Digital Comminication and Communication Network

8/14/2019 Digital Comminication and Communication Network

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VII ID IG IT A L C O M M U N I C A T I O N

A N D C O M M U N I C A T I O NN E T W O R K S

ar t me nt o f E le c tr i ca l andC om put e r Engi ne e r ing ,Universi ty o f I l l inois a t Ch icago,Chicago, I ll inois , U SA

Lucent Technologies ,Napervil le, Il l inois, USA

c o n c e r n e d w i t h dig i ta l communica t ion a n d dataommunica t ion ne tworks . I n t h e f i rs t p a r t o f th e s e c t io n w e

a l s o f d i g it a l c o m m u n i c a t i o n a n d i n t h eu s s i m p o r t a n t a s p ec t s o f d a ta c o m m u n i -

I n a d i g i t a l c o m m u n i c a t i o n s y s t e m , t h e i n f o r m a t i o n i s p r o -d s o t h a t i t c a n b e r e p r e s e n t e d b y a s e q u e n c e o f d i sc r e t e

T h e d i g it a l s o u r c e m a y b e t h e r e s u l t o f s a m p l i n g a n du c h a s th e c o n t e n t s o f a c o m p u t e r m e m o r y . W h e n

1 a re ca l l ed b i t s fo r b inary sou rce .A d i g it a l s i gn a l c a n b e p r o c e s s e d i n d e p e n d e n t l y o f w h e t h e r i t

a n e s s e n t ia l ly u n l i m i t e d r a n g e o f s ig n a l c o n -g a nd p ro cess ing op t i ons i s ava i lab le to the des igner .

e n d i n g o n t h e o r i g i n a t i o n a n d d e s t i n a t i o n o f t h e i n f o r -source cod ing , encryp-

fo r spec t ra l con t ro l , f orw ard e rror correct ion(FEC) coding, special modulat ion t o s p r e a d t h e s i g n a l s p e c t r u m ,n d equalization t o c o m p e n s a t e f o r c h a n n e l d i s t o r t i o n .

I n m o s t c o m m u n i c a t i o n s y s t e m d e s i g n s , a g e n e r a l o b j e c t i v eto u se as e f f i c ien t ly as poss ib le the re sou rces o f ban dw id th

a n d t r a n s m i t t e d p o w e r . T h u s , w e a r e i n t e r e s t e d i n b o t h b a n d -width e f f ic iency, def ined as the ra t io o f da ta ra te to s igna lb a n d w i d t h , a n d power e f f ic iency, c h a r a c t e r i z e d b y t h e p r o b a b i l -i t y o f m a k i n g a r e c e p t i o n e r r o r a s a f u n c t i o n o f s ig n a l - t o - n o i s er a t i o ( S N R ) .

M o d u l a t i o n p r o d u c e s a c o n t i n u o u s - t i m e w a v e f o r m s u i t ab l ef o r t r a n sm i s s i o n t h r o u g h t h e c o m m u n i c a t i o n c h a nn e l, w h e r e a sd e m o d u l a t i o n i s to ex t rac t the da ta f rom the rece ived s igna l . Wed i s c u s s v a r i o u s m o d u l a t i o n s c h e m e s .

Spread spectrum r e f e r s t o a n y m o d u l a t i o n s c h e m e t h a t p r o -d u c e s a s p e c t r u m f o r t h e t r a n s m i t t e d s i g n a l m u c h w i d e r t h a nb a n d w i d t h o f t h e i n f o r m a t i o n t o b e t r a n s m i t te d i n d e p e n d e n tl yo f t h e b a n d w i d t h o f th e i n f o r m a t i o n - b e a r i n g s i g n al . W e d i s-cuss the code d iv is ion mu l t ip l e access (C D M A) s p r e a d s p e c t r u ms c h e m e .

