M J Robers - //8 Answers o Exercises in Chaper 7 - Correlaion Funcions 7- (from Papoulis and Pillai) The random variable C is uniform in he inerval (,T ) Find R, ()= u( C), ()= C (Use R (, )= R,, < or < R (, )=, > T and > T min(, )/ T, oherwise R (, ) T = in ) if 8 6 - - - 5 5 5 R (, )= T 7- The random variable C is equally likely o have any value in {,,,3,} Find R [ n,n ] if n = u n C, n = n C, n < or n < R n,n =, n and n min( n,n )+, oherwise 5 Soluions 7-
M J Robers - //8 Uni Sequence Process 8 R[n,n ] 6 8 6 n - R n,n =, n < or n < or n > or n > or n n / 5, n = n and n and n - n 6 8 Uni Impulse Process 5 R[n,n ] 5 8 6 n - - n 6 8 7-3 (from Papoulis and Pillai) The random variables A and B are independen and Gaussian wih zero mean and sandard deviaion and p is he probabiliy ha he sochasic CTCV process ()= A B crosses he axis in he inerval (,T ) Show ha p = an ( T ) (Hin: p = P ( A / B T )) Proof 7- (from Papoulis and Pillai) The sochasic CTCV process Find P () 3 933 78535 Find E ( + ) ( ) () is WSS and Gaussian wih R = e 7-5 The sochasic DTDV process n is WSS and geomerically disribued wih R m [ ]= ( 8 )m Find P [ n] 5 Find E( [ n + ] [ n ] ) 9 7 Soluions 7-
M J Robers - //8 () wih R Find E( ZW ) and E ( Z + W ) 7-6 (from Papoulis and Pillai) Given a Gaussian sochasic CTCV process random variables Z = ( + ), W = f ZW ( z, w) 733, 865, find f Z z = e, we form he, P Z < and P Z < = 695 f ZW ( z,w)= exp z / 95zw + ( w /) 99933 53 f Z ( z)= ez /8 7-7 (from Cooper and McGillem) A WSS sochasic CTCV process having sample funcions () has an auocorrelaion funcion R ( )= 5e 5 Anoher WSS sochasic process has sample funcions Y()= ()+ b( ) Find he value of b ha minimizes he mean-squared value of Y -665 Find he value of he minimum mean-squared value of Y 368 (c) If b, find he maximum mean-squared value of Y 665 7-8 Consider a WSS sochasic CTDV process having sample funcions illusraed in Figure E7-8 () A T +T Figure E7-8 A sample funcion from a WSS sochasic DTDV process In his process a pulse of heigh A and widh T is equally likely o occur or no occur a imes ± nt The probabiliy of occurrence of he pulse is independen of he probabiliies of occurrence of all oher pulses The ime is random and uniformly disribued over = o =T and he pulse widh T is less han half he pulse spacing T Find he auocorrelaion funcion for his process R ( )= A T ri T T + A T nt ri T T n= Soluions 7-3
M J Robers - //8 A T A T T A T T T -T R (τ) T τ 7-9 Consider a WSS sochasic DTDV process having he sample funcions illusraed in Figure E7-9 [n] A B n n [n] A B A B n n 3[n] n n Figure E7-9 A DTCV bipolar-pulse sochasic process In his process bipolar pulse pairs occur every ineger muliple of in discree ime wih probabiliy p The probabiliy of occurrence of each pulse pair is independen of he occurrence of any oher pulse pair The posiions n of he firs pulse pair afer n = (wheher or no he pulse pair acually occurs) are uniformly disribued on {,,3, } Find he auocorrelaion funcion for his process ( R m = p p) ( A + B ) m + AB m + + m + p ( A + B ) m k + AB m k + m k 3 k ={ } If p = /, A = and B = graph he auocorrelaion funcion R[m] 5 m - -5 7- Which of he following funcions could or could no be an auocorrelaion funcion of a real WSS sochasic process and why? 3sin( ) Canno be 3cos( 5 ) Can be Soluions 7-
