Theory of Optimum Noise Immunity

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THE THEORY OF OPTIMUM ROISE

THE THEORY OE OPTIMUM NOISE IMMUNITY

THE THEORY OF OPTIMUM NOISE IMMUNITY

by V. A. KoteFnikov TRANSLATED FROM THE RUSSIAN

by R. A. Silverman

DOVER PUBLICATIONS, INC. New York

C opyright © 1960 by R. A. Silverm an. All rights reserved u n d er Pan A m erican and In te r­ n ational C opyright Conventions.

P ublished in C anada by G eneral P u blishing Com ­ pany, Ltd., SO Lesmill R oad, Don Mills, T o ro n to , O ntario. Published in the U nited K ingdom by C onstable and Com pany, L td., 10 O range Street, London WC 2.

T h is Dover edition, first published in 1968, is an unabridged rep u blication of the work originally p u b ­ lished in 1960.

Library of Congress Catalog Card Number: 68-20594 M anufactured in the U nited States of A m erica Dover Publications, Inc. 180 Varick Street New York, N. Y. 10014

AUTHORS PREFACE

T h i s b o o k i s t h e a u t h o r ' s d o c t o r a l d i s s e r t a t i o n , p r e s e n t e d in J a n u a r y , the a c a d e m i c c o u n c i l of the M o lo to v E n e r g y I n s t i t u t e in M o s c o w .

1947, b e f o r e

D e s p i t e the f a c t th a t

m a n y w o r k s d e v o t e d to n o i s e i m m u n i t y h a v e a p p e a r e d i n t h e t i m e t h a t h a s e l a p s e d s i n c e t h e w r i t i n g of t h i s d i s s e r t a t i o n , n o t a l l of t h e t o p i c s c o n s i d e r e d in it h a v e a s y e t a p p e a r e d in p r i n t .

C o n s i d e r i n g th e g r e a t i n t e r e s t sh o w n in t h e s e m a t t e r s , a n d a l s o the n u m b e r of

r e f e r e n c e s m a d e t o t h i s w o r k i n t h e l i t e r a t u r e , t h e a u t h o r h a s d e e m e d i t a p p r o p r i a t e to publish it, w ithout in tro d u c in g any s u p p le m e n ta ry m a te r ia l.

H o w e v e r , in p r e p a r i n g the

m a n u s c r i p t f o r p u b l i c a t i o n , it w a s s o m e w h a t c o n d e n s e d , a t the e x p e n s e of m a t e r i a l of secondary interest.

M o r e o v e r , C h a p t e r Z, w h i c h c o n t a i n s a u x i l i a r y m a t h e m a t i c a l m a t e r i a l ,

h a s b e e n r e v i s e d s o m e w h a t , to m a k e i t e a s i e r r e a d i n g , a n d s o m e o f t h e m a t e r i a l h a s b e e n r e l e g a t e d to t h e a p p e n d i c e s .

The author

EXPLANATORY NOTE T he study of p r o b a b ility th e o r y a n d its a p p lic a tio n s h a s had a long an d illu s tr io u s h i s t o r y in R u s s i a ,

b e g i n n i n g i n t h e e a r l i e s t d a y s of t h e 18 t h c e n t u r y , a n d c o n t i n u i n g i n a n

u n b r o k e n l i n e d o w n to t h e p r e s e n t g e n e r a t i o n .

In o u r t i m e w e h a v e s e e n a r e a l i z a t i o n t h a t

i n m a n y a s p e c t s o f s c i e n c e , t e c h n o l o g y , a n d h u m a n b e h a v i o r t h e e l e m e n t of r a n d o m n e s s i s s o f u n d a m e n t a l t h a t o f t e n o n e c a n h a r d l y d e f i n e a m e a n i n g f u l p r o b l e m , m u c h l e s s s o l v e i t, w ithout using p ro bability theory.

D u r i n g the r a p i d t e c h n o l o g i c a l d e v e l o p m e n t s of the

W o r l d W a r II p e r i o d , t h e c o m m u n i c a t i o n a n d d e t e c t i o n a r t s u n d e r w e n t s u c h a r e a l i z a t i o n , a n d a s w o u ld be e x p e c t e d ,

s t a t i s t i c a l c o m m u n i c a t i o n t h e o r y (or i n f o r m a t i o n th e o ry ) h a s

o c c u p i e d s o m e of the b e s t m i n d s a m o n g m a t h e m a t i c i a n s a n d e n g i n e e r s in the Sovie t U nion j u s t a s it h a s e l s e w h e r e . O n e of t h e m o s t i m p o r t a n t S o v i e t c o n t r i b u t i o n s , a n d o n e t h a t w a s u n t i l r e c e n t l y v i r t u a l l y u n k n o w n o u t s i d e t h e U . S . S . R . , w a s t h e 1 9 4 7 d o c t o r a l d i s s e r t a t i o n of V. A . K o t e l ’n i k o v ,

at that tim e a 4 0 - y e a r - o l d c o m m u n ic a tio n s e n g in e e r ,

who had a l r e a d y

i n h i s y o u n g e r d a y s ( 1 9 33 ) b e c o m e w e l l k n o w n f o r h i s w o r k o n s a m p l i n g t h e o r e m s f o r b a n d -lim ite d functions.

K o t e l ’n i k o v ’ s d i s s e r t a t i o n c o n s t i t u t e d a n e x t e n s i v e a n a l y s i s of

th e e f f e c t s of a d d i t i v e g a u s s i a n n o i s e on c o m m u n i c a t i o n s y s t e m s , a n d of w h a t c o u l d be done a t the r e c e i v e r to m i n i m i z e t h e m .

U nlike S h a n n o n ’s i n f o r m a t i o n t h e o r y , he did

n o t go e x t e n s i v e l y i n to th e i m p l i c a t i o n s of a f r e e d o m to c h o o s e c o m p l i c a t e d t r a n s m i t t e r signals. M a n y S o v ie t c o n t r i b u t i o n s to the s t a t i s t i c a l c o m m u n i c a t i o n a r t a r e f a i r l y w e ll know n to u s .

E v e r y stu d e n t of th e s e m a t t e r s know s the n a m e s K hinchin and K o lm o g o r o v a s

p a r t n e r s w ith W e s t e r n m a t h e m a t i c i a n s ( n o ta b ly N o r b e r t W i e n e r ) in the e a r l y d e v e l o p m e n t of s p e c t r a l a n d f i l t e r i n g t h e o r i e s f o r r a n d o m f u n c t i o n s .

Y et fe w of us h a v e b e e n a w a r e

t h a t t h e r e e x i s t e d i n 1947 i n t h i s d i s s e r t a t i o n a s t a t i s t i c a l a n a l y s i s of c o m m u n i c a t i o n p r o b l e m s u s in g w h a t we now c a l l d e c i s i o n t h e o r y t e c h n i q u e s a n d a n t i c i p a t i n g by s e v e r a l y e a r s m u c h of th e w o r k of W o o d w a r d , D a v i e s ,

Siegert and others,

w ith w h ic h we a r e

m ore conversant. T h i s b o o k i s a v e r b a t i m t r a n s l a t i o n of " T h e T h e o r y o f P o t e n t i a l N o i s e I m m u n i t y ” , p u b l i s h e d b y t h e S t a t e P o w e r E n g i n e e r i n g P r e s s i n 1 95 6 . indicated,

A s the A u t h o r ’s P r e f a c e h a s j u s t

i t i s e s s e n t i a l l y i d e n t i c a l t o t h e 1947 d i s s e r t a t i o n .

In p r e p a r i n g t h i s E n g l i s h

e d i t i o n , no t e c h n i c a l e d itin g h a s b e e n done o t h e r th a n the c o r r e c t i o n of m i s p r i n t s .

The

p r e s e n t v o l u m e th u s r e t a i n s the e x a c t f l a v o r of the o r i g i n a l , a llo w in g one to se e f r o m h i n d s i g h t w h i c h o f K o t e l ’n i k o v ' s m a n y h i g h l y o r i g i n a l i d e a s h a v e b e e n d e v e l o p e d f u r t h e r a n d w hich have not.

v

By n o m e a n s a l l o f K o t e l ' n i k o v ' s r e s u l t s h a v e s i n c e b e e n o b t a i n e d i n d e p e n d e n t l y b y o t h e r s , a nd th u s the v o l u m e s h o u ld be of m u c h m o r e than j u s t

historical interest.

P e r h a p s the r e a d e r w ill be a i d e d by the fo llo w in g few c o m m e n t s w h ic h s h o u l d m a k e t h e u n f a m i l i a r t e r m i n o l o g y a l i t t l e e a s i e r to f o l l o w , a n d s h o u l d c l a r i f y t h e r e l a t i o n s h i p b e t w e e n this a nd o t h e r w o r k s in the c o m m u n i c a t i o n t h e o r y f ie ld . F i r s t of a ll t h e r e is the q u e s t i o n of j u s t w h a t is m e a n t by " n o i s e i m m u n i t y " . h e r e , it is a g e n e r ic t e r m w ith a d if f e r e n t m e a n in g f o r d if f e r e n t s itu a tio n s .

As used

F o r P a r t II,

t h e c a s e i n w h i c h c o m m u n i c a t i o n t a k e s p l a c e by t r a n s m i t t i n g o n e o u t o f a f i n i t e n u m b e r o f p o s s i b l e s i g n a l s t h e t e r m r e f e r s to p r o b a b i l i t y o f n o e r r o r . would now c a ll the " m u l t i p l e - a l t e r n a t i v e d e c i s i o n " p r o b l e m . )

( P a r t II d i s c u s s e s w h a t w e P a r t II I t r e a t s t h e s i t u a t i o n

i n w h i c h a c o n t i n u u m o f t r a n s m i t t e d s i g n a l s i s a s s u m e d (a p a r a m e t e r A

ranging over som e

i n t e r v a l ta k in g the p l a c e of th e p r e v i o u s d i s c r e t e in d e x of th e p o s s i b l e s i g n a l s ) ; i. e. , the p r o b l e m of

"p a ra m e te r estim ation".

H e r e a g r e a t e r n o i s e i m m u n i t y r e f e r s to a d e c r e a s e

in m e a n s q u a r e v a l u e of th e e r r o r b e t w e e n the v a l u e of A i n d i c a t e d by th e r e c e i v e r a n d t h a t actually tran sm itted .

A nd then w h e n the a u t h o r t r e a t s in P a r t IV the c a s e of s i g n a l l i n g

u s in g w a v e f o r m s (the p a r a m e t e r now b e in g r e p l a c e d by a f u n c tio n of t i m e in s o m e t i m e i n t e r v a l ) a n i n c r e a s e d n o i s e i m m u n i t y r e f e r s to a d e c r e a s e i n t h e a v e r a g e n o i s e p o w e r t h a t a d d itiv e ly c o r r u p t s the r e c e i v e r o u tp u t.

In o t h e r w o r d s , t h e a u t h o r i s d e a l i n g w i t h t h e

m e a n - s q u a r e e r r o r b etw een the m o d u la tin g sig n a l e n te r in g the t r a n s m i t t e r , and that r e p r o ­ d u c e d by the r e c e i v e r .

N o te t h a t the a u t h o r n e v e r s a y s t h a t n o i s e i m m u n i t y " i s " o n e of

t h e s e t h r e e t h i n g s , but r a t h e r "is c h a r a c t e r i z e d b y " one of t h e m .

T h is u s a g e p e r s i s t s in

the l a r g e n u m b e r of S o v ie t p a p e r s th a t h a v e c o n ti n u e d K o t e l ' n i k o v ' s w o r k . T h e d e v e l o p m e n t p r e s e n t e d h e r e is n o t a b l e in it s a b s e n c e of a n y d e p e n d e n c e on an advanced m ath em atical background.

T he r e a d e r p o s s e s s i n g a p a s s i n g f a m i l i a r i t y with

F o u r ie r s e r i e s , d is c r e te and continuous p ro b a b ilitie s and p ro b a b ility d e n sitie s (sim p le , j o i n t , a n d c o n d itio n a l) a n d th e n o tio n of s t a t i s t i c a l i n d e p e n d e n c e w ill h a v e no t r o u b l e .

At

s e v e r a l points s o m e known r e s u l t s of p ro b a b ility th e o ry a r e invoked w ithout r e f e r e n c e o r proof.

(One th a t the b e g in n in g r e a d e r m i g h t n o t be f a m i l i a r w ith is the C e n t r a l L i m i t

T h e o re m , Equation 2 -3 3 .)

H o w e v e r , t h e s e i n s t a n c e s a r e r a r e ; by and l a r g e the t r e a t m e n t

is c o m p letely self-su fficien t. K o te l'n ik o v m a d e e x t e n s i v e u s e of g e o m e t r i c m o d e l s of the s i g n a l l i n g a n d d e t e c t i o n p r o c e s s e s a s o p e r a t i o n s on v e c t o r s i n m u l t i - d i m e n s i o n a l s p a c e , a n a r t i f i c e t h a t S h a n n o n introduced la te r.

T he r e a d e r w ill find t h e s e g e o m e t r i c i n t e r p r e t a t i o n s v e r y h e l p f u l .

m a t e r i a l of e a c h c h a p t e r i s r e v i e w e d in t e r m s of th e g e o m e t r i c m o d e l a t t h e e n d of t h e chapter.