I n t h e s e c o n d p a r t o f t h e s e c t i o n , o u r f o c u s i s o n d a t ac o m m u n i c a t i o n s n e t w o r k u s u a l l y i n v o l v i n g a c o l l e c t i o n o fc o m p u t e r s a n d o t h e r d e v i c e s o p e r a t i n g a u t o n o m o u s l y a n di n t e r c o n n e c t e d t o p r o v i d e c o m p u t i n g o r d a t a s e r v i c e s . T h ei n t e r c o n n e c t i o n o f c o m p u t e r s a n d d e v ic e s c a n b e e s t a b li s h e dw i t h e i t h e r w i r e d o r w i r e l e s s t r a n s m i s s i o n m e d i a . T h i s p a r ts t a rt s w i t h a n i n t r o d u c t i o n t o f u n d a m e n t a l c o n c e p t s o f d a t ac o m m u n i c a t i o n s a n d n e t w o r k a r c h i t e c t u r e . T h e s e c o n c e p t s


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50 Vijay K. G arg and Yih-Chen W ang

ys t e m c om p one n t s a r e w or k ing toge the r t o bu i ld a da t a c om -u n i c a t i o n n e t w o r k . M a n y d a t a c o m m u n i c a t i o n a n d n e t -

t o un de r s t a n d a dv a nc e d ne tw or k ing t e c hno log ie s.pe n S ys t e m I n t e r c onne c t ion ( O S I ) a nd TC P / I P a r c h i t e c tu r e .

OSI i s a ne two rk re fe rence mode l tha t i s r e fe renced by a ll typeso f ne tw or ks w h i l e t he TC P / I P a r c h i t e c tu r e p r ov ide s a f r a m e -

o r k w h e r e t h e I n t e r n e t h a s b e e n b u i l t u p o n . T h e n e t w o r ke vo lu t ion c a nn o t be so suc c e ss f u l w i th ou t t he w ide sp r e a d u se

o f l oca l ne tw or k ing . The loc al ne tw o r k t e c hno logy inc lud ingthe w i re l es s Loc al A r e a N e tw o r k ( W LA N ) w a s the n in t r odu c e dfol lowed by var ious wire less ne twork access technologies , l ikeF D M A , T D M A , C D M A , a n d W C D M A . T h e c o n v er g e n ce o fva r ie t ie s o f ne tw or ks un de r t he I n t e r ne t ha s be gun . O ne im -po r t a n t d r ive r ha s be e n the i n t r o duc t io n o f S e s sion I n i ti a t i onP r o toc o l ( SI P ) t ha t c ou ld p r ov ide a f r a m e w o r k f o r I n t e g r a t e dM u l t im e d ia N e tw o r k S e r vi c e ( I M S ) . The c onve r ge nc e w on ' t beposs ib le w i th ou t e x t ens ive u se o f a S o f f sw i tc h , t he c r e a t ion o fA LL- I P a rc h i t e c tu r e a nd p r ov id ing b r oa d ba nd a cc es s u t il i z ingop t i c a l ne tw or k ing t e c hno log ie s .


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1S i g n a l T y p e s , P r o p e r t i e s ,

a n d P r o c e s s e sK . G a r g

Depa rtment o f Electrical andCom puter Engineering,University of Illinois at Chicago,Chicago, Illinois, U SA

i h - C h e n W a n gLucent Technologies,

Naperville, Illinois, USA

1 .1 S i g n a l T y p e s . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 9 5 11 .2 E n e r g y a n d P o w e r o f a S i g n a l . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . 9 5 11 .3 R a n d o m P r o c e s s e s . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . 9 5 2

1 . 3. 1 S t a t i s t i c a l A v e r a g e o f a R a n d o m P r o c e s s 1 . 3. 2 S t a t i o n a r y P r o c e s s 1 . 3. 3 P r o p e r t i e s o fa n A u t o c o r r e l a t i o n F u n c t i o n 1 . 3. 4 C r o s s c o r r e l a t i o n F u n c t i o n s 1 . 3. 5 E r g o d i c P r o c e s s e s

1 .4 T r a n s m i s s i o n o f a R a n d o m S i g na l T h r o u g h a L i n e a r T i m e - I n v a r i a n t F i lt e r . .. . 9 5 31 .5 P o w e r S p e c t r a l D e n s i t y . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 9 5 4

1 .5 .1 P r o p e r t i e s o f p s d1 .6 R e l a t i o n B e t w e e n t h e p s d o f I n p u t V e r s u s t h e p s d o f O u t p u t . . . . . . . . .. . . . . . . . .. . . . . 9 5 5

