (worst case), the effect of the counter is to produce a maximum
delay of up to T seconds, where the delay time T is given by

N. rB/, ref
Thus, in the frequency domain we represent the feedback counter by
a ea)
Nrp Nes

PLACE calculates this phase shift and notifies the user of the resulting
phase margin degradation.

The PLL system bandwidth is the frequency for PLL unity open
loop gain; it is found by solving | 6(jw)G(jw) | = 1 for w. We have the
following:

For the third-order active loop filter, the user may specify the desired
PLL bandwidth, or, alternatively, the value defaults to frer/50. PLACE
uses this value to optimize the loop for best phase noise performance
(see Appendix C).

The loop filter time constants are calculated from the undamped
natural frequency and the damping by using the equations derived in
Appendix C. Alternatively, the user may directly specify the loop filter
time constants.

PLACE assumes 0.1-uf capacitors and calculates the resistors from
the definition of the time constants. Note that the user can linearly
scale the resistors and capacitors to any desired value (e.g., if the user
desires 0.01-uf capacitors, multiply the calculated resistor values by
10).

To maintain real values of resistors and capacitors, certain con-
straints must be met for the lag-lead loop filter case. PLACE first
asks for the desired value of w,. Then, when it asks for the desired
value of damping, the entered value for damping must satisfy two
conditions:

Hence, after the user specifies the desired value for w,, PLACE asks
the user to enter a value for damping that satisfies

The frequency at which F(s) is down 3 dB is found by solving
| F(jw) | = 0.707 for w. We have the following (iff +; > V2r2, else F(s)
is never down 3 dB):

For the second and third active loop filter cases, Fgw is undefined
because F(s) has a pole at the origin. For these cases PLACE calculates
the zero frequency.

The hold range is defined as the maximum input frequency range
that the PLL will track once it is in lock. It may be found experimen-
tally by slowly varying the input frequency to a PLL that is in lock
and noting when lock is lost. (The word “slowly” is emphasized because
if the input frequency has a step change, the transient behavior of the
loop may cause the PLL to loose lock, even though the step is within
the hold range.) .

The hold range is independent of the type of loop filter F(s) and is
given by numerous authors (e.g., see Ref. 6) as K,Kv,-F(0)/Nrs, where
F (0) is the de gain of the loop filter. (Note that the error voltage is a
constant dc value when the loop is in lock.) For passive loop filters
F(0) = 1, and accounting for the feed-forward counter, PLACE com-
putes the hold range as

—— Hz. (6)
For active loop filters F(0), and hence the hold range, can be made
arbitrarily large (until the VCO or operational amplitude satu-

When recovering timing from Pulse Code Modulated (PCM) data
lines, the hold (and capture) range may be asymmetrical due to the
density of ones.”

Nonsinusoidal phase detectors theoretically extend the hold and
capture ranges. For example, triangular phase detectors have a so-
called “extension factor” of 7/2, while sawtooth phase detectors have
a factor of 7 (see Ref. 6). PLACE leaves it to the user to modify the
phase detector gain constant K, (if the designer is so inclined) when
PLACE requests the value.

The capture range is defined as the maximum input frequency range
for which a PLL will acquire phase lock without skipping cycles (i.e.,
the phase detector voltage monotonically drives the VCO toward lock,
without any beating). (Some authors call this the lock-in range.) The
pull-in range is defined as the maximum input frequency range for
which a PLL will acquire lock (typically with skipping cycles), even if
it takes minutes or hours to lock up. (In this case the phase detector
output voltage may be a beat note swinging the VCO above and below
fin-)

Whereas eq. (6) gives an accurate value for the hold range, the
determination of the capture and pull-in range is extremely difficult
because the loop starts (by definition) out of lock, and thus a nonlinear
analysis is required. The solution cannot be expressed in closed form
(except for the F(s) = 1 case), and only a graphical phase-plane
trajectory procedure®® will yield true results. PLACE approximates
the capture and pull-in ranges by using the approximating equations
found in the literature and modifying them to account for the feedback
and feed-forward counters:

1 K, Kv,
Fesptare = frota = foun = a Nyr fo Hz
(see Refs. 8 and 9).
RC loop filter: If Kr; < 0.25,
1 K,Kvr
feaotaxe = faota = foun = oe N, FF a Hz

For the second- and third-order active loop filter cases, the hold and
pull-in ranges will be as large as the maximum frequency range of the
VCO, assuming the operational amplifier output does not saturate at
the supply rail, and assuming negligible loop delay.

