. .

A STUDY OF DECISION REGION RECEIVERS FOR THE SUNBLAZER

SPACE EXPERIMENT

James L. Mannos CSR T-67-5 June 1 9 6 7

https://ntrs.nasa.gov/search.jsp?R=19670021621 2018-08-07T08:08:05+00:00Z

t

ii.

A S'ILDY OF DECISION REGION RECEIVERS

FOK T U SUNBLAZER SPACE EXPERIMENT

JAWS LEE ?fAiWOS

Submitted t o the Department of Electrical Engineering on Nay 1 9 , 1967,

i n p a r t i a l f u l f i l l m e n t of t h e requirements f o r t h e degree of Master of

Science.

ABSTRACT

For a Rayleigh channel with a d d i t i v e bandl imited no i se and time d e l a y , s e v e r a l types of optimum r e c e i v e r s are developed. These inc lude t h e 3-ary case f o r no ise , and forward and backward Barker codes, and t h e optimum binary case f o r s i g n a l and noise . p r o b a b i l i t y i n each case are derived, and t h e r e s u l t s are compared f o r v a r i o u s va lues of s i g n a l t o no i se r a t i o . i o r t h e s e r e c e i v e r s r e q u i r e s a n i n f i n i t e number of c o r r e l a t o r s ; and s i n c e any phys ica l system can have only a f i n i t e number, an i n v e s t i g a t i o n i n t o t h e e f f e c t of t h i s r e s t r i c t i o n on r ece ive r performance i s made. The concept of receiver "guessing" f o r small s i g n a l t o n o i s e r a t i o s i s also explored i n some detail .

I n t h e optimum binary case the express ion f o r t h e e r r o r p r o b a b i l i t y cannot be eva lua ted without t h e a i d of a piecewise l inear approximation t o t h e dec i s ion curves. The e f f e c t of varying t h e number of l i n e a r seg- ments i n the approximation is examined.

advantages of block coding as compared t o b i t by b i t s igna l ing .

Expressions f o r t h e e r r o r

The t h e o r e t i c a l r e a l i z a t i o n

F i n a l l y , a simple coding scheme i s used t o i l l u s t r a t e some of t h e

Thesis Supervisor: W. B. Davenport T i t l e : P rofessor of E l e c t r i c a l Engineering.

a

i f I *

iii .

The au thor would l i k e t o thank p a r t i c u l a r l y h i s t h e s i s adv i so r ,

I’rof. Wilbur 8 . Davenport, f o r his cons tan t advice and encouragement;

anu also Prof , Harry L. Van Trees and l l r , Gordon R. G i l b e r t f o r advice

on c e r t a i n a s p e c t s of t he problem.

V. l i a r r ing ton both f o r sugges t ing t h e problem and then suppor t ing t h e

work a t t h e Center f o r Space Research.

I am a l s o indebted t o Prof . John

c

iv

TABLE OF CONTENTS

Chapter 1 Introduction

1.1 A Problem in Space Communications

1.2 Sunblazer Requirements - A MDre Specific Formulation of t h e Problem

1.3 h e Eetimation Problem

Ckapter 2 The Cammunications System

2.1 The Transmitter

2.2 '2he Channel

2.3 The Receiver R o n t &d

Chapter 3 Derivation of t he N- Signal Rules

3.1 Representation of t h e Signals a s Vectors

3.2 The N-Signal Rule

3.3 The Binary Decision Rule for N Signal8

Chapter 4 Dis t r ibu t ions for t h e Rayleigh Channel

2 Tlerivation of t h e S t a t i s t i c s of x i ChaDter 5

Chapter 6 Sunblazer Signal Analysis

6.1 3-Ary Signal Rule

6.2 h o b a b i l i t y of mor Given No S i m a l Present

6.3 h o b a b i l i t y of Error Given m l was "ran mn i t t ed

4

5

5

7

9

13

13

14

17

19

31

35

35

37

39

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6.4

6.5

6.6

6.7

6.8

6.9

Probabi l i ty of Error Given m2 Transmitted

%tal Probabi l i ty of Rror f o r Case 1 with Qi 1

P(E) When 0 9 1, and .Orrmaary of Results

Ettdif ied Binary Decision Rule - Care 2

Optimum Binary Decision Rule - Case 3 P(Vnoise) , Q S 1/2

6.10 P(FJmi), Q S 1/2

6.11 P(E) for Q 2 1/2, and Summary of Case 3 Re su 1 t s

Chapter 7 Analysis of Receiver Performance Vs. S i m a l t o Noise and Time Delay

7.1 Case 1, The 3-Ary Receiver

7.2 Case 2, The Modified Binary Case

7.3 Case 3 , me optimum Binary Case

7.4 %me Further Comments on -cision Region Receiver P(E) as a Function of Signal to Noise

7.5 P(B) Vs. Time Delay Error

7.6 A Simple Cod€ng Scheme

Chapter 8 Summary and Final Remarks

8.1 Future Ideas and IbRanrions

43

44

45

48

50

53

55

59

61

61

67

73

81

82

87

88

88

Bib1 iography 90

Figure Number

1

Page -bet

3

vi

Fie(ure m b e r

2a, 2b 3a, 3b 4 5 6a,6b 7a, 7b 8 9 lo

13 14 15,16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

11.12

Page Number - -.. 6 8 11 15 21 26 29 36 38 40 46 49 51 62 63 64 65 68 69 70 71 72 74 7 5 77 78 79 80 83 84 85 86

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CHAPTER 1

INTRODUCTION

1. A Problem i n Space Communications

The design of an "optimum" r e c e i v e r f o r a communications problem i s

always dependent on t h e na tu re of t h e channel through which the messages

must be sen t .

s i n c e a d e t a i l e d knowledge of t h e channel mechanisms are gene ra l ly un-

I n most caoes a p r o b a b i l i s t i c d e s c r i p t i o n must be chosen

known. This is p a r t i c u l a r l y t r u e of space communications where the cor-

rup t ing e f f e c t s of t h e channel may inc lude g a l a c t i c no i se , d i spe r s ion ,

and s c a t t e r i n g , t o n a m e j u s t a few. L i t t l e of even a p r o b a b i l i s t i c

n a t u r e is known about many of t hese phenomena, as only a few deep space

probes have been launched, and none of t hese were concerned p r imar i ly

wi th the deep space environment.

M. I. T. is involved i n the cons t ruc t ion and launching of a sa te l l i t e ,

Sunblazer, whose purpose is t o ga the r information about t h e space me-

dium i t s e l f .

des ign and eva lua t ion of t h e d i f f e r e n t types of r e c e i v e r s f o r t h e

Sunblazer P r o j e c t .

The Center f o r Space Research a t

This r e p o r t is d i r e c t l y concerned wi th t h e t h e o r e t i c a l

P a r t of t h e complexity of t h i s problem is due t o t h e f a c t t h a t t h e

i n i t i a l Sunblazer r e c e i v e r s w i l l n o t have t h e b e n e f i t of any previous

experimental da ta . Thus we are faced wi th t h e case of designing a re-

c e i v e r which is f a i r l y v e r o o t i l e i n i t s a b i l i t y t o adapt t o unexpected

s i g n a l condi t ions .

1.2 Sunblazer Requirements - A More S p e c i f i c Formulation of t he Problem

To be more s p e c i f i c , the Sunblazer sa te l l i te w i l l b roadcas t two

-2-

s i g n a l s a t d i f f e r e n t f requencies . By measuring t h e t i m e de lay between

these s i g n a l s , information may be i n f e r r e d about such phenomena as t h e

e l e c t r o n dens i ty of t h e Suns corona, Furthermore, each t i m e a o igna l

is t ranomit ted i t may t ake on two forms r ep resen t ing e i t h e r a binary

one o r zero, The Sunblazer sa te l l i t e w i l l use "forward" and "backward"

Barker codes f o r t hese s i g n a l s . A forward Barker code and i t s auto-

c o r r e l a t i o n func t ion are shown i n F igure 1. (For a backward Barker

code simply r eve r se t h e n-axis i n Figure 1).

I n order t o approach t h i s problem it is f i r s t neceosary t o develop

a gene ra l r ece ive r "philosophy" rn

The receiver should dec ide whether ml o r m2 was t rano-

mi t ted and a l s o t h e s i g n a l de lay t i m e ,

amount of information which can be obta ined ,

If t h e p r o b a b i l i t y of making a mistake i n de tennin ing the d a t a

f o r part (1) becomes unacceptable , can we look a t only t h e

s i g n a l de lay (ice., throw away t h e t e l eme t ry information)

and thus reduce our p r o b a b i l i t y of e r r o r ?

whether w e can reduce the p r o b a b i l i t y of e r r o r by n o t re-

q u i r i n g as much da ta .

F ina l ly , i f t h e p r o b a b i l i t y of determining any d a t a a t a l l

from t h e s i g n a l g e t s very small, can we a t least make a deci-

s i o n as t o whether o r n o t signal is rece ived? i r e , , f o r ex-

tremely weak s i g n a l cases poss ib ly t h e b e s t t h a t can be de-

termined io whether any s i g n a l energy is p r e s e n t i n t h e no i r e .