1 .1 S ign a l Typ ess a f u n c t i o n t h a t c a r r ie s s o m e i n f o r m a t i o n . M a t h e m a t -

ca l ly , a s i g n al c a n b e c l a s s if i e d as d e t e r m i n i s t i c o r r a n d o m . F o r ae t e r m i n i s t i c s i gn a l, t h e r e is n o u n c e r t a i n t y w i t h r e s p e c t t o i ts

t i m e . D e t e r m i n i s t i c s i g n a ls a r e m o d e l e d b y e x p l i c ita t h e m a t i c a l e x p r e s s i on s . F o r a r a n d o m s ig n a l, th e r e i s s o m ee g r e e o f u n c e r t a i n t y b e f o r e t h e s i g n a l o c c u r s . A p r o b a b i l i s t i co d e l i s u s e d t o c h a r a c t e r i z e a r a n d o m s i g na l . N o i s e i n a

o m m u n i c a t i o n s y s te m i s a n e x a m p l e o f t h e r a n d o m s ig na l.A s i g n a l x ( t ) i s p e r i o d i c i n t i m e i f t h e r e e x i s t s a c o n s t a n t

T o > O , su c h th a t :x ( t ) = x ( t + T o ) fo r - o o < t < c x ), (1 . 1 )

t i s t h e t i m e .T h e s m a l l e s t v a l u e o f T o t h a t s a t i s f ie s e q u a t i o n 1 .1 is c a l l e d

p e r i o d o f x ( t ) . T h e p e r i o d T o d e f in e s t h e d u r a t i o n o f o n ex ( t ) . A s i g n a l f o r w h i c h t h e r e i s n o s u c hf T o t h a t s a t i s f i e s e q u a t i o n 1 .1 i s r e f e r r e d t o a s a n o n -

1 .2 E n e r g y a n d P o w e r o f a S i g n a ln t a n e o u s p o w e r o f a n e l e c tr i c a l s ig n a l is g iv e n a s:

p ( t ) = x Z ( t ) , ( 1 . 2 )

w h e r e x ( t ) i s e i t h e r a v o l t a g e o r c u r r e n t o f t h e s i g n a l.T h e e n e r g y d i s s i pa t e d d u r i n g t h e t i m e i n t e r v a l ( - T / 2 ,T / 2 ) b y a r e a l s i g n a l w i t h i n s t a n t a n e o u s p o w e r g i v e n b y

e q u a t i o n 1 .2 is w r i t t e n a s :T/2

Erx = I x2 (t)d t" ( 1 . 3 )- r / 2

T h e a v e r ag e p o w e r d i s s i p at e d b y t h e s i g n a l d u r i n g i n t e r v al Tw i l l b e a s f o l l o w s :

r/ 2p r l J~ . x 2 ( t ) d t . ( 1 . 4 )

- T / 2I n a n a l y z i n g c o m m u n i c a t i o n s i g n al s , i t i s o f t e n d e s i r a b l e t o

d e a l w i t h w a v e f o r m e n e r g y . W e r e f e r t o x ( t ) a s a n e n e r g ys i g n a l o n l y i f it h a s n o n z e r o b u t f i n i te e n e r g y 0 < E ~ < o cf o r a l l t i m e , w h e r e :

ix = r~oolim J x 2 ( t ) d t = x 2 ( t ) d t . ( 1 . 5 )- T/2 -oo

A s i g n a l i s d e f i n e d t o b e a p o w e r s i g n a l o n l y i f i t h a s f i n it eb u t n o n z e r o p o w e r 0 < P x < o c f o r al l t i m e , w h e r e :

2005 by Academ icPress. 9 5 1


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52 V i j a y K . G a r g a n d Y i h - C h e n W a n gT/2

Px = T-~lim -T1 I x 2 ( t ) d t " (1 . 6 )-T/2

1 .3 R a n d o m P r o c e s s e sr a n d o m p r o c es s X ( t , s ) c a n b e v i e w e d a s a f u n c t i o n o f tw o

s a m p l e s p a c e s, a n d t i m e t. W e s h o w N s a m p l eo f ti m e , { X k ( t ) } ( s e e F i g u r e 1 . 1) . E a c h o f t h e s a m p l e

n s i d e r e d a s t h e o u t p u t o f a d i f f e re n t n o i s eo r . F o r a f ix e d s a m p l e p o i n t Sk, t h e g r a p h o f t h e f u n c -