Paul E. G reen, Jr. M . I . T. L in c o ln L a b o r a t o r y

The

TABLE OF CONTENTS

A U T H O R 'S P R E F A C E EXPLANATORY NOTE

PART I AUXILIARY M A T E R IA L CHAPTER 1 INTRODUCTION

1-1.

M e th o d s of c o m b a t i n g n o is e

1-2.

C lassificatio n of noise

1-3. 1-4.

................................................................................................

1

..............................................................................................................

1

M e s s a g e s a n d s i g n a l s ..................................................................................................................

2

T he c o n t e n t s of this book

.......................................................................................................

3

CHAPTER 2 AUXILIARY M A T H E M A T IC A L M A TERIA L 2-1. 2-2.

Som e definitions

............................................................................................................................

5

R e p r e s e n t a t i o n o f a f u n c t i o n a s a l i n e a r c o m b i n a t i o n of o r t h o n o r m a l f u n c t i o n s .......................................................................................................

6

2-3.

N o rm a l fluctuation noise

..........................................................................................................

2-4.

R e p r e s e n t a t i o n of n o r m a l f l u c tu a tio n n o i s e a s a F o u r i e r s e r i e s

2-5.

L i n e a r f u n c t i o n s of i n d e p e n d e n t n o r m a l r a n d o m

2-6.

T h e p r o b a b i l i t y t h a t n o r m a l f l u c t u a t i o n n o i s e f a l l s in a g i v e n r e g i o n . . .

16

2-7.

G e o m e t r i c i n t e r p r e t a t i o n of o u r r e s u l t s

18

variables

8

..............

13

...........................

15

.......................................................................

P A R T II TRANSMISSION OF D IS C R E T E MESSAGES CHAPTER 3 T H E I D E A L R E C E I V E R F O R D IS C R E T E SIGNALS 3-1.

D i s c r e t e m e s s a g e s a n d s i g n a l s ............................................................................................

3-2.

T h e i d e a l r e c e i v e r ..........................................................................................................................

3-3.

G e o m e t r i c i n t e r p r e t a t i o n of the m a t e r i a l of C h a p t e r 3

Vll

20 21 24

CHAPTER 4 NOISE IM M U N ITY F O R SIGNALS W ITH T W O D I S C R E T E V A L U E S

4-1.

P r o b a b i l i t y o f e r r o r f o r t h e i d e a l r e c e i v e r ..................................................................

25

4-2,

Influence of the ra tio

P ( A ^ ) / P ( A ^ ) ....................................................................................

27

4-3.

O p t i m u m n o i s e i m m u n i t y f o r t r a n s m i s s i o n w i t h a p a s s i v e s p a c e ................

29

4-4.

O p t i m u m n o i s e i m m u n i t y f o r t h e c l a s s i c a l t e l e g r a p h s i g n a l ............................

31

4-5.

N o ise im m u n ity fo r the c l a s s i c a l t e l e g r a p h sig n a l and r e c e p tio n w i t h a s y n c h r o n o u s d e t e c t o r ............................................................................................

31

N o i s e i m m u n i t y f o r the c l a s s i c a l t e l e g r a p h s i g n a l an d r e c e p t i o n w i t h a n o r d i n a r y d e t e c t o r ..................................................................................................

34

4-7.

R e s u l t s on t h e n o i s e i m m u n i t y o f s y s t e m s w i t h a p a s s i v e s p a c e ..................

36

4-8.

T h e o p t i m u m c o m m u n i c a t i o n s y s t e m w i t h a n a c t i v e s p a c e ................................

38

4-9.

N o i s e i m m u n i t y f o r f r e q u e n c y s h i f t k e y i n g ....................................................................

39

O ptim um n o ise im m u n ity for n o r m a l fluctuation n o is e with f r e q u e n c y - d e p e n d e n t i n t e n s i t y ........................................................................................

42

4-6.

4-10. 4-11.

G e o m e t r i c i n t e r p r e t a t i o n o f t h e m a t e r i a l o f C h a p t e r 4 .................

44

CHAPTER 5 NOISE IMMUNITY FO R SIGNALS WITH MANY D IS C R E T E V A L U E S

5-1.

G e n e r a l s t a t e m e n t o f t h e p r o b l e m .......................................................................................

46

5-2.

Optimum noise im m unity for orthogonal equiprobable signals w i t h t h e s a m e e n e r g y ...........................................................................................................

46

5-3.

E x a m p l e o f t e l e g r a p h y u s i n g 32 o r t h o g o n a l s i g n a l s ...............................................

48

5-4.

O p t i m u m n o i s e i m m u n i t y f o r c o m p o u n d s i g n a l s .......................................................

50

5-5.

E x a m p l e o f a f i v e - v a l u e d c o d e ...............................................................................................

52

5-6.

T h e o p t i m u m s y s t e m f o r s i g n a l s w i t h m a n y d i s c r e t e v a l u e s ..........................

53

5-7.

A p p r o x i m a t e e v a l u a t i o n o f o p t i m u m n o i s e i m m u n i t y ............................................

56

5-8.

E x a m p l e o f t h e t r a n s m i s s i o n o f n u m e r a l s b y M o r s e c o d e ................................

58

P A R T III TRANSMISSION OF S E P A R A T E P A R A M E T E R V A L U E S

CHAPTER 6 G E N E R A L T H E O R Y O F T H E I N F L U E N C E O F N O I S E ON T H E T R A N S M I S S I O N OF S E P A R A T E P A R A M E T E R VALUES

6 - 1.

G e n e r a l c o n s i d e r a t i o n s ..............................................................................................................

62

6-2,

D e t e r m i n a t i o n of t h e p r o b a b i l i t y o f t h e t r a n s m i t t e d p a r a m e t e r .....................................................................................................................................

63

viii

6-3.

The function

(X) n e a r t h e m o s t p r o b a b l e v a l u e X .............................. x ^ vxm E r r o r a n d o p t i m u m n o i s e i m m u n i t y in th e p r e s e n c e of low i n t e n s i t y n o i s e ..............................................................................................................................

6-4. 6-5.

P

65 66

S e c o n d m e t h o d of d e t e r m i n i n g the e r r o r a n d o p t i m u m n o i s e i m m u n i t y in t h e p r e s e n c e o f l o w i n t e n s i t y n o i s e ..............................................

68

6-6.

S u m m a r y o f C h a p t e r 6 ...................................................................................................................

71

6 - 7.

G e o m e t r i c i n t e r p r e t a t i o n o f t h e m a t e r i a l o f C h a p t e r 6 .......................................

72

CHAPTER 7 T H E O P T IM U M NOISE IM M U N ITY O F VARIOUS SY STEM S F O R T R A N SM IT T IN G S E P A R A T E P A R A M E T E R V A L U E S IN T H E P R E S E N C E O F L O W I N T E N S I T Y N O I S E

7-

A m p l i t u d e m o d u l a t i o n ....................................................................................................................

73

7-2.

L i n e a r m o d u l a t i o n .............................................................................................................................

74

7-3.

G e n e r a l c a s e o f p u l s e t i m e m o d u l a t i o n ............................................................................

74

7-4.

S p e c ia l c a s e of p u ls e tim e m o d u la tio n (o p tim u m n o is e im m u n ity )

...............

76

7-5. 7-6.

1.

S p e c i a l c a s e of p u l s e t i m e m o d u l a t i o n ( n o is e i m m u n i t y f o r the f i r s t m e t h o d o f d e t e c t i o n ) .............................................................................................................

78

S p e c i a l c a s e of p u l s e t i m e m o d u l a t i o n ( n o i s e i m m u n i t y f o r the s e c o n d m e t h o d o f d e t e c t i o n ) .............................................................................................................

80

7-7.

F re q u e n c y m odulation (general case)

...............................................................................

84

7-8.

F req u en cy m odulation (special case)

.........................................................................

86

R a i s i n g the n o i s e i m m u n i t y w ith o u t i n c r e a s i n g the e n e r g y , le n g th , o r b a n d w i d t h o f t h e s i g n a l .....................................................................................................

87

7 - 9.

CHAPTER 8 NOISE IM M UNITY F O R TRANSMISSION O F S E P A R A T E P A R A M E T E R VALUES IN T H E P R E S E N C E O F S T R O N G NOISE

8 - 1.

D e r i v a t i o n of the g e n e r a l f o r m u l a f o r e v a l u a t i n g the e f f e c t of high i n t e n s i t y n o i s e .............................................................................................................................

90

8-2.

C o m p a r i s o n of the f o r m u l a s f o r w e a k a n d

........................................

92

8-3.

P u ls e tim e m odulation

.....................................................................................................................

93

8-4.

F re q u e n c y m odulation

.....................................................................................................................

96

T h e s y s t e m f o r r a i s i n g the n o i s e i m m u n i t y w ith o u t i n c r e a s i n g the e n e r g y , l e n g t h , o r b a n d w i d t h o f t h e s i g n a l .............................................................

96

8-5. 8-6.

G e o m e t r i c i n t e r p r e t a t i o n of t h e r e s u l t s

strongnoise

ofC h a p t e r

8 ................................................. 98

P A R T IV TRANSMISSION OF W A V E FO R M S

CHAPTER 9 G E N E R A L T H E O R Y O F T H E I N F L U E N C E O F W E A K N O I S E ON T H E T R A N S M I S S I O N OF WAVEFORMS

9-1.

G e n e r a l c o n s i d e r a t i o n s .................................................................................................................

100

9-2.

T h e i n f l u e n c e o f w e a k n o i s e on t h e m o d u l a t i n g w a v e f o r m s ...............................

101

9-3.

C o n d i t i o n s f o r t h e i d e a l r e c e i v e r .........................................................................................

103

9-4.

M e a n s o f r e a l i z i n g t h e i d e a l r e c e i v e r ..............................................................................

104

9-5.

T h e e r r o r f o r i d e a l r e c e p t i o n .................................................................................................

106

B r i e f s u m m a r y of C h a p t e r 9

107

9-

6.

.................................................................................................

C H A P T E R 10 DIRECT M ODULATION SYSTEMS

1 0 - 1. D e f i n i t i o n ..............................................................................................................................................

108

10-2,

D e r i v a t i o n o f b a s i c f o r m u l a s .................................................................................................

108

10-3.

O ptim um noise im m u n ity for a m p litu d e and lin e a r m o d u latio n

.................

109

10-4.

O p t i m u m n o i s e i m m u n i t y f o r p h a s e m o d u l a t i o n .......................................................

Ill

10-5.

N oise im m u n ity for a m p litu d e m o d u latio n and o r d i n a r y re c e p tio n

10-6.

N o i s e i m m u n i t y f o r p h a s e m o d u l a t i o n a n d o r d i n a r y r e c e p t i o n ....................

113

7. N o i s e i m m u n i t y f o r s i n g l e - s i d e b a n d t r a n s m i s s i o n .................................................

114

10-

....

Ill

C H A P T E R 11 PU L SE M ODULATION SYSTEMS

1 1 - 1. D e f i n i t i o n ...............................................................................................................................................

115

11-2.

A w a y o f r e a l i z i n g t h e p u l s e m o d u l a t i o n s y s t e m .......................................................

115

11-3.

O p t i m u m n o i s e i m m u n i t y f o r the p u l s e m o d u l a t i o n s y s t e m

117

11-4.

N o i s e i m m u n i t y f o r the r e c e i v e r a n a l y z e d in s e c t i o n 11-2

11-5.

Optim um noise im m unity for pulse am plitude m odulation

............................

122

11-6.

O p t i m u m n o i s e i m m u n i t y f o r p u l s e t i m e m o d u l a t i o n ............................................

123

11-7.

Optim um noise im m unity for pulse frequency m odulation

124

x

............................

119

...............................

C H A P T E R 12 IN TEG R A L MODULATION SYSTEMS

12-1.

D e f i n i t i o n ...........................................................................................................................................

125

12-3.

O p t i m u m n o i s e i m m u n i t y f o r i n t e g r a l m o d u l a t i o n s y s t e m s ........................

125

l'2-3.

O p t i m u m n o i s e i m m u n i t y f o r f r e q u e n c y m o d u l a t i o n .........................................

127

C H A P T E R 13 E V A L U A T I O N O F T H E I N F L U E N C E O F S T R O N G N O I S E ON T H E TRANSMISSION OF W A V EFO RM S

13-1.

G e n e r a l c o n s i d e r a t i o n s .............................................................................................................

129

13-2.

M axim um

d i s c r i m i n a t i o n of t r a n s m i t t e d w a v e f o r m s

........................................

129

13-3.

M axim um

d i s c r i m i n a t i o n f o r p h a s e m o d u l a t i o n ......................................................

130

13-4.

M axim um

d i s c r i m i n a t i o n f o r w e a k n o i s e ......................................................................

131

13-5.

M axim um

d i s c r i m i n a t i o n f o r w e a k n o i s e a n d p h a s e m o d u l a t i o n ..................

133

A PPENDICES

A p p e n d ix A.

T h e s p e c i f i c e n e r g y o f h i g h - f r e q u e n c y w a v e f o r m s .............................

135

A p p e n d i x B.

R e p r e s e n t a t i o n of n o r m a l f l u c t u a t i o n n o i s e by two a m p l i t u d e - m o d u l a t e d w a v e s ........................................................................

135

A p p e n d i x C.

