APPLICATION OF SEQUENTIAL DECISION
THEORY TO VOICE COMMUNICATIONS
By
RAY HOWARD PETTIT
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
June, 1964
ACKNOWLEDGMENT
The author wishes to acknowledge his great debt to
his advisor, Professor T. S. George, whose advice and
encouragement made this dissertation possible. He wishes
to thank Professor W. H. Chen for his assistance throughout
the period of graduate study at the University of Florida,
and the other members of his committee for their aid.
The author also wishes to thank the Martin Company,
Orlando, Florida, for providing the interesting work
assignments which led him to return to graduate school and
to pursue the research reported herein.
A special acknowledgment is due the LockheedGeorgia
Company of the Lockheed Aircraft Corporation, where the
author was employed during the course of this research.
Assistance was provided by the Advanced Research Organization
in forms too numerous to mention, most important of which was
a research climate which facilitated the completion of this
dissertation.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENT . . . . . . . ii
LIST OF FIGURES. . . . . . . . . .. v
KEY'TO SYMBOLS . . . . . . . . x
ABSTRACT . . . . . . . . . . xiv
Chapter
I. INTRODUCTION . . . ... ..... 1
The Beginnings of Sequential Analysis. . 1
Description of the Sequential Test . . 2
Radar Application. ....... . 6
Data Application .... .... . . 12
Voice Systems. ...... . . ... 14
II. CONVENTIONAL AND SEQUENTIAL SYSTEMS WITHOUT
IMPULSE NOISE CONSIDERATION. . . . 20
Introduction . . . . . . .. 20
QPPM, Double Threshold, Speech Statistics. 21
QPPM, Single Threshold, Conventional . 32
QPPM, Double Threshold, Largest of s . 35
QPPM, Largest of N . . . . . 40
Discussion of Results. . . . . 45
III. CONVENTIONAL AND SEQUENTIAL SYSTEMS WITH
IMPULSE NOISE CONSIDERATION. . .. . 82
Introduction . . . . . 82
iii
Chapter Page
QPPM, Four Threshold, Speech Statistics . 83
QPPM, Four Threshold, Nearest to Reference. 90
QPPM, Single Threshold. . . . . . 96
QPPM, Largest of N. . . . . . . 96
QPPM, Double Threshold, NonSequential. . 97
QPPM, Single Threshold, Largest of N. .. 100
Discussion of Results . . . . . 103
IV. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE
RESEARCH. . . . . . .. . . 146
Conclusions . . . . . . . . 146
Future Research . . .. . . . 147
APPENDICES. . . . . . . . . . . 149
A. Speech Probability Functions. . . . 150
B. Error Bound for Approximate Evaluation of
the Integral in P8(cor) . . . . 154
BIBLIOGRAPHY. . . . . . . . . . . 156
BIOGRAPHICAL SKETCH . . . . .. . . 158
LIST OF FIGURES
Figure Page
1. Functional Diagram, QPPM, Double Threshold,
Speech Statistics System . . . . 23
2. Functional Diagram, QPPM, Single Threshold,
Conventional System. .... ...... .. 34
3. Functional Diagram, QPPM, Double Threshold,
Largest of s System. . . . . . . 37
4. Functional Diagram, QPPM, Largest of N System. 42
5. Probability of Error for Input SignaltoNoise
Ratios Between 4.0 and 14.0 Decibels, Without
Impulse Noise.. . . . . . 47
6. Probability of Error for Double Threshold,
Speech Statistics System for K2 Between 0 and
1.0, and for SignaltoNoise Ratio 5.9
Decibels . . . . . . . 50
7. Probability of Error for Double Threshold,
Speech Statistics System for K2 Between 0 and
1.0, and for SignaltoNoise Ratio 9.0
Decibels . . . . . . . . 52
8. Probability of Error for Double Threshold,
Speech Statistics System for K2 Between 0 and
1.0, and for SignaltoNoise Ratio 11.0
Decibels . .. . . . . . . . 54
9. Probability of Error for Double Threshold,
Speech Statistics System for K2 Between 0 and
1.0, and for SignaltoNoise Ratio 12.1
Decibels . . . . . . . . . 56
10. Probability of Error for Double Threshold,
Speech Statistics System for K2 Between 0 and
1.0, and for SignaltoNoise Ratio 13.0
Decibels . . . . . . . . . 58
11. Values of K1 and K2 for Double Threshold,
Speech Statistics System for SignaltoNoise
Ratios Between 4.0 and 14.0 Decibels, with
Minimum Probability of Error . . . . 60
12. 9, andr s2 for Double Threshold, Speech
Statistics System for SignaltoNoise Ratios
Between 4.0 and 14.0 Decibels, with Minimum
S Probability of Error. . .. .. . 63
13. Probability of Error for Single Threshold,
Conventional System for K1 Between 0 and 1.0,
and for SignaltoNoise Ratios Between 5.9 and
16.0 Decibels. . . . . . . . . 65
14. Value of K1 for Single Threshold, Conventional
System for SignaltoNoise Ratios Between 4.0
and 14.0 Decibels, with Minimum Probability of
Error. . . . . . . . . . 67
15. Probability of Error for Double Threshold,
Largest of s System for K2 Between 0 and 1.0,
and for SignaltoNoise Ratio 9.0 Decibels . 70
16. Probability of Error for Double Threshold,
Largest of s System for K2 Between 0 and 1.0,
and for SignaltoNoise Ratio 11.0 Decibels. 72
17. Probability of Error for Double Threshold,
Largest of s System for K2 Between 0 and 1.0,
and for SignaltoNoise Ratio 12.1 Decibels. 74
18. Probability of Error for Double Threshold,
Largest of s System for K2 Between 0 and 1.0,
and for SignaltoNoise Ratio 15.0 Decibels. 76
19. Values of K1 and K2 for Double Threshold,
Largest of s System for SignaltoNoise Ratios
Between 4.0 and 14.0 Decibels, with Minimum
Probability of Error . ......... 78
Page
Figure
2
20. K, s, and. sr for Double Threshold, Largest
of s System for SignaltoNoise Ratios
Between 4.0 and 14.0 Decibels, with Minimum
Probability of Error *. . . . .. 80
21. Functional Diagram, QPPM, Four Threshold,
Speech Statistics System, with Impulse Noise 85
22. Functional Diagram, QPPM, Four Threshold,
Nearest to Reference System, with Impulse
Noise. . . . . . . . . . 92
23. Functional Diagram, QPPM, Double Threshold,
NonSequential System, with Impulse Noise. . 99
24. Functional Diagram, QPPM, Single Threshold,
Largest of N, NonSequential System, with
Impulse Noise. . . . ... . ... . 102
25. Probability of Error for Input SignaltoNoise
Ratios Between 9.0 and 16.0 Decibels, with
Impulse Noise. . . . . . . ... 106
26. Probability of Error for Four Threshold,
Speech Statistics System, with a and b at
Optimum Settings, for SignaltoNoise Ratio
12.1 Decibels. . . . . . . . . 109
27. Probability of Error for Four Threshold,
Speech Statistics System, with a and c at
Optimum Settings, for SignaltoNoise Ratio
12.1 Decibels. . . . . . . . 111
28. Probability of Error for Four Threshold,
Speech Statistics System, with a and d at
Optimum Settings, for SignaltoNoise Ratio
12.1 Decibels. . . . . . . . . 113
29. Values of K1, K2, K3 for Four Threshold,
Speech Statistics System for SignaltoNoise
Ratios Between 9.0 and 16.0 Decibels, with
Minimum Probability of Error . . . .. 116
30. Probability of Error for Four Threshold,
Nearest to Reference System, with a and b at
Optimum Settings, for SignaltoNoise Ratio
12.1 Decibels. . . . . . .. . 118
vii
Figure
Page
Figure
31. Probability of Error for Four Threshold,
Nearest to Reference System, with a and c at
Optimum Settings, for SignaltoNoise Ratio
12.1 Decibels. . . . . . . . . 120
32. Probability of Error for Four Threshold,
Nearest to Reference System, with a and d at
Optimum Settings, for SignaltoNoise Ratio
12.1 Decibels. . . . . . . . . 122
33. Values of K1, K2, K3 for Four Threshold,
Nearest to Reference System for Signalto
Noise Ratios Between 9.0 and 16.0 Decibels
with Minimum Probability of Error. . . . 124