The lockup time is defined for PLACE as the time it takes the PLL
to acquire phase and frequency lock from an initial frequency offset
equal to the pull-in range. For the no loop filter case and low gain
(Kr; < 0.25) passive loop filter case, PLACE calculates the lockup
time as Tiock = 1/K seconds.®

For second-order active loop filters and high-gain (Kz, > 0.25)
passive loop filters, the lockup time is approximately given by Tio, =
Aw?/2fw? (see Refs. 6 and 8). PLACE lets Aw = ([feapture + fpun]/
2.Nyr)2z7, which is where the approximation is most accurate; for the
second-order loop filter, PLACE lets Aw = 27(2fcapture). PLACE does
not calculate the lockup time for the third-order loop filter.

In the discussion that follows, the term “reference signal” is the
signal entering the phase detector; thus, if Nrr = 1, the reference
signal is the incoming signal itself. Also, the terms jitter and phase
noise are equivalent.

The output jitter of a PLL may be conveniently separated into two
main components: jitter generation and jitter propagation. Jitter prop-
agation refers to the output jitter due to jitter on the incoming signal
(or the reference signal for frequency synthesizer applications). Jitter
generation may be subdivided into five components: output jitter due
to leakage of the reference signal through the phase detector; output
jitter due to VCO phase noise; and output jitter due to phase noise of
the phase detector, frequency divider, and loop filter (if an active loop
filter is employed). PLACE analyzes each component of jitter sepa-
rately.

The most important result of this section may be summarized
(without mathematics) as follows: A PLL acts as a high-pass filter to
VCO phase noise, and acts as a low-pass filter to incoming reference
signal jitter.

4.4.1.1 Jitter bandwidth. It may easily be shown that the PLL’s
response to jitter on the incoming reference signal is given by the
closed loop gain transfer function H(s), eq. (4). Merely repeat the
analysis of Section 4.1.1 with ¢; now representing an instantaneous
phase deviation of the incoming reference frequency. Equation (4) is
repeated here for convenience:

Thus, above the frequency for unity closed loop gain H(s), the input
signal jitter will be attenuated. Thus a PLL acts as a low-pass filter
to jitter on the incoming reference signal. Also, notice from eq. (4)
that within the PLL system bandwidth, a PLL multiplies the reference
noise by the feedback counter divisor Nrg. That is, the feedback
counter effectively adds 20 log Nrg dB/Hz phase noise to the reference
signal.

The jitter bandwidth is the frequency at which H(s) is down 3 dB
from its dc value. If we neglect any parasitic VCO or operational
amplifier poles, we have the following:

When there are parasitic poles (which often reduce the bandwidth
values calculated above), PLACE uses an iterative search to find the
jitter bandwidth.

4.4.1.2 Frequency and magnitude of jitter peaking. The peak jitter gain
| H(Jwpeax) | is equivalent to the peak magnitude of H(s). Jitter peaking
is the difference (in decibels) between the closed loop gain peak
magnitude | H(jwpeax) | and the de value of H(s). To find it we first
find the frequency of peak jitter gain wpea, by setting the derivative of
| H(jw) |? to zero and solving for wpeax. Then we substitute wpea, back
into H(jw). Neglecting any parasitic VCO or operational amplifier
poles, we get the following results:

For the lag-lead and third-order active loop filters, PLACE uses an
iterative search to find the jitter peaking. The above equations do not
reveal the jitter peaking due to parasitic poles. PLACE uses an iterative
search to find the jitter peaking in such cases.