X ( t , s k) v e r s u s t i m e t i s c a l l e d a r e a l i z a t i o n o r s a m p l eo f t h e r a n d o m p r o c e ss . T o s i m p l i f y t h e n o t a t i o n , w e

X k ( t ) = X ( t , S k ) . (1 . 7 )W e h a v e a n in d e x e d e n s e m b l e ( f a m i ly ) o f r a n d o m v a r i a b le s

X ( t , s ) } , w h i c h i s c a l l e d a r a n d o m p r o c e s s . T o s i m p l i f y t h ee su p p r e s s t h e s a n d u s e X ( t ) t o d e n o t e a r a n d o m

A r a n d o m p r o c e s s X ( t ) i s a n e n s e m b l e o f t i m e f u n c t i o n sg f u l e v e n t a s s o c ia t e d w i t h a n o b s e r v a t i o n o f o n e

h e r a n d o m p r o ce s s.

1 .3 . 1 S t a t i st i c a l A v e r a g e o f a R a n d o m P r o c e s sp r o b a b i l i t y d e n s i t y

( p d f ). W e d e fi n e t h e m e a n o f t h e r a n d o m p r o c es sas :

E { X ( t k ) } = i ( x . p X k ) d X = IXx(tk), (1 . 8 )

w h e r e X ( t k ) i s t h e r a n d o m v a r i a b l e a n d pXk(X) i s t h e p d f o fX ( t k ) a t t i m e tk .

W e d e f in e t h e a u t o c o r re l a t io n f u n c t i o n o f th e r a n d o m p r o -cess X ( t ) t o b e a f u n c t i o n o f t w o v a r i ab l e s , h a n d t2 , a s :

R x ( h , t2 ) = E { X ( h ) X ( t 2 ) } , (1 . 9 )w h e r e X ( h ) a n d X ( t 2 ) a r e r a n d o m v a ri ab le s o b t a i n e d b y o b -s e r v i n g X ( t ) a t t ime s h an d t2, re s pec t ive ly .

T h e a u t o c o r r e l a t i o n f u n c t i o n i s a m e a s u r e o f t h e d e g re e t ow h i c h t w o t i m e s a m p l e s o f t h e s a m e r a n d o m p r o c e s s ar e re l at e d .

1 . 3 .2 S t a t i o n a r y P r o c e s sA r a n d o m p r o c e s s X ( t ) i s s a i d t o b e s t a t i o n a r y i n t h e s t r i c ts e n s e i f n o n e o f i ts s t a t is t i cs i s a ff e c t e d b y a s h i f t i n t i m e o r i g i n .A r a n d o m p r o c e s s is s a id t o b e w i d e - s e n s e s t a t i o n a r y ( W S S ) i ft w o o f i ts st a ti st ic s ( m e a n a n d a u t o c o r r e l a t i o n f u n c t i o n ) d on o t v a r y w i t h s h i f t i n t im e o r i g in . T h e r e f o r e , a r a n d o m p r o c e s si s WSS i f :

E { X ( t ) } = tXx = co ns tan t , (1 . 10)a n d

R x ( h , t2) = R x ( h - t2) = Rx(T), (1 .11 )w he re "r = h - t2 .

In a s t r i c t s ens e , s ta t io n a ry i m p l i e s W S S , b u t W S S d o e s n o ti m p l y s t a t i o n a ry .

~6

X l ( t ~ .

. / " 3 X 2 ( t )

:~(t)

XN(t)

. T i m e

F I G U R E 1.1 Random Noise Process

1 .3 .3 P r o p e r t i e s o f a n A u t o c o r r e l a t i o n F u n c t i o nA n a u t o c o r r e l a t i o n f u n c t i o n r e a d s a s :

R x ( T ) = E [ X ( t + ' r ) X ( t ) ] fo r a l l t . (1 . 12) T h e m e a n s q u a r e v a lu e o f a r a n d o m p r o c e s s c a n b e

o b t a i n e d f r o m R x ( T ) b y s u b s t i t u t i n g "r = 0 i n e q u a t i o n1.12:

Rx(O) = E [ X 2 ( t ) ] . ( 1 . 1 3 ) T h e a u t o c o r r e l a t i o n f u n c t i o n R x ( 'r ) is a n e v e n f u n c t i o n o f

-r; that is:Rx('r) = R x( - "r). (1 . 14)