34. Probability of Error for Single Threshold,
Impulse Noise System for K1.K2*K3 Between 0
and 2.0 and for SignaltoNoise Ratios Between
9.0 and 16.0 Decibels. . .. . .. 126
35. Value of Kl*K2*K3 for Single Threshold,
Impulse Noise System for SignaltoNoise
Ratios Between 9.0 and 16.0 Decibels, and with
Minimum Probability of Error . . . . 128
36. Probability of Error for Double Threshold,
Impulse Noise System for Kl*K2*K3*K4 Between
0 and 1.0 and K1 K2 Between 1.0 and 3.6, and
for SignaltoNoise Ratio 9.0 Decibels . . 130
37. Probability of Error for Double Threshold,
Impulse Noise System for K1*K2*K3*K4 Between
0 and 1.0 and K1'K2 Between 1.0 and 3.6, and
for SignaltoNoise Ratio 11.0 Decibels. . 132
38. Probability of Error for Double Threshold,
Impulse Noise System for K1*Kp2*K*K4 Between
0 and 1.0, and K1.K2 Between 1.0 and 3.6, and
for SignaltoNoise Ratio 12.1 Decibels. . 134
viii
Page
39. Probability of Error for Double Threshold,
Impulse Noise System for K1*K2*K3*K4 Between
0 and 1.0 and K *K2 Between 1.0 and 3.6, and
for SignaltoNoise Ratio 13.0 Decibels. . 136
40. Probability of Error for Double Threshold,
Impulse Noise System for Kl*K2*K3*K4 Between
O and 1.0, and K1*K2 Between 1.0 and 3.6, and
for SignaltoNoise Ratio 16.0 Decibels. . 138
41. Values of K1*K2 and K1K2 K3*K4 for Double
Threshold, Impulse Noise System for Signal
toNoise Ratios Between 9.0 and 16.0 Decibels,
with Minimum Probability of Error. .. . 140
42. Probability of Error for Single Threshold,
Largest of N System for K1~K2 Between 0.3 and
3.5, and for SignaltoNoise Ratios Between
9.0 and 13.0 Decibels. . . . . 143
43. Value of K1'K2 for Single Threshold, Largest
of N System for SignaltoNoise Ratios Between
9.0 and 16.0 Decibels, and with Minimum
Probability of Error . ... . . . . 145
Figure
Page
KEY TO SYMBOLS
Symb ol
a
a,
a
1
A
A
b
/
B
B
c
d
erf(x)
Ho
H1
Io(x)
K
K1
K2
Description
A threshold.
Input signaltonoise ratio.
A preset value of a for the sequential test.
Probability of false alarm in radar.
An upper threshold number.
Amplitude of signal pulse at the receiver.
A threshold.
Probability of false dismissal in radar.
A lower threshold number.
Amplitude of impulse at the receiver.
A threshold.
A threshold.
x
Error function, e Lu..
The hypothesis of noise alone.
The hypothesis of signal and noise.
Modified Bessel Function of first kind, zerot
order.
Average value for the number of slots for
termination.
Ratio b/A for systems of Chapter II, d/A for
systems of Chapter III.
Ratio a/b for systems of Chapter II, c/d for
systems of Chapter III.
Symbol
K 3
K4
Am
Am
11n
n(t)
N
p
Pl(R1)
p2(R2)
p (R3)
P(K)
P(s)
P(err)
Pl(cor)
P2(cor)
P (cor)
P4(cor)
P5(cor)
Description
Ratio b/c for systems of Chapter III.
Ratio a/b for systems of Chapter III.
Average number of impulses in N1 noise slots.
Probability ratio for m observations.
Likelihood ratio at the mE stage.
Number of impulses in N1 noise slots.
Input noise function.
Number of pulse slots per speech sample frame.
A priori probability that H is true.
Probability density of noise envelope.
Density of signal and noise envelope.
Density of impulse and noise envelope.
Probability of termination on the KI slot.
Probability that the number of slots with
envelopes in the "possible signal" zone is
s, given that none are in the "signal" zone.
Probability of error, lP(cor).
Probability of correct decision for QPPM,
Double Threshold, Speech Statistics system.
Probability of correct decision for QPPM,
Single Threshold system.
Probability of correct decision for QPPM,
Double Threshold, Largest of s system.
Probability of correct decision for QPPM,
Largest of N system.
Probability of correct decision for QPPM,
Four Threshold, Speech Statistics system,
with impulse noise.
Symbol
P6(cor)
P7(cor)
P8(cor)
P (cor)
P10(cor)
PB(n)
PB (n)
Ps(j)
pr(y)
P (j/f)
Q(x,y)
R
s
s(t)
2
n
s
Description
Probability of correct decision for QPPM,
Four Threshold, Nearest to Reference system,
with impulse noise.
Probability of correct decision for QPPM,
Single Threshold system, with impulse noise.
Probability of correct decision for QPPM,
Largest of N system, with impulse noise.
Probability of correct decision for QPPM,
Double Threshold, NonSequential system,
with impulse noise.
Probability of correct decision for QPPM,
Single Threshold, Largest of N system,
with impulse noise.
Probability of n impulses in Nl noise slots.
Probability of n impulses in j1 noise slots.
Probability of speech pulse in the ji slot.
Probability of reference pulse in the Ib slot.
Conditional probability that signal pulse in
ji if reference is )t slot.
Number of impulses closer than signal to
reference.
V2 yA+ XIL
The "Q" function, fv.e' r.(xv). Jv,
An envelope.
Average number of slots with "possible signal"
envelope.
Input signal function.
Average noise power at the IF.
Variance of the number of slots with "possible
signal" envelopes.
xii
Symbol
wm(x,9)
W1(x)
W2(x1/x2 ,z)
I
Description
Unknown parameter of a probability function.
Joint probability density function of
Xlx2,...xm when 9 is true.
First probability density of the instantaneous
speech amplitudes.
Conditional density of speech amplitude x, when
the value T seconds before was x2.
xiii
ABSTRACT
Wald's Sequential Decision Theory is applied to voice
communication systems of the quantized pulse position modu
lation type. When interference is narrowband Gaussian
noise, two sequential systems are considered. The "Double
Threshold, Speech Statistics" system divides the speech
sampled frame into N discrete slots. The three decision
zones for each slot are: (1) signal is present, (2) signal
is possibly present, and (3) signal is not present. If the
end of the frame has produced no decision for the first
zone, the receiver chooses one from the group representing
the second decision zone. This choice is made on the basis
of the one most likely to be the signal, as indicated by
speech statistics.
The "Double Threshold, Largest of S" system is the
other sequential type. It is similar to the previously des
cribed technique, with the exception that when the choice
must be made from the group representing the second decision
zone, it is made on the basis of the largest one. When
compared to the conventional, single threshold method, the
sequential systems provide a reduction of about 1.5 decibels
in required signaltonoise ratio for a specified probability
of error. Their performance is almost as good as a "Largest
xiv
of N" receiver, which considers all N slots and chooses the
largest as signal. The sequential methods seem to represent
a compromise between these two techniques, having some
features of each.
When interference is narrowband Gaussian noise, with
occasional impulselike large amplitude pulse noise, the
sequential systems have four thresholds. Thus the "signal
is possibly present" and the "signal is not present" decision
zones consist of two separate parts for each. The two
sequential types are the speech statistics and the nearest
to reference systems. The reference referred to is the
decision zone representing signal present. Analysis is made
on the basis of a mathematical model consisting of a Poisson
distribution for the number of impulses in a frame, with an
average number of one, and a modified Rayleigh distribution
for the envelope of the impulselike interference, with
amplitude of the pulse three times the signal pulse.
