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English Pages [166] Year 1960
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)