T h e a u t o c o r r e l a t i o n f u n c t i o n Rx('r) h a s i t s m a x i m u mm ag ni t ud e a t ~- = 0 ; tha t i s:

IRx('r) I _< Rx(O) . (1 . 15)


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1 9 5 3ignal Types, Propert ies , and ProcessesT o p r o v e t h i s p r o p e r t y , w e c o n s i d e r t h e n o n - n e g a t i v e q u a n -

E [ { X ( t + w ) - 4 - X ( t ) } 2 1 > 0 . ( 1 . 1 6 )

E [ X 2 ( t + " r)] i 2 E [ X ( t + ' r ) X ( t )] + E[X 2(t)] >_ O. ( 1 . 1 7 )2Rx(O) 2Rx(~ ) >_ O. (1.18)

- R x ( 0 ) < R x ( T ) < R x ( 0 ) . ( 1 . 1 9 )IRx('r)l _< Rx(O). ( 1 . 2 0 )

1.3.4 Crosscorrelation Functionsr t w o r a n d o m p r o ce s se s X ( t ) a n d Y ( t ) w i t h a u t o -R x ( t , u ) a n d R r ( t , u ) , r e s p e c t i v e l y . T h e

w o c r o s s co r r e l a ti o n f u n c t i o n s o f X ( t ) a n d Y ( t ) a r e d e f in e d a s:R x y ( t , u ) = E [ X ( t ) Y ( u ) ] , ( 1 . 2 1 )

n dR r x ( t , u ) = E [ Y ( t ) X ( u ) I . ( 1 . 2 2 )

R ( t , u ) = [ R x ( t , u ) R x r ( t , u ) l ( 1 . 2 3 )L R r x ( t , u ) R y ( t , u ) '

r

R ( ' r ) = r R x ( , r ) R x r ( ' r ) ] ( 1 . 2 4 )[ R y x ( ' r ) R r ( ' r ) "

T h e c r o s s c o r r e l a ti o n f u n c t i o n i s n o t g e n e r al l y a n e v e n f u n c -f % a n d i t do e s n o t h a v e a m a x i m u m v a l u e a t t h e o r ig i n .

R X y( T ) = R r x ( - ' r ) . ( 1 . 2 5 )1 . 3 .5 E r g o d i c P r o c e s s e s

l e f u n c t i o n x ( t ) o f a s t a ti o n a r y p r o c e s sw i t h o b s e r v a t i o n i n t e r v a l - T / 2 < t < T /2 . T h e t i m el b e a s w r i t t e n h e r e :

T /2, Jx~ ( T ) = ~ x ( t ) d t . ( 1 . 2 6 )-T /2

T h e t i m e a v e r a g e t X x ( T ) i s a r a n d o m v a r i a b l e , a n d i t s v a l u eo n o f t h e r a n d o m p r o ce s s X ( t ) i s s e l e c t e d f o r

e q u a t i o n 1 .2 6 . S i n c e X ( t ) i s a s s u m e d t o b e s t a t i o n a r y ,e a n o f t h e t i m e a v e r a g e I X x (T ) is g i v e n a s:

T/2 T/2l J 1 IE [ t X x (T ) ] = T E [ x ( t ) ] d t = T

- T / 2 - T / 2t xxd t = tXx , ( 1 . 2 7 )

w h e r e ~ x i s t h e m e a n o f p r o c e s s X ( t ) .W e s a y t h a t t h e p r o c e s s X ( t ) i s e r g o d i c i n m e a n i f t w o

c o n d i t i o n s a r e s a t i s f i e d :( 1 ) T h e t i m e a v e r a g e ~ x ( T ) a p p r o a c h e s t h e e n s e m b l e

a v e r a g e ~ x i n l i m i t a s t h e o b s e r v a t i o n t i m e T g o e s t oi n f i n i t y :

l i m ~ x ( T ) = ~ x . ( 1 . 2 8 )T~oo( 2 ) T h e v a r i a n c e o f ~ x ( T ) , t r e a t e d a s a r a n d o m v a r i a b l e ,

a p p r o a c h e s z e r o in l i m i t a s t h e o b s e r v a t i o n i n t e r v al Tg o e s t o i n f i n i t y .