This is t h e maximum

Here w e are ask ing

problem i s now t o de r ign receivers i l l u o t r a t i n g t h e charac te r -

*

i s t i c e of each of t h e above t h r e e cases, and then t o compare t h e i r per-

formancei.

t \

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f

n I C

I I

I 1

I I

1 I

I I

I

2 9 b 8 I. ia

-- I 4 I

a ) The Eleven-hit Barker Code

t

b ) The Correspondincl Autocarrelation h n c t ion

r C 0 I O 1% K I .c I

I I I 1 I I I I I 1

--

FISlRB 1

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1.3 The Estimation Problem

The most s t r a igh t fo rward method f o r es t imat ing t h e a r r i v a l t i m e of

t h e s i g n a l is t o d i v i d e the t i m e scale i n t o many l i t t l e "baskets" and

then check t o see i n t o which "basket" t h e received s i g n a l f a l l s . With

an i n f i n i t e number of incrementa l ly small baske ts w e could expect psr-

f a c t r e so lu t ion . However, s i n c e no real r e c e i v e r w i l l be a b l e t o use

more than a f i n i t e number of i n t e r v a l s , i t is necermary t o check what

e f fec t t h i s r e s t r i c t i o n has on r e c e i v e r performance. It should a l s o

be noted t h a t t h i s ec t ima t ion method i s a c t u a l l y a d e t e c t i o n problem;

s i n c e w e are r e a l l y asking, hae t h e r e c e i v e r f o r baske t number 1 de-

t ec t ed a s igna l?

I n the next chapter w e w i l l begin t o develop t h e model6 neces-

s a r y t o proceed wi th t h e a n a l y s i s .

-5-

CHAPTER 2

THE CONMUNICATIOEU'S SYSTEM

Before any f u r t h e r r e s u l t s can be obtained i t is f i r s t necessary t o

model t h e t r a n s m i t t e r , t h e channel, and t h a t p a r t of t h e receiver which

demodulates t h e incoming s i g n a l s , Then the remaining po r t ion of t h e re-

c e i v e r can be designed t o g ive "optimum" performance wi th r e s p e c t t o

some c r i te r ia , as mentioned i n the previous chap te r ,

2.1 The Transmi t te r

We wish t o t ransmi t any one of mi d i f f e r e n t messages, Each mes-

sage has a c e r t a i n p r o b a b i l i t y P i of being t r ansmi t t ed , These messages

are f ed i n t o an encoder (see Figure 2) which is e s s e n t i a l l y a device

which makes a one-to-one t ransformation between t h e messages and t i m e -

l i m i t e d , lowpass s igna l s ' ,

are shown i n F igure 2A,

ampli tude modulation scheme is used which amounts t o a frequency t rans-

l a t i o n of t h e lowpass s i g n a l (Figure 2B).

A sample s i g n a l and i t s frequency spec t rum

A double-sideband suppressed carr ier (DSB-SC)

For ease of a n a l y s i s i t i s

T assumed t h a t

and ETi is t h e t r ansmi t t ed energy. Thus t h e s i g n a l a t the antenna is

cf t h e form

sO(t) = JlETi S i ( t ) cos W O t

Since a s t r i c t l y bandl imited, t i m e l i m i t e d s i g n a l doesn ' t e x i s t , a

d e f i n i t i o n f o r bandwidth t h a t r e q u i r e s t h a t 90% of the s i g n a l energy

be between * w is used.

I JZTT, cos

PTflJRE ?. Fbclel of t h e "ransnitter

S$t\ T

FIGURE 2a. Sample Signal and i.ts !3pectrum

PIGURE 2b. @ectrum of Transmitted Signal

-7-

where w o = 2n f o

2.2 The Channel

The channel modal used here a t t e n u a t e s the s i g n a l by a f a c t o r of

a, i n t roduces a random phase angle 8 i n t o t h e carrier, and de lays t h e

received s i g n a l by an amount T. Both 8 and a are random v a r i a b l e s

which are assumed t o be t ime-invariant over t h e per iod of s i g n a l t r ans - I

mission' . Furthermore, t h e channel a l s o in t roduces a d d i t i v e Gaussian

no i se , n ( t ) . A block diagram is shown i n F igure 3.

The obvious ques t ion as t o what are real is t ic d i s t r i b u t i o n s f o r a,

8 , and n ( t ) is very hard t o answer. Over t h e f requencies a t which

Sunblazer is opera t ing (75 m c and 225 mc) t h e a d d i t i v e n o i s e may be

modeled w e l l by band l imi t ed white Gaussian noise2 . See F igure 3A.

However, l i t t l e seems t o be known about t he form of e i t h e r t h e

phase o r ampli tude d i s t r i b u t i o n . The fol lowing assumptions were made;

(1) Random Phase - Slowly vary ing i n t i m e wi th r e spec t t o s i g n a l

l eng th wi th a uniform d e n s i t y from 0 t o 2n. This assump-

t i o n is t h e r e s u l t of many phys ica l models, and p a r t i c u l a r l y

of t he s ~ a t t e r i n g and long-path-length models.

Random Amplitude - There is a c t u a l l y very l i t t l e t o go on ( 2 )

here. S c a t t e r i n g models lead t o Rayleigh d i s t r i b u t i o n s ,

which have allowed closed form express ions f o r the receiver

e r r o r s . Resu l t s of t h e assumption of a Rayleigh d i s t r i b u -

t i o n show t h a t rece iver p r o b a b i l i t y of e r r o r i s approxi-

mately d i r e c t l y propor t iona l t o l / ( s i g t o no i se r a t i o ) .

I f t h i s assumption doesn ' t hold then t h e whole s igna l ing scheme becomes

i n e f f e c t i v e as w e s h a l l la ter see.

G i l b e r t , Some Communication Aspects of Sunblazer, p . 2

-8- .

E c

c

(L 0

!2 f s 0 3

1 m

I f we assume no s c a t t e r i n g of t he s i g n a l , t h e p r o b a b i l i t y of

e r r o r decreases exponent ia l ly wi th s i g n a l t o n o i s e r a t i o , The

a c t u a l d i s t r i b u t i o n probably g ives r e s u l t s which l i e somewhere

between these two cases, Also, t h e r e s u l t s from the Rayleigh

case don ' t d i f f e r s i g n i f i c a n t l y from the Rayleigh p lus specu-

lar ( d i r e c t ) component case u n t i l t h e specu la r component be-

comes 4 t o 5 times l a r g e r than t h e s c a t t e r e d component1.

Thus f o r low s i g n a l t o no i se r a t i o s our r e s u l t s should be

reasonably va l id .

t i o n a l lows a comparative s tudy of t h e d i f f e r e n t r ece ive r

I n any case, assumption of t h i s d i s t r i b u -

conf igu ra t ions t o be made.

Since t h e t i m e de lay T i s a l s o cons t an t f o r each t ransmission,

and w e are es t ima t ing i t s a r r i v a l , t h e d e t e c t i o n scheme does n o t need

any p r o b a b i l i t y in format ion concerning it. However, If an average pro-

b a b i l i t y of e r r o r (with r e spec t t o t i m e d e l a y ) is des i r ed then c e r t a i n

assumptions about a d i s t r i b u t i o n f o r t must be made, But s i n c e t h i s

is n o t c r i t i c a l f o r t h e de t ec t ion problem, i t can wait u n t i l a f t e r the

f i rs t launch when b e t t e r d a t a w i l l be a v a i l a b l e .

2.3 The Receiver Front End

I d e a l l y t h e receiver output should be a lowpase s i g n a l of t he same

form as t ransmi t ted . However, the t r ansmi t t ed r i g n a l has been corrupted

t y t h e channel so t h i s is gene ra l ly n o t poss ib le . The r e c e i v e r i npu t i s

of the form

Van Trees, Detec t ion and Est imat ion Theory

-10-

From Figure 4 w e see t h a t t h e receiver f r o n t end c o n s i s t s f i r s t of a

bandpass f i l t e r wiiose output i s of t h e form

where

I n order t o s impl i fy (2.1) w e express n b p ( t ) as a Four ie r series. Q1

nbp( t ) = 1 (xcn cos nut + xsn s i n n u t ) 11=1 T

Xcn = 2 T n b p ( t ) cos n u t d t 0

T

Xsn = ' T 1 n b p ( t ) s i n n u t d t 0

Now, rewr i t ing nu0 a s (nu-ug) + "0 w e Zet

n b p ( t ) = n c ( t > E cos ugt + n s ( t > 5 s i n c u g t

m L where n c ( t ) = fi 1 [xcn cos (nu-uglt + xsn s i n ( n ~ - u o > t l n= 1

1 " n s ( t ) = E 1

n= 1

and [xsn cos (nw-ug)t - xcn s i n (nw-wo>tl

The only nonvanishing terns i n tile sums above are t h e ones f o r which nf

f a l l s between - w + f g and f g + W. Thus n c ( t ) and n s ( t ) a r e l o w

pass waveforms wi th a frequency spectrum of width 2w centered about

zero. Therefore,

r ( t > = /T cos ugt [ a JETi s i ( t + T ) cos e + n C ( t > l

+ 5 s i n ugt [ a JETi s i ( t + r ) s i n e + n s ( t ) l

Nest, using the synchronous demodulation scheme. shown i n r i p r e 4 , mult i -

plying by sin ugt and cos ug t , and then low p a s s f i l t e r i n g , w e have

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f o r t h e r ece ive r f r o n t end output , two s i g n a l s

r c ( t > = a s i ( t + t ) s i n e + n s ( t )

r s ( t ) = a & si ( t+r) s i n 0 + n s ( t )

On the b a s i s of r c ( t ) and rs(t) w e want t o dec ide which of t he

m i w a s sen t . I n o the r words w e would l i k e t o set the es t imated message

&, equal t o t h e t r ansmi t t ed message m i .

