• TABLE OF CONTENTS
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 Title Page
 Acknowledgement
 Table of Contents
 List of Figures
 Key to symbols
 Abstract
 Introduction
 Conventional and sequential systems...
 Conventional and sequential systems...
 Conclusions and recommendations...
 Appendices
 Bibliography
 Biographical sketch














Title: Application of sequential decision theory to voice communications
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Title: Application of sequential decision theory to voice communications
Physical Description: xvi, 159 leaves : illus. ; 28 cm.
Language: English
Creator: Pettit, Ray Howard, 1933-
Publication Date: 1964
Copyright Date: 1964
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Subject: Mathematical statistics   ( lcsh )
Communication and traffic   ( lcsh )
Voice   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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Thesis: Thesis - University of Florida.
Bibliography: Bibliography: leaves 156-157.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
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Volume ID: VID00001
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Table of Contents
    Title Page
        Page i
        Page i-a
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
    List of Figures
        Page v
        Page vi
        Page vii
        Page viii
        Page ix
    Key to symbols
        Page x
        Page xi
        Page xii
        Page xiii
    Abstract
        Page xiv
        Page xv
        Page xvi
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
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        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
    Conventional and sequential systems without impulse noise consideration
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
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        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
    Conventional and sequential systems with impulse noise consideration
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
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        Page 139
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        Page 142
        Page 143
        Page 144
        Page 145
    Conclusions and recommendations for future research
        Page 146
        Page 147
        Page 148
    Appendices
        Page 149
        Page 150
        Page 151
        Page 152
        Page 153
        Page 154
        Page 155
    Bibliography
        Page 156
        Page 157
    Biographical sketch
        Page 158
        Page 159
        Page 160
        Page 161
Full Text











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 Lockheed-Georgia

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, Non-Sequential. . 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 Signal-to-Noise
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 Signal-to-Noise Ratio 5.9
Decibels . . . . . . . 50

7. Probability of Error for Double Threshold,
Speech Statistics System for K2 Between 0 and
1.0, and for Signal-to-Noise Ratio 9.0
Decibels . . . . . . . . 52

8. Probability of Error for Double Threshold,
Speech Statistics System for K2 Between 0 and
1.0, and for Signal-to-Noise Ratio 11.0
Decibels . .. . . . . . . . 54

9. Probability of Error for Double Threshold,
Speech Statistics System for K2 Between 0 and
1.0, and for Signal-to-Noise Ratio 12.1
Decibels . . . . . . . . . 56

10. Probability of Error for Double Threshold,
Speech Statistics System for K2 Between 0 and
1.0, and for Signal-to-Noise Ratio 13.0
Decibels . . . . . . . . . 58










11. Values of K1 and K2 for Double Threshold,
Speech Statistics System for Signal-to-Noise
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 Signal-to-Noise 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 Signal-to-Noise Ratios Between 5.9 and
16.0 Decibels. . . . . . . . . 65

14. Value of K1 for Single Threshold, Conventional
System for Signal-to-Noise 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 Signal-to-Noise 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 Signal-to-Noise 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 Signal-to-Noise 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 Signal-to-Noise Ratio 15.0 Decibels. 76

19. Values of K1 and K2 for Double Threshold,
Largest of s System for Signal-to-Noise 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 Signal-to-Noise 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,
Non-Sequential System, with Impulse Noise. . 99

24. Functional Diagram, QPPM, Single Threshold,
Largest of N, Non-Sequential System, with
Impulse Noise. . . . ... . ... . 102

25. Probability of Error for Input Signal-to-Noise
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 Signal-to-Noise Ratio
12.1 Decibels. . . . . . . . . 109

27. Probability of Error for Four Threshold,
Speech Statistics System, with a and c at
Optimum Settings, for Signal-to-Noise Ratio
12.1 Decibels. . . . . . . . 111

28. Probability of Error for Four Threshold,
Speech Statistics System, with a and d at
Optimum Settings, for Signal-to-Noise Ratio
12.1 Decibels. . . . . . . . . 113

29. Values of K1, K2, K3 for Four Threshold,
Speech Statistics System for Signal-to-Noise
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 Signal-to-Noise 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 Signal-to-Noise Ratio
12.1 Decibels. . . . . . . . . 120

32. Probability of Error for Four Threshold,
Nearest to Reference System, with a and d at
Optimum Settings, for Signal-to-Noise Ratio
12.1 Decibels. . . . . . . . . 122

33. Values of K1, K2, K3 for Four Threshold,
Nearest to Reference System for Signal-to-
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 Signal-to-Noise Ratios Between
9.0 and 16.0 Decibels. . .. . .. 126

35. Value of Kl*K2*K3 for Single Threshold,
Impulse Noise System for Signal-to-Noise
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 Signal-to-Noise 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 Signal-to-Noise 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 Signal-to-Noise 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 Signal-to-Noise 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 Signal-to-Noise Ratio 16.0 Decibels. . 138

41. Values of K1*K2 and K1-K2 K3*K4 for Double
Threshold, Impulse Noise System for Signal-
to-Noise 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 Signal-to-Noise Ratios Between
9.0 and 13.0 Decibels. . . . . 143
43. Value of K1'K2 for Single Threshold, Largest
of N System for Signal-to-Noise 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 signal-to-noise 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 N-1 noise slots.