T h e i n s t a n t a n e o u s v a l u e o f n o r m a l f l u c t u a t i o n n o i s e ...........................

137

A p p e n d i x D.

N o r m a l f l u c t u a t i o n n o i s e m a d e up o f a r b i t r a r y p u l s e s

138

.....................

PART AUXILIARY

I

M A TERIA L

CHAPTER 1 INTRODUCTION

1-1

M e th o d s of c o m b a tin g n o is e O r d i n a r i l y , a r a d i o r e c e i v e r i s a c t e d u p o n n o t o n l y by d i s t u r b a n c e s ( s i g n a l s ) p r o d u c e d

b y t h e r a d i o t r a n s m i t t e r , b u t a l s o b y d i s t u r b a n c e s ( n o i s e ) p r o d u c e d by a l a r g e v a r i e t y o f sources.

T h e n o i s e c o m b i n e s w ith the s i g n a l s a nd c o r r u p t s th e m ; in the c a s e of t e l e g r a p h i c

r e c e p t i o n t h i s l e a d s to e r r o r s , a n d i n t h e c a s e o f t e l e p h o n i c r e c e p t i o n to b a c k g r o u n d n o i s e , s ta tic , etc.

W h e n t h e s i g n a l s a r e t o o s m a l l c o m p a r e d to t h e n o i s e , r e c e p t i o n b e c o m e s

im possible. T h e follow ing m e th o d s of c o m b a tin g n o ise a r e used; 1.

D e c r e a s i n g t h e s t r e n g t h o f t h e n o i s e by t a k i n g a c t i o n a g a i n s t t h e i r s o u r c e s .

Z.

I n c r e a s i n g t h e r a t i o o f t h e s t r e n g t h o f t h e s i g n a l s to t h a t o f t h e n o i s e b y i n c r e a s i n g

the t r a n s m i t t e r p o w e r a n d by u s i n g d i r e c t i o n a l a n t e n n a s . 3.

I m p r o v i n g the r e c e i v e r s .

4.

C h a n g i n g the f o r m of the s i g n a l s w h ile k e e p in g t h e i r p o w e r f ix e d .

(T his is done

w i t h th e a i m of f a c i l i t a t i n g th e c o m b a t i n g of n o i s e in th e r e c e i v e r . ) T h e f i r s t t w o m e t h o d s a r e n o t c o n s i d e r e d i n t h i s b o o k , w h i c h i s d e v o t e d r a t h e r to t h e l a s t t w o m e t h o d s , a n d h a s a s i t s g o a l to e x a m i n e w h e t h e r i t i s p o s s i b l e to d e c r e a s e t h e e f f e c t o f n o i s e by i m p r o v i n g t h e r e c e i v e r s , g i v e n t h e e x i s t i n g k i n d s o f s i g n a l s . w h a t c a n b e a c h i e v e d i n c o m b a t i n g n o i s e by c h a n g i n g t h e f o r m o f t h e s i g n a l s ?

In p a r t i c u l a r , W hat f o r m of

s ig n a ls is o p t i m u m f o r this p u r p o s e ?

1-2

C l a s s i f i c a t i o n of n o is e We can c l a s s i f y the n o i s e w h ic h i m p e d e s r a d i o r e c e p t i o n into the fo llo w in g c a t e g o r i e s ; A.

S i n u s o i d a l n o i s e c o n s i s t i n g of one o r a f in ite n u m b e r ( u s u a lly s m a l l ) of s i n u s o i d a l

oscillations.

T h i s c a t e g o r y o f n o i s e i n c l u d e s i n t e r f e r e n c e f r o m t h e p a r a s i t i c r a d i a t i o n of

o n e o r m o r e r a d i o s t a t i o n s o p e r a t i n g a t f r e q u e n c i e s n e a r th a t of the s t a t i o n b e in g r e c e i v e d . B.

I m p u l s e n o i s e c o n s i s t i n g of s e p a r a t e i m p u l s e s w h ic h follow on e a n o t h e r a t s u c h

l a r g e t i m e i n t e r v a l s th a t the t r a n s i e n t s p r o d u c e d in the r e c e i v e r by one i m p u l s e h a v e s u b ­ s t a n t i a l l y d i e d o u t by t h e t i m e t h e n e x t i m p u l s e a r r i v e s .

T his c a te g o ry of n o ise in c lu d e s

s o m e kinds of a t m o s p h e r i c n o ise and n o ise f r o m e l e c t r i c a l a p p a r a tu s .

1

C.

N o r m a l f l u c t u a t i o n n o i s e 1- o r , a s i t i s s o m e t i m e s c a l l e d ,

sm oothed-out noise.

T h i s a l s o c o n s i s t s of s e p a r a t e i m p u l s e s , o c c u r r i n g a t r a n d o m t i m e i n t e r v a l s , but th e i m p u l s e s f o l l o w o n e a n o t h e r s o r a p i d l y t h a t t h e t r a n s i e n t s p r o d u c e d in t h e r e c e i v e r by t h e i n d i v i d u a l i m p u l s e s a r e s u p e r i m p o s e d in n u m b e r s l a r g e e n o u g h to w a r r a n t t h e a p p l i c a t i o n of the la w s of l a r g e n u m b e r s of p r o b a b i l i t y t h e o r y .

T h is c a t e g o r y of n o i s e i n c l u d e s v a c u u m

t u b e n o i s e , n o i s e d u e to t h e t h e r m a l m o t i o n o f e l e c t r o n s in c i r c u i t s , a n d s o m e k i n d s o f a t m o s p h e r i c n o ise and n o ise f r o m e l e c t r i c a l a p p a r a t u s .

At v e r y high f r e q u e n c i e s this

kind of n o i s e is e n c o u n t e r e d a l m o s t e x c l u s i v e l y . D.

I m p u l s e n o is e of an i n t e r m e d i a t e t y p e , w h ic h o c c u r s w hen the t r a n s i e n t s p r o d u c e d

in t h e r e c e i v e r b y t h e i n d i v i d u a l i m p u l s e s a r e s u p e r i m p o s e d , b u t n o t in n u m b e r s l a r g e e n o u g h to w a r r a n t t h e a p p l i c a t i o n w i t h s u f f i c i e n t a c c u r a c y o f t h e l a w s o f l a r g e n u m b e r s . T h i s k in d of n o i s e i s i n t e r m e d i a t e b e t w e e n c a t e g o r i e s B a n d C. M e th o d s of stu d y in g the a c tio n of s i n u s o i d a l an d i m p u l s e n o i s e on r a d i o r e c e i v e r s a r e at p r e s e n t quite w ell d ev e lo p e d .

T h e stu d y of i m p u l s e n o i s e of the i n t e r m e d i a t e ty p e ,

w h e n t h e t r a n s i e n t s p r o d u c e d by t h e i n d i v i d u a l i m p u l s e s a r e j u s t b e g i n n i n g to b e s u p e r ­ im p o s e d , is m u c h m o r e difficult.

M o r e o v e r , in t h i s c a s e , w e n e e d to k n o w n o t o n l y t h e

s h a p e s of the s e p a r a t e i m p u l s e s , but a l s o the p r o b a b i l i t y of s u p e r p o s i t i o n of i m p u l s e s which have v a rio u s s h a p e s , and which obey v a r io u s tim e d is tr ib u tio n s .

In m o s t c a s e s w e

do n o t h a v e t h i s i n f o r m a t i o n a b o u t t h e n o i s e , a n d i t s e e m s to b e q u i t e d i f f i c u l t to o b t a i n . F o r t h e s e r e a s o n s , a n d a l s o b e c a u s e n o i s e o f c a t e g o r y C i s o f t e n e n c o u n t e r e d , in w h a t follow s we s h a l l c o n s i d e r o n ly n o i s e of this l a t t e r c a t e g o r y ; we s h a l l often d e s i g n a t e n o rm a l fluctuation noise sim ply as noise.

1-3

M e s s a g e s and sig n a ls By a m e s s a g e w e s h a l l m e a n t h a t w h i c h i s to b e t r a n s m i t t e d .

The m e s s a g e s with

w hich we s h a l l be c o n c e r n e d c a n be d iv id e d into t h r e e c a t e g o r i e s . A.

D iscrete m essages. M e s s a g e s in t h e f o r m o f s e p a r a t e n u m b e r s (p a r a m e t e r s ) , w h i c h c a n t a k e o n a n y

v a l u e s in c e r t a i n r a n g e s . C*

M e s s a g e s in th e f o r m of w a v e t r a i n s , w h i c h c a n a s s u m e a c o n t i n u o u s i n f i n i t y of

different w aveform s. T h e m e s s a g e s w h i c h a r e t r a n s m i t t e d in t e l e g r a p h y b e l o n g to t h e c a t e g o r y o f d i s c r e t e m essages.

In t h i s c a s e , t h e y c o n s i s t o f d i s c r e t e l e t t e r s , n u m e r a l s , a n d c h a r a c t e r s ,

w h i c h c a n t a k e on a f i n i t e n u m b e r o f d i s c r e t e v a l u e s .

M o r e o v e r , in m a n y i n s t a n c e s , t h e

m e s s a g e s t r a n s m i t t e d i n r e m o t e - c o n t r o l s y s t e m s b e l o n g to t h i s c a t e g o r y .

1.

T h e u s e o f t h e w o r d " n o r m a l " a l l u d e s to t h e f a c t t h a t w e d e a l h e r e w i t h o n e o f a v a r i e t y of p o s s i b l e f lu c tu a tio n p r o c e s s e s .

Z

In t h e c a s e o f t h e t r a n s m i s s i o n o f i n d i v i d u a l m e a s u r e m e n t s w i t h t h e a i d o f t e l e m e t e r i n g , t h e m e s s a g e s c o n s i s t o f t h e v a l u e s o f c e r t a i n p a r a m e t e r s ( e. g. , t e m p e r a t u r e ; p r e s s u r e , etc* ) m e a s u r e d a t g i v e n t i m e i n t e r v a l s . a r b i t r a r y v a l u e s lying w ithin c e r t a i n r a n g e s .

T h e s e q u a n t i t i e s u s u a l l y ta k e on

T h u s , in th i s c a s e w e c a n n o t r e s t r i c t o u r ­

s e l v e s to a f i n i t e n u m b e r o f p o s s i b l e d i s c r e t e m e s s a g e s .

M e s s a g e s o f t h i s k i n d b e l o n g to

c a t e g o r y B. In t h e c a s e of t e l e p h o n y , t h e m e s s a g e s a r e a c o u s t i c a l v i b r a t i o n s , o r t h e e l e c t r i c a l v i b r a t i o n s t a k i n g p l a c e i n t h e m i c r o p h o n e , w h i c h c a n t a k e o n a n i n f i n i t e n u m b e r of d i f f e r e n t form s.

T h e s e m e s s a g e s b e l o n g to c a t e g o r y C .

In t e l e v i s i o n , t h e o s c i l l a t i o n s a c t i n g on

t h e t r a n s m i t t e r c a n b e t a k e n a s t h e m e s s a g e ; t h i s m e s s a g e a l s o b e l o n g s to t h e l a s t c a t e g o r y . We s h a l l a s s u m e t h a t s o m e v a r i a t i o n i n v o l t a g e , p r o d u c e d b y t h e o p e r a t i o n of t h e t r a n s ­ m i t t e r , a c t s upon the r e c e i v e r in p u t.

We h a v e c a l l e d t h i s v a r i a t i o n i n v o l t a g e a s i g n a l .

C l e a r l y , t h e r e w i l l be a s i g n a l c o r r e s p o n d i n g to e a c h p o s s i b l e t r a n s m i t t e d m e s s a g e .

The

r e c e i v e r m u s t u s e t h i s v o l t a g e w a v e f o r m (i. e. , s i g n a l ) to r e p r o d u c e t h e m e s s a g e to w h i c h the s i g n a l c o r r e s p o n d s .

1-4

T h e c o n te n ts of this book In t h i s b o o k w e c o n s i d e r t h e i n f l u e n c e o f n o r m a l f l u c t u a t i o n n o i s e o n t h e t r a n s m i s s i o n

of m e s s a g e s .

The p ro b le m

w h ic h w ill c o n c e r n us is the follow ing:

We a s s u m e t h a t w h e n

t h e n o i s e p e r t u r b a t i o n i s n o t s u p e r i m p o s e d on t h e s i g n a l , t h e n t h e r e c e i v e r w i l l r e p r o d u c e the t r a n s m i t t e d m e s s a g e ex a c tly .

If n o i s e i s a d d e d to t h e s i g n a l , t h e n t h e s u m o f t w o

v o l t a g e s w i l l a c t u p o n t h e r e c e i v e r i n p u t , i. e. , t h e s i g n a l v o l t a g e p l u s t h e n o i s e v o l t a g e . In t h i s c a s e , d e p e n d i n g o n t h e s u m v o l t a g e , t h e r e c e i v e r w i l l r e p r o d u c e s o m e m e s s a g e o r o t h e r , w h ic h in a given i n s t a n c e m a y be d i f f e r e n t f r o m the o n e th a t w a s t r a n s m i t t e d . C l e a r l y , e a c h s u m v o l t a g e w h ic h a c t s upon the r e c e i v e r p r o d u c e s the p a r t i c u l a r m e s s a g e w h i c h c o r r e s p o n d s to i t .