The error performance of the conventional single
threshold and the "Largest of N" receivers is shown to be
completely inadequate under these conditions. A conventional
receiver modified to include an upper threshold for discrimi
nation against impulses gives a reasonable performance. The
sequential methods, however, give an improvement of about
2.0 decibels over this. Roughly equivalent to this improve
ment is that given by a "Largest of N" system which is
modified to include an upper threshold for impulse removal.
xv
The conclusion reached is that sequential techniques
in the form described provide an improvement over conventional
or modified conventional methods. The improvement is probably
not large enough under most circumstances to warrant their
use. However, specific situations may allow their beneficial
application. It is felt that further study should be made of
sequential techniques for other modulation methods in voice
communications, possibly in other forms from those described
here.
xvi
CHAPTER I
INTRODUCTION
The Beginnings of Sequential Analysis
The voice communication systems discussed in this
report have resulted from efforts to apply the ideas of
sequential analysis to this field of study. It is there
fore appropriate to relate in this initial chapter some of
the background information on sequential analysis, and to
describe some earlier applications of this theory.
Whether in radar, data, or pulsecoded voice
communications the basic problem is the detection of a
signal in noise. When the examination of the random process
is to provide a positive or negative indication as to the
presence of a signal the problem may be considered a test
of statistical hypotheses. That is, based on the available
information a choice is to be made between hypothesis H :
signal is not present, and H1: signal is present. This may
involve a statistical test of a mean, a signaltonoise
ratio, etc. Sequential analysis, including the sequential.
probability ratio test, was devised by Abraham Wald1 for
hypothesis testing. He and Wolfowitz2 showed that this test
requires on the average fewer samples to terminate than do
tests of fixed sample size, for equal probabilities of error.
A distinguishing feature of the test is that it is conducted
in stages, with the length not specified in advance, but
determined by the progress of the test. Thus the sample
size is a random variable. Although the length of the test
is smaller on the average than that for any other test, it
may be very large for a particular trial. Stein' showed
that not only is the expected length of a sequential test
finite, however, but all moments of the length are finite
if the samples are independent. Thus the question was
resolved as to whether sequential tests terminate.
Because the length of a particular test may be longer
than can be tolerated, many practical situations require
that a test be terminated prematurely. At the point of
truncation, some alternate criterion is adopted so that a
choice of hypothesis may be made. A truncated test requires
on the average a smaller number of samples, but the error
performance is worse than for the untruncated sequential
test. Truncation represents a compromise between the
completely sequential test and the fixedsample test.
Description of the Sequential Test
We let X. be the random variable representing a
sample of a random process for which a statistical decision
problem is formulated. Sample Xi is independent of any
other sample Xj for i / j. The successive samples of X are
denoted by X1 X2, ... The joint probability density
function for the samples is considered known, except for
some parameter 9 (or a set of parameters 5 in the general
case), and is represented by Wm(X;9). The subscript indi
cates that the experiment is at the mt stage. Based on the
sequential test a choice is to be made between two
hypotheses: Ho, the value of 9 is g ; and H1, the value of
9 is 1. Therefore, if hypothesis H is true the joint
probability density function of the m samples is W (X;G );
if hypothesis H1 is true the density is Wm(X;Q1). If the
a priori probability that H is true is p, then the a priori
probability that H1 is true is 1 p. More generally, of
course, there could be more than two hypotheses from which
to choose. Here we are interested in the binary case only.
The likelihood ratio at the mt stage of the experi
ment is defined by
(1 p)Wm(X;Ql)
,m (1.1)
Am= Wm(X; ,)
The carrying out of the sequential test requires this
likelihood ratio to be computed at each stage of the experi
ment. Two threshold numbers AZ > 1 and B' < 1 are chosen.
At the mt stage the decision is made to continue the test if
B < Am AK for m = 1,2,...,n 1 (1.2)
The test is terminated and hypothesis H accepted if at the
n1 trial
An< B (1.3)
Similarly, hypothesis H1 is accepted at the nt trial and
the, test terminated if
An > A (1.4)
Observe that if we set B = (1 p) B and A = (1 p) A
p P
the sequential test can be set up in terms of the
conditional probabilities, independent of the a priori
probabilities p and 1 p. This can be seen by considering
the probability ratio
W (X;,1)
Am = m (1.5)
m = o
Note that Am is 1 Pm The test procedure is thus
slightly modified. At the mtb stage the decision is made to
continue the test if
B< m< A for m = 1,2,...,n 1 (1.6)
The test is terminated and hypothesis H accepted if at the
ntb trial
n B (1.7)
Similarly, hypothesis H1 is accepted at the nt trial if
Xn> A (1.8)
The standout feature of the sequential test is the
dividing of the decision zone into three parts (by means of
the two thresholds). These are: (1) a zone of acceptance
for H (2) a zone of acceptance for H1, and (3) a zone of
indifference. The conventional fixednumberofsamples
test has associated with it only the two zones of acceptance
and one threshold. As pointed out above, the sequential
test provides a savings, on the average, in the required
number of samples for given probabilities of error.
The two types of error possible are the acceptance
of Ho when H1 is true and the acceptance of H1 when H is
true. Wald1 has showed that the thresholds A and B depend
on these probabilities of error. The procedure followed
in the conventional sequential test is to select acceptable
error probabilities, thereby determining the thresholds.
Of course, the smaller the error probabilities, the larger
the average test length. It is emphasized at this point
that a modified form of the conventional sequential test is
utilized in the voice communications applications discussed
in the body of this report. The several variations from
the conventional will be explained in detail in the following
chapters.
Radar Application
There has been much work in applying the sequential
decision theory to radar systems. In the radar problem, the
system decides on the presence or absence of a target in a
particular region of space during a particular time period.
Based on the signal received at the set, it may not be
clear as to whether this is from the radar pulse returned
from a target or whether it is noise alone. In this case,
the radar continues to "look" until a decision can be made.
Wald's theory, when considered from the point of view of
detecting a radar signal in noise, offers a saving in the
time required to make a decision within certain tolerable
error limits as to the presence or absence of a target.
Sequential detection of signals in noise falls into
the general domain of Statistical Decision Theory. Van Meter
and Middleton studied the general application to the
reception problem of decision theory. Blasbalg,5 Fox, and
Bussgang and Middleton7 have made valuable contributions
with their works involving sequential signal detection.
Much of the discussion here of the radar application will
follow the results of Bussgang and Middleton.7
For radar, the parameter for which the sequential
test is performed is the input signaltonoise ratio a.
Thus the two hypotheses being considered are H : the value
of a is zero, and H : the value of a is greater than zero.
Acceptance of Ho (rejection of H1) is a decision that no
target is present. This is a "dismissal." Acceptance of
H1 (rejection of Ho) is a decision that a target is present.
This is an "alarm." As expected, the decision to accept H1
is more likely to be made as a is increased. A preset
signaltonoise ratio a is used to set up the sequential
test. The value a is such that for a less than al, an
acceptance of H is not too objectionable. The test involves
selecting tolerable values for o< the probability of a false
alarm, and for/ the probability of a false dismissal.
The quantity/3 is determined by the probability of dismissal
for a = a1. Sometimes oc and 3 are called the probabilities
of error of the first and second kind, respectively. The
"strength of the test" refers to these two probabilities of
error and is denoted by (or ). The test is regarded as
stronger as o< ,3 are made smaller.
At this point we will consider the incoherent se
quential detection of the radar signal in noise. Following
this will be a discussion of coherent sequential detection.