4.4.1.3 Noise bandwidth. The noise bandwidth of a PLL is the band-
width of an equivalent rectangular filter that would yield the same
output noise power (variance) as the PLL, if they both have white
noise inputs of equal density. PLACE calculates the one-sided noise
bandwidth, where New = fo | H(jw) |?dw. Various authors have eval-
uated the integrals, and we have the following:

For the third-order active loop filter, the noise bandwidth is approxi-
mately given by the PLL system bandwidth defined earlier.

The output noise of the PLL, given a white noise input with power
Pi, is given by Pour = PinNpw. Similarly, the output signal-to-noise
ratio of the PLL is inversely proportional to the noise bandwidth.

4.4.2 Output jitter due to leakage of the reference through the phase
detector

4.4.2.1 Frequency and magnitude of first sideband. Any feedthrough of
the reference frequency through the loop filter phase modulates the
VCO, producing sidebands at the VCO output. The frequency of the
first sideband is the frequency leaving the phase detector: twice the
reference frequency for exclusive-or gate phase detectors, the reference
frequency for other phase detectors. The magnitude of the first side-
band is found as follows: the modulating frequency is fm = frer, and the
Modulation Index (MI) is MI = Af/fm = Af/frer. Using conventional
frequency modulation/phase modulation analysis, we can express the
resulting VCO output voltage u(t) by using Bessel functions as

where we have neglected the higher-order sidebands. Now, if the
modulation index is small (MI < 0.3), we have (J;(MI))/(Jo(MD) =
MI/2. For example, (J;(0.2))/(Jo(0.2)) = 0.0995/0.9900 = 0.1 = MI/2.
Hence,

When we realize that most of the signal power is in the carrier for
small modulation indices, we see that the above equation is the
definition of the “(f) (see Refs. 12 and 13) used in specifying oscillator
phase noise.

where V,, is the peak amplitude of f,-; at the phase detector output.
(For exclusive-or gate phase detectors, the phase detector output
frequency is twice fer.)

Thus, the magnitude of the first sideband (relative to the carrier) at
the PLL output is given by

where F(jw) is evaluated at rer. (This result agrees with Ref. 14.)
PLACE asks the user for the value of V,,, the peak voltage output
from the phase detector.

4.4.2.2 Peak frequency deviation and phase jitter. It can be shown that
the peak phase deviation is related to “(/) by the following relation:

The magnitude of &(f) is given by eq. (7); PLACE 2.0 solves the
above equation for the peak phase deviation in radians. Then, the
peak phase jitter in degrees is 57.3 times as large. The peak frequency
deviation is found by using Afpeak = fmA@peak-

As long as the peak phase jitter is less than approximately 28
degrees, the PLL will generally stay in lock (see Ref. 16).

4.4.2.3 Loop filter’s attenuation of reference frequency. PLACE calcu-
lates the magnitude of F(s) at the reference frequency and informs the
user of the result.

4.4.3 Output jitter due to PLL components
4.4.3.1 Output jitter due to VCO phase noise. PLACE calculates the

PLL’s relative response to VCO phase noise (i.e., it calculates the
PLL’s attenuation of VCO phase noise); PLACE does not calculate
the absolute value of the VCO-generated phase noise itself. (References
9, 12, and 17 explain how to measure the absolute VCO phase noise.)
A VCO may be modeled as a pure signal source corrupted by an
instantaneous phase fluctuation ¢vco(t). We modify the PLL model
by adding this noise source to the VCO of Fig. 1. The total PLL output
phase is now @out(t) = dioop(t) + évco(t), where ¢roop(t) is the output
phase for a PLL with an ideal VCO. We now have the following:

but
PourlS) = de(s)F(s)K,Ky/s + dvco(s),
so that
ee Pout — Pvco
F(s)K,Ky/s
Algebraic manipulations yield
Nyp KF(s)/s 1
out = Pin 1 + KF(s)/s + dvco 1+ KF(s)/s
Pout = Pin ae + ¢vco Ee TENG (9)
Thus, the PLL response to VCO phase noise is given by
: (10)