T h is c o r r e s p o n d e n c e m a y be d i f f e r e n t f o r d i f f e r e n t r e c e i v e r s .

D e p e n d i n g on t h i s c o r r e s p o n d e n c e , a r e c e i v e r w i l l b e m o r e o r l e s s s u b j e c t to t h e i n f l u e n c e of n o i s e f o r a given kind of t r a n s m i s s i o n .

We s h a l l find o u t w h a t this c o r r e s p o n d e n c e

o u g h t to b e f o r t h e m e s s a g e c o r r u p t i o n to b e t h e l e a s t p o s s i b l e .

The r e c e i v e r which has

th is o p t i m u m c o r r e s p o n d e n c e w ill be c a l l e d i d e a l . N e x t w e s h a l l d e t e r m i n e the m e s s a g e p e r t u r b a t i o n w h ic h r e s u l t s when n o i s e is a d d e d to t h e s i g n a l s , a n d w h e n t h e r e c e p t i o n i s w i t h a n i d e a l r e c e i v e r ; t h e m e s s a g e p e r t u r b a t i o n o b t a i n e d i n t h i s w a y w i l l b e t h e l e a s t p o s s i b l e u n d e r t h e g i v e n c o n d i t i o n s , i. e. , f o r r e a l r e c e i v e r s u n d e r the s a m e c o n d i t i o n s , the m e s s a g e p e r t u r b a t i o n c a n n o t be l e s s .

The noise

i m m u n i t y c h a r a c t e r i z e d by th is l e a s t p o s s i b l e m e s s a g e p e r t u r b a t i o n w ill be c a l l e d the o p tim u m noise im m unity.

T h is n o i s e i m m u n i t y c a n be a p p r o a c h e d in r e a l r e c e i v e r s if the

r e c e i v e r i s c l o s e to b e i n g i d e a l , b u t i t c a n n o t b e e x c e e d e d .

By c o m p a r i n g the o p t i m u m

n o i s e i m m u n i t y w ith the n o i s e i m m u n i t y a f f o r d e d by r e a l r e c e i v e r s , we c a n ju d g e how c l o s e t h e l a t t e r a r e to p e r f e c t i o n , a n d h o w m u c h t h e n o i s e i m m u n i t y c a n b e i n c r e a s e d b y

3

i m p r o v i n g t h e m , i. e. , to w h a t e x t e n t i t i s a d v i s a b l e to w o r k o n f u r t h e r i n c r e a s i n g t h e n o i s e i m m u n i t y f o r a given m e a n s of c o m m u n i c a t i o n .

K now ledge of the o p tim u m n o ise

i m m u n i t y m a k e s i t e a s y to d i s c o v e r a n d r e j e c t m e t h o d s o f c o m m u n i c a t i o n f o r w h i c h t h i s n o i s e i m m u n i t y is low c o m p a r e d w ith o t h e r m e t h o d s .

This can be done w ithout r e f e r e n c e

to t h e m e t h o d o f r e c e p t i o n , s i n c e r e a l r e c e i v e r s c a n n o t a c h i e v e n o i s e i m m u n i t y g r e a t e r than the o p t i m u m .

By c o m p a r i n g the o p t i m u m n o i s e i m m u n i t y f o r d i f f e r e n t m e a n s of

c o m m u n i c a t i o n , we can e a s i l y e x p la in (as will be s e e n s u b s e q u e n t l y ) the b a s i c f a c t o r s on w hich the i m m u n i t y d e p e n d s , a n d t h e r e b y i n c r e a s e the i m m u n i t y by c h a n g i n g the m e a n s of com m unication.

In t h e b o o k , t h e s e m a t t e r s a r e i l l u s t r a t e d b y a w h o l e s e r i e s o f e x a m p l e s

which have p r a c tic a l in te r e s t.

H o w e v e r, the e x a m p le s c o n s id e r e d a r e f a r f r o m e x h a u stin g

a l l p o s s i b l e c a s e s in w h i c h o n e c a n a p p l y th e m e t h o d s of s t u d y i n g n o i s e i m m u n i t y d e v e l o p e d here. In t h e b o o k , a l l q u e s t i o n s a r e d i s c u s s e d in c o n n e c t i o n w i t h r a d i o r e c e p t i o n , i n t h e i n t e r e s t o f g r e a t e r c l a r i t y ; h o w e v e r , a l l t h a t i s s a i d i s d i r e c t l y a p p l i c a b l e to o t h e r f i e l d s , like, for ex a m p le , cable c o m m u n ic a tio n , a c o u s tic a l and h y d ro a c o u s tic a l sig n alin g , etc. M o r e o v e r , in t h e b o o k , a l l s i g n a l a n d n o i s e d i s t u r b a n c e s a r e c o n s i d e r e d to b e o s c i l l a t i o n s of v o lta g e ; h o w e v e r , n o th in g is c h a n g e d if w e c o n s i d e r i n s t e a d o s c i l l a t i o n s of c u r r e n t , a c o u s t i c a l p r e s s u r e , o r o f a n y o t h e r q u a n t i t y w h i c h c h a r a c t e r i z e s t h e d i s t u r b a n c e a c t i n g on the r e c e i v e r . T h is book d o e s not c o n s i d e r c e r t a i n i r r e g u l a r p e r t u r b a t i o n s of the s i g n a l s , w h ic h c a n s t r o n g l y a f f e c t both the o p e r a t i o n of r a d i o r e c e i v e r s a n d t h e i r n o i s e i m m u n i t y . of such p e r tu r b a tio n s a r e fading, echo p h e n o m e n a , etc.

Exam ples

M o r e o v e r , i t s h o u l d b e k e p t in

m i n d t h a t in t h i s b o o k t h e w o r d n o i s e i s h e n c e f o r t h ( f o r b r e v i t y ) u n d e r s t o o d to r e f e r to n o r m a l fluctuation no ise; in d e e d , this is the only kind of n o i s e w hich w ill be c o n s i d e r e d .

4

CHAPTER Z AUXILIARY M A T H E M A T IC A L M A TERIA L

Z- 1 S o m e d e f i n i t i o n s We now i n t r o d u c e s o m e d e f i n i t i o n s w h ic h s i m p l i f y the s u b s e q u e n t a n a l y s i s . th a t a ll w a v e f o r m s u n d e r c o n s i d e r a t i o n lie in the i n t e r v a l a lw a y s the c a s e for su ffic ie n tly l a r g e T he m e a n v a lu e of a w a v e f o r m

We a s s u m e

- T / Z , + T / Z , w hich is o bviously

T.

A( t)

o v e r the i n t e r v a l

T

i s d e s i g n a t e d by

+ T/Z (2-D

B y t h e s c a l a r p r o d u c t o f t wo f u n c t i o n s th e ir p r o d u c t o v e r the i n t e r v a l

A( t)

and

-T /Z , +T/Z.

B(t), we u n d e r s t a n d the m e a n v a l u e of

T h u s , the s c a l a r p r o d u c t i s

+ T/Z A( t) B(t ) dt

(Z-Z)

-T/Z It i s c l e a r f r o m t h e d e f i n i t i o n t h a t A( t) B(t ) =

B(t ) A( t)

(2-3)

F ur th e rm o r e A ( t ) l f i( t ) + C ( t n =

A( t) B ( t) + A ( t ) C ( t )

(2-4)

and [ a A ( t ) J l b S ( t T J - a b A ( t) fe(t) w here

a

and

b

a re a r b itr a r y constants.

(2-5)

T h u s , the s c a l a r p r o d u c t of f u n c tio n s h a s the

s a m e p r o p e r t i e s a s the s c a l a r p r o d u c t of v e c t o r s ; i n s t e a d of s c a l a r s we h a v e c o n s t a n t s , a n d i n s t e a d of v e c t o r s we h a v e f u n c t i o n s . We w r i t e + T /2 (2 - 6 )

In w h a t f o l l o w s , w e s h a l l o f t e n e n c o u n t e r t h e q u a n t i t y + T /2 (2-7) -T /2

5

T h is quan tity will be c a lle d the sp e c ific e n e r g y of the w a v e f o r m e x p e n d e d in a r e s i s t a n c e of 1 o h m a c t e d upon by the v o l t a g e

A(t).

A(t).

It e q u a l s th e e n e r g y

The quantity

V A 2 (t)

(2-8)

w ill be c a l l e d the e f f e c tiv e v a lu e of the w a v e f o r m

A(t).

A fu n ctio n with e ffe c tiv e v a lu e

u n i t y i s s a i d to b e n o r m a l i z e d . If two f u n c tio n s d i f f e r only by a c o n s t a n t , th e y a r e s a id to c o i n c i d e in d i r e c t i o n . n o r m a l i z e d f unc tion w h ic h c o i n c i d e s in d i r e c t i o n w ith a given f u n c tio n

A(t)

is obviously

----- -----------------

l / We s h a l l s a y t h a t t h e f u n c t i o n s

A ^ t ) , A 2 ( t) , . . .

2-2

1 < i, X < n ? except when

(2-9)

A 2 (t) , A^(t)

A . ( t ) A '(t7 for all

The

a r e ( m u t u a l l y ) o r t h o g o n a l , if

= 0

(2-10)

i =X •

R e p r e s e n t a t i o n of a f unc tion a s a l i n e a r c o m b i n a t i o n of o r t h o n o r m a l f u n c tio n s If the s y s t e m of f u n c tio n s C ^ t ) , C 2 (t), . . .

, C n (t)

(2-11)

s a t i s f i e s the e q u a tio n s C 2 (t)

=1

(2-12)

C k (t) Cx (t) = 0 w here

1 < k, £
%

T h e d e p e n d e n c e o f th e e x p r e s s i o n in c u r l y b r a c k e t s on

(cj^ “ ^ 2 ^ o

*S s ^ o w n *n F i g u r e 4 - 6 .

We c a n d r a w the fo llo w in g c o n c l u s i o n s f r o m an e x a m i n a t i o n of th is f i g u r e . 1.

F o r the kind o f o p e r a t i o n in q u e s t i o n , the l a r g e s t o p t i m u m n o i s e i m m u n i t y is

obtained for the fre q u e n c y d iffe re n c e (o ^-o i,)

0.7 (4-45)

=~ o

F o r s m a l l e r d if f e r e n c e s , the o p tim u m n o is e im m u n ity b e c o m e s s m a l l e r .

This c i r c u m ­

s t a n c e a l l o w s o n e to d e t e r m i n e t h e m i n i m u m f r e q u e n c y b a n d w i d t h b e l o w w h i c h o n e s h o u l d n o t go i f o n e w i s h e s to a v o i d l o s s o f n o i s e i m m u n i t y .

40

41 Fig. 4 -6 .

T h e t e r m in c u r l y b r a c k e t s in E q .

(4-44).

Z.

F o r th e k in d of o p e r a t i o n in q u e s t i o n , a n d f o r th e o p t i m u m f r e q u e n c y d i f f e r e n c e ,

the v a l u e of

a

7

7

2

i s 1. 2 Q /CT , i . e. , 2 . 4 t i m e s l a r g e r t h a n t h e v a l u e o b t a i n e d f o r t r a n s -

m i s s i o n w ith a p a s s i v e s p a c e , if in both c a s e s th e s p e c i f i c s i g n a l e n e r g y

Q

2

is identical.

T h u s , the o p tim u m n o is e i m m u n i t y fo r f r e q u e n c y shift key in g is not m u c h l a r g e r than the o p t i m u m n o i s e i m m u n i t y o b t a i n e d f o r the o p e r a t i o n w ith a p a s s i v e s p a c e a n a l y z e d in Section 4 -4 .

M o r e o v e r , i f w e b e a r i n m i n d t h a t i n t h e l a t t e r c a s e , a c c o r d i n g to S e c t i o n s

4 - 5 a n d 4 - 6 , w e c a n c o m e v e r y c l o s e to t h e o p t i m u m n o i s e i m m u n i t y , t h e n w e a r e l e d to the c o n c l u s i o n th a t w e c a n n o t get a p p r e c i a b l y m o r e n o i s e i m m u n i t y w ith f r e q u e n c y s h if t k e y i n g (in t h e c a s e o f u n d i s t o r t e d s i g n a l s a n d n o r m a l f l u c t u a t i o n n o i s e ) t h a n w i t h c l a s s i c a l am plitude m odulation.

T h e gain in n o i s e i m m u n i t y w h ic h is o b s e r v e d w h e n c h a n g i n g

f r o m a m p l i t u d e to f r e q u e n c y m o d u l a t i o n ( f o r s h o r t w a v e o p e r a t i o n ) m u s t e v i d e n t l y b e a s c r i b e d to s i g n a l d i s t o r t i o n p r o d u c e d b y f a d i n g .

4-10

O p tim u m n o is e i m m u n i t y fo r n o r m a l flu c tu a tio n n o is e with f r e q u e n c y - d e p e n d e n t intensity Until this s e c tio n , we h a v e c o n s i d e r e d n o r m a l flu c tu a tio n n o is e c o n s i s t i n g of a l a r g e

n u m b e r of v e r y s h o r t p u l s e s w h ic h h a v e a c o n s t a n t i n t e n s i t y .