The key t o analyzing t h i s problem l i e s i n express ing t h e s i g n a l s and

no i se i n terms of a f i n i t e dimensional vec to r space. By us ing f i n i t e d i -

mensional vec to r s we have t o worry only about t he s ta t i s t ics f o r a f i n i t e

number of components, r a t h e r than f o r an i n f i n i t e number of times. I n

the nex t c h a p t e r vec to r dec i s ion r u l e s f o r t h e r e c e i v e r s w i l l be developed.

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CHAPTER 3

DERIVATION OF THE N-SIGNAL DECISION RULES

I n t h i s s e c t i o n gene ra l r u l e s are developed f o r implementing the

f i r s t two cases of t h e receiver philosophy. The t h i r d case is an exten-

s i o n of case two us ing a block coding scheme and w i l l be considered i n

more d e t a i l later on in Chapter 7 ,

3.1 Representat ion of S igna l s as Vectors

It can be shown1 t h a t any time varying set of s i g n a l s can he repre-

sen ted by v e c t o r s i n an appropr ia te s i g n a l space. Furthermore, i f we

have N s i g n a l s , then an or thogonal b a s i s f o r t h i o vec to r space c o n s i s t i n g

of a maximum of K vectors can be found. S t o c h a r t i c processes may a l s o be

represented as i n f i n i t e dimension v e c t o r s us ing the Karhunen-Loeve expan-

s i o n 2 *

and on the b a s i s of t hese vec to r s make t h e "best" dec i s ion as t o which

s i g n a l w a s a c t u a l l y s e n t .

t h e assumptions used t o gene ra t e t h e dec i s ion making rule ,

of one class of dec i s ion c r i t e r i a , t h e minimum p r o b a b i l i t y of e r r o r

case, fol lows.

Thus w e would like t o r ep resen t t he rece ived s i g n a l s as vec to r s ,

O f course, t he "best" cho i r e is a func t ion of

An example

Wozancraf t & Jacobs, Pr inc iDles of Communications Eng inee r in , ,

ppo 266-273.

Davenport 6 Root, Random Signals and Noise, P. 96.

p rocess r e q u i r e s an i n f i n i t e dimensional vec to r f o r complete charac te r -

i z a t i o n , w e are gene ra l ly faced w i t h having t o use only t h e f i n i t e num-

ber of components l y i n g along t h e N s i g n a l vec to r s .

easier d e s c r i p t i o n f o r t h e noise than as a time-varying s i g n a l ,

Although the n o i s e

This means an

3.2 The N-Signal Rule

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Assume t h a t t he c o s t of saying message i was s e n t when message j

w a s a c t u a l l y s e n t is cij.

minimize the t o t a l r i s k , R, def ined as

Next w e wish t o design a r e c e i v e r which w i l l

N N R ,, 1 1 Prob [ j s e n t ] c i j Prob [ say i / j s e n t ]

i l l j-1

Using a dec i s ion space model1 (see Figure 5 ) , w e can r ep resen t

P [ say i / j s e n t ] as

- where r is the rece ived vec tor . Ca l l ing Prob [j s e n t ] I'd, we have

f o r t h e r i s k :

Now l e t s choose t h e c o s t equal t o one if w e make a mis take (i.e., say is

when j was s e n t ) and zero if w e don ' t make a mis take ( say i, when i

was s e n t ) . 0 i = j L e t c i j -

Then

A

i+ j

But t h i s may be recognized as Prob [making a n e r r o r ] - P[E]

Van Trees, p 12.

-1 5-

r ---- -7 I I

I I

143 I - 1

I

Y

I

L - - --- -J

1

L 0 P 2 6 or d 0 CL

d c)

m 2 0 L

-16-

j #i

To minimize P[E] w e should a s s i g n each p o i n t i n z t o t h e zi

which minimizes express ion (3.1) f o r P[E]. Thus w e have t h e d e c i s i o n

r u l e :

. . . . - Ia(E) I l ( E ) and I2 (g ) and . . . Ip+ i (R) , Choose %

Hence, t h e dec i s ion r u l e is t o choose mi as t h e message t rans-

m i t t e d i f

, Expanding and canceling ou t common f a c t o r w e have

- - P- (A) 'k - I m i f o r a l l k + i R

m m

and d i v i d i n g by Pi(:). , The d e c i s i o n r u l e s i m p l i f i e a t o choosing m e s -

sage m j i f and on ly i f -

, O r , t o s t a t e t h i s more compactly, set

t o m j i f and only i f Pi Pm (T) is mi - .* r

m, t h e e s t ima ted message, equa l

a maximum f o r i = j.

A

-17- . Although t h i s model does lead t o the smallest p o s s i b i l i t y of making

an e r r o r i t i s somewhat misleading, s i n c e t o real ize t h i s performance w e

need t o know t h e a p r i o r i p r o b a b i l i t i e s , P j , and t h e c o s t s c i j exac t ly .

Th i s r e s t r i c t i o n , however, can be minimized by varying both PJ and c i j

and c a l c u l a t i n g t h e i r e f f e c t on P [ E ] . A l t e rna te ly , t h e r e are o t h e r test

procedures which do n o t r e q u i r e t h i s information, such as minimax tests o r

Neyman-Pearson tests', bu t t h e s e u s u a l l y produce l a r g e r va lues f o r

o r maximize o t h e r q u a n t i t i e s of i n t e r e s t such as p r o b a b i l i t y of de t ec t ion .

IIowever, s i n c e P[E] i s a good measurement of t h e performance of a com-

munication system t h e f i r s t approach w i l l be used here' . 3.3 The Binary Decision Rule f o r N-signals

P[E] ,

Next w e would l i k e t o s p e c i a l i z e t h e r e s u l t t o t h e case where no

pena l ty is incu r red i f we confuse any of m 2 through m, with any

o t h e r m2 through q,. The only c o s t occurs i f we mistake rn l f o r

one of t h e messages m2 t o mn. Th i s w i l l a l low us t o i n v e s t i g a t e

case 2 of t h e r e c e i v e r philosophy; s i n c e by s e t t i n g m i equa l t o no i se

then we are ask ing the ques t ion is t h e r e s i g n a l o r n o i s e present .

We now have a binary dec i s ion problem. That t h i s w i l l reduce t h e

p robab iq i ty of e r r o r can be seen i f w e observe t h a t a d d i t i o n a l

are now set equa l t o zero , and so even i f w e f i x t h e

c i j ' s

I ( E ) ' s , t h e r i s k ,

Hancock 6 Wentz, S ipna l Detection Theory, pp. 35, 40, 43.

We s h a l l see la te r on t h a t by developing t h e express ion f o r t h e minimum

P[E] o t h e r cases i f w e d e s i r e .

case w e have p r a c t i c a l l y a l l t h e t o o l s needed t o eva lua te t h e

-18-

w i l l decrease.

a d d i t i o n a l c i j ' s zero, but a l s o by a d j u s t i n g the I(E)'s t o make t h e

i n t e g r a l i n each nonzero t e r m a minimum.

The optimum r u l e does even b e t t e r by n o t on ly making t h e

L e t c i j - o i = j o r {, > > simultaneously)

theref o r e

f '2

Then t h e dec i s ion r u l e t o minimize PIE] is t o choose;

Both equat ions (3.2) and (3.3) depend upon P j which w e have, and - R P i (q) which must be ca l cu la t ed .

m - I n the next s e c t i o n we s h a l l c a l c u l a t e P- (A) 5. m i

for t h e channel

m model developed i n Chapter 2.

-19-

CHAPTER 4. ii

P- (0) & m i FOR THE RAYLEIGH CHANNEL

As mentioned earlier t h e key t o ana lyz ing t h i s problem l i e s i n ex-

p res s ing t h e s i g n a l s and n o i s e i n terms of a f i n i t e dimensional vec to r

space. The ith s i g n a l may be represented as - N

k= 1 s i ( t ) 1 s k i $ k ( t ) o r s i (sli, * * S N i )

where T T

s i j = J s i ( t ) 4 j ( t ) d t and 4 k ( t ) 9 j ( t ) d t 6kj 0 0

W e would a l s o l i k e t o r ep resen t r c ( t ) and r s ( t ) as v e c t o r s so

w e may apply t h e r e s u l t s of t h e previous sec t ion . Taking rc(t) f i r s t ,

w e f i n d - rc - (rcl, r c 2 . . , rcn) by

T

-20-

I n a s imi l a r manner w e may f i n d the components of rsj of t he vector

Expressing these equat ions

vec to r s , the following

i n terms of mat r ix n o t a t i o n w e have, f o r the

It is i n t e r e s t i n g t o observe t h e e f f e c t t h a t t h e channel and demodu-

l a t o r have On the " t ransmit ted vector".

i n Figure 6A may be mapped I n t o those of 6B.

p l i t u d e simply cause a s c a l e r m u l t i p l i c a t i o n of t h e rece ived vec tor while

t he time delay T gives rise t o a l i n e a r t ransformation. This t r ans fo r -

mation a c t u a l l y r o t a t e s t he vec to r i n the s i g n a l space.