l i r a v a r [ ~ x ( T ) ] = 0 . (1 . 2 9 )L i k e w i s e , t h e p r o c e s s X ( t ) i s e r g o d i c i n t h e a u t o c o r r e l a -

t i o n f u n c t i o n i f t h e f o l lo w i n g t w o l i m i t i n g c o n d i t i o n s a r efu l f i l l e d :

( 1 ) l i m R x( ' r , T ) = R x ( 'r ). ( 1 . 3 0 )T---~( 2 ) l i m var[Rx( 'r , T )] = 0 . (1 . 3 1 )r~cx)

E x a m p l e 1A r a n d o m p r o c e s s is d e s c r i b ed b y x ( t ) = A c o s ( 2 w f t + ( b ) ,w h e r e ~ b i s a r a n d o m v a r i a bl e u n i f o r m l y d i s t r i b u t e d o n( 0 , 2 -r r) . F i n d t h e a u t o c o r r e l a t i o n f u n c t i o n a n d m e a n s q u a r ev a l u e .

S o l u t i o n s :R x ( h , t2 ) = E[ A co s (2z rf i l q - (b )Aco s (2"rrfi2 q - qb)]

= A 2 E { ~ c o s 2 7 r f ( t 1 - - t 2 ) + ~ c o s [ 2 ~ T f ( h - t2) + 2qb]}A 2= - - c o s 2 ~ f ( t l - - t2).2A 2 A 2Rx f i r) = - - cos 27rf(t1 - t 2 ) = - - COS2Trfx.2 2A 2R x ( 0 ) = - - .2

1 .4 T r a n s m i s s io n o f a R a n d o m S i g n a lT h r o u g h a L i n ea r T i m e - I n v a r i a n tF i l t e r

F i g u r e 1 . 2 s h o w s a r a n d o m i n p u t s i g n a l X ( t ) p a s s i n g t h r o u g h al i n e a r t i m e - i n v a r i a n t f i l t e r . T h e o u t p u t s i g n a l Y ( t ) c a n b e


6/8

9 5 4 Vijay K. Garg and Y ih-Chen W ang

x(t)Impulse response function h ( t )

* Filter . Y(t )I G U R E 1 .2 R a n d o m S i g na l T h r o u g h a L in e a r T i m e - i n v a r i a n t

Filter

r e l a t e d t o t h e i n p u t t h r o u g h a n i m p u l s e r e s p o n s e f u n c t i o nh(T1) a s :

Y ( t ) = i h ( % ) X ( t - % ) d 'r l .- - 0 0

( 1 . 3 2 )

T h e m e a n o f Y ( t ) i s a s f o l l o w s :

D y ( t ) = E [Y (t)] = E [ ~__J ('rl)X (t - ' r l ) d ' r l ( 1 . 3 3 )

T h e s y s t e m i s s t a bl e p r o v i d e d t h a t t h e e x p e c t a t i o n E [ X ( t ) ] i sf i n i te f o r a ll t. W e m a y i n t e r c h a n g e t h e o r d e r o f e x p e c t a t i o na n d i n t e g r a t i o n i n e q u a t i o n 1 . 3 3 , a n d t h e f o l l o w i n g r e s u l t s :

l a y ( t ) = i h ( ' r l ) E [ X ( t - 1 " l ) ] d ' r = i h ( ' r l ) ~ x ( t - " r l ) d ' r l "- - 0 0 - - 0 0

( 1 . 3 4 )F o r a s t a t i o n a r y p r o c e s s X ( t ) , t h e m e a n b~x(t) i s c o n s t a n t

l a x, s o w e m a y s i m p l i f y e q u a t i o n 1 .3 4 as :

[ g Y = l a X ' i h ( ' r l ) d ' r l = D x " H ( 0 ) ,- - ( X 3

( 1 . 3 5 )

w h e r e H ( 0 ) i s t h e z e r o - f r e q u e n c y ( d c ) r e s p o n s e o f t h e s y s t em .E q u a t i o n 1 .3 5 s t a te s t h a t t h e m e a n o f t h e r a n d o m p r o c e s sY ( t ) p r o d u c e d a t t h e o u t p u t o f a l i n ea r t i m e - i n v a r i a n t s y s t e m

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

T h e a u t o c o r r e l a t i o n f u n c t i o n o f t h e o u t p u t r a n d o m p r o c e s sY ( t ) i s a s fo l lo w s :