In A p p e n d i x D i t i s s h o w n

t h a t n o i s e c o n s i s t i n g of p u l s e s of a r b i t r a r y s h a p e c a n b e w r i t t e n a s

W*

(t)

-z

2*

i t + 0? . c o s 2i

i f w e t a k e i n t o a c c o u n t c o m p o n e n t s w i t h f r e q u e n c i e s f r o m JUL I T (mutually) independent n o r m a l ra n d o m v a r i a b l e s .

to

(4-46)

V / T; h e r e t h e

0*

are

T h is e x p r e s s i o n d i f f e r s f r o m (2 -5 4 ) in

t h a t h e r e t h e a m p l i t u d e o f a n o i s e c o m p o n e n t d e p e n d s on i t s f r e q u e n c y .

We no w e x p la in

h o w t h e c a s e o f t h e n o i s e ( 4 - 4 6 ) c a n b e r e d u c e d to t h e c a s e c o n s i d e r e d p r e v i o u s l y . S u p p o s e t h a t t h e r e c e i v e d s i g n a l c a n a g a i n t a k e o n t wo v a l u e s s u p p o s e t h a t to t h e s i g n a l i s a d d e d t h e n o i s e with the f r e q u e n c y . Figure 4-7a.

We u se the r e c e i v e r

In th i s s c h e m e

B

A ^(t)

and

A ^(t), and

W* (t) w i t h t h e i n t e n s i t y 0**(f), w h i c h v a r i e s yZA, V p r e p a r e d a c c o r d i n g to t h e s c h e m e s h o w n i n

R

d e s i g n a t e s a n e q u a l i z e r , i. e. , a l i n e a r d e v i c e w h i c h h a s k k(f) =

(4-47) )

R

V v_

(c) Fig. 4 -7 . A ^g ( t)

and

R q is the i d e a l r e c e i v e r f o r the s i g n a l s A^^(t)

and n o i s e with c o n sta n t in t e n s it y

0“ ; R i s t h e i d e a l r e c e i v e r f o r t h e s i g n a l s o and A^( t) a nd n o i s e w i t h i n t e n s i t y CT*(f); t he f o u r - p o l e w i t h t r a n s f e r c o e f f i c i e n t

A (t) 1 B is

K, where

| K | *= C~ /0~*(f); B " 1 i s t h e f o u r - p o l e w i t h ° _l tr a n s fe r coefficient K

43

i . e. , c o n s t a n t i n t e n s i t y . l e t t h e m h a v e the f o r m s

T h e s i g n a l s a l s o c h a n g e t h e i r f o r m in g o in g t h r o u g h th e e q u a l i z e r ; A 1Q(t)

and

A 2Q(t)

d u c e s an e r r o r if a n d o nly if the r e c e i v e r of e r r o r of the r e c e i v e r

a n d th a t the s p a c e b e t w e e n a d ot a n d a d a s h in o n e n u m e r a l

We s h a l l a s s u m e th a t the p r o b a b i l i t y of t r a n s m i s s i o n is the

s a m e fo r the

various n u m erals. W e d e n o t e t h e s i g n a l c o r r e s p o n d i n g to t h e n u m e r a l 0 b y A^(t), . . .

, to 9 by

A^(t).

A ^ ( t ) , to 1 b y

A ^ ( t ) , to 2 b y

T h e n , a s can e a s i l y be v e r i f i e d , if we s u b t r a c t the v a lu e of the

s i g n a l c o r r e s p o n d i n g to th e n u m e r a l

j

f r o m t h e v a l u e o f t h e s i g n a l c o r r e s p o n d i n g to t h e

n u m e r a l i , and if we a s s u m e th at the in itia l t i m e s of the s ig n a ls c o in c id e and that the f r e q u e n c y of the w a v e f o r m is m u c h g r e a t e r than

T

w h e r e the

1 / *£ , w e o b t a i n

( A . ( t) - A . ( t ) ) 2 ■

a r e giv e n in T a b l e 5 - 1 .

V sj U 2 r o

T h u s , a c c o r d i n g to E q .

(5-55)

(5-51), for this c a s e we

have V •• U % ij o uo

a ..

- V V li J

i f

af

w here a 1 = V 7^ 7 T

Uo / a -

(5-56)

w hence it follows that

P ij and

P„

= 0, a s a l r e a d y p o i n t e d out.

= V(-v/ v ij

a ')

for

j /

i

(5-57)

On t h e b a s i s o f t h i s d a t a w e c a n c o n s t r u c t T a b l e 5 - 2 ,

w here we have w ritten Pn

T h e n , k e e p in g in m i n d t h a t in th is c a s e Eq.

= V (V ^

a ’)

P(A q ) = P(A j)

(5-58)

= ...

= P (A ?)

= 0.1

and applying

(5-5), we obtain 0 .8 P 1 + 0.2 P 2


exp

(6-14) It sh o u ld be m e n t i o n e d th a t in this c a s e o n l y to t h e e x t e n t t h a t t h e q u a n t i t y

^ X(A)

A xm

d e p e n d s on t h e r e c e i v e d w a v e f o r m

d e p e n d s on

X(t).

In t h e s e c a l c u l a t i o n s , w e a s s u m e d f o r s i m p l i c i t y t h a t E q . of

A lying betw een In f a c t , p x a >

- «

and

+ » .

(6-11) is v a lid for all v a lu e s

H o w e v e r, this will not a lw a y s be t r u e ,

m u st always vanish for

(6 -1 3 ) a n d (6-1 4 ) can give a big e r r o r w h e n

A

< -1 A xm

X( t)

and

even fo r s m a l l

A > + l , which m e a n s that E q s. near

i

!•

T h e r e f o r e , the r e s u l t s

o b t a i n e d in t h i s s e c t i o n a n d in s u b s e q u e n t s e c t i o n s b a s e d on t h i s o n e , r e q u i r e a m p l i f i c a t i o n in the c a s e w h e r e

6-4

A 'x m

is n e a r

+ 1. —

E r r o r a n d o p t i m u m n o i s e i m m u n i t y in th e p r e s e n c e o f lo w i n t e n s i t y n o i s e S up p o se that when the w a v e f o r m

ideal, reproduces a p a ra m e te r

A, A+ dA .

X(t )

We now d e t e r m i n e

A s a l r e a d y r e m a r k e d in S e c t i o n 6 - 2 , P ^ Q ) dA

is the

is r e c e i v e d , the t r a n s m i t t e d p a r a m e t e r l i e s in th e i n t e r v a l

T h i s i s a l s o th e p r o b a b i l i t y t h a t t h e v a l u e of th e p a r a m e t e r r e p r o d u c e d b y th e

r e c e i v e r h a s an e r r o r ly i n g in t h e i n t e r v a l c a s e , the m e a n s q u a r e e r r o r

2

5m

a r r i v e s , the r e c e i v e r , w hich is not n e c e s s a r i l y

, w hich is a function of the w a v e f o r m .

the r e s u l t i n g m e a n s q u a r e e r r o r . p r o b a b ility th at if

X (t )

I

£ m

A - Ax; A+ dA

-

Ax*

T h e r e f o r e , in t h i s

is given by the e x p r e s s i o n

( X - A x )2 p x (A> dA

i

A 2 p x (A) dA - 2A x 66

J

dA + * x2

since +

1

j* p x a > d A

= i.

Xv a r i e s w ith the c h o i c e of

As is ev id en t f r o m this f o r m u l a ;

m p a ra b o lic law , and has a m in im u m for som e value r e s p e c t to

A ” ,Xx o »

\

ina c c o r d a n c e w ith a 'x ~ D ifferentiating m 'with

A x ,> a n d s e t t i n g t h i s d e r i v a t i v e e q u a l to z e r o , w e o b t a i n a n e q u a t i o n f o r

of the f o r m +1

|d(< 0 / d V

i

- 1

- - 2f

APx a > < a

■

o

A XO

whence

J

+

\ o o r , w h a t a m o u n t s to t h e s a m e t h i n g , u n d e r the c u r v e

p

X

(A) .

If t h e w a v e f o r m

Axo

We s h a l l c a l l X( t)

=

^

1 A P X(A) d A

(6-15)

i s t h e a b c i s s a o f t h e c e n t e r of m a s s of t h e a r e a

XO

the o p t i m u m v a l u e of the p a r a m e t e r A . "

i s r e c e i v e d , th e n the m i n i m u m v a l u e of t h e m e a n s q u a r e e r r o r ,

w h ic h is o b ta in e d if the r e c e i v e r r e p r o d u c e s the v a lu e

A

xqj

i s given by the e x p r e s s i o n

+1

z m °X m

= 1J ( A - AAx o) z p x (A) d A

It s h o u l d be r e m a r k e d t h a t in the e a s e w h e r e

-^X(A)

'( 6 - i 6 )7

is a s y m m e t r i c c u r v e w ith a sin g le

m a x i m u m , th e n th e a b c i s s a of the c e n t e r of m a s s of the c u r v e o b v i o u s l y c o i n c i d e s w ith the a b c i s s a of th e m a x i m u m , w h ic h m e a n s t h a t in th is c a s e

A x o “ 'A x m

(' 6 - 1 7 )

T h u s , a c c o r d i n g to t h e r e s u l t o f t h e p r e c e d i n g s e c t i o n , w e c a n a s s e r t t h a t w h e n t h e n o i s e i s s u f f i c i e n t l y w e a k , in w h i c h c a s e s y m m e t r i c ) , then square e rro r.

A xo

an(^

U sing E q s .

A xm

P^^)

obeys a G a u s s ia n d istrib u tio n (which is

a r e ecl u a ^

and the id e a l r e c e i v e r gives the l e a s t m e a n

(6-16) an d (6 -1 4 ), we c an find that this e r r o r is

i L

■ --------°~,2

21 T h is is the l e a s t p o s s i b l e e r r o r fo r s u ffic ie n tly s m a l l

.

It is o b t a i n e d w ith the id e a l

r e c e i v e r a n d o b v i o u s l y d e t e r m i n e s the o p t i m u m n o i s e i m m u n i t y in the p r e s e n c e of w e a k

67

noise.

H e r e , a n d in w h a t fo llo w s ,

wg

uncd.0rst3.nd w es , k n o i s e to b e n o i s e t h a t h a s a n

i n t e n s i t y l o w e n o u g h to m a k e t h e c o n s i d e r a t i o n s o f S e c t i o n 6 - 3 v a l i d * Eq.

As is evident f r o m

( 6 - 1 8 ) , t h e o p t i m u m n o i s e i m m u n i t y f o r t r a n s m i s s i o n o f a p a r a m e t e r i s p r o p o r t i o n a l to

th e s p e c i f i c e n e r g y of the w a v e f o r m

A^ ( A x m , t ) , i . e* , o f t h e d e r i v a t i v e o f t h e s i g n a l w i t h

r e s p e c t to t h e t r a n s m i t t e d p a r a m e t e r . Using Eq. intensities.

(6 -1 6 ), we can a ls o d e t e r m i n e the m e a n s q u a r e e r r o r for l a r g e n o i s e

H o w e v e r , i t i s d i f f i c u l t to u s e t h i s e r r o r to e v a l u a t e t h e o p t i m u m n o i s e

im m unity.

The point is that for la r g e

d e p e n d on the r e c e i v e d s i g n a l a l s o d e p e n d s on

X(t).

6” , th e c h a r a c t e r of t h e f u n c t i o n

X(t), and t h e r e f o r e the q u a n tity

b e g i n s to

given by E q.

(6-16)

In t h i s c a s e , i n o r d e r to e v a l u a t e t h e n o i s e i m m u n i t y , w e m u s t a l s o

e v a l u a t e the p r o b a b i l i t y of the v a r i o u s v a l u e s of difficulties*

6 mm

^^(A)

X ( t ) , w h i c h l e a d s to a s e r i e s o f m a t h e m a t i c a l

In C h a p t e r 8 w e s h a l l r e t u r n to t h e p r o b l e m o f t h e e v a l u a t i o n o f t h e o p t i m u m

n o is e i m m u n i t y when the n o is e i n t e n s i t y is l a r g e . We now find the p r o b a b i l i t y t h a t , in th e p r e s e n c e of w e a k n o i s e , th e i d e a l r e c e i v e r r e p r o d u c e s the v a lu e of the t r a n s m i t t e d p a r a m e t e r w ith an e r r o r e x c e e d in g value*

e in a b s o l u t e

O b v i o u s l y , t h i s p r o b a b i l i t y i s e q u a l to Axm~ €

P(tfl > e

Using Eq.

+1 P x 0 ) dX

) =

+

P X (A) dA

( 6 - 1 4 ) , a n d k e e p i n g in m i n d th e n o t a t i o n u s e d in E q . ( 2 - 4 7 ) , w e o b t a i n

2T A ^ ( >

P ( l £ I > e)

2V

xrrV t)

2 V (-

(6-19) 1m m

6-5

S e c o n d m e t h o d of d e t e r m i n i n g th e e r r o r a n d o p t i m u m n o i s e i m m u n i t y in t h e p r e s e n c e of low i n t e n s i t y n o i s e T h e r e i s a s e c o n d m e t h o d of f in d in g the s i z e of th e e r r o r f o r the c a s e of t r a n s m i s s i o n

of a p a r a m e t e r in the p r e s e n c e of low i n t e n s i t y noise*

Although this m e th o d gives a r e s u lt

w hich c o in c id e s with th a t a l r e a d y o b ta in e d , w e s h a ll e x a m i n e it a n y w a y , s in c e th is m e th o d is i n t e r e s t i n g in i t s own r i g h t , a n d s i n c e w e s h a l l u s e i t l a t e r , a l b e i t in a m o r e c o m p l i c a t e d form* noise

As b e f o r e , le t the sig n al W

(t)

A(^\,t)

rep resen t som e transm itted p a ra m e te r

X*

The

m a y o r m a y n o t b e a d d e d to t h e s i g n a l , w i t h t h e r e s u l t t h a t a w a v e f o r m X(t )

a c t s upon the r e c e i v e r ; th is w a v e f o r m is in the p r e s e n c e o f n o i s e .