(This result is confirmed in Refs. 5, 8, and 13.) The above equation
tells us that the —3 dB point of the PLL response to VCO phase noise
occurs when |{8(s)G(s)| = 1; but this occurs at the PLL system
bandwidth (defined in Section 4.1.6). Hence, the PLL will pass through
the VCO phase noise unattenuated above the PLL system bandwidth.
Below the PLL system bandwidth, the output jitter due to VCO phase
noise is reduced by the loop gain 8(s)G(s).

4.4.3.2 Output jitter due to frequency divider phase noise. In Section
4.4.1.1 it was shown that the feedback counter effectively adds 20 log
Nrs GB/Hz phase noise to the reference signal; this section discusses
the jitter generated in the frequency divider itself.

internal active devices. Typical noise floors for Transistor-Transistor
Logic (TTL), Emitter-Coupled Logic (ECL), and Metal Oxide Semi-
conductor (MOS) dividers are —120 to —140 dB. TTL dividers have
the lowest noise, followed by ECL, and then MOS. (Typical values
may be found in Ref. 9.)

As in the case for VCO phase noise, PLACE does not need to know
the absolute value of the divider noise. The PLL response to this noise
is the same as for jitter on the incoming signal, because the PLL
cannot tell whether the noise comes from the divider or signal inputs
to the phase detector. Thus, the PLL response to divider noise is
identical to the low-pass filter shape computed for incoming line jitter.
[The PLL response to noise from an active loop filter is also given by
eq. (4).]

In timing recovery circuits, the divider noise is negligible compared
with the input signal jitter. In frequency synthesizers, however, the
two may be equal, and then the resultant noise equals their rms sum.

4.4.3.3 Output jitter due to phase detector phase noise. Typical noise
floors for phase detectors are —150 to —160 dB/Hz (see Ref. 15). The
PLL response to this phase noise is again given by eq. (4). The phase
detector noise floor improves 3 dB per octave of reference frequency
reduction.

PLACE asks the user for the tolerance of the VCO and phase
detector gain constants, and the tolerance on the loop filter resistors
and capacitors. It then performs a Taylor series expansion and retains
the first-order terms. Then it calculates upper and lower bounds for
the hold range, capture and pull-in ranges, and lockup time. This
allows the designer to observe the effect of component variation on
the PLL.

PLACE 2.0 features an interactive routine that allows the designer
to optimize PLL performance. The user specifies either a larger hold
range, larger capture and pull-in range, faster lockup time, or less
output jitter. After the user specifies a desired item to optimize,
PLACE automatically adjusts the PLL damping and natural frequency
to achieve the desired goal.

An interactive program to aid in the development of PLLs has been
written. PLACE has been successfully used in the design of PLLs for
digital channel banks, muldems, and radio transmission systems. In

addition, routine use of PLACE on several existing PLL designs
revealed problems that were subsequently corrected before the product
was placed in the field. The program has saved many PLL designers
from tedious calculations, allowing a better understanding of PLL
performance.

The author wishes to thank M. Luniewicz for his encouragement
and inspiration.

13. L. Martin, “Noise Property Analysis Enhances PLL Designs,” Elec. Des. News, 26,
No. 18 (September 1981), pp. 91-8.

17. J. Payne, “Measure and Interpret Short-Term Stability,” Microwaves, 15, No. 7
(July 1976), pp. 34-45.

18. A. Przedpelski, “Optimize PLL’s to Meet Your Needs,” Electron. Des., 26, No. 23
(September 1978), pp. 134-7.

F(s) Loop filter transfer function
Faw Loop filter 3-dB frequency
Ee Frequency of modulation
fret Reference frequency

Magnitude of jitter peaking
Lumped gain constant, K = (K,Kv,)/Nre
Phase detector gain constant in V/rad
VCO gain constant in Hz/V
VCO gain constant in rad/s/V
Noise bandwidth in hertz
Feedback counter divisor
Feed-forward counter divisor
PLL bandwidth

Your VCO pole at 10.0 Hz reduced the phase margin by 7.41 degrees, and reduced the
PLL bandwidth by 0.1 Hz.