Gaussian noise e f f e c t i v e l y adds a random v e c t o r t o t he r o t a t e d s i g n a l

vec to r as shown i n Figure 6B.

For example, t h e v e c t o r s ehown

The random phase and am-

F i n a l l y t h e

R One f u r t h e r s t e p remains before w e can a c t u a l l y c a l c u l a t e P; (;) - - 0 Q

and that is t o c a l c u l a t e t h e s t a t i s t i c s of nc and nsa

It may be shown1 t h a t n c ( t ) and n 8 ( t ) are Gauooian proceeses

with:

Wozancraft 6 Jacobs, p. 4 9 6 .

-21-

1

-22-

'W

I -W

= o [ 0 elsewhere

are Gaussian random v a r i a b l e s . In o rde r t o f i n d the joint d i e t r i b u t i o n I

(4 . 3)

-23-

Since n c ( t ) and n s ( t ) have a power d e n s i t y spectrum which i s equal

t o f o r - w < f < w, t h e s i g n a l spectrum, w e can assume t h e power

spectrum of t h e no i se uniform f o r a l l f because of t he f a c t t h a t out-

of-band no i se doesn ' t a f f e c t t h e performance of an optinlum rece ive r ' .

This makes Rc(t-u) = Rs(t-u) = uo(t-u) - N o and s i m p l i f i e s (4.3) consid- 2

Thus nsi, n c i (i = 1,2,...,A) are s t a t i s t i c a l l y independent random

v a r i a b l e s and have a j o i n t d i s t r i b u t i o n .

o r , i n vec to r n o t a t i o n

(4 .4)

Ti P- (-) I m i

m

We are now i n a p o s i t i o n t o c a l c u l a t e o r , f o r t h i s s p e c i f i c

case PF, p, m

Theorem of I r r e l evancy , See Wozencraft & Jacobs, p . 220. S u f f i c i e n t

S t a t i s t i c s . See Van Trees, po 33.

-24-

Introducing the fol lowing d e f i n i t i o n t o s impl i fy n o t a t i o n

w e have :

- - - - Now i f rc = a and r8 = B on a given t r ia l , then

- - nc = a - a JETi cos e Gi Z ( T ) ns = B - a JETi s i n e a i E ( T ) - -

and w e may write

- - where the l a s t l i n e is j u s t i f i e d by t h e assumption t h a t

t i s t i c a l l y independent of m, 8, a. S u b s t i t u t i n g equat ion (4.4) i n t o

nc, nB are sta-

(4.3) w e have

P 7

- 25-

t

I

Expanding and f a c t o r i n g o u t terms which do no t depend on

f o r t he r i g h t s i d e of equat ion (4.6):

i w e have,

The p r o b a b i l i t y d i s t r i b u t i o n s f o r a and 0 a r e Rayleigh and uniform

r e s p e c t i v e l y ( see Figure 7). A 2

P,(A) - + e - 7 A 2 o

L e t t i n g k(a,?i) - k, and averaging with respect t o 8 and a g ives ,

Transforming t o Car tes ian coord ina tes by the fol lowing change of v a r i a b l e s ,

x - A COS e y = A s i n 0 dxdy = AdAd0

w e have;

-26-

1 .

-27-

I

These i n t e g r a l s are modified forms of t h e Gaussian d i s t r i b u t i o n s and

may be evaluated t o g ive

A

xz has its maximum v a l u e when t, the re la t ive de lay between t h e

incoming s i g n a l and t h e c o r r e l a t e d s i g n a l , i s zeroa

estimate t h e a r r i v a l t i m e of a s i g n a l as the moment when x2 i

l a r g e s t , At t h i s i n s t a n t x: - (si a)2 + (si 0 E ) 2 and

This a l lows u s t o

i s t h e -

-28-

T - - where s i u is t he c o r r e l a t i o n of si and a; s i ( t ) a ( t ) d t i n

terms of the t i m e vary ing s igna l s . 0

Once we assume a s i g n a l has been de tec t ed w e a loo wish t o determine

which of the m j ' s It was. By sub8 t i tU t ing (4.8) i n t o t h e dec io ion

r u l e s of the prev ious s e c t i o n w e see t h a t N-ary dec io ion r u l e becomes:

choose m = mj, i f f PiAi e Bixf is a max f o r i = j where

A i = p i

While t h e modified b inary r u l e becomes;

choose m2 o r m3 o r . . , nn

B1 +: iff 1 P, AJ eBj x~ > P I A I e N 2

otherwise choose ml. A r e c e i v e r which w i l l gene ra t e t h e es t imated 188-

sage is shown i n F igure 8. The f a c t t h a t in p r a c t i c e only a f i n i t e

number of c o r r e l a t i o n device6 are used, may l e a d t o a came where t h e

2 maximum value of x i w i l l n o t occur when T - 0. Note t h a t by uoing

m u l t i p l e c o r r e l a t o r s t h e va lue of T is now used t o i n d i c a t e t h e m i s s

t i m e by c l o s e s t c o r r e l a t o r . Th i s is j u s t another way of say ing , what

-30-

happens t o the probability of error when the receiver misoes locking onto

the received input by T seconds. Numerical values of error probability

for various T'S are presented i n Chapter 7 .

-31-

CHAPTER 5.

DERIVATION OF THE STATISTICS OF x:

Af t e r developing the dec is ion r u l e f o r an "optimum" r e c e i v e r , the

An

f o r d i f f e r -

ques t ion arises as t o j u s t how optimum the r e c e i v e r r e a l l y is.

answer t o t h i s ques t ion r e q u i r e s t h e c a l c u l a t i o n of

e n t va lues of ETi, No, and T . The only a d d i t i o n a l information needed

t o make these c a l c u l a t i o n s i s the s t a t i s t i c s f o r t h e x i .

P[E]

2

2 We have two cases, the s t a t i s t i c s of x i when message mi w a s

s e n t , and the s t a t i s t i c s when message

Case 1 - given m i w a s s e n t ;

m j ( j # i) w a s s en t .

-

are independent zero mean Gaussian random v a r i a b l e s . Since where ncj - (g * si)

Gausoian random v a r i a b l e . Thus i n order t o s p e c i f y u s i completely

i s t h e sum of Gaussian random v a r i a b l e s , then i t a l s o is a - -

w e need only f i n d i t s mean var iance

-32-

2 NO Since w e have, from equat ion ( 4 . 3 A ) , t h a t E [ n c i rick] = a i 6 i k 7 6 i k

w h e r e iJ(m, a*) i s def ined as a Gaussian random v a r i a b l e wi th mean m

and va r i ance u2

Simi la r ly : -

-T c i = a JETi s i n , e si Z(T) si

Papoulis, P robab i l i t y . etc, p. 195.

de x COS e where n = J 'n2 1 + n: and Io(x) = 21 0

We want z = w2, t h e r e f o r e

2 - - - - L e t t i n g x - u si, y - 8 * si , z = x i s n1 - b i , and n2 = c i s we

may use (5.1). Then

2 So, i f mi was t r ansmi t t ed , t h e p r o b a b i l i t y d i s t r i b u t i o n f o r t he x i

was s e n t ; mj Case 2 - given -

- B = a J s i n e Sj R ( T ) + iis

ETj

Proceeding i n t h e same manner as be fo re w e can d e r i v e t h e d i s t r i b u t i o n s

-34-

with

2 Thus, when mj is s e n t , the p r o b a b i l i t y d e n s i t y func t ion f o r x i i s

2 7 0

I t i s e a s i l y seen from t h e d e r i v a t i o n of t hese r e s u l t s t h a t t h e

P 2 (z/m,) a r e independent whenever an or thogonal s i g n a l set i s chosen. xi/m

Furthermore, t h e above equat ions are v a l i d f o r unequal s i g n a l energy,

random c a r r i e r phase and amplitude, and a r b i t r a r y s i g n a l c tos sco r re l a -

t i o n func t ions . Thus, i n theory anyway, one can set up t h e d e c i s i o n

r eg ions and eva lua te t h e P[E] f o r any or thogonal N-signal r ece ive r ' .

The nex t chapter i s a s p e c i a l i z a t i o n of t h e s e r e s u l t s t o t h e Sunblazer

s i g n a l format of two messages and noise .

~~ ~

Many times t h e express ion f o r P[E] c o n t a i n s i n t e g r a l s which r e q u i r e

a numerical so lu t ion .

-35-

CHAPTER 6 .

SUNBLAZER SIGEAL ANALYSIS

The Sunblazer receiver is faced with making a decislon as to whether

ml (a binary zero), m2 (a binary one) of m3 (nothing) was transmitted.

Using orthogonal signals we have

m2 m3 (noise case) ml

ETI = zT ETp ET

The signal space is ehown in Figure 9. Since, to a good approximation

the forward and backward Barker codes are uncorrelated we have for the

crosscorrelation matrix, the form

R(r) - Next lets look at the rules for this case.