( 1 . 3 6 )A s s u m i n g t h a t m e a n s q u a r e E [ X 2 ( t ) ] i s f i n i t e f o r a l l t a n d

t h e s y s t e m i s s t a b l e , t h e n :

Rv( t , u )=

Rr( t , u )=

i h ( % ) d ' r l i h ( ' r 2 ) d ' r 2 E [ X ( t - T 1 ) " X ( u - T 2 ) ] .- ( x ) - o o

i h ( # r ] ) d ' r l i h ( ~ 2 ) d ~ 2 R x ( t - ~ l , U - ~ 2 ) .- - 0 0 - - ~

( 1 . 3 7 )W h e n t h e i n p u t X ( t ) i s a s t a t i o n a r y p r o c e s s , t h e a u t o c o r r e -

l a t io n f u n c t i o n o f X ( t ) i s o n l y a f u n c t i o n o f d if f e re n c e b e t w e e nt h e o b s e r v a t i o n t i m e t - % a n d u - % . T h u s , w i t h -r = t - u ,w e h a v e :

Ry( 'r )=ii[h( ' r l )h( ' r2) .Rx( ' r -%+'r2)](d~l)(d' r2 ) ( 1 . 3 8 )- o o - o o

R y ( O ) = E [ y 2 ( t ) ] = i i h ( ~ l ) h ( T 2 ) R x ( ~ 2 - % ) ( d % ) ( d % ) ( 1 .3 9 )- o o - o (3

R y ( O ) i s a c o n s t a n t .1 .5 Pow er Spec tra l D en s i tyL e t H ( f ) d e n o t e t h e f r e q u e n c y r e s p o n s e o f th e s y s t e m ; t h e n:

h ( % ) = i [ H ( f ) . e J 2 ~ r f ~ l l d f . ( 1 . 4 0 )- - O O

T h e f o l l o w i n g a l s o a p p l i e s :

E [ y 2 ( t ) ] = [ H ( f ) " e j 2 f '' ] . h ( T 2 ) R x ( ' r 2 - ' 1 " 1 ) ( d ' r l ) ( d ' r 2 )- o o - o c

= i H ( f ) d f i h ( " r i) d 'r 2 [ R x ( 'r e - 'r l )e J 2 ~ r f 'q ] d " r l .o o o o

( 1 . 4 1 )L e t x = T2 - - T 1, a n d t h e f o l l o w i n g r e s u l t s :

S [ y 2 ( t ) ] = i S ( f ) d f ' i [ h ( T 2 ) d 7 2 e J 2 ~ x f ~ 2 ] d T 2 " i [ g x ( T ) c - J 2 a f f T ] d T- (3 o - ~ - o o

= i I H ( f ) 1 2 d f " [ R x ( ~ ) e ' 2 ~ f ] d ' r , ( 1 . 4 2 )- - ( 3 O - - O O

w h e r e ] H ( f ) ] i s t h e m a g n i t u d e r e s p o n s e o f th e f il te r.W e d e f i n e :

Sx (f) = i [Rx(~)e-J2~f~]d~"- - 0 0


7/8

1 S i gna l Type s , P roper t i e s , an d P roces ses 95 5S x ( f ) is called the p o w e r s p e c t r a l d e n s i t y o r p o w e r s p e c -

of t he s t a t i ona r y p r oc e s s X ( t ) ,Now, we can rewr i te equa t ion 1 .42 as :

E [ y 2 ( t ) ] = i [ I H ( f ) 1 2 ' S x ( f ) ] d f .o o

(1.43)

The m e a n squa r e va lue o f a s t a bl e l i ne a r t im e - inva r i a n t f i l te rp o w e r s p e c t r a l d e n s i t y o f

p r oc e s s m u l t i p l i e d by the squa r e d m a gn i tud e r e -of the f il te r.