Jitter Analysis:
1. Output Jitter Due to Incoming Reference Signal Jitter:

Frequency of Magnitude of Noise Jitter
Jitter Peaking Jitter Peaking Bandwidth Bandwidth
(Hz) (dB) (Hz) (Hz)
1.6 1.272 22.5 3

2. Output Jitter Due to Reference Leakage Through the Phase Detector:
Frequency of Magnitude of Peak Phase Peak Frequency Ref. Freq.

1st Sideband 1st Sideband Jitter Deviation Attenuation
(Hz) (dBc) (degrees) (Hz) (dB)
8000 —52 0.278 19.4 23
3. Output Jitter Due to VCO Phase Noise:
Your PLL attenuates VCO phase noise BELOW 1.2 Hz.
Sensitivity Analysis:
Hold Range Capture Range Pull-In Range Lockup Time
(+/- Hz) (+/- Hz) (+/— Hz) (sec)
Lower Bound: 539.0 36.1 539.0 0.30792615
Nominal: 568.0 39.6 568.0 0.32447433
Upper Bound: 597.0 43.1 597.0 0.34102252
APPENDIX C

Note that the phase margin is 90 degrees and the loop is theoretically
unconditionally stable; but, if we account for any parasitic VCO pole
(or any other parasitic poles lumped into the VCO pole), PLACE
shows that the PLL can be unstable for very high loop gain K. No
loop filter results in a first-order type-1 PLL.

We can get the denominator into the classical second-order control
system form of s? + 2fw,s + w2 if we let w2 = K/r and 2tw, = 1/r.
Then,

Note that the phase starts off at —90 degrees and approaches —180
degrees at high frequencies. The larger the value of 7, the lower the
damping and the smaller the phase margin. The RC loop filter results
in a second-order type-1 PLL.

We can get the denominator into the classical second-order control
system form of s? + 2fwps + w% if we let w2 = K/r, and 2fw, =
(1 + Kr2)/7;. Then,
1+ s(2¢/w, — 1/K 1 + s(2¢/w, — 1/K
8° + Wlans + wr Ss Ss
—}]} +2;—41
Wn Wn

Note that the phase starts off at —90 degrees, approaches —135
degrees midway between 1/7, and 1/72, and then approaches —90
degrees again for high frequencies.

The lag-head loop filter results in a second-order type-1 PLL.
(For large gain K, eq. (7) reduces to eq. (8), i.e., the lag-lead loop filter
is an approximation of the second-order active loop filter.)

If it is desired to account for the operational amplitude pole, the
designer may lump this pole into the VCO parasitic pole. Try to keep
the operational amplitude bandwidth much larger than the PLL
bandwidth (defined in Section 4.2).

s/F(s) + K  s?x7, + K(1 + ros) 9 8? + sKt0/7, + K/71
We can get the denominator into the classical second-order control
system form of s? + 2fw,s + w? if we let w2 = K/7, and 2tw, =

the open loop gain magnitude is
: . K
| B(ja)G(jw) | = —G V1 + (wr2)’,
1
and the open loop gain phase is
6(jw) = —180 + arctan(wre).

Note that the phase starts off at —180 degrees and approaches —90
degrees at high frequencies. Thus, the second-order PLL exhibits a
large phase margin, even with parasitic (VCO) poles. Ensure 7; < 72
for good stability.

This loop filter results in a second-order type-2 PLL.
C.5 Case 5. Third-order active loop filter: F(s) = (1 + 72s)/[71s(1 + 735)]

Note that the phase starts off at —180 degrees, approaches —135
degrees midway between 72 and 73, and then approaches —180 degrees
again for high frequencies.