6.1 3-Ary Signal Rule

From Chapter 4 the decision rule is set m, if 2 2

p i Ai e Bi xi < Pj Aj e j xJ for i + j

-36-

A 6 2 0 9

ar F 0

1

L a 2 3 Ln

-37-

P3 A3 k ( 1 - 2P) B3 = 0

Defining B 1 = B2 - B, A1 = A2 = A, and Q = PA/l-p w e now can so lve 2

equat ion (6.1) f o r x j .

2 2 I n P j A j + B j x j > I n P i A i + B i x i

These r u l e s can b e s t be summarized by t h e diagram shown in Figure 10.

Note t h a t t h i s diagram a p p l i e s only t o the case f o r

1 (see Figure 10) is g r e a t e r than zero. More w i l l be s a i d about t h e case

Q e 1 where p o i n t

where Q > 1 later on. In order t o eva lua te t h e p r o b a b i l i t y of e r r o r w e

use equat ions (5.2) and (5.3). The c a l c u l a t i o n i s most convenient ly done

i n t h r e e s t e p s by c a l c u l a t i n g the ind iv idua l p r o b a b i l i t y of e r r o r s f o r

t h e cases when ml,'ma and rn3 are sen t .

6.2 P r o b a b i l i t y of Erro r Given No S igna l Present, 5 3 - (0, 0) = 0

2 2 (XI, x p ) The receiver makes a m i s t a k e i n the event t h a t t he p o i n t

lies o u t s i d e t h e shaded area i n F igure 11. When no s i g n a l i s p resen t ,

2 2 note : x l and xp are independent

2

2 X l = nc; + n s l

2 2 x2 = "c2 + "s2

-38-

I I

/

CHOOSL

L SLOPE * I

I '

-39-

and from equat ion (5.2)

then

PIE/,] = Yrob [Error i s made/given n o i s e only]

= 1 - Prob [Correct dec i s ion /g iven no i se on ly ]

p [I - ~ l / B N o ] 2

Defining D = %N*

P ~ / n ] = 1 - (1 - aD)2

= 2QD - Q2D f o r Q 1 (6.2)

6.3 P r o b a b i l i t y of Er ror Given m1 was Transmit ted, Si (1, O), ET1 p ET

2 2 If m l was t r ansmi t t ed then we make an e r r o r i f x l and x2 l ead t o

a p o i n t o u t s i d e t h e shaded a r e a i n Figure 12. With s i g n a l mi presen t

t hen

2 xl = (a 6 cos e ~ ( t ) + nC1)2 + (a JET s i n e R ( T ) + ns l )2

2 2 2 x2 - "c2 + "s2

2 2 Note t h a t x l and x2 are independent random v a r i a b l e s , S u b s t i t u t i n g

i n t o (5.2) f o r x: and (5.3) f o r x2, 2

-40-

a

I I I

4 I I I.

P c r * %

c E

-41 -

Then,

ready

averaging over the random amplitude (note the phase terms have a l -

cancelled out) g ives

a r 1

where

-42-

m . a= 0

We would l i k e t o eva lua te t h i s i n t e g r a l and then s u b s t i t u t e i t back i n t o

But Io(x) can be expressed i n terms of an i n f i n i t e series of t h e form

Interchanging t h e i n t e g r a l and summation s i g n s of (6.7) m

T t i s i n t e g r a l is s t r a igh t fo rward and t h e expres s ion may be evalua ted t o

g ive

S u b s t i t u t i n g w i n t o ( 6 , 8 ) , mult ip ly ing ou t terms and then recombining

ET wi th c = 2(g) y2 R2(T) + 1

Using t h i s r e s u l t wi th equat ion (6.5) and rear ranging g i v e s

which may be eva lua ted , wi th D l / ~ ~ o n as

c r e Q < 1 P [E/m1] = 1 - QD’C + C+l

6.4 P r o b a b i l i t y of Error Given m2 t r ansmi t t ed s p = ( 0 ..

Z L The r e c e i v e r i s i n c o r r e c t whenever t h e p o i n t (XI , x2) is i n s i d e

t h e shaded and do t t ed areas i n F igure 12. When m 2 w a s t ransmi t ted

2 2 X l = nc; + "Sl

2 x2 = (a COS e K ( T ) + nc2)2 + (a 6 s i n 8 R ( T ) + n a 2 l 2

and proceeding as i n t he last s e c t i o n

Uut i t may be observed t h a t t h i o is t h e same problem w e j u s t f i n i s h e d

so lv ing i n the prev ious s e c t i o n i f x: and x$ are interchanged. Thus,

6 . 5 Total P r o b a b i l i t y of Er ror f o r Case 1 wi th Q

The t o t a l p r o b a b i l i t y of e r r o r is def ined as

1 - 3

1- 1

(6;11) P[EI 1 ~ [ m i I P [ ~ / m i I

S u b s t i t u t i n g our prev ious r e s u l t s into(b.\ l \ we have f o r t h e case 1 prob-

a b i l i t y of e r r o r t h e fo l lowing express ion

P[E] - (1 - 2P)(2QD - Q2D) + 2P + c + l Q -1 Q 1 1

(6.12)

where

As mentioned previous ly t h i s r e s u l t i s on ly v a l i d when Q 5 1. I n t h e

next s e c t i o n w e look a t t h e case when Q > 1.

-45- . 6.6 P[El When Q > 1 and Summary of Resul t s

When Q < 1 po in t one i n Figure 10 moves t o the o r i g i n ( i t cannot

go nega t ive s i n c e x: > 0) and the dec i s ion space is shown i n Figure 13.

Note t h a t t he r ece ive r w i l l never say no i se when (z > 1. This i n t e r e s t i n g

phenomena w i l l be examined i n more d e t a i l i n Chap te r 7. Using the same

d e f i n i t i o n s f o r P[E/mi] as i n the l as t s e c t i o n w e s ee immediately t h a t

W / p l = 1 2 Nhen m l is s e n t , t he s t a t i s t i c s of xf and x2 a r e given by (6.9)

and (6.10). An e r r o r occurs when w e say m 2 r a t h e r than m l , even

22 1 - - NO

21 - - e N o dz2 dz l e NOC -

By u t i l i z i n g symmetry i t can be shown t h a t

1 p = p [ E / q l = c+l

l h e r e f o r e , t he p r o b a b i l i t y of e r r o r when Q > 1 is

i= 1

1 = 1 - 2 P + 2 P ( x ) Q > 1

-46-

\

\

li U

I I

47-

When (1 > 1 we have

> 1 P ET NO

(1 - 2 P ) ( 2 - y2 + 1)

or ET

2 T Y 2 + 1

p ' ET 4 - y 2 + 3 NO

ET - y 2 A l l the previous expressions can b e s implif ied by observing that No '

i s actual ly the average received s ignal t o noise ra t io , which w i l l here-

@ Summarizing the resu l t s for case 1 w e have, a f t er be designated as - ER NO

when ER NO ER NO

2 - + l

4-+ 3 P I (6.13)

P [E] = (1 - 2P)(2QD - Q2D) + 2 P ( 1 - QD" + Q -1 \ c + l I

or when

1 P [ E ] (1 - 2P) + 2P (x)

where

P Q = ER

( 2 Ng + 1) (1 - 2P)

D = 1 + - 1 ER 2 - NO

-4a=

6.7 Plodified Binary Decision Rule - Case 2

By removing t h e l i n e sepa ra t ing t h e dec i s ion spaces of m l and m2

i n F igure 10 w e can develop a b inary dec i s ion r u l e .

We know t h a t the optimum receiver w i l l do as w e l l as o r b e t t e r than t h i s

case s i n c e an optimum r e c e i v e r , as t h e name impl ies , i o t h e b e s t one can

(See F igure 1 4 ) .

do. The reason, however, f o r examining t h i s case is twofold; f i r s t , i t

a l lows a check on the cons iderably more complicated optimum case and

second, i t presents a c o n t r a s t t o he lp dec ide what f a c t o r s are most im-

p o r t a n t f o r optimum performance.

We w i l l fo l low t h e same gene ra l procedure f o r eva lua t ing t h e per for -

mance of case 2 as we d id f o r case 1.

For Q I1 i t is e a s i l y seen t h a t P[E/n] i s e x a c t l y t h e same

f o r t h i s case as f o r case 1 and is given by ( 6 . 2 )

P [E/nI 2QD Q 2D f o r Q c 1

When i s s e n t t he re ce iver makes an e r r o r i f the p o i n t

l ies i n s i d e the square of F igure 12. Then

where Px:(z) and Px$(z) are given by ( 6 . 3 ) and (6.4).

I

\

.

-30-

E (1 - Q D / C ) ( l - QD)

Again i t can be seen by symmetry t h a t P[E/m2] w i l l be the same as

P I E / m I l a

P ( 1 - QDlc) (1 - QD)

F i n a l l y when Q > 1 then the r e s u l t is f a i r l y obvious, s i n c e t h e

receiver only guesses s i g n a l and t h e r e f o r e w i l l be wrong anytime only

no i se is present . Therefore ,

P [E/n] 1 f o r Q > 1 3

i= 1

Using the f a c t t h a t P [ E ] = 1 P[mi] P[E/,i] and c o l l e c t i n g re-

s u l r s we have, when

P z 4 (9 + 3

P I E ] - ( 1 - 2P)(2QD - Q2D) + 2P(1 - Q D ) ( l - Q W C )

or i f 2 (2) + 1

2 (2) + 3 p 1

P[E] 1 - 2P

where

P Q =

6.8 Optimum Binary Decision Rule - Case 3

I n t h i s s e c t i o n we deoign t h e optimum r e c e i v e r t o dec ide whether Only

no i se is present o r whether e i t h e r message ml o r m2 was t r ansmi t t ed .