For incoherent detection no phase information is available
at the receiver. The random variable of interest is the
envelope of a narrowband Gaussian noise plus a sine wave
at the center frequency of the noise band. The appropriate
probability density function for this envelope R is shown
by Rice8 to be
8
W(R;A ) = I  for R > 0 (1.9)
en 2n n
2
where an is the meansquare value of noise at the if
filter and I (u) is the modified Bessel function of the
first kind and of order zero. A is the peak amplitude of
the sine wave. For the envelope of noise alone (A = 0),
the density function is
R R
W(R;0) =  2 for R > 0 (1.10)
C 2 o
n n
These are the Rayleigh distributions. The true signalto
noise ratio a, which is the unknown parameter of these
densities, is
A
a = o (1.11)
~c
If, for convenience, we change the random variable from
S to X, the densities become
2 2
W(x;a) = 2x e a2 + x ) I(2ax) x o
(1.12)
=0 x< .
From this the probability ratio Am can be found, and the
test carried out in terms of the threshold numbers A and B.
An alternate procedure involves the consideration of the
test in terms of the logarithm of the probability ratio.
For this case, the zone of indifference is bounded by log A
and log B.
Under the usual condition of independent samples,
the two error probabilities are
Sb R. n2
0 =e d .[) o(e(( R ] (1.13)
o a
07
o a2.
RRL
e 2 J e3 /, 0_
o[Ir R(1.14)
0
In the above, oe and/3 are expressed in terms of the actual
signal level thresholds a and b. Even though n, the final
stage of the test, is a discrete random variable assuming
integer values, it is convenient to consider it as a con
tinuous variable. Thus p(n/Hk;a) represents the conditional
probability density function of n under the condition that
Hk (k = 0.1) is accepted. The true value of the signalto
noise ratio a is also required to specify this density.
Simplification of the error probability expressions
yields
oO
0 0 62'
r ( Y '/ H 0) z , '( 1 1 5 )
(3= pd p 4){Q( r)a. r),] +
.[iQ(^7^))J
(1.16)
where Q(x,y) is defined by Marcum9 as
00 v + X2
Q(x,3) = v.e I,(xv) v (1.17)
Bussgang and Middleton' have studied this case ex
tensively. They show how the average number of samples
required increases as the probabilities of error are lowered.
The savings in average number of observations is paid for in
the sample length's becoming random. The larger the variance
of the sample length, the greater the savings in average
length. As the variance gets smaller the savings is reduced
and the sequential test approaches a nonsequential.
Similarly, Bussgang and Middleton7 have thoroughly
studied the sequential coherent detection case. Here, of
course, it is assumed that the receiver has complete signal
phase information. Statistically, the problem is the testing
of the mean. Under H the true mean is zero; for H1 the
mean is the value of the signal at the sampling time. The
general technique for the test does not change. The
probability ratio is constructed and tested against the two
threshold numbers as before. The density functions for
this case are Gaussian. Bussgang and Middleton' treat the
general case of correlated samples, and give as an example
the case with an exponential autocorrelation function.
It is interesting to note the results of the compari
son between the sequential coherent and sequential non
conherent detectors. For signals smaller than the preset
value al, the coherent detector is less likely to give a
false dismissal than the noncoherent. In many cases this
difference is very great. For signals larger than the
preset value, the coherent detector is more likely to give
a false dismissal.
In the comparison of the average number of samples,
it is found that in the weak signal case this average number
depends on a for the coherent test and on a2 for the non
coherent. As an example of how this affects the average
sample number, Bussgang and Middleton7 plotted al2 n(a) for
coherent and al n(a) for noncoherent versus a/a. The
peaks of the two curves are the same but occur at different
values of a/a1. Since the case considered is for a1 very
small, the coherent sequential detector therefore requires
fewer average samples at this peak value.
For the radar case, any truncation of the test re
quired would normally be set for some large number of
samples. The result would be a test very close to the
untruncated one. When a test is truncated the average
number of samples required to terminate is reduced. How
ever the strength of the test ( o ,3 ) is reduced to (oc ,1).
Complicated expressions relating c to c' and/3 to 13 have
been derived, but they will not be repeated here.
Data Application
Data applications of the sequential theory are dif
ferent in several respects from the radar case. The most
conspicuous difference is the use of a feedback channel for
data. The receiver employs three decision zones as before:
a zone of acceptance of Ho, a zone of acceptance of H1, and
a zone of indifference. Here H represents the hypothesis
that one of the binary data symbols was transmitted, while
H1 represents the hypothesis that the other symbol was sent.
The feedback channel is used whenever there is a
reception falling in the zone of indifference. Whenever
this situation occurs, the receiver notifies the transmitter
over the feedback channel that it is not clear which symbol
was sent. The transmitter repeats this symbol and the
decision process starts over. The transmitter then either
goes on to the next symbol or repeats the same symbol,
depending on the decision at the receiver. With probability
unity, the transmitter can eventually go on to the next
symbol, although in specific instances, a long time may be
required.
When compared to the method of repetition a fixed
number of times, it is seen that the sequential technique
provides a savings which can be realized in increased data
rate or reduced necessary signal power for a given
probability of error.
The receiver of the standard sequential system
utilizes all past samples in deciding which signal is
present and if another sample is required to give more
accuracy. A modified form of this has received much
emphasis in the literature. The technique referred to is
the sequential test without memory, usually called null
zone detection.
In nullzone detection, the same three choices of
decision are available to the receiver. However, only the
most recent sample is available on which to base the
decision. For this situation there is a nonzero
probability that the receiver would continue asking for a
repeat of a particular signal, and thus never move on to
the rest of the message. Consideration of truncation
becomes very important here.
Harris, Hauptschein, and Schwartzl0 have extensively
studied the nullzone detection problem. Among other
aspects of the problem they studied the manner in which the
threshold levels should be adjusted in order to minimize
communication costs. These costs depend on power,
bandwidth, and transmission time. For nontruncation the
uniform null (no adjustment) is optimum. For truncation,
however, the levels should be adjusted for each sample so
that the probability of the sample falling in the null
zone is reduced over the previous sample. They show that
the loss is small, however, if no adjustments are made.
In these studies the feedback channel is usually
considered errorfree. This is reasonable since normally
only a small amount of information compared to the channel
capacity is transmitted over it. The probability of error
can therefore be made very small.
Boorstynll has also studied nullzone detection,
concentrating his efforts on the case of extremely small
average number of samples; i.e., between one and two. For
the most part he deals with situations in which a repeat
is required only a small fraction of the time, and only
one repeat is allowed. He demonstrates that even for these
small average sample sizes a significant improvement is
obtained.
Voice Systems
With the above providing the background a description
will now be given of the voice communication systems which
have been devised from consideration of the sequential
techniques. Just as there are differences between the radar
and data sequential systems, so are the voice systems dif
ferent from them in several respects.
Search of the literature shows no previous successful
efforts along these lines. The problem is of a somewhat
different nature from the radar or data problems. With
speech there is not enough time to allow for many samples
before making a decision, as in the radar case. But,
instead, a decision must be made during every speech sample
interval. Similarly, no provision can be made for a feed
back communications path, as in the data case, for the
purpose of requesting a retransmission. (Of course, in
duplex communications the listener can simply request that
a word or sentence be repeated, but here a particular
sample of the voice wave is being referred to in the
preceding statement.)
All of the discussion here will be concerned with
systems of the quantized pulse position modulation type
(QPPM); i.e., information regarding the speech amplitude at
the sampling instant is denoted by transmitting the pulse
in a particular one of the N discrete pulse position slots
in the speech sample interval. The pulse will be detected
incoherently. It is believed that techniques similar to
those described here can also be applied to other types of
modulation, but this will be left for future study.
Two cases will be discussed. First, the interference
will be considered to consist of a narrowband Gaussian
noise. A double threshold QPPM sequential system is des
cribed in Chapter II which is effective against this type
of interference. Second, the interference is a narrowband
Gaussian noise and an impulsive noise. In Chapter III is
described the fourthreshold level QPPM sequential system
which effectively combats this kind of interference.
