. .
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
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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
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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
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5
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7
9
13
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19
31
35
35
37
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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
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Bib1 iography 90
Figure Number
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Page -bet
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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
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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.
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I 1
I I
1 I
I I
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2 9 b 8 I. ia
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a ) The Eleven-hit Barker Code
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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 .
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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
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
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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.
-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 .
-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;
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
-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
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
-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
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).
-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 .
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
-62-
$ 3 Q cr
0
0 . . . e
e 0 0
e 0 e 0
e e 0
0 0
0 e 0 0
0 e e
0
e e 0 e
0
0
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0
0 e 0
e 0
0 e e
e 0 0
0 e e e
0 0
0 0
0
0
e e 0
e 0
0 e
0
0
0 e
e 0 0
0
0 e 0
e 0
e e
e 0
0 0
il 0
0 e
0
0 0
0 0
0 0
3 b
1 u r n n v s
-63-
a
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dl:
0
0
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0
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0
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0
0
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0
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0
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0
0 . . . 0
-64-
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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 .
-69-
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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 ~
-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).
can Le ob-
(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
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