-51-

t

The f i r s t s tep involves c a l c u l a t i n g the dec i s ion r u l e and space using

t h e r e s u l t s of Chap te r 4,

The dec i s ion r u l e i s

Where t h e no ta t ion i n d i c a t e s t o choose s i g n a l when the g r e a t e r than

e q u a l i t y holds, and noise when t h e less than does,

2 Solving t h i s equat ion f o r x2 g ives

PA As before w e l e t Q - 1-2p

and so t he dec is ion r u l e becomes

Since t h i s funct ion has a p o s i t i v e second d e r i v a t i v e i t is concave down-

wards, The dec is ion space is shown i n Figure 15,

f i n e the i n t e r c e p t of t h e curve and the xl a x i s as G which makes

For convenience we de-

2

G I - - l 1 Q

Thus the optimum rece ive r f o r t h e b inary r e c e i v e r "guesses" when

r a t h e r than when

1 . Q LT,

Q 2 1, as was the case f o r t h e previous r e c e i v e r ,

.Unfortunately, applying d i r e c t l y methods of s o l u t i o n used i n the

last two cases l e a d s t o i n t e g r a l s which cannot be evalua ted in c losed

form.

prec i se - l inea r approximation t o the curve and then al lowing

piecewise linear approximation is shown in Figure 16 ,

more c lose ly on the n t h

Refer r ing again to Figure 16, w e s ee

Bpe apprpacb I s tg " l i n e a r i z e " the problem by making an N-segment

The N * OD.

Next l e t ' o focus

segment and determine Its s l o p e and i n t e r c e p t .

-53-

n 1 1 B Q

B 0

a = - l n ( - - G : )

n+l - c = - I ~ ( - - G 1 1 N )

Using t h e s tandard s lope- in te rcept formulas t h e nth segment’s equat ion

2 2 is of t h e form x2 0 m(n) xl + k(n) where

m(n) = - Fi I n G -

1 Q 1 - - G N Q

n+l ( 6 . 15)

and

(6.16)

6.9 Case 3 - P [E/noise] Q 112

Refer r ing t o Figure 1 5 w e see t h a t t h e r e c e i v e r is i n c o r r e c t when-

2 2 ever noise only is p resen t and the po in t

shaded area. The s t a t i s t i c s for XI and x2 are the same as the n o i s e

only case i n t h e las t sec t ion .

(XI , x2) l i es ou t s ide the 2 2

The expression for P[E/noiee]

(see Figure 16) which is t h e sum of a l l t h e con t r ibu t ions ly ing above t h e

l i n e a r segments; and area 2 which i s t h e region above the x l -ax is extend-

ing from x1 = - 111 G t o Q). With P[E/noise , N] denot ing t h e N-segment

approximation t o P [E/noise] w e have

can be broken up i n t o two terms, area 1

2

2 1 B

Substituting ( 6 . 1 7 ) into (6.18) w e have, for (6.18)

(6.18)

Evaluating the f i r s t double in tegra l , and the inner integral of the second,

equation (6 .19) reduces t o

p *’ - N e

(6.20)

The integral i n (6.20) can b e evaluated straightforwardly not ic ing that

w e have two cases , when m(n) + 1 = 0 and when m(n) + 1 # 0

-55-

n+l I)n - ($ - G N )

Defining T(n) = - n D(n+l) 1 Q (- - G P!)

Then (6.20) becomes r

M- \

(6.21)

where D and G are def ined as before .

6.10 Case 3 - P[ E/,i], Q 2 -1/2

Since PIE/ml] = P[E/m2] by symmetry arguments only PIE/,l] w i l l

be derived. I t i s somewhat e a s i e r t o eva lua te t h e I n t e g r a l s f o r t h i s

case i f w e f i n d , i n s t e a d of P[E/m,] ,

P[c/ml] = P [ c o r r e c t dec is ion/mlsent ] = 1 - PIE/ml] Since w e are f ind ing P[C/m 3 t he space over which w e i n t e g r a t e is

2 tile same as t h e last sec t ion . The d i s t r i b u t i o n s f o r x: and x2 are

-56-

(6.22)

The express ion f o r P,:(zl)a i s g iven i n s t e a d of Px:(zl) s i n c e

i t can be shown t h a t any d e r i v a t i o n f o r P(E) u ses Px:(zi) only i n t h e

form Px:(zi) Breaking the i n t e g r a t i o n i n t o two r eg ions t o t ake i n t o

account t h e con t r ibu t ions of area above t h e curve and above t h e

a

2 axis,

and then averaging with respect t o t h e random amplitude w e have,

Rearranging t h e i n t e g r a l s and us ing t h e d e f i n i t i o n of Pxi(zi)a

g ives

+ m= o

( 6 . 2 3 )

Substituting equations ( 6 . 2 2 ) into ( 6 . 2 3 ) and collecting terms

Evaluating the first double integral and the inner integral of the second

leaves us with

( 6 . 2 4 )

-58-

The remaining integral i n ( 6 . 2 4 ) has two forms depending on the value of

m(n) + l / c .

L

h

Using the above resul t and equations (6.16). (6 .21) , and (6.24) g ives

-59-

I ~

I . 2 (D) + 1

4 (2) + 4 P L

6.11 P[E] for Q > 1 / 2 and Summary of Case 3 Results

1 2 When Q > - the receiver w i l l always say s ignal . However, wfth

probability 1 - 2P only noise i s present a t the receiver input. Thus,

the probabil i ty of error is 1 - 2P. ??[E[ = 1 - 2P

Summarizing the case 3 r e s u l t s we have;

for

t m= 0

while €or

where

P [ E ] 1 - 2P P

Q' ER (1 - 2P)(2 + 1)

n e 1 0 - G N

N Q n+l 1 -

- - G N Q

m(n) - - - In In G

In the next cahpter these rather involved equations are graphed and

analymd,

-61-

CHAPTEK 7.

ANALYSIS OF RECEIVER PEKFORbIANCE

VS. SIGNAL TO TXOISE AND TIME DELAY

A f t e r cons ide rab le mathematical a n a l y s i s w e are now i n a p o s i t i o n

t o answer ques t ions about t h e development of a r e c e i v e r "philosophy",

comparison of P ( E ) f o r t h e d i f f e r e n t cases w i l l reveal f o r which va lues

of We also have not iced

t h e f a c t t h a t t h e r e are times when a d e c i s i o n r eg ion r e c e i v e r simply

A

E K / i ~ o a par t icu lar receiver performs the bes t .

throws i n t h e towel" and guesses. A c l o s e r look a t t h e reasons f o r t h i s I f

behavior and i t s e f f e c t on r ece ive r performance w i l l be made, F i n a l l y ,

t h e r e s u l t s of our bit-by-bit a n a l y s i s w i l l be extended by an example,

t o t h e more p r a c t i c a l case of an e r ro r - co r rec t ing block code. The expres-

s i o n s developed i n t h e prev ious chapter were q u i t e complicated involv ing

loga r i thms and non-integer va lues r a i s e d t o non-integer powers. There-

f o r e , i n o rde r t o eva lua te then f o r many p o i n t s a computer program w a s

w r i t t e n , The program a l s o con ta ins a graphing r o u t i n e which d i s p l a y s t h e

d a t a both s i n g l y and f i v e graphs t o a page.

T h e d i f f e r e n t cases a r e s tud ied i n t h e next s e c t i o n s i n t h e o rde r

i n which they were der ived ,

7 . 1 Case 1, The 3-Ary Receiver

Observing t h e graphs of F igures 17-20, w e n o t i c e t h a t t h e expres-

s i o n f o r P I E ] dec reases monotonically wi th inc reas ing s i g n a l t o no i se

r a t i o

s i o n f o r P [ L ]

r ange of 'R/i< from ,1 t o 100 by a func t ion of t h e form

( E R / ~ { o ) . It is i n t e r e s t i n g t o n o t e t h a t t h e complicated expres-

€o r t h i s case can be f a i r l y a c c u r a t e l y modeled over a

0

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-66-

a + b b The f a c t t h a t t h e e r r o r p r o b a b i l i t y decreases only in-

v e r s e l y w i t h t he s i g n a l t o no i se r a t i o even f o r l a r g e va lues of

i s due pr imar i ly t o the assumption of a s c a t t e r i n g model i n the channel.