In each case, the decision zones are of the three
usual types: signal present, signal absent, and the zone
of indifference. The receiver must examine the N pulse
position slots sequentially during each speech sample
interval, beginning with the first slot and continuing
through the Nt slot or through the termination slot if
termination occurs before the end of the frame. During the
time a slot is being examined, only one sample is available
from which hypothesis H or hypothesis H1 is decided if
possible. At this stage, the receiver is acting like the
nullzone detector of the data case (the sequential test
without memory).
If hypothesis H1 is accepted for the slot under
examination (thedecision is made that this slot contains
the signal) the test is terminated until the next speech
frame begins. Acceptance of H for a slot does not termi
nate the test. This is a difference from the previous cases
discussed.
The receiver has memory of those slots for which the
sample falls into the zone of indifference. Use is made of
this information for those frames in which no slot sample
calls for acceptance of H1. That is, at the end of the NT
slot, if no slot has been chosen as definitely having the
signal, a choice will then be made from only those slots
which were not rejected by choice of hypothesis H The
test is therefore truncated after N slots.
There are alternative criteria available for use in
choosing the signal slot from the several "possible signal"
slots. Use may be made of speech statistics for this
choice, for example. Samples of the speech wave represent
a Markov process. The probability density function for a
sample is conditional upon the amplitudes of the preceding
samples. Therefore, for example, based on the position of
the pulse in the previous frame, that one of the "possible
signal" slots could be chosen which represented the most
probable slot. The mathematical model used for the speech
marginal and conditional probability density functions is
12
based on the experimental results of Davenport.12 A more
complete and detailed discussion of this model is given in.
Appendix A.
For the double threshold level situation, an alternate
criterion is to make the choice from the "possible signal"
slots on the basis of the one with the largest amplitude
pulse.
For the four threshold level situation, a possible
criterion is to make the choice on the basis of the slot
whose pulse amplitude is nearest to the range of amplitudes
representing acceptance of H1. Other criteria are possible,
but these are the ones considered in this study.
For all of these there is a nonzero probability that
all slots in a frame represent acceptance of H When this
occurs the receiver chooses the center slot (the Nt) and
proceeds to the next frame.
Of primary interest is how these systems compare to
conventional QPPM systems. The two standard QPPM methods
are the single threshold, incoherent detection type, and
the largest of N type, without a threshold. In the first,
the receiver selects as the signal slot the first slot
whose pulse amplitude exceeds the threshold. In the second,
the receiver selects as the signal slot that one whose pulse
amplitude is the largest.
For comparison purposes there are many criteria
possible, such as: (1) output signaltonoise ratio, (2)
bandwidth required, (3) average cost in the Bayes sense,
(4) complexity, and (5) probability of correctly choosing
the proper slot. For mathematical tractability, the latter
one is used in this study.
In addition to the comparisons reported here, para
metric studies of the systems are also given. For various
input signaltonoise ratios, the threshold settings are
allowed to vary and the effect noted on the probability of
correct choice. The difference between an adaptive and a
nonadaptive system can also be seen. The sense in which
the systems are adaptive is that the threshold settings are
selfadjusting with changes in the noise level so as to be
optimum.
Other quantities of interest are the mean and the
variance of the number of "possible signal" slots from
which a choice is to be made if no "signal" slot is
available. These have a bearing on the complexity of that
part of the system which performs this task. As the
thresholds are varied, the report describes the variation
of these quantities with particular input signaltonoise
ratios. Also of interest is the mean of the number of slots
before termination of the test.
With this chapter providing the introduction, the
remainder of this report will concern itself with more
details of the system descriptions and with a complete
discussion of the results.
CHAPTER II
CONVENTIONAL AND SEQUENTIAL SYSTEMS WITHOUT
IMPULSE NOISE CONSIDERATION
Introduction
In this section is a discussion of four QPPM systems,
with the interference taken to be narrowband Gaussian
noise. Two of the systems are conventional types while the
other two have been devised from a consideration of
sequential principles.
It should be noted that there have been made several
modifications of the sequential techniques as used in radar
and data for these voice applications. The radar receiver
can afford a very large number of samples before choosing
a hypothesis. The voice receiver, however, must make a
decision no later than the end of each speech sampling
frame. Therefore, for voice we are dealing with a moderate
number of samples and the maximum number allowed is fixed.
Minimization of the average number of samples is not
the primary consideration for voice. Instead of fixing the
values of the two types of error, and thereby fixing the
thresholds so as to minimize the average number of samples,
we now vary the thresholds to obtain a minimum probability
of error. Of course, a test with a truncation point does
not have fixed thresholds for minimization of average
samples. The thresholds instead should move closer together
in some manner, not yet clear, as the test proceeds. Thus,
fixing the errors for fixed thresholds is not appropriate.
The data application allows a feedback channel for
requesting retransmission of a questionable reception. For
the same reason that the voice receiver must make a decision
no later than the end of each frame, no feedback channel can
be used with it.
Another difference which will be made clear in the
system descriptions is that the test terminates only with
the acceptance of the signal hypothesis. Acceptance of
the noise alone hypothesis does not stop the test.
QPPM, Double Threshold, Speech Statistics
A functional diagram of this system is given in
Figure 1. The signal and narrowband Gaussian noise are
input to an IF filter and then envelope detected. The
optimum decision thresholds will be shown later to be
adaptive ones in the sense that they change with changing
noise level. This system utilizes two thresholds, a and b,
with the b larger.
During each of the N pulse slots, the receiver is
called upon to attempt a choice of hypothesis. If the
envelope during the slot is greater than b, the signal
hypothesis H1 is chosen. Thus the receiver has selected a
Figure 1
Functional Diagram, QPPM, Double Threshold, Speech Statistics System
DECISIONS
ADAPTIVE IF ABOVE b CHOOSE H
s F n) ENVELOPE THRESHOLDS IF BELOE a, CHOOSE H H0
DETECTOR a, b IF BETWEEN a AND b, _
a<b CAN NOT DECIDE
NOTE TIME OF BLANK STORE SIGNAL REMOVE FROM STORAGE
HI OCCURENCE. FOR SLOT AS A THOSE SLOTS FOR
THIS IS SIGNAL REMAINDER REFERENCE FOR WHICH SIGNAL WAS
SLOT OF FRAME FOLLOWING FRAME QUESTIONABLE
H THIS IS A NOISE AT END OF FRAME, IF
SLOT. REMOVE FROM ALL DECISIONS WERE H0,
CONSIDERATION AS CHOOSE CENTER SLOT
POSSIBLE SIGNAL SLOT AS SIGNAL SLOT.
AT END OF FRAME, IF NOT
ABLE TO CHOOSE HI BEFORE,
? NOTE AND STORE THE CHOOSE THE ONE OF THE
SLOT BEING CHECKED STORED SLOTS WHICH IS
NEAREST TO THE PREVIOUS
FRAME'S SIGNALSLOT
slot as the signal slot. The remainder of that frame may
now be blanked out and possibly used for the reception of
nonrealtime data, for example. The receiver stores this
slot as a reference for the following frame to use if
necessary. All previous slots for the frame which were in
the'"possible signal" category are removed from storage at
this time.
If the envelope is below a, the noise alone hypothesis
H is chosen. This does not halt the test for the frame, but
o
removes the tested slot from consideration as a "possible
signal" slot. At the end of the frame, if all slots have Ho
associated with them, the receiver selects the center slot
as the desired one.
Whenever the envelope is between a and b for a slot,
the receiver stores this as a "possible signal" slot, and
proceeds to the next slot. If some following slot has the
signal hypothesis accepted, this information is removed from
storage. If, however, at the end of the frame there has
been no selection of a signal slot, this slot is compared
with any other stored "possible signal" slots. By using
speech statistics the slot most likely to be the signal slot
is then chosen. This is the slot nearest to the position
occupied by the signal of the preceding frame, as stored in
the receiver.