1.5.1 Prop erties of psd

S x ( f ) = i [ R x ( 7 ) e - J 2 ~ ' f f * ] d %- - 0 0

n d7R x ( 7 ) = I [ S x ( f ) e J 2 ~ f ; ] d f"

- - 0 0

P r o p e r t y 1Fo r f = 0, we get:

S x (O ) = [ R x ( v ) d v . (1.44)o x )

P r o p e r t y 2The m e a n squa r e va lue o f a s t a t iona r y p r oc e s s e qual s t heto t a l a r e a unde r t he g r a ph o f p sd :

R x ( O ) = E [ X 2 ( t ) ] = [ S x ( f ) d f . (1.45)

P r o p e r t y 3The p sd o f a s t a t i ona r y p r oc e s s is a lw a ys non - ne ga t ive :S x ( f ) >_ 0 for a ll f . (1 .46)

P r o p e r t y 4The p sd o f a re a l - va lue d r a nd om p r oc e s s is a n e ve nf unc t ion o f f re que nc y :

S x ( - f ) = S x ( f ) . ( 1 .47 )S x ( - f ) = ~[ [ R x ( 7 ) e J e v f ~ ] d 7 .

- - O C

Because R x ( 7 ) i s equa l to R x ( - 7), we can write:

7S x ( - f ) = ] [ R x (T ) eJ 2 ~ f~ ] d v = S x ( f ) .o o P r o p e r t y 5

The psd , a pp r op r i a t e ly no r m a l i z e d , ha s t he p r ope r t i e susua l ly a ssoc i at e d w i th a p r oba b i l i t y de ns i ty f unc t ion :

S x ( f )P x ( f ) - ~ >_ 0 for a ll f . (1.48)f S x ( f ) d f- - 0 0

1 . 6 R e l a t i o n B e t w e e n the p s d o f I n p u tV e r s u s t h e p s d o f OutputS y ( f ) = I [ R y ( ' r ) e j 2 r f f * ] d ' r .

o o

[ __ T 1 +

Let % = "r + % - %:S y ( f ) = H ( f ) H * ( f ) S x ( f ) = I n ( f / I 2 . S x ( f ) . (1.49)

T h e p s d o f th e o u t p u t p r o ce s s Y ( t ) equa ls the power spec t ra ld e n s i ty o f t h e i n p u t X ( t ) m ul t ip l i e d by the squa r e d m a gn i tuderesponse of the f il te r.E x a m p l e 2N um be r s 1 a nd 0 a re r e p r ese n te d by pu lse o f a m p l i t ude A a nd- A vol ts , r e spec tive ly , and du ra t ion T sec. The pulses a re no tsync h r on iz e d , so t he s t a r t i ng t im e t a o f the f ir s t c om ple t e pu l sefor pos i t ive t ime is equa l ly l ike ly to l ie anyw here be tween 0 andTse c (see Figures 1.3 and 1.4). Tha t is, ta is the s am ple valu e ofa un i f o r m ly d i s t r i bu t e d r a ndo m va r i a b l e Ta w i th i t s p r oba b i l -i t y de ns i ty f unc t ion de f ine d by :

( ~ O < t d < T .f r . ( t . ) = - -elsewhere.T /2A 2 [R x ( % ) = ~ - X ( t ) X ( t - %)d' r .

- T / 2

A

F I G U R E 1 .3- A

Pulse of Amplitude A

X ( t )


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956 V i ja y K . G a r g a n d Y i h -C h e n W a n gA

" ~ 1 ~- A

FIGURE 1.4 Sh iftedPulse of A mpli tude A

x ( t - ~O

Rx(0) = Tota l average power .R x ( 'r ) = A 2 ( 1 - ~ -~ -) , ' r , < T

= 0 I T I _ > r .S X ( f ) = A 2 ' r ( S i n i r ' 2

\ ~rfr ji S x ( f ) d f = Tota l average power

- - O ( 3

TJ ( ~ ) (sin'rrfT~ecrfTx ( f ) = A 2 1 - e - J 2 W d . r = A g T \

-TA2 T2 ;( ' m { g ( f )T s ' n c i ' a * ' - - T

F I G U R E 1 . 5 Autocorrelation Function

X ( t - ' q )

T h es e ex am p l e eq u a t io n s s h o w t h a t f o r a r a n d o m b i n a ry w av ei n w h i ch b i n a ry n u m b er s 1 an d 0 a re r ep re s en t ed b y p u ls e g ( t )a n d - g ( t ) , respect ive ly , the psd Sx(f ) i s equal to the energyspect ra l dens i ty ~g(f ) o f the sy mb ol shap ing pu l se g ( t ) , d i v i d edb y t h e s y m b o l d u ra t i o n T . N o t e t h e au t o co r r e l a t i o n fu n c t i o n isshown in Figure 1 .5 .

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