E R / ~ o ,

For a s c a t t e r i n g model g ives rise t o a Rayleigh d i s t r i b u t i o n f o r t h e re-

ceived s i g n a l amplitude; and t h i s means t h a t on any given t ransmiss ion ,

no matter how l a r g e the t r ansmi t t ed power, w e may r e c e i v e no s i g n a l ,

Although i t is not obvious from the graph, the case 1 r e c e i v e r , f o r

t h e P - .45 case , has begun t o guess t h a t n o i s e i s never s e n t once

ER/so f a l l s below about 1.8.

r ece ive r t o t ake s i n c e say ing n o i s e i s never rece ived w i l l add

This i s a c t u a l l y a l o g i c a l s t e p f o r t h e

.1 t o

P[E],

t ake t o be made between t h e weak s i g n a l s and n o i s e more o f t e n than a

whereas us ing t h e poor (low ER/N ) rece ived d a t a causes a m i s - 0

t e n t h of t h e t i m e .

of ER/~o, in t he graphs of case 1 i s obscured by t h e f a c t t h a t t h e

major por t ion of t h e receiver e r r o r occurs no t from mistaking s i g n a l

Actua l ly , t h e r e c e i v e r ' s "guessing" f o r small v a l u e s

and noise , but r a t h e r from confusing s i g n a l s one and two. A much more

profound e f f e c t of r e c e i v e r "guessing" on P [ E ] w i l l be seen later when

w e e l imina te t h e pena l ty f o r confusing 81 and 82.

It is worth no t ing t h a t s e t t i n g P = .50 l e a d s t o t h e s imple case

of choosing between one of two equal energy, e q u a l l y l i k e l y s i g n a l s ;

express ion 6.13 f o r t he case one then reduces t o t h e s t anda rd

r e s u l t ' f o r t h i s problem,

P[E]

Wozencraft & Jacobs, p. 533.

- 67-

7.2 Case 2, The Modified Binary Case

As mentioned earlier, t h i s i s a suboptimum d e c i s i o n r u l e involv ing

t h e cho ice of e i t h e r s i g n a l or noise. O f primary i n t e r e s t w i l l be t o

v e r i f y t h a t t h i s case does indeed g i v e a n express ion f o r P[E] which is

always g r e a t e r than o r equa l t o t h e optimum binary case (case 3).

The f i r s t i n t e r e s t i n g phenomena w e n o t i c e i s t h a t f o r t h e P = .5

case (Figure 21) t h e r e c e i v e r has P[E] i d e n t i c a l l y equa l t o zero f o r

a l l LR/N . l edge t h a t a s i g n a l w i l l be t ransmi t ted wi th p r o b a b i l i t y one. Since

Th i s is due t o t h e r ece ive r us ing only t h e a p r i o r i know- 0

P = 1 / 2 i s always g r e a t e r than

4 (+) + 3 0

2 t h e r e c e i v e r "guesses" s i g n a l r e g a r d l e s s of t h e va lues of

a s i g n a l is always t r ansmi t t ed , i t never makes a mistake'.

xi and s i n c e

'.loving on t o a more i n t e r e s t i n g case w e n o t i c e t h a t t h e graph f o r

t h e P = .45 case no t on ly shows guess ing f o r ER/Wo

but a l s o is n o t monotone decreasing. It i s obvious t h a t t h i s is n o t an

less than 1.9

optimum r e c e i v e r s i n c e one could do b e t t e r by simply guessing s i g n a l a t

eve ry p o i n t where P[E] is g r e a t e r than .1. It i s i n t e r e s t i n g t o ob-

s e r v e t h a t t h e graphs of P[E] f o r cases 1 and 2 start t o converge f o r

l a r g e s i g n a l t o no i se r a t i o s ( for example, see Figure 25, curves l abe led

T h i s i s a c t u a l l y an u n i n t e r e s t i n g case because i f w e knew beforehand

t h a t a s i g n a l is broadcas t with p r o b a b i l i t y equa l 1, w e ha rd ly need t o

des ign a r e c e i v e r t o t e l l u s whether or n o t a s i g n a l w a s t r ansmi t t ed .

..

-68-

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-73-

1 and 2) .

p r i n c i p a l e r r o r i s t o mistake s i g n a l s 1 and 2, but t h a t a s the s i g n a l s

g e t s t ronge r t h i s mistake i s made less o f t e n and t h e l i m i t i n g f a c t o r i s

the confusion of s i g n a l s wi th noise . It i s a l s o worthwhile t o n o t i c e

t h a t f o r l a r g e E R / N ~

of each o t h e r ( t o wi th in .00001 fo r E R / ~ o = 100).

c e i v e r ope ra t ing i n t h i s range ga ins noth ing by using t h e s i m p l e r deci-

s i o n r u l e . We w i l l see t h a t t h i s r e s u l t ho lds t r u e a l s o f o r t h e optimum

binary case of t he next s ec t ion .

7.3 Case 3, The Optimum Binary Case

This would i n d i c a t e t h a t f o r small s i g n a l t o n o i s e r a t i o s t h e

t h e converging graphs a c t u a l l y l i e almost on top

That is t o say, a re-

An eva lua t ion of t h e performance of t h e case 3 receiver i s f u r t h e r

complicated over t h e prev ious cases by t h e need f o r l i n e a r i z i n g t h e de-

c i s i o n r u l e . A computer subrout ine w a s written which eva lua ted P[E]

whi le vary ing t h e number of segments from 1 t o 50. A very s u r p r i s i n g

r e s u l t w a s obtained; f o r any number of segments g r e a t e r than 2 t h e

w a s witi i in one p l ace i n lo5 of t h e 50-segment case!

of t h e d e c i s i o n r u l e r e v e a l s t h a t f o r l a r g e s i g n a l t o no i se r a t i o s the

d e c i s i o n r u l e equat ion (6.14A) approaches very c l o s e l y a square ( see

F igu re 26). Since a two-segment approximation w i l l f i t a square as

p e r f e c t l y as a f i f t y , and s i n c e t h e dec i s ion r u l e is p r a c t i c a l l y a

it i s then reasonable t o expect f o r l a r g e t h a t t h e dec i s ion r u l e

w i l l be independent of t he number of segments (above 2) . For the s m a l l

s i g n a l t o n o i s e r a t i o s w e f i n d t h a t t h e main c o n t r i b u t i o n t o t h e va lue

of P[E] comes from i n t e g r a t i n g over area 2 of Figure 27. Since t h e

c o n t r i b u t i o n s from area 1 a r e smal1,the exac t shape of i t , be i t curved

(F igu re 27A), q u a d r a l a t e r a l (Figure 27B), o r otherwise i s r e l a t i v e l y

P[E]

A c l o s e r examination

square,

E R / N ~

-74-

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x:

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6 ,

-76-

unimportant. A combination of t hese two f a c t o r s then he lps t o exp la in

t h e surpt&.fqg r e s u l t s of t he computer program.

This resul t raises an i n t e r e s t i n g quest ion. Why, i f case 2 i s a

square, and the optimum case can be c l o s e l y approximated by a square, is

case 2 (as we have seen) n o t optimum? The answer t o t h i s ques t ion is t h a t

t h e case 2 decis ion r u l e has an xi- intercept a t I/Q whi le the case 3

r u l e has t h e i n t e r c e p t a t

2

l / q - 1. It would then appear t h a t t h e c r i t i -

cal f a c t o r i n t h e des ign of a n optimum binary receiver i s n o t t h e exac t

shape of t h e dec ie ion curve bu t r a t h e r t he i n t e r c e p t on t h e x i axes. 2

Case 3 with P = 1 / 3 ( see F igure 30) i l l u s t r a t e s a clear example of

r e c e i v e r guessing. The va lue of t h e p r o b a b i l i t y of e r r o r express ion

starts t o r i s e f o r decreas ing s i g n a l t o n o i s e r a t i o , F i n a l l y , when

P[E]

f o r guessing, t h e r e c e i v e r does guess and t h e

based upon t h e input d a t a i s about t o exceed t h e e r r o r p r o b a b i l i t y

P[E] curve l e v e l s o f f .

From Chapter 6

(1 - 2P)(2% R ( r ) + 1) P

- I 1 NO Q

and, as E R / ~ , becomes l a r g e , l / ~ - 1 a l/ ~ . Thus, f o r t hese s i g n a l ,

t o n o i s e ratios t h e case 1 and 2

From our previous d i scuss ion i t was shown t h a t equal i n t e r c e p t s g ive rise

t o approximately equal express ions f o r Checking Figure 25 (curves

2, 3) w e see t h a t f o r E R / N ~ g r e a t e r than 4.7 t h e carves d i f f e r by a

van i sh ing ly small amount.

s u l t t h a t f o r l a r g e s i g n a l t o no i se r a t i o s , cases 1, 2, and 3 a l l behave

approximately t h e same, and t h e r e f o r e w e should use t h e case 1 r e C e f V e r

which e x t r a c t s t h e most information.

2 x i - i n t e r c e p t s are about t h e same.

P[E] .

This graph a l s o p o i n t s o u t t h e i n t e r e s t i n g re-

-77-

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-81- t

The l a s t po in t t o observe from Figure 25 i s t h a t case 3 n o t only

begins t o guess earlier than c a s e 2 ( l / ~ - 1 as compared t o l /Q) , but

also dec reases more r a p i d l y f o r small s i g n a l t o n o i s e r a t i o s .

F i n a l l y , t h e 2-segment case 3 express ion was compared wi th case 2 ,

f o r a l l v a l u e s of "/pio from .01 t o 100, f o r v a l u e s of P from

.01 t o .50, and f o r all va lues of T between 0 and 75 usec. A t every

p o i n t t h e case 3 P [ E ] was less than o r equal t o t h e case 2 P[E] , which

indeed i t should be i f t h e previous t h e o r i e s were c o n s i s t e n t .