The speech probability expressions required are
developed in Appendix A. The first is the a priori
probability that the signal is in the jb slot. This is
Ps (J) 0.3 3 e + e
+o.i [erf 24 )} erM2) (2.1)
where
X z
er (x) = e du (2.2)
The upper sign is to be used for 1 j N and the lower
sign for 2 + 1 < j < N
The second required expression is the probability
that the speech sample's amplitude corresponds to the jt
position, given that the previous position was thetmb.
This is given by
AJ C e.:!L .iA
Je NL+ e + ('
2 e7 7 .. ( =m ((2.3)
The desired expression for the probability of correct
decision during a frame is composed of three parts:
Pl(cor) = P1 (correct, signal in "signal" zone)
+ P1 (correct, signal in "possible signal" zone)
+ P1 (correct, signal in "noise alone" zone)
(2.4)
The probability of a correct decision, given that the
signal is in the jtb slot and that its envelope exceeds b,
is the probability that the j1 preceding slots with noise
alone do not exceed b. The probability densities of the
envelopes of narrowband Gaussian noise alone and sine wave
plus noise are Rayleigh and Modified Rayleigh, respectively,
as.discussed in Chapter I. Therefore,
P1 (correct, signal in "signal" zone)
N ba
J=l
where
00 XZ 2+
Qo/3, )= x. 1o (cx)x (2.6)
The second part of the expression is more complicated.
Given that the signal envelope is in the "possible signal"
region and that none of the "noise alone" envelopes exceed
b, we can develop the probability of a correct decision
with the reference slot being the jb. We have to consider
only z/ and double the resulting expression, since there
is an equal contribution for) > N due to the symmetry of
the speech first probability density. Under these condi
tions, the probability of correct decision is the probability
that none of the noise slots closer to the reference than the
signal slot is, are in the "possible signal" zone. For the
special case where a noise slot which is "possible signal"
is the same distance from the reference as the signal slot
is, the receiver chooses the one nearer to the center of
the frame. The range of values for j, the signal slot, must
be separated into three parts: jj 9 j + 1 4 j ~ 21 ,
and 21 + 1 j < N The result is
Pi (correct, signal in "possible signal" zone)
=Q( ) b" NI
the probability of correct decision is the probability that
P1 (correct, signal in "noise alone" zone)
(. L e
/ 1I
For the case of signal in the "noise alone" region,
the probability of correct decision is the probability that
all of the noise envelopes are below a and that the signal
slot is the center one. Therefore,
P, (correct, signal in "noise alone" zone)
S Q iif eLj"" (2.8)
Combining the three contributing parts gives the
final result
)N bL .J
P,(o ( ',)* P(r i [i' Ie 2
j=1
an NPr )
r r .2 2 r a i
Ps 7 TO' S
a 2. i.
S ,a I )., e 
ff"*[*Q )I e (2.9)
The expression derived above for the probability of a
correct decision has considered only two frames at any one
time. Higher order speech statistics were not available to
allow more than the preceding frame to be used for reference
purposes. This would seem to have only a small effect and
therefore could be safely neglected. Furthermore, the fact
that the reference position is taken to be correct each
time is acceptable since, for the practical situation, the
probability of error for a particular frame is small.
Another quantity of some interest is the average
number of slots tested before termination occurs. This is
K = K P(K) (2.10)
K=I
where P(K) is the probability of termination on the Kb slot.
The three cases to be considered are: signal slot is the
Ki,,signal slot is before the Kb, and signal slot is after
the Ki. A termination occurs on the Kt slot for the first
case if the signal envelope exceeds b and none of the
envelopes of the preceding K1 slots exceed b. For the
second case, the signal envelope is below b, as well as the
envelopes of the first K2 noise slots. The Kt slot con
taining noise alone does exceed b. The last case requires
the first Kl slots to have noise envelopes below b, with
the Kb above b. Special consideration must be given to the
termination on the last slot. This can occur from the
envelope of the last slot, whether signal or noise, exceed
ing b and the envelopes of the other slots being below b.
It also can occur due to the test truncation requirement
when there are no envelopes exceeding b during the entire
frame. Consideration of these facts gives the result
zi bK KI A
+. 62[ PQ (
2, a, S
j=
bI N 30
j= K+(
+ e 
I ^
SI Q ^ (2.11)
Other quantities of interest are the mean and the
variance of the number of slots with envelopes in the
"possible signal" zone, when no slot's envelope exceeds b
during the entire frame. These are given by
N
s= ZS. P(S) (2.12)
5=1
and
a:= Ts ()T
() (5) (2.13)
5=1
where P(S) is the probability that the number will be s,
given that the test did not terminate due to an envelope
being above b. The two cases to be considered are that the
signal slot makes up one of the s, and that it does not make
up one of the s slots. It is thus seen that
(' [, )].[er, c I P e
L sLe Ql2 AVs
S b s NiIs
(2.14)
Therefore,
a2 Z. 
a b a2 NI
') a 7()' e eH.et]  []"'1
SS
3 N1 S
s) '. IJ N t (2.15)
[* (, .)J [ i e J J
and a2 b sI NS
1, )J. ( azAS 1 L Ns
dsz b Q^s He'^ e'^ ]3 [ l e 3"'
zA b .
((2.16)
QPPM, Single Threshold, Conventional
The functional diagram for this system is given in
Figure 2. The output of the envelope detector is compared
to a single threshold b in this case. The slot whose
envelope is the first to exceed b is selected as the signal
slot. That is, for each slot, either the signal hypothesis
H1 or the noise hypothesis Ho is selected. As soon as a
slot has H1 associated with it, the test is stopped and the
remainder of the frame could be blanked out. Should Ho be
selected for a slot, the receiver proceeds to examine the
next one. As before, if all slots in the frame lead to
choices of Ho, the receiver chooses the center slot as the
signal position.
Figure 2
Functional Diagram, QPPM, Single Threshold, Conventional System
s(t)+n(t) ENVELOPE ADAPTIVE
IF DETECTOR THRESHOLD
b
V
DECISIONS
IF ABOVE b, CHOOSE HI
IF BELOW b, CHOOSE HO
BLANK OUT REMAINING
SLOTS IN FRAME
e HI
 H0
NOTE TIME OF
OCCURENCE WITHIN
FRAME OF THE SAMPLE.
THIS IS SIGNAL SLOT
H
H0
GO TO NEXT SLOT
FOR REPEAT OF
DECISION PROCESS
IF LAST SLOT STILL
LEADS TO Ho,
CHOOSE AS SIGNAL
SLOT THE CENTER ONE
The probability of correct decision within the frame
is then seen to be the probability that signal envelope is
above b and all preceding noise envelopes are below b, or
that all envelopes are below b and signal is the center
slot. Therefore,
3
j=i
+ )** (2.17)
QPPM, Double Threshold, Largest of s
This system is shown in Figure 3. As can be seen,
it is very similar to the Speech Statistics system of
Figure 1. In fact, they are identical up to the point where
a hypothesis has been chosen. When H1 is chosen for a slot,
that slot is selected as the signal slot. The remainder of
the frame is blanked, and all slots stored as "possible
signal" slots are removed from storage. Note that it is
not required that the signal slot be stored as a reference,
because a new criterion is used for resolving the "possible
signal" slots. Choice of H for a slot removes that slot
from any consideration as signal, except when all slots are
Ho. In this case the center slot is used as signal. One
additional storage requirement when a slot's envelope is in
Figure 3
Functional Diagram, QPPM, Double Threshold, Largest of s System
s(t)+n() ENVELOPE
DETECTOR
ADAPTIVE
THRESHOLDS
a, b
a_ b
NOTE TIME OF BLANK
OCCURENCE. FOR
THIS IS SIGNAL REMAINDER
SLOT OF FRAME
NOTE AND STORE THE
SLOT BEING CHECKED.