A (X ,t)

if t h e r e i s no n o i s e , a n d

A (A , t)+W y t )

We r e p r e s e n t the w a v e f o r m by

X|t> ■ kT *= 1, \ c k"> 68

(6 - 2 0 )

w h e r e the

C k (t)

by the v a lu e s

a r e given o r t h o n o r m a l fu n c tio n s.

x ^, . . . , x^.

X(t)

D e p e n d i n g on the r e c e i v e d w a v e f o r m

d u c e s s o m e v a lu e of the p a r a m e t e r tra n s m itte d value.

Then

is c o m p l e t e l y c h a r a c t e r i z e d X (t), the r e c e i v e r r e p r o ­

^ , a v a l u e w h ic h m a y o r m a y n o t c o i n c i d e with the

W e a s s u m e t h a t to e a c h w a v e f o r m

X(t )

a c t i n g upon the r e c e i v e r

c o r r e s p o n d s a s p e c i f i e d v a l u e of the p a r a m e t e r , w h ic h is r e p r o d u c e d by the r e c e i v e r . C l e a r l y , fo r e v e ry r e c e i v e r the r e p r o d u c e d p a r a m e t e r equals s o m e function

A

=F(xp

...

, xn)

(6-21)

w hich c h a r a c t e r i z e s its o p e ra tio n . S u p p o se the r e c e i v e d w a v e f o r m r e c e i v e s an i n c r e m e n t

dX(t)

=

n y dx k = 1

C (t)

(6-Z2)

O b v i o u s l y , in th is c a s e th e p a r a m e t e r v a l u e r e p r o d u c e d by the r e c e i v e r a l s o r e c e i v e s an i n c r e m e n t , e q u a l to

■

t,

K= 1

-# ■ % K

(6-23)

-rrtrd x m

w here we have designated dF

L(t)

a s f o llo w s f r o m (Z-Z2).

(6-24)

c k (t)

S uppose that the t r a n s m i t t e d p a r a m e t e r is changed by

dX

and

s u p p o s e t h a t n o n o i s e i s a d d e d to t h e s i g n a l ; t h e n t h e w a v e f o r m a r r i v i n g a t t h e r e c e i v e r c h a n g e s by an a m o u n t dX(t) = A ^ ( A , t ) d X

(6-25)

A ^A ,t) = -

-

J

Ai

=

jt

(1 - c o s 2 , ( t. - .; ) * ] d* 0

' SI

-fla

sinil(t»

2 n

- t!.) *]

° A s i K - ‘V 1

i

(7-28)

S u b s t i t u t i n g th is v a l u e in ( 7 - 2 7 ) , w e o b ta in

6

l

'S I

2Cr-

2n

sinJK t’ - t ' ) o 1

1

-n-i*; - 'I '

J

dUm (t)’ — 5 t~

--

(7-29)

t = t* o

As w e s e e ; in th is c a s e a l s o th e e r r o r is a r a n d o m v a r i a b l e w h ic h o b e y s a G a u s s i a n law . It fo llo w s f r o m the f o r m u l a j u s t o b t a i n e d t h a t the m e a n s q u a r e e r r o r f o r the given m e a n s o f r e c e p t i o n i s e q u a l to T

ssi inn/ L i L( t( 't 1-- tt*l ) ) *1

/ L1' A(t;°- »;> J

2 1 m

(7-30)

T2 [ J V 1 1 2 »* .

at

J ,_

,,

( w hich by the s y m m e t r y of

U ^jt)

equals

( 7 - 3 0 ) to f i n d t h e q u a n t i t y

A>J the p ro b a b ility that when 'a is t r a n s m i t t e d , the r e c e i v e r , a s a r e s u l t of the a d d itio n of n o is e

we d e n o te the p r o b a b i l i t y th a t w hen t h e p a r a m e t e r v a l u e

is t r a n s m i t t e d , the r e c e i v e r , a a s a r e s u l t o f t h e a d d i t i o n o f n o i s e to t h e s i g n a l , r e p r o d u c e s a p a r a m e t e r v a l u e s a t i s f y i n g the c o n d itio n

A


\

+

c) +

We s h a l l a s s u m e th a t the t r a n s m i t t e d p a r a m e t e r + 1, w i t h e q u a l p r o b a b i l i t y .

Ai

T h e n the p r o b a b i l i t y th a t

A 21 an d that a t the s a m e tim e

(A < A

px

A

1*1 > €

[px

u

j

j

-

e)

c a n t a k e on a n y v a l u e in the r a n g e Aj

-1,

s a t i s f i e s the i n e q u a l i t y

< A 2 + dA2

y i s e q u a l to

> \ + c) +

a

dx? < a 2 - e» - j i

H e n c e , the p r o b a b i l i t y th a t the e r r o r e x c e e d s € in a b s o l u t e v a l u e , w h e n a p a r a m e t e r v a l u e

'Xl

(not know n in a d v a n c e ) i s t r a n s m i t t e d , e q u a l s

J -1

ip x a > A

2 + e> + p

c

a


e) -

j

p

- 11 +, e

2

+1 - *

_ea

*

> A C)

A°

2

1( -U

dA

(X < x

p

]

^ o +e

)— 2 °

2

J v < a>A,+v * a < x “"

is we a s s u m e that the a p r i o r i p r o b a b ility of tr a n s m i t t i n g e ith e r sig n a l is the s a m e .

How­

e v e r , t h i s p r o b a b i l i t y o f e r r o r c a n n o t b e l e s s t h a n t h e p r o b a b i l i t y of e r r o r ( g i v e n b y E q . (4 -8 )) w h i c h d e t e r m i n e s the o p t i m u m n o i s e i m m u n i t y f o r the s i g n a l s in q u e s t i o n , i. e.

\ w here

V(a)

V + P ». ♦

(2-47);

[A(V 6 >l) - A(V

is d e f i n e d b y E q . (4-4) a n d in th is c a s e e q u a ls

e't)]2

[A (* 0 + e,t) - A(Xq -

e , t ) ] 2 dt

( 8 - 1)

F r o m this we o b ta in the u n i v e r s a l f o r m u l a 1-e

p(i€)> j - d - e )

91

v(ai)dA0

(8- 2)

f o r c a l c u l a t i n g the p r o b a b i l i t y of e r r o r s g r e a t e r than d e p e n d on

€ .

In m a n y c a s e s

doesnot

If t h i s i s t h e c a s e , t h e q u a n t i t y u n d e r t h e i n t e g r a l i s c o n s t a n t , a n d w e

obtain P(|«f| > €

) > 2 (l-6 )V(Oj)

(8-3)

It follow s f r o m th e s e e q u a tio n s th at the s m a l l e r the d i s t a n c e

V

| A C \ 0 + 6 , t ) - A ( X o - 6 , t ) I2

b e t w e e n the p o i n t s of the s i g n a l c u r v e c o r r e s p o n d i n g to p a r a m e t e r v a l u e s w h ic h a r e s e p a r a t e d f r o m o n e a n o t h e r by the a m o u n t exceeding

8-2

2 c , the l a r g e r the p r o b a b i l i t y of o b ta in in g an e r r o r

c.

C o m p a r i s o n of the f o r m u l a s f o r w e a k a n d s t r o n g n o i s e We now c o m p a r e the r e s u l t o b t a i n e d in th e p r e c e d i n g s e c t i o n w ith th e r e s u l t o b t a i n e d

in C h a p t e r 6 f o r t h e c a s e o f w e a k n o i s e ; t h e r e w e d e r i v e d E q . a b ility that the e r r o r £ noise.

i s g r e a t e r th an

(6 -19), w hich gives the p r o b ­

c , f o r th e i d e a l r e c e i v e r in the p r e s e n c e of w e a k

T his f o r m u l a is valid fo r a given m o s t p r o b a b le v a lu e

^

a r e e q u a lly p r o b a b l e , then when ^ xm xm in g e x p r e s s i o n f o r the p r o b a b i l i t y in q u e s t i o n :

P(\S\ > € )

X

. If w e a s s u m e t h a t a l l xm is n o t know n in a d v a n c e , w e o b ta in th e f o l l o w -

J* V ( a )

dX

(8-4)

w here

v

a

2T A ' {\ ,t) ______ X xm; '

(8-5)

L e t us c o m p a r e th is r e s u l t w ith th e r e s u l t g iv e n b y E q . s u ita b le both f o r s tro n g and w eak n o is e .

F o r sm all

A U o + € ,t) - A (X Q S u b s t i t u t i n g th i s v a l u e in E q .

€,t)

(8-2), which is u n iv e r s a l and is

€ , we can take

= A £ \ 0 ,t) 2 €

(8-1), we obtain

V

a l■

2TAx < V ‘>

92

(8- 6 )

T h i s q u a n t i t y i s to b e s u b s t i t u t e d i n E q . of e r r o r . Eqs.

(8 -2 ), w hich gives a lo w e r bound for the p ro b a b ility

F r o m th e se f o r m u l a s , we see that

a

= a ^ , w h ic h m e a n s th a t the r i g h t s i d e s of

( 8 - 2 ) a n d ( 8 - 4 ) d i f f e r o n l y b y t h e i r l i m i t s o f i n t e g r a t i o n , a d i f f e r e n c e w h i c h g o e s to

z e ro as

€ ->0.

It f o l l o w s f r o m t h e s e e x p r e s s i o n s t h a t i f t h e i n e q u a l i t y ( 8 - 2 ) i s c h a n g e d to

an e q u a l i t y , the n it g iv e s the v a l u e of th e p r o b a b i l i t y of the s m a l l e r r o r s p r o d u c e d by the i d e a l r e c e i v e r in th e p r e s e n c e of w e a k n o i s e .

8-3

P u ls e tim e m odulation F o r a m p l i t u d e a n d o t h e r l i n e a r m o d u l a t i o n , the f o r m u l a s o b t a i n e d in C h a p t e r 7 a r e

v a l i d f o r n o i s e of a r b i t r a r y i n t e n s i t y , a n d t h e r e f o r e t h e r e i s no p o i n t in i n v e s t i g a t i n g t h e s e k i n d s o f m o d u l a t i o n u s i n g t h e m e t h o d s of S e c t i o n 8 - 1 . of p u l s e t i m e m o d u la tio n .

T h e s i t u a t i o n i s d i f f e r e n t in the c a s e

F o r t h i s k i n d o f m o d u l a t i o n , a c c o r d i n g to E q s .

(2 -2 6 ), (7-9) and

( 8 - 1), w e o b t a i n

t -

sin_A_

2

I1

(X o - « >m

£

1-------- 1

1

K°

A[

’

e)

a , b 1, c , d ! , e .

w hich c h a r a c t e r i z e s the id e a l r e c e i v e r m u s t lie above

In t h e c a s e o f w e a k n o i s e , f o r s m a l l v a l u e s o f t h e q u a n t i t y

€ and for id eal re c e p tio n , we can d e te r m in e

e), using E q s.

T h i s q u a n t i t y i s r e p r e s e n t e d i n F i g u r e 8 - 1 by t h e c u r v e a t i o n of t h e f i g u r e , t h a t f o r J L 'Z Qe < 2 . 7 , t h e c u r v e s close together. w ith the c u r v e

H ow ever, for a"

T h u s , the

c >

a n.

(6-19) and (8-5).

It i s a p p a r e n t f r o m an e x a m i n ­

a , b 1, c , d ' , e

and

a"

a r e quite

2 . 7 , we obtain a d r a s t i c d iv e r g e n c e b etw een th e m ,

going below the c u r v e

a , b * , c , d*, e , w h i c h i s i m p o s s i b l e , a s r e m a r k e d .

I t f o l l o w s f r o m t h i s t h a t f o r - f l T e > 2. 7 . t h e f o r m u l a f o r w e a k n o i s e a n d s m a l l e r r o r s i s o ’ co m p letely inapplicable. We now c la r if y th e se r e s u l t s .

F o r the give n m e a n s of c o m m u n i c a t i o n and f o r the

m e t h o d s of r e c e p t i o n d e s c r i b e d in S e c t i o n s 7 - 5 a n d 7 - 6 , s m a l l e r r o r s a r e c a u s e d by w e a k n o i s e , w h ic h p r o d u c e s a d i s p l a c e m e n t of the s i d e s of the p u l s e . of e r r o r f a l l s off s h a r p l y a s the e r r o r is i n c r e a s e d . no ise w a v e f o r m e x c e e d s the th r e s h o ld v o lta g e a l m o s t equal p r o b a b ility at any tim e .

U .

T he p r o b a b ility of this type

L a r g e e r r o r s a r e o b t a i n e d w hen the It i s c l e a r th a t th is c a n h a p p e n w ith

T h e r e f o r e , the p r o b a b i l i t y of l a r g e e r r o r s d o e s n o t

f a l l off m u c h w h e n the e r r o r is i n c r e a s e d .