7.4 Some Fur the r Comments on Decision Region Receiver

P[E] as a Function of E R / ~ o ~~

A s mentioned previous ly t h e r ece ive r guesses when it f i n d s t h a t t h e

i n p u t d a t a i s of such poor q u a l i t y t h a t b e t t e r r e s u l t s can be obta ined

by simply choosing s i g n a l , It should be noted t h a t when t h e r e c e i v e r

guesses , i t always chooses s i g n a l - never no i se , Th i s i m p l i e s t h a t a

necessary cond i t ion f o r optimum rece ive r guess ing is f o r t h e sum of t h e

s i g n a l p r o b a b i l i t i e s t o be g r e a t e r than l / 2 . Other cond i t ions f o r

guess ing are placed upon

s t r u c t u r e . For example,

t h e a p r i o r i p r o b a b i l i t i e s by t h e r e c e i v e r

case 3 r e q u i r e s

-'PIT 1 1 4 -

d i i l e case 2 r e q u i r e s

F i n a l l y , i t appears t h a t , except f o r those t i m e s when t h e r ece ive r

is guess ing , an optimum receiver has a P [ E ] func t ion which i s monotone

dec reas ing wi th s i g n a l t o n o i s e r a t i o 1 . It should be noted t h a t t h i s is n o t always t r u e f o r =-optimum re- c e i v e r s , See case 2 graphs f o r an example.

-8 2-

This concludes t h e examination of t h e e f f e c t s of varying s i g n a l t o

n o i s e r a t i o s on receiver performance. I n t h e next s e c t i o n w e s h a l l s tudy

t h e r e s u l t s of having only a f i n i t e number of c o r r e l a t o r s a v a i l a b l e f o r

use i n t h e rece iver .

7.5 P[E] Vs. Time Delay Error

From Chapter 4 w e remember t h a t t he e s t ima t ion r u l e is t o w a i t un-

2 t i l X i

of t he s i g n a l s , i.e., s e t 'I, t h e r e l a t i v e s i g n a l de lay equal t o zero.

However, s ince we have only a f i n i t e number of c o r r e l a t o r s w e are l i k e l y

to choose a maximum f o r which T # 0. The e f f e c t of t h i s nonzero T,

i s shown i n t h e fol lowing Figures , 32 through 35. For a low s i g n a l t o

n o i s e r a t i o (.1) it a c t u a l l y makes l i t t l e d i f f e r e n c e what T i s - t he

performance of t he r ece ive r i s about equa l ly bad f o r a l l va lues of t i m e

delay. For s igna l t o n o i s e r a t i o of 1 t h e e r r o r i n c r e a s e s l i n e a r l y

with T reaching a maximum a t t h e baud l eng th of t h e Barker code. As

t h e s i g n a l t o no i se r a t i o f u r t h e r i nc reases the curves f o r P [ E ] start

becoming more concave. For E R / N ~ equal t o 100 w e can even m i s s by 2/5

of a band l eng th and s t i l l have e r r o r p r o b a b i l i t i e s of less than

It a l s o may be no t i ced t h a t f o r any 'I, P[E] f o r case 3 is always

less than P[E] f o r case 2, as expected.

i s a maximum and then choose t h i s i n s t a n t as t he a r r i v a l time

1 fjy using coding, e r r o r p r o b a b i l i t i e s of less than

t a ined even a f t e r missing by 315 a band l eng th .

curve 5).

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(See F igure 35,

. -83-

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7.6 A Simple Coding Scheme

By us ing a coding scheme1 i t i s p o s s i b l e t o decrease the message

e r r o r s over t h e b i t by b i t c a s e s j u s t s tud ied . For example, curve 5 on

t h e prev ious f i g u r e s r e p r e s e n t s a coding scheme having e i g h t code words

of l e n g t h 1 2 b i t s and minimum hamming d i s t a n c e 7. A 7-distance code

a l lows f o r c o r r e c t i o n of t h r e e o r less e r r o r s . The i n d i v i d u a l b i t by

b i t e r r o r s used i n t h e c a l c u l a t i o n s f o r case 5 are given by case 4

(curve 4 on t h e g raphs ) , an equal energy, equa l p r o b a b i l i t y optimum re-

c e i v e r . Note t h a t t h e block p r o b a b i l i t y of e r r o r i s t i n y i n comparison

t o t h e b i t by b i t case u n t i l t h e p e r b i t e r r o r s become moderate. Once

t h i s happens P[E] f o r t h e block s i g n a l i n g case rises extremely r a p i d l y

( exponen t i a l ly ) t o a much g r e a t e r va lue than t h e b i t by b i t e r r o r .

P [ e r r o r / b i t ] = PIS then t h e block e r r o r p r o b a b i l i t y can be expressed as

I f

12

This coding can supply an exponent ia l decrease i n r e c e i v e r probabi l -

i t y of e r r o r . flowever, t h i s i s a t t h e expense of reduced t ransmiss ion

rates s i n c e w e must t r ansmi t 1 2 b i t s t o r ep resen t t h e same messages w e

could represent wi th 3 b i t s were i t n o t f o r e r r o r co r rec t ion .

Abramson, Informat ion Theory and Coding

CHAPTER 8.

SUMMARY AND FINAL REMARKS

I n t h e preceding c h a p t e r s w e have looked i n d e t a i l a t t h e problem

of communicating through a noisy , s c a t t e r i n g channel wi th time de lay .

It was observed t h a t t h i s channel r o t a t e d t h e " t r ansmi t t ed vec tor" i n

s i g n a l space, s ca l ed i t i n amplitude and added t o i t a n o i s e v e c t o r ,

Severa l d e c i s i o n r u l e s , us ing t h e rece ived vec to r as a b a s i s , were de-

veloped. The two most important cases were the 3-ary d e c i s i o n r u l e

(case 1) f o r choosing between message 1, message 2, o r n o i s e , and t h e

optimum binary r u l e (case 3) f o r choosing between any message and noise .

Whicii c a s e should be used f o r t h e Sunblazer r e c e i v e r w a s found t o be

determined p r imar i ly by t h e s i g n a l t o n o i s e r a t i o - f o r l a r g e E R / ~ o

use case 1, while f o r small E R / ~ J ~ use case 3.

It was also no t i ced t h a t both t h e case 1 and case a receivers

r e s o r t e d t o "guessing" under c e r t a i n c o n d i t i o n s of low s i g n a l t o n o i s e

r a t i o .

t he c r i t i c a l f a c t o r s f o r optimum recep t ion were explored .

The optimum binary case was a l s o examined i n more d e t a i l and

F i n a l l y , a simple coding scheme w a s used t o i l l u s t r a t e t h e advan-

t ages which coding can r e a l i z e over simple b i t by b i t s i g n a l l i n g systems.

t.1 Future Ideas and Extensions

There a r e numerous ex tens ions which can be made t o t h i s problem.

These inc lude t h e e f f e c t of d i s p e r s i o n on t h e s i g n a l , non-orthogonal

s i g n a l se ts , and d i f f e r e n t channel models.

channel model would be t he Rici.an, s i n c e i t i s probably a more rea l i s t ic

nlodel of t h e space environment through which Sunblazer must t r a n s m i t ,

Fur ther work might also be done on r e c e i v e r "guessing1' and t h e e f f e c t Of

A p a r t i c u l a r l y i n t e r e s t i n g

-89-

the shape of t h e dec is ion region on the expressions f o r P[E]. Final ly ,

i t might be interest ing to examine other coding schemes besides the for-

ward arid backward Barker codes, particularly those with non-zero off

diagorral terms i n &(?).

-90-

BIBLIOGRAPHY

1.

2.

3.

4.

5 .

6.

7.

8.

9.

10.

11.

12.

Abramson, Information Theory - and Coding, Mc Qaw-Hill, Nsw York, 1963

Balakrishman, Space Communications, lk maw-Hill, New York, 1963

Davenport and Root, Random Signals and Noise, Mc maw-Hill, New York, 1958

--

Gilbert , Some Communication Aspects of - Sunblazer, Center for Space Research, M.T.T., TSR-TR-66-9, 1966

-

Ooulomb, 9 i g i t a l Cowunications with Space Applications, Prentice- Hall, Ehglewood Cl i f f s , N. J . , 1 9 6 4

Helstrom, S t a t i s t i c a l Theory of - Signal Detection, Pergamon Press, New York, 1960

Hancock and Wintz, Sicpal Detection I Theory, I!& maw-Hill, New York, 1966

Harrington, Study of a &all Solar Probe, *ace Research, M.1 .T., PR-5255-5, 1965

Sunblazer, Center f o r -----

Middleton, S t a t i s t i c a l Camnunication Theory, Pk G r a w - H i l l , New York, 1960

Papoulis, Probabili ty, Random Variables - and Stochast ic ---' Processes W maw-Hill, New York, 1965

Van Pees, Detection - and &timation -- Theory, To be published September 1967

Wozencraf t and Jacobs, Pr inc ip les - of Communication Engineering, Mc Qraw-Hil l , New York, 1965