M J Robers - //8 (c) sinc( /) Can be (d) rec( ) Canno be (e) ri( /) Can be 8 (f) ri + ri + 8 Canno be 6 (g) ri + ri + 6 Canno be (h) cos( )ri( /3) Can be (i) sin ( ) Canno be (j) ( 5)+ ( + 5) Canno be (k) ( 5)+ + ( + 5) Can be (l) ( 5)+ 5+ ( + 5) Canno be 7- A WSS sochasic CTCV process has an auocorrelaion funcion R ( )= cos( )ri( /3)+ cos( 6 )+ Find is mean value, mean-squared value and sandard deviaion 6, 37 Does he auocorrelaion have a sinusoidal componen and, if so, wha is is frequency? Yes, a 3 Hz (c) Wha is he smalles posiive value of a which () and ( + ) are uncorrelaed? = 9 { ()} has he auocorrelaion funcion R ( ) graphed below A sample 7- An ergodic, sochasic CTCV process funcion () of ha random process is sampled a a rae f s = /T s A DTCV random variable is formed according o he following formula Y = + ( T s )+ T s Y = 3T s Y k = 3kT s + ( T s )+ 5T ( s ) + ( ( 3k + )T s )+ ( 3k + )T s R (τ) If he sampling rae f s is 5 Hz, wha is he variance of he random variable Y? 3/8 - τ (s) Soluions 7-5
M J Robers - //8 If he sampling rae f s is Hz, wha is he variance of he random variable Y? 3/ 7-3 An ergodic, deerminisic sochasic CTDV process has sample funcions each of which is a sequence of coniguous recangular pulses of widh second The pulses can have only wo possible ampliudes and wih equal probabiliy The pulse ampliudes occur randomly and are all saisically independen of each oher Skech he auocorrelaion funcion for his random process () () 6 8 6 8 3() 6 8 6 8 () 6 8 6 8 R (τ) 6 8 6 8 9 7- Le he funcion - τ g( )= / a, <, oherwise For wha values of a could his funcion be an auocorrelaion funcion? a = 7-5 Generae samples from a sample funcion of an ergodic sochasic CTCV process wih he insrucion sequence N = ; % Number of ime samples = randn(n,) ; % Gaussian disribued random values [b,a] = buer(3,5) ; % Third-order Buerworh lowpass filer = filfil(b,a,) ; % Compue filered signal Then compue and graph an esimae of he auocorrelaion funcion using he approximaion given in he ex 3 R (nt s ) - - -8-6 - - 6 8 n 7-6 Find he mean and variance of sochasic CTCV processes having he following auocorrelaion funcions Soluions 7-6
M J Robers - //8 R ( )= 8e,8 R ( )= 6 + 3 + ±9,3 (c) R ( )= e cos( ), 7-7 Two independen WSS sochasic CTCV processes wih sample funcions () and Y() have auocorrelaion funcions 5 R ( )= 3sinc( 8 ), R Y ( )= e Find he auocorrelaion funcions of Z s ()= ()+ Y() and Z d ()= () Y() R Zs 5 ( )= 3sinc( 8 )+ e R Zd 5 ( )= 3sinc( 8 )+ e (c) Find he cross-correlaions R Zs Z d saed above in he ex R Zs Z d and R Zd Z s and compare heir maximum values o he upper limi ( )= 3sinc( 8 )+ e 5 = R Zs Z d R Zs Z d ( ) R Zs R Zs, R Zs = 5 and R Zd = 5, R Zs Z d 5 R Zs Z d (τ) 6 - - τ 7-8 A WSS sochasic CTCV process has an auocorrelaion funcion Find R Find R ( ) ( ) R R R ( )= 5ri /, ( )= 5, <, <, > ( )= 5 ( + )+ ( ) Soluions 7-7
M J Robers - //8 7-9 A common echnique for opimally deecing a sinusoidal signal in he presence of broadband noise is o muliply he signal plus noise by a sinusoid of he same frequency and le he produc excie a lowpass filer The muliplying sinusoid is generaed by wha is called he local oscillaor The expeced value of he filer response indicaes he ampliude of he sinusoid Le he sinusoidal signal be ()= sin 5 + and le he auocorrelaion of he noise be R N ( )= e 8 Le he local-oscillaor sinusoid be of he form, y()= cos( 5 + ) Then he produc of he incoming signal and he locally-generaed sinusoid is Z()= y() ()+ y()n () Find he expeced value of Z(), wih a consan E Z() = 5 sin( + + )+ sin( ) Wha value of maximizes E( Z() )? = /± n, n =,,,3, Soluions 7-8