STORE PULSE AMPLITUDE
DURING THIS SLOT
HI
Ho
DECISIONS
IF ABOVE b, CHOOSE HI
IF BELOW a, CHOOSE HO
IF BETWEEN a AND b,
CAN NOT DECIDE
~ H
*.** H 0
REMOVE FROM STORAGE
THOSE SLOTS FOR
WHICH SIGNAL WAS
QUESTIONABLE
AT END OF FRAME, IF
ALL DECISIONS WERE H0,
CHOOSE CENTER SLOT
AS SIGNAL SLOT.
AT END OF FRAME, IF NOT
ABLE TO CHOOSE HI BEFORE,
CHOOSE THE ONE OF THE
STORED SLOTS WITH
LARGEST AMPLITUDE
?i
THIS IS A NOISE
SLOT. REMOVE FROM
CONSIDERATION AS
POSSIBLE SIGNAL SLOT
*
the "possible signal" zone is that the envelope amplitude,
as well as the slot position, is stored. At the end of the
frame, if there has been no selection of H1, the receiver
selects as the signal slot the one of the s stored whose
envelope is largest.
The probability of correct decision for this case is
also made up of three parts.
P (cor) = P3 (correct, signal in "signal" zone)
+ P3 (correct, signal in "possible signal" zone)
+ P3 (correct, signal in "noise alone" zone)
(2.18)
The first and last of these are identical to the first and
last terms of Pl(cor). The second probability is the
probability that j of the noise envelopes, in the "possible
signal" zone, are less than the signal envelope, also in
the "possible signal" region, while the remaining noise
envelopes are in the "noise alone" region. That is,
P3 (correct, signal in "possible signal" zone)
R i a [^~ j ( 2 1 9 )
J=O a. a o
Now,
,C) e J d = I (2.20)
o 6
K a ^
J *?}'d^ = ^ "'
aa R
I~ca,i~ e 2 "
By the binomial expansion
,e Z
R j
e J
 e<7~
J
2.
a
(II)
It follows that
P3 (correct, signal in "possible signal" zone)
No a2 a R+A2
j=0 Cl
e" INrJ i
e
Rs+A
J *e
'e e" '` O R
If now we let +1'R be replaced by x, the integral becomes
2 A2.
2 A x
101 41 0 n j
dOx
V*
Az A
.
( 0;
e
a~7r7a
e A.
e ; 0
Ia)I
bHa,37 
and
(2.21)
(2.22)
N1
j=0
e )20
r.(i) *g
(2.23)
(2.24)
Rt
A
;rat
e "
(J}
\nl
f7
OL &I
YI)
wnt+
C __x
Ab i
4,
The final expression is
N bL 'j.
'(CID r) ( p5(j) [I
given in equations 2.11, 2.15, and 2.16, respectively.
j=0o t=o
QPPM, Largest of N
Z C"' (2.25)
ThThe receiver stores the N envelopes received during system frame.
At the end of thare identity selects as thderived previougnal slot the
given in equations 2.11, 2.15, and 2.16, respectively.
QPPM, Largest of N
This system is functionally described in Figure 4.
The receiver stores the N envelopes received during a frame.
At the end of the frame it selects as the signal slot the
one whose envelope is largest.
A correct decision is made by the receiver whenever
all of the noise envelopes are below the signal envelope.
This is represented by
Figure 4
Functional Diagram, QPPM, Largest of N System
s()+n(t) ENVELOPE __ STORE AMPLITUDE FOR
W DETECTOR EACH OF N SLOTS
CHOOSE AS THE SIGNAL SLOT
THE LARGEST OF THE N
P or) =
00 R
SP, (R J [<(F, J.". A
a 2.
00 R +A
CR ~ <^
 
o
(,
AR
R
e
NI
R li dK
00 2 2
= e "
'0(7A)
11
'4I
e dt R1
R *A
:= ..e" I.
O
co R (+i) +AZ
o
S R Z'~~~
0 h
l(AR R
) d/ K
(AR~)
(N K)
KN
<=0o
(2.26)
K 2
. 2 e2 dR
Replacing RifK by x, the integral becomes
oO .x .
A A 00
0
\(K I A
+/0%
4; *
leC
K+1
(4 K / X
2 AL
e .7*To dX
K Az
K+l
Thus, we have
NI
P+ r)= o.
K=o
K A2A
(K+1/2i (2.28)
K()
(K.)
K 71
f
)O)
(2.27)
I.
r
Discussion of Results
The primary result of this chapter is depicted in
Figure 5. This allows a direct comparison to be made for
the systems discussed. The criterion of comparison is
minimum probability of error for a given signaltonoise
input ratio. Thus, the curves in this figure are based on
optimum threshold settings for the given noise conditions.
It is interesting to note that the sequential system
based on speech statistics is approximately 1.5 decibels
better than the conventional, single threshold system,
throughout the range of interest of signaltonoise ratios.
The sequential system based on a largest of s selection is
about 1.0 decibel better than the conventional. Slightly
better than these is the system based on selecting the
largest amplitude out of N possible choices. This is a
more complex system, however, and therefore has its
advantage offset to some degree. It will be shown that the
average value of s, with optimum thresholds, is about 1 or
2. Therefore, it would be relatively simple to select the
most probable signal slot out of s under these conditions.
Thus, the sequential systems offer a compromise between the
simplicity of the conventional technique and the higher
theoretical performance of the largest of N system. The
differences noted above would appear to be insignificant
under most circumstances. However, there are, foreseeably,
FIGURE 5
Probability of Error for Input SignaltoNoise Ratios
Between 4.0 and 14.0 Decibels, Without Impulse Noise
100
P] (err), N = 16
 P2(err), N = 16
 P3(err), N = 16
 P4(err), N = 16
0 P (err); N = 8,
0, P2(err); N =8,~
s O0 P3(err); N =8,
\\\ \\
\\
10 
10 \ \
4 6 .8 10 12 14
A2, DECIBELS
2,2
2on2
situations in which use of the sequential systems would be
advantageous.
The calculations for the curves of Figure 5 were
based on a value of 16 for N. To obtain an indication of
what changes might result for other values, single points
were obtained for each system for values of 8 and 32 for N.
The relative changes between systems in the probabilities
of error are about the same amount. As a result, the
conclusions reached above are unchanged.
The results of a comprehensive parametric study for
the speech statistics system are given in Figures 6, 7, 8,
9, and 10. These are for signaltonoise ratios of 5.9, 9.0,
11.0, 12.1, and 13.0 decibels, respectively. Within the
range of values shown, the effect on the probability of
error is seen for any combination of threshold settings.
The optimum thresholds can be obtained from these curves.
As can be seen, the optimum thresholds are not particularly
critical, in the sense that variation in the vicinity of
the optimum point does not increase the probability of error
by a large amount.
The variation of the optimum thresholds with signal,
tonoise ratio is given in Figure 11. For the practical
range of interest K2 has the approximately constant value
of 0.7, with K1 slightly greater. For a reasonably narrow
range of signaltonoise ratios, K1 and K2 could be fixed
FIGURE 6
Probability of Error for Double Threshold, Speech Sta
tistics System for K2 Between 0 and 1.0, and for Signal
to Noise Ratio 5.9 Decibels
SK1 =0.6__
0.7
I
0.9
1.0
A2
 5.9 db
2an2
I 1
0.4 K2 0.6
2
1
0.2
I
0.8
1.0
FIGURE 7
Probability of Error for Double Threshold, Speech Sta
tistics System for K2 Between 0 and 1.0, and for Signal
toNoise Ratio 9.0 Decibels
0 0.2 0.4 0.6
0.8
FIGURE 8
Probability of Error for Double Threshold, Speech Sta
tistics System for K2 Between 0 and 1.0, and for Signal
toNoise Ratio 11.0 Decibels
0.7
A= 11 db
20\2
0.2 0
FIGURE 9
Probability of Error for Double Threshold, Speech Sta
tistics System for K2 Between 0 and 1.0, and for Signal
toNoise Ratio 12.1 Decibels
0.2 0.4 0.6 0.8
FIGURE 10
Probability of Error for Double Threshold, Speech Sta
tistics System for K2 Between 0 and 1.0, and for Signal
toNoise Ratio 13.0 Decibels
0.2
0.4
0.6
0.8
FIGURE 11
Values of K1 and K2 for Double Threshold, Speech Sta
tistics System for SignaltoNoise Ratios Between 4.0
and 14.0 Decibels, with Minimum Probability of Error
60
1.4
1.2
1.0 K
I KI
0.8
K2
0.6
0.4
0.2
0
4 6 8 10 12 14
A2 DECIBELS
2 n 2
beforehand. The degradation would be slight from the
variable threshold situation.