For the third-order active filter case, it is possible to define an
equivalent damping and natural frequency if we let 1/73 be much
higher than 1/72. Rather than making this approximation, PLACE
solves directly for the phase margin using an algorithm (see Ref. 18)
that optimizes phase noise performance. The point of minimum phase
shift (i.e., the inflection point of the open loop gain phase) is placed
at exactly the frequency w, for unity open loop gain. (The user can
specify w,, or else PLACE defaults the value to f,, = frer/50.) This will
occur when the loop filter time constants obey the following relations:

PLACE asks the user for the desired phase margin and computes the
above values for 71, 72, and 73. Later the user has the opportunity to
enter his own values for these variables.

Jeffrey H. Saunders, B.S.E.E., 1979, The University of Massachusetts at
Amherst; M. Engineering (Electrical), 1980, Cornell University; AT&T Bell

Laboratories, 1979—. Mr. Saunders’ earliest involvement at AT&T Bell
Laboratories was with the integrated circuit design of CMOS microprocessors.
He is currently involved in the design of software-controlled digital channel
banks and muldems. Mr. Saunders’ major responsibilities have been system

synchronization, carrier recovery, and jitter characterization. Member, Eta
Kappa Nu, Tau Beta Pi, Phi Kappa Phi, Phi Eta Sigma, IEEE.

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Chi G. C., Nakahara S., The Microstructure of Thin Aluminum Copper (4.5
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Hexatic-B to Smectic-A Phase Transition in Liquid-Crystal Films. Phys Rev

Concentration Cell Using Room-Temperature Chloroaluminate Ionic Liq-

Conduction-Band Discontinuities and Band-Gap Differences of InGaAsP/InP

Gossard A. C., Quantum Well Structures and Superlattices and Their Applica-

Gurvitch M., Target Erosion and Deposition Rates in Planar Magnetron Sput-

to-Monoclinic Phase-Transition in Antifluorite and Related Structure

Intense Electron-Precipitation Spike Over the Southern Polar Cap. J Geo R-
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Inoue A., Chen H. S., Krause J. T., Masumoto T., Young Modulus Sound-Velocity
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Iye Y., Berglund P. M., McNeil L. E., The Magnetic-Field Dependence of the
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Jacobs L., Jose J. V., Novotny M. A., First-Order Reentrant Transition in Gran-
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Joy D. C., Beam Interactions, Contrast and Resolution in the SEM. J Mi-
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Joy D. C., High-Resolution Lithography and the Role of Secondary Elec-
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Karlicek R., Long J. A., Donnelly V. M., Thermal-Decomposition of Metalorganic
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Kauzlarich S. M., Averill B. A., Teo B. K., Structural Studies of FeOC, Intercalated
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Kleiman R. N., Bishop D. J., Pindak R., Taborek P., Shear Modulus and Specific
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Kramer L., Hohenberg P. C., Effects of Boundary Conditions on Spatially Peri-
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Levine B. F., Near Room-Temperature 1.3-um Single Photon Counting With an
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Lundberg H., Macklin J. J., Silfvast W. T., Wood O. R., High-Gain Soft-X-Ray-
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Mahajan S., Dutt B. V., Temkin H., Cava R. J., Bonner W. A., Spinodal Decompo-
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Manzione L. T., Predictive Parameters in Reactive Processing of Polyure-
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Marcus M., Tsai C.-L., EXAFS Measurements of Lattice Vibrations of Ti and Cu
in Ti. Sol St Comm 52(5):511-513, Nov 1984.

Miller D. A. B., Optical Bistability and Differential Gain Resulting From Ab-
sorption Increasing With Excitation. J Opt Soc B 1(6):857-864, Dec 1984.