Probability ratio for m observations.

Likelihood ratio at the mE stage.

Number of impulses in N-1 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, l-P(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, Non-Sequential system,
with impulse noise.

Probability of correct decision for QPPM,
Single Threshold, Largest of N system,
with impulse noise.

Probability of n impulses in N-l noise slots.

Probability of n impulses in j-1 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 narrow-band 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 signal-to-noise 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 narrow-band Gaussian noise, with

occasional impulse-like 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 impulse-like 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 pulse-coded 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 signal-to-noise

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 fixed-sample 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 fixed-number-of-samples

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 signal-to-noise 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

signal-to-noise 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 narrow-band 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 a-n is the mean-square value of noise at the i-f
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 signal-to-
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. n-2

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 signal-to-

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-),] +



.[i-Q(^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 non-sequential.
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 non-coherent. 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 non-coherent 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 null-zone 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 non-zero

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 null-zone 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 non-truncation 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 error-free. 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 null-zone 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 narrow-band 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 narrow-band

Gaussian noise and an impulsive noise. In Chapter III is

described the four-threshold 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

null-zone 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 non-zero 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 signal-to-noise 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 signal-to-noise 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

non-adaptive system can also be seen. The sense in which

the systems are adaptive is that the threshold settings are

self-adjusting 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 signal-to-noise

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 narrow-band 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 narrow-band 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 SIGNAL-SLOT









slot as the signal slot. The remainder of that frame may

now be blanked out and possibly used for the reception of

non-real-time 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 .-i-A
J-e 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 j-1 preceding slots with noise

alone do not exceed b. The probability densities of the

envelopes of narrow-band 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" N-I














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 N-Pr )


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 K-1 slots exceed b. For the
second case, the signal envelope is below b, as well as the
envelopes of the first K-2 noise slots. The Kt slot con-

taining noise alone does exceed b. The last case requires
the first K-l 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 -K-I 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 sL--e Ql2 AVs
S b s Ni-I-s



(2.14)



Therefore,





a2- Z. -
a b a2 N-I-
-') a -7()-' e e-H.--et] - []"'1
S-S





3 N-1- S

s)- -'. IJ N- t (2.15)
[*- (, .)J [ i- e- J J








and a2 b s-I N-S
1, )J. ( azAS- 1 L- N-s
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)


N-o a2 a R+A2

j=0 Cl


e" IN-rJ 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)


N-1

j=0


e )20


r.(-i) *g


(2.23)


(2.24)


Rt
-A
;rat
e "


(J}
\nl


f-7
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


N-I
R li dK


00 2 2

= e "


'0(7A)


11-


'4-I
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 Rif-K 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


N-I
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 signal-to-noise

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 signal-to-noise 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 Signal-to-Noise 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
2o-n2









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 signal-to-noise 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,

to-noise 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 signal-to-noise 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-

to-Noise 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-

to-Noise 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-

to-Noise 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-

to-Noise 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 Signal-to-Noise 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 a-2 under optimum threshold

conditions. This information is given in Figure 12. For a

particular signal-to-noise 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 non-real 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-

to-noise ratios.

The variation of the optimum threshold with signal-

to-noise 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 Signal-to-Noise 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 Signal-to-

Noise Ratios Between 5.9 and 16.0 Decibels






100 65
10













12
0\411.5K1 db0


12.1 db




10-2 \13 db/




16 db





10-3
10---0.20.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 Signal-to-Noise 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 signal-to-noise 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 signal-to-noise 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

signal-to-noise 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 Signal-to-

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 Signal-to-

Noise Ratio 11.0 Decibels








100-














10-1














10-2,














10-3
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 Signal-to-

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 Signal-to-

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 Signal-to-Noise Ratios Between 4.0 and

14.O Decibels, with Minimum Probability of Error



















































4 6 8 10 12


2
DECIBELS
2o-n





















FIGURE 20

- 2
K, s, and s2s for Double Threshold, Largest of s

System for Signal-to-Noise 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
2o-n2








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 narrow-band Gaussian

noise and an impulse-like 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 N-l noise slots in a frame, n

will be considered to have impulses of the type described.

The remainder have only narrow-band 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...= N-I (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 j-1

noise slots has an envelope in the "signal" zone. Here we

are interested in the number of noise impulses in the j-1




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