The other quantities of interest for the speech
statistics system are C, s, and a2 under optimum threshold
conditions. This information is given in Figure 12. For a
particular signaltonoise ratio, the difference between
the length of a frame and the value of K gives the number
of slots which perhaps could be used in some other manner;
for example, in the transmission of nonreal time data.
The value of 5, along with s2, gives an indication of the
complexity of that part of the receiver which chooses the
one of the s possible slots as the signal slot. As 5 gets
larger, the complexity increases. As is seen in Figure 12,
s is between 1 and 2 for the range of interest for the
optimum settings, with small variance. The indicated
complexity is not great.
The parametric study of the conventional system is
depicted in Figure 13. Here, of course, only one threshold
is involved. The effect on probability of error is seen
for a change in the threshold, for a wide range of signal
tonoise ratios.
The variation of the optimum threshold with signal
tonoise ratio is given in Figure 14. It is very close
to the variation of K1 shown in Figure 11 for the speech
statistics system.
FIGURE 12
K, s, and a2 for Double Threshold, Speech Statistics
System for SignaltoNoise Ratios Between 4.0 and 14.0
Decibels, with Minimum Probability of Error
10 2.0
8 1.6
b
6 1.2
4 0.8
2 0.4
0 I 0 i 0
4 6 8 10 12 14
A2
.A DECIBELS
2a,
FIGURE 13
Probability of Error for Single Threshold, Conventional
System for K1 Between 0 and 1.0, and for Signalto
Noise Ratios Between 5.9 and 16.0 Decibels
100 65
10
12
0\411.5K1 db0
12.1 db
102 \13 db/
16 db
103
100.20.j 060..
0 0.2 0.4 K1 0.6 0.8 1 0
FIGURE 14
Value of K1 for Single Threshold, Conventional System
for SignaltoNoise Ratios Between 4.0 and 14.0 Deci
bels, with Minimum Probability of Error
67
I I
4 6 8 10 12 14
A2, DECIBELS
2 2
n
Similarly, a parametric study of the largest of s
system was carried out, and the results plotted in Figures
15, 16, 17, and 18 for signaltonoise ratios of 9.0,
11.0, 12.1, and 13.0 decibels, respectively. As before,
the effect on probability of error of change in thresholds
is easily seen. It is to be noted that the curves dip for
values of 0 and about 0.7 for K2. It is to be expected
that probability of error would decrease as the upper
threshold increases and the lower threshold approaches
zero, since the largest of s system approaches the largest
of N system. However, the system performance is almost as
good if the optimum threshold is taken as the one for the
second dip in the curve. This allows the value of s to be
small, with a corresponding decrease in complexity. The
variation of these threshold settings is shown in Figure
19. Again, these could be fixed beforehand for a reasonably
narrow range of signaltonoise ratios, with only small
degradation in error performance.
For this system also, in Figure 20, the variations
of K, s, and 2 are given. For the practical range of
signaltonoise ratios, the values are not significantly
different from the speech statistics system.
This completes the report in this chapter of the
study of the two sequential systems, the conventional
system, and the largest of N system, for the case of no
FIGURE 15
Probability of Error for Double Threshold, Largest of
s System for K2 Between 0 and 1.0, and for Signalto
Noise Ratio 9.0 Decibels
1.0
0 0.2 0.4 0.6 0.8
FIGURE 16
Probability of Error for Double Threshold, Largest of
s System for K2 Between 0 and 1.0, and for Signalto
Noise Ratio 11.0 Decibels
100
101
102,
103
L.
0 0.2 0.4 0.6 0.8
2
= 11 db
2 2
1.0
FIGURE 17
Probability of Error for Double Threshold, Largest of
s System for K2 Between 0 and 1.0, and for Signalto
Noise Ratio 12.1 Decibels
1.0
0 0.2 0.4 0.6 0.8
I
FIGURE 18
Probability of Error for Double Threshold, Largest of
s System for K2 Between 0 and 1.0, and for Signalto
Noise Ratio 13.0 Decibels
76
00
K1 =0.3
1.0
0.4 /
0.9
1.0
~ 0.8 0.9
0.5 9
0.8
co0.7
2 0.6 ~ 0.
2
13db
2 22
3
I I I i I
0 0.2 0.4 0.6 0.8 1.0
Ko
FIGURE 19
Values of K1 and K2 for Double Threshold, Largest of
s System for SignaltoNoise Ratios Between 4.0 and
14.O Decibels, with Minimum Probability of Error
4 6 8 10 12
2
DECIBELS
2on
FIGURE 20
 2
K, s, and s2s for Double Threshold, Largest of s
System for SignaltoNoise Ratios Between 4.0 and 14.0
Decibels, with Minimum Probability of Error
80
2 $2.4
2
0 2.0
8 1.6
CN
b
6 1.2
S
4  0.8
2 0.4
2
0
O s 
4 6 8 10 12 14
A2 DECIBELS
2on2
impulse noise. Included have been fairly comprehensive
studies of the error performance of each system under
various conditions. The 1.5 decibel improvement over the
conventional, for the speech statistics system, would
appear to be small under most circumstances. However,
there are possible situations where this would be justi
fiably enough. The sequential systems provide other
choices, other than the conventional and largest of N.
They apparently offer a compromise between the simplicity
of the former and the error performance of the latter.
CHAPTER III
CONVENTIONAL AND SEQUENTIAL SYSTEMS WITH IMPULSE
NOISE CONSIDERATION
Introduction
The QPPM voice systems discussed in this chapter are
considered to be under the influence of narrowband Gaussian
noise and an impulselike noise. The impulse noise consists
of pulses of amplitude B, greater than the signal pulse
amplitude A. It arises from sources external to the
receiver. The noise pulse carrier frequency is the same as
the signal carrier. Of the Nl noise slots in a frame, n
will be considered to have impulses of the type described.
The remainder have only narrowband Gaussian noise. The
probability distribution of n will be taken to be Poisson
for purposes of this discussion. That is,
X
,X n=v o,i, z...= NI (3.1)
where X is the average number of impulses per frame. This
is felt to be representative of certain special communication
situations. Descriptions of the systems considered and the
derivations of the expressions for the probability of correct
decision are given in the following.
QPPM, Four Threshold, Speech Statistics
This system is functionally described in Figure 21.
The similarities between Figure 1 and Figure 21 should be
noted. In this case, the output of the envelope detector
is compared to the adaptive thresholds a, b, c, and d.
Herb, a is less than b, b less than c, and c less than d.
If the envelope is between b and c, the receiver chooses
the signal hypothesis H1. If the envelope is below a, or
above d, the noise hypothesis H is selected. If between a
and b, or c and d, neither hypothesis is chosen at this
point. The remainder of the receiver processing is identical
to the QPPM, Double Threshold, Speech Statistics system of
Chapter II, shown in Figure 1. Note that the difference is
the additional noise zone and the additional zone of
indifference.
The probability of correct decision is again a sum
of three contributing terms. That is,
P5(cor) = P5 (correct, signal in "signal" zone)
+ P5 (correct, signal in "possible signal" zone)
+ P5 (correct, signal in "noise alone" zone)
(3.2)
If the signal, in the ji slot, is in the "signal" zone, a
correct decision will be made if none of the preceding j1
noise slots has an envelope in the "signal" zone. Here we
are interested in the number of noise impulses in the j1