T h i s p r o p e r t y , w h i c h i s e a s y to e x p l a i n f o r t h e

m e t h o d of r e c e p t i o n in q u e s t i o n , i s (as sh o w n by F i g u r e 8-1) a n e c e s s a r y f e a t u r e of the g i v e n m e a n s of c o m m u n i c a t i o n , r e g a r d l e s s of w h i c h m e a n s o f d e t e c t i o n w e u s e .

The la rg e

e r r o r s , f o r w h ic h th e f o r m u l a s d e r i v e d in C h a p t e r 6 f o r w e a k n o i s e a r e n o t v a l i d , w ill b e called anom alous.

As we s e e f r o m F i g u r e 8-1, a n o m a lo u s e r r o r s m u s t begin at l e a s t f r o m

o n . F o r e x a m p l e , i t i s c l e a r f r o m t h e f i g u r e t h a t f o r 0.1 CT = 2 , ° _2 the p r o b a b i l i t y t h a t an a n o m a l o u s e r r o r o c c u r s , m u s t be g r e a t e r than 6 x 1 0 . This m e a n s

the v a lu e

€ = 2.7

t h a t in m o r e tha n 6 p e r c e n t of the c a s e s , on the a v e r a g e , a n o m a l o u s e r r o r s o c c u r f o r th e g iv e n v a l u e of

Q Iqt.

In g e n e r a l , t h e p r o b a b i l i t y o f o c c u r r e n c e o f a n o m a l o u s e r r o r s c a n b e

found using the fa c t that they begin when

e > 2.7.

T h u s , a c c o r d i n g to ( 8 - 8 ) , t h e s e

e r r o r s begin for

a

2

Q —

yi s i n 2. 7 N . ( 1 --------- z 7 T ) ~

Q

w h ic h m e a n s th a t th e ir p r o b a b ility is P( 2V( —) ~ CT

(8-9)

F o r low i n t e n s i t y n o i s e , the p r o b a b i l i t y of a n o m a l o u s e r r o r s is v e r y s m a l l , so th a t th e y n e e d n o t be c o n s id e r e d and the w eak n o ise th e o ry can be applied.

95

8-4

F re q u e n c y modulation W e n o w a p p l y t h e r e s u l t s o b t a i n e d i n t h i s c h a p t e r to t h e c a s e o f f r e q u e n c y m o d u l a t i o n ,

c o n s i d e r e d in S e c tio n 7 - 8 .

We h a v e a s ig n a l given by E q.

(7-37).

A p p l y i n g E q . ( 8 - 1 ) to

this s ig n a l, and taking (2-26) into a c c o u n t , we obtain

- -—

TU

2

| c o s [ (a>Q+ XL

a l

■ ■ ■— ' ■ —

+XL c)t + 0]

i

"

j

X Q - XL e )t + 0 ] 3

- c o s [(c^o +

2 -

D j(t)

o |(t) since,

a s w e h a v e e x p l a i n e d , D ^( t )

obviously a m in im u m for

B^(t)

(9-17)

+ B-2 (t)

and

B^(t)

m u s t be orth o g o n al*

This e x p re ssio n is

= 0, w h e n c e it f o llo w s th a t f o r the i d e a l r e c e i v e r

D g U) (9-18)

Li(t) w here

9-4

Dj>(t)

^(7)

is defined by E q. (9-10).

M e a n s of r e a liz in g the ideal r e c e i v e r We now show th a t the r e c e i v e r w h i c h , w h e n the w a v e f o r m

X(t)

is r e c e iv e d , r e p ro d u c e s

the v a l u e of the f u n c tio n w h ic h m i n i m i z e s the e x p r e s s i o n

R =

(X(t) - A F ( t ) ] Z

is i d e a l in the s e n s e f o r m u l a t e d in the p r e c e d i n g s e c t i o n . form

F Q(t)

(9-19)

In f a c t , w h e n a m o d u l a t i n g w a v e ­

i s t r a n s m i t t e d in the a b s e n c e of n o i s e , we o b v i o u s l y h a v e

X(t)

=

(t)

a f

o and Eq.

(9-19) h a s its l e a s t p o s s i b l e v a lu e of z e r o f o r the c a s e w h e r e

A

(t)

and

A

(t) o

104

c o in c id e , and the w a v e f o r m

F(t)

r e p r o d u c e d by the r e c e i v e r is

in q u e s t i o n d o e s n ot i n t r o d u c e e r r o r s in the a b s e n c e of n o i s e . function of the p a r a m e t e r s .

We s t i p u l a t e d th a t the w a v e f o r m

r e c e i v e r i s to g i v e t h e m i n i m u m v a l u e o f t h e e x p r e s s i o n d e r i v a t i v e s of

R

w i t h r e s p e c t to A n

m u s t vanish.

SR

F^(t).

T h u s , the r e c e i v e r

F(t), and th e re fo re F(t)

R .

R , is a

r e p r o d u c e d by t h e

T h e r e f o r e , the p a r t i a l

We o b t a i n t h e c o n d i t i o n

- 2 [ X(t) - A p (t)

D^(t)]

(9-

a 0

20)

w here 5A

(t)

Dx (t)

If th e r e c e i v e d w a v e f o r m r e c e i v e s a s m a l l i n c r e m e n t and

AX(t), t h e n , o b v i o u s l y ,

A ^ m u s t a l s o r e c e i v e i n c r e m e n t s if the e x p r e s s i o n f o r

S u p p o s e the p a r a m e t e r s

Ajj

r e c e i v e the i n c r e m e n t s

R

A ^ ( t ) , F(t)

i s a g a i n to b e a m i n i m u m .

a A ^ >then

A^(t)

r e c e i v e s the

instrum ent

A A F (t)

=

£

Dg(0

(9-21)

w hich m e a n s that we have R

[ X( t) +

T h e v a lu e s of the i n c r e m e n t s value.

AX(t) - A F (t) -

2

^e(t)

A ^ m u s t be s u c h th a t the e x p r e s s i o n

T h e r e f o r e the p a r t i a l d e r i v a t i v e s o f

dR

£

[ X( t) +

M o r e o v e r , tak in g into a c c o u n t E q.

R

w i t h r e s p e c t to

a A*

(9-22)

R

again has a m in im u m

m u s t v a n i s h , so tha t

D ^ ( t ) A ^ ^ ] D^(t)

AX(t) - A F (t) -

(9-20) a nd the f a c t th a t the

Dj>(t)

0

with d if f e r e n ti in d ic e s

a r e o rth o g o n al, we obtain

AX( t ) D g ( t ) - D j ( t )

= 0

whence A X ( t ) Djj (t) (9-23) o |(t)

105

The sm a lle r

A X ( t)

and

A ^ ,

the m o r e e x a c t E q .

(9-21) is .

to z e r o , w e a r r i v e at th e c o n d i t i o n c h a r a c t e r i z e d b y E q s . r e c e i v e r w hich r e p r o d u c e s the w a v e f o r m

F(t)

L e t t i n g t h e s e q u a n t i t i e s go

(9-8) and (9-18).

m inim izing Eq.

T h e r e f o r e , the

( 9 - 1 9 ) h a s no e r r o r in the

a b s e n c e of n o i s e , a nd g iv e s the m i n i m u m p o s s i b l e e r r o r in the p r e s e n c e of w e a k n o i s e . T h u s , t h i s r e c e i v e r i s i d e a l i n t h e s e n s e e s t a b l i s h e d in S e c t i o n 9 - 3 .

9-5

The e r r o r for ideal reception We now d e t e r m i n e the a m o u n t of e r r o r given by the id e a l r e c e i v e r w hen w e a k

f lu c tu a tio n n o i s e is a d d e d to th e s i g n a l.

Suppose a w av efo rm

(9-24) F ° (t) was tran sm itted .

■ i ? i .

v )

T h e n , in the a b s e n c e of n o i s e , the r e c e i v e d w a v e f o r m is

X(t) =

(t), o

and the id e a l r e c e i v e r r e p r o d u c e s the w a v e f o r m When the w e a k n o i s e an a m o u n t

d X (t )

W

= W

(t)

F^(t)

d e t e r m i n e d by the p a r a m e t e r s

i s a d d e d to t h e s i g n a l , t h e r e c e i v e d w a v e f o r m i s c h a n g e d b y

(t), a n d , a c c o r d i n g to E q s .

An , &

(9-8) and (9 -1 8 ), th e p a r a m e t e r s

w hich c h a r a c t e r i z e the w a v e f o r m r e p r o d u c e d by the id e a l r e c e i v e r , r e c e i v e i n c r e m e n t s

Ufc(t) d X ( t ) (9-25) ° X (t) It s h o u l d b e n o t e d t h a t t h e r a n d o m v a r i a b l e s sin ce the

D ^( t )

0^

V 2T

D^(t)

with d if f e r e n t in d ic e s a r e i n d e p e n d e n t,

with d if f e r e n t in d ic e s a r e o rth o g o n a l.

T h u s , the w a v e f o r m r e p r o d u c e d by

the id e a l r e c e i v e r h a s the f o r m

m

4 ■ ih

A

f I. I,(t)

V ‘> ■ 1

1

Y2T

D 2 (t)

= F Q(t) + W* ( t)

(9-

w here

W* ( t ) =

i-i. v V

(0 Z i - 1 s i n T

D ‘ .(t)

106

it + 0 ^ cos

it)

26 )

C o m p a r in g this e x p r e s s i o n with Eq.

( D - 3 ) , w e s e e t h a t d u e to t h e a c t i o n o f t h e n o i s e w h i c h i s

a d d e d to t h e s i g n a l t h e r e c e i v e r a d d s to t h e m o d u l a t i n g w a v e f o r m n o i s e w ith an i n t e n s i t y a t f r e q u e n c y

< r * (T >

i/T

^

n o rm a l fluctuation

e q u a l to

a-

,

F Q(t)

-------------

V

(9-27)

D2 iW

w here 5A

(t)

V c) We s h a l l h e n c e f o r t h c a l l th is n o r m a l f l u c t u a t i o n n o i s e th e n o i s e at the r e c e i v e r o u t p u t .

This

i n t e n s i t y of the n o i s e a t th e r e c e i v e r o u tp u t is the m i n i m u m p o s s i b l e a n d c h a r a c t e r i z e s the o p tim u m n o is e im m u n ity for a given m o d u la tio n s y s te m . n o t d e p e n d on

9-6

i, we o m it this index and w r ite

In t h e c a s e w h e r e

g-*(i/T)

does

or*.

B r ie f s u m m a r y of C h a p te r 9 We c a l l i d e a l the r e c e i v e r w h ic h e x a c t l y r e p r o d u c e s the m o d u l a t i n g w a v e f o r m in the

a b s e n c e o f n o i s e , a n d g i v e s t h e b e s t a p p r o x i m a t i o n to t h e m o d u l a t i n g w a v e f o r m i n t h e p r e s e n c e of w e a k n o i s e . the quantity

R

T he id e a l r e c e i v e r r e p r o d u c e s the w a v e f o r m

given by (9-19).

F(t)

which m in im iz e s

When r e c e p t i o n is w ith the id e a l r e c e i v e r and the n o i s e is

w e a k , th e r e p r o d u c e d w a v e f o r m d i f f e r s f r o m the m o d u l a t i n g w a v e f o r m by the f l u c tu a tio n n o is e with in te n s ity given b y E q . the functions

D ^ (t )

that

=

D ^ U )

=

(9-27).

S A ^ ( t a r e

In d r a w i n g t h e s e c o n c l u s i o n s , i t w a s a s s u m e d t h a t o r t h o g o n a l f o r a n y p a i r of d i f f e r e n t i n d i c e s , a n d

D^.(t).

107

C H A P T E R 10 DIRECT MODULATION SYSTEMS

10-1

Definition By d i r e c t m o d u l a t i o n s y s t e m s w e s h a l l u n d e r s t a n d s y s t e m s i n w h i c h t h e m o d u l a t i n g

w aveform (m essage) m itted signal.

F( t )

e n t e r s d i r e c t l y as a p a r a m e t e r into the e x p r e s s i o n f o r the t r a n s ­

In t h i s c a s e , w e c a n w r i t e t h e g e n e r a l f o r m o f t h e s i g n a l a s A F (t) = A

( 1 0 - 1)

[F(t), t ]

E x a m p l e s of d i r e c t m o d u l a t i o n s y s t e m s a r e a m p l i t u d e m o d u l a t i o n , w h e r e the s i g n a l c a n be written as A F (t) = UQ [ 1 + M F ( t ) ]

c o s ( u ) o t + 0q )

p h a s e m o d u la tio n , w h e r e the sig n a l can be w r itte n as

A

etc.

t

(t)

= U

o

c o s [co t + m F ( t ) + 0 ]

o

o

F r e q u e n c y m o d u la tio n , w h e r e the t r a n s m i t t e d signal is w r itte n as

d o e s n o t b e l o n g to t h e d i r e c t s y s t e m s i n t h e s e n s e o f t h e t e r m i n o l o g y o f t h i s b o o k . modulating w aveform in teg ral m odulation.