Miller D. A. B., Chemla D. S., Damen T. C., Gossard A. C., Wiegmann W., Wood T.
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The Quantum-Confined Stark Effect. Phys Rev L 53(22):2173-2176, Nov 26 1984.
Mills A. P., Crane W. S., Emission of Band-Gap-Energy Positrons From Surfaces
of LiF, NaF, and Other Ionic Crystals. Phys Rev L 53(22):2165-2168, Nov 26
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Miraglia S., Hodeau J. L., Marezio M., Ott H. R., Remeika J. P., Relationship Between
the Crystal Structure and the Reentrant Superconducting Properties of
(Sn,_,Er,)EraRheSnig. Sol St Comm 52(2):135-137, Oct 1984.

Mitchell J. W., Selective Substoichiometric Separation of Fluoride by Solvent-

Murarka S. P., Interactions in Metallization Systems for Integrated-Circuits.
J Vac Sci B 2(4):693-706, Oct—Dec 1984.

Murarka S. P., Vaidya S., Cosputtered Cobalt Silicides on Silicon, Polycrystalline
Silicon, and Silicon Dioxide. J Appl Phys 56(12):3404-3412, Dec 15 1984.

Naqui A., Kumar C., Ching Y. C., Powers L., Chance B., Structure and Reactivity
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Odagaki T., Lax M., Day R. S., Dispersive Hopping Conduction in Quasi-One-
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Ondrias M. R. et al., A Resonance Raman Study of the Temperature Dependence
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1984.

Tone Location by Cyclotomic Filters
D. Hertz, R. P. Kurshan, D. Malah, and J. T. Peoples

Recognition of Isolated Digits Using Hidden Markov Models With
Continuous Mixture Densities
L. R. Rabiner, B.-H. Juang, S. E. Levinson, and M. M. Sondhi

Maximum-Likelihood Estimation for Mixture Multivariate Stochastic
Observations of Markov Chains
B.-H. Juang

Some Properties of Continuous Hidden Markov Model
Representations
L. R. Rabiner, B.-H. Juang, S. E. Levinson, and M. M. Sondhi

Variable-Length Packetization of u-Law PCM Speech
R. Steele and F. Benjamin

Architectural Overview
D. L. Carney, J. I. Cochrane, L. J. Gitten, E. M. Prell, and
R. Staehler

Hardware Design
J. C. Borum, J. Janik, Jr., R. F. Metz, J. Russell, D. P. Smith,
and KE. J. Theriot

Physical Design/Hardware
F. N. Graff, Jr., C. E. Jeschke, C. R. Komp, B. E. Nevis, and
D. W. Zdan

Operations, Administration, and Maintenance Capabilities
P. T. Fuhrer, L. J. Gitten, B. A. Newman, and B. E. Snyder

System Test, First Office Application, and Early Field Experience
H. A. Bauer, L. M. Croxall, and E. A. Davis

System Development Environment
R. G. Basinger, J. A. Herndon, B. Kaskey, J. A. Lindner, and
J. M. Milner

In the March 1985 issue (Vol. 64, No. 3) on Assuring High Reliability
of Lasers and Photodetectors for Submarine Lightwave Cable Systems,
the list of editorial staff on the inside front cover should have included
the following: “R. L. Hartman, Coordinating Editor of the submarine
lightwave cable reliability issue.”

In the paper by Z. L. Budrikis and M. Hatamian, “Moment Calcula-
tions by Digital Filters,” AT&T Bell Laboratories Technical Journal,
Vol. 68, No. 2 (February 1984), the following corrections are noted:

Page 219: equation (3): The upper limit on the
summation terms should be “n” instead of “N”.

Page 220, the equation in the middle
of the page between eq. (5) and eq. (6):
u(n) should be u(n-1).

In the paper by M. Hatamian and E. G. Bowen, “Homenet: A Broad-
band Voice/Data/Video Network on CATV,” AT&T Technical Jour-
nal, Vol. 64, No. 2, Part 1 (February 1985), the following corrections
are noted:

magazine :: Bell System Technical Journal :: BSTJ V64N05 198505