F(t)

Since the

a p p e a r s b e h i n d th e i n t e g r a l , w e s h a l l c a l l t h i s k i n d of m o d u l a t i o n

Single sid e b a n d t r a n s m i s s i o n is a lso not a d i r e c t s y s t e m ,

s i n c e in t h i s

c a s e a l s o the s i g n a l c a n n o t b e e x p r e s s e d a n a l y t i c a l l y in t e r m s of th e m o d u l a t i n g w a v e f o r m F(t).

In C h a p t e r 11 w e s h a l l s t u d y p u l s e m o d u l a t i o n s y s t e m s , w h i c h a r e a l s o n o t c l a s s i f i e d

as d ire c t sy ste m s.

10-2

D e r i v a t i o n of b a s i c f o r m u l a s Since b y h y p o th e s is the m o d u la tin g w a v e f o r m

F(t)

a d ir e c t m odulation s y s te m we can w r ite the signal as

108

can be e x p r e s s e d by Eq.

(9-1), for

whence »A_(t)

V We a ls o a s s u m e that the function f r e q u e n c ie s g r e a t e r than

dA

'

*

[dA p(t)/ dF ]

s ir - W

2

c o n t a i n s o n ly s i n u s o i d a l c o m p o n e n t s with

. i ^ / T , i. e. , g r e a t e r th a n t w i c e t h e m a x i m u m f r e q u e n c y of th e

s i n u s o i d a l c o m p o n e n t s of t h e m o d u l a t i n g w a v e f o r m T h e n , a c c o r d i n g to E q .

(t)

F(t); th is c o n d itio n is u s u a l l y s a t i s f i e d .

(Z-Z6), w e o b ta in Dj(t)

= [d A p ( t ) / 9 F ] 2 lj(t)

= [dA F ( t ) / a F ] 2

---------------------- 2-----------------. . ^ ( D D k (t) * [ d A F ( t ) / a F ] Z I ^ ( t ) Ik (t) = 0

(10-2)

I t f o l l o w s f r o m t h e s e e q u a t i o n s t h a t t h e c o n d i t i o n s ( 9 - 1 5 ) w h i c h w e r e i m p o s e d on t h e a r e s a t i s f i e d in this c a s e , a n d w e can u s e E q s .

(9-26) and (9-27).

D ^t)

It i s a c o n s e q u e n c e o f

t h e s e e q u a t i o n s t h a t , f o r th e k i n d of m o d u l a t i o n s y s t e m in q u e s t i o n , w e h a v e a t th e o u t p u t of the id e a l r e c e i v e r not only the m o d u latin g w a v e fo rm a d d e d to i t .

F (t), but a lso n o r m a l fluctuation n o ise

T h i s n o i s e h a s a u n i f o r m s p e c t r u m a n d an i n t e n s i t y w h i c h , a c c o r d i n g to E q .

( 1 0 - Z ) , i s e q u a l to cr

Q/ 2 ir , s o t h a t t h e r e s t r i c t i o n i m p o s e d

[dA F ( t ) / d F ] 2

=

-

U2 M 2

so t h a t, a s a r e s u l t of n o i s e , a t the o u t p u t of th e i d e a l r e c e i v e r w e h a v e n o r m a l f l u c t u a t i o n n o ise with in te n sity

e

k

- kX)

is a n o r m a l r a n d o m v a r i a b l e , and

w e c a n d e t e r m i n e t h e q u a n t i t y

j u s t a s in S e c tio n 13-3 we d e t e r m i n e d the q u a n t i t y

V (a),

u s i n g c u r v e 1.

f i g u r e , t h e t wo c u r v e s a r e c l o s e t o g e t h e r o n l y w h e n t h e m o d u l a t i o n i n d e x where

m > 2, the v a lu e of

g i v e n b y c u r v e 2.

V( 2,

it i s m a s k e d a t the r e c e i v e r o u tp u t only w hen the n o i s e w a v e f o r m is so l a r g e a t the t i m e w he n F^(t)

is tr a n s m i t t e d that Eq.

(13-13) an d the w e a k n o is e t h e o r y a r e not v a lid .

134

APPENDICES

A p p e n d ix A.

T h e s p e c i f i c e n e r g y of h i g h - f r e q u e n c y w a v e f o r m s

A s is w e l l - k n o w n , a h i g h - f r e q u e n c y s i g n a l c a n be r e p r e s e n t e d q u ite g e n e r a l l y a s

A( t) =

U m (t) c o s [ cDQt + 0 ( t ) ]

( A - 1)

T h e s p e c if ic e n e r g y of this s ig n a l is

Q2

=

T

U ^ ( t) c o s 2 \a)ot +

A 2 (t) = T

0(t) ]

CaJ q i s s o l a r g e t h a t t h e f r e q u e n c i e s w h i c h

Now if we a s s u m e , a s is u s u a lly the c a s e , that

e f f e c t i v e l y m a t t e r i n t h e e x p r e s s i o n c o s ( 2 Oj t + 20 ( t ) a r e all h i g h e r th an th e f r e q u e n Z ° c i e s c o n t a i n e d in t h e f u n c t i o n U ^ ( t ) , a n d t h a t t h e c o n s t a n t c o m p o n e n t of c o s [ ZuJ^t + 20(t)] c a n b e s e t e q u a l to z e r o f o r t h e s a m e r e a s o n , t h e n b y E q . ______

Q2

-

T

(2 -2 6 ), we obtain +T/ 2

A 2 (t)

= -T

U ^(t)

J

= -

U ^ ( t ) dt

(A-2)

-T/2

A p p e n d i x B.

R e p r e s e n t a t i o n o f n o r m a l f l u c t u a t i o n n o i s e b y t wo a m p l i t u d e - m o d u l a t e d w a v e s

We c o n s id e r n o r m a l fluctuation n o is e with fre q u e n c ie s f r o m

^

/T

to

V /T

and con­

stan t in te n s ity , and w rite

„

v

k Let

jb

and

w

n

be i n t e g e r s .

- fr

-r r

i = -n L

■

.2

(B -l)

z

T hen the w a v e f o r m (2-54) c an be w r itte n as

2 [ 02jjo + 2 i - l

. (t ) =

/*■>*

n

J

”^ e2/

0+2i+1

2k

+ 6 2j& +2i o L

COS • y

SU1

T

+ l ) t + 9 2 £ + 2i COS o

£ Sln "“T" lt COS T .

lt

2k

C O S — ^r-

135

j ^ Qt - s i n

T

(A>+l)t]J

^ot + cos T " lt Sln ~T~ *o‘]

it sin

^

(B-2)

Setting

w

( 2 r t / T ) JlQ =

(t) = -----

£

VT*

i = -n

+ —~ -/T"

and fa c to rin g out cos

(02jfc

+2i- 1 sin O

c ^ t

it + 9 ^

+ 2i C° s i f O

L e 2 A + 2 i - 1 COS “ T " U ‘ e 2 i i = -n

+2i s i n

F i n a l l y , a d d in g t e r m s with the s a m e a b s o l u t e v a l u e of

W

.(t)

/I,'*)

-fr~

(02 i

i = l

(02 1 n

o

o

o

-2i> c o s ^ f

U) Sln

( B-3)

o1

T U

c o s m) t o

it

r

+ ^0 2 j i - 2 i " 0 2 i ' +2i^ Sln o o H e r e we h a v e n e g l e c t e d the t e r m s w ith T

T

+ 2 i - 1 " 0 2£, - 2i - 1^ s t n o

+2 i + 0 2 ^

U) C° S a V

i, we obtain

(9 2j^ + 2 i - 1 + 9 2 £ - 2i - 0 o

e n o u g h , s i n c e if

cJqt, we o b ta in

and sin

i

c o s ^ f it

T

(B-4)

s i n u) t o

U

= 0 , w hich is p e r m i s s i b l e if we take

is i n c r e a s e d w h ile the f r e q u e n c i e s

JLU / T

and

"O / T

T

large

a r e kept the

s a m e , the n u m b e r of t e r m s w ill i n c r e a s e , w h ile e a c h t e r m b e c o m e s a r b i t r a r i l y s m a l l , now in tr o d u c e the no tatio n

6 2.& + 2 i - 1 " 0 2 i - 2 i - 1 ~ ^ o o

02 i - l

0 2 JZ/ + 2 i - 1 + 0 2j& - 2 i - 1 = ^ o o

0 2i ( B-5)

0 2 £ +2i + 9 lSb - 2 i “ O o

e 2, i., Q- 2 i - e_2 i-o . +2i ,.=

w h e r e , a c c o r d i n g to S e c t i o n 2 - 5 , t h e

0 2i ~fz

^

0' 2 i-1

© ^ i - F 0 2 i - l > 0 2 i , 0 2i

are

(mutually) i n d e p e n d e n t

r a n d o m v a r i a b l e s . S u b s t i t u t i n g t h e s e q u a n t i t i e s in ( B - 4 ) , w e o b t a i n

136

We

^

^

(t)

" W^ o ' n ^ o +n (t ) =

^

W 'l,n (t)sin

cOQt +

-n

W ' ^ n (t) c c S ( J o t

(B-6)

w here 2* u + " z i cos t

. E i «e 2 i - i s i n

;= r

w y n (t> = — —

2 rt . n 2 rt . . s m -Tjr- i t + 0!>\ c o s —tj- i t )

.Z , (e2 i-l i = 1

-fr~

u)

n < > '- > > a r e i n d e p e n d e n t n o r m a l f l u c t u a t i o n p r o c e s s e s w i t h f r e q u e n c i e s f r o m z e r o to — = 2T The quantity process F

tcj

O

=

_L

X

O

= i

J.

+ —5 —JUL)

—V

*

J.

is the m e a n a n g u l a r f r e q u e n c y of the

W ft) /c,V v

A p p e n d i x C.

T he in s ta n ta n e o u s v a lu e of n o r m a l flu ctu atio n n o ise

W e now find the v a l u e of n o r m a l flu c tu a tio n n o i s e w ith c o n s t a n t i n t e n s ity at s o m e i n s t a n t of t i m e

W

Jt. ^

t

= t^.

(t.) 1

a

A c c o r d i n g to E q s .

J L JJL

(2-54) and (2-74), we have

(e2jt-i s i n ^ r

sm

2

2------

T

i l i + e2* ^

0 1 *

1

+ cos

2

cos^

i

2* /i » ------ J / t .)

T

1

ti) Q 0,

1

(C -l)

0

where

0^

is a n o rm a l ra n d o m v a ria b le .

Introducing

= \) / T

f^

and

f^

- /l/T ,

of th e f r e q u e n c y b a n d of the p r o c e s s u n d e r c o n s i d e r a t i o n , we find th a t f o r l a r g e

= < rvfv T he r m s val ue of

W

u

( t , ) is t

6

v

/

0,

, which a g r e e s with (2-57).

137

the lim its

T

(C-2)

A p p e n d i x D.

N o r m a l f l u c t u a t i o n n o i s e m a d e up of a r b i t r a r y p u l s e s

We c o n s i d e r the p a s s a g e of n o r m a l f l u c tu a tio n n o i s e t h r o u g h a l i n e a r s y s t e m . process

W

^(t)

given by E q s .

L e t the

(2-54) and (2 -2 7 ), and c o n s i s t i n g of the v e r y s h o r t p u l s e s

( 2 - 2 8 ) , a c t upon the input of the s y s t e m .

This p r o c e s s can be w ritten as

H -1

sin

2 it

it + e

2 * 2i

it)

w h e r e \) c a n b e a r b i t r a r i l y l a r g e , i f t h e p u l s e s a r e t a k e n to b e s h o r t e n o u g h .

The p ro c e ss

a t the ou tp u t of the s y s t e m is

w*(t)

-

£

Z M lL D .

JL =i

I

*/T"

, sm 72 i - l

i t + 0(J-)

+ %z£/

cos

^

i t

+ 0 (/)

(D -l) w here

k ( ^ / T ) exp [ j 0 ( i / T )

frequency

W* ( t)

Jb / T .

~

]

i s th e c o m p l e x t r a n s f e r c o e f f i c i e n t of the s y s t e m a t the

E x p a n d in g the s i n e a n d c o s i n e t e r m s in th is e x p r e s s i o n , w e o b ta in

g k ^ - T)

|

t0 2 i - 1 c o s 0 ( i / T ) - 0 2 i s i n 0 ( i / T ) ]

sin L fr- ^ t

+ [ e 2j^ _ 1 s i n 0 t i / T ) + 0 2 ^ c o s 0 ( i / T ) ] c o s ^ i t ^

A c c o r d i n g to E q s .

(D-2)

(2-74) and (2-75), we h a v e

62 i - l cos

" e 2 i sin

= V

c o s2 0 (i/T ) + sin2 0 (i/T )

9 * ^

= 6 * ^

and 62 i - l sin

where

6 ^ _q

and

+ e 2/

0*^

c o s ^ ( i / T ) = *n/ s i n 2 0 ( i / T ) + c o s 2 0 ( i / T )

9 * ^ = 9* ^

a r e i n d e p e n d e n t n o r m a l r a n d o m v a r i a b l e s , s i n c e t he c o n d i t i o n ( 2 - 7 6 )

i s s a t i s f i e d , i. e. , c o s 0 ( i / T ) sin 0 ( i / T ) - sin 0 ( / / T ) c o s 0 ( i / T ) A c c o rd in g ly , we obtain

138

=0

w *'»

•

+

P -3)