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EL6303 Prof Rappaport HW 9 Due 3pm, Nov 27, 2013 1. Let ( )X t be a wide-sense stationary real random process whose

autocorrelation function is periodic with a fundamental period T. Show in this case that the least-mean-square-error linear estimate of the value of a sample function at a time which is an integral number of periods away from an observation instant is equal to the observed value. That is, show that ˆ ( ) ( )X t kT X t+ = for any t, k an integer.

Hint: Let Y be estimated by aX b+ i.e. Y aX b= + , then a and b which

minimize the mean square error are given by 2

Cov( , )ˆ { } ( { })X

X YY E Y X E Xσ

= + − .

(Prove 2

Cov( , )ˆ { } ( { })X

X YY E Y X E Xσ

= + − as part of your solution.)

2. Let the stochastic processes ( )X t and ( )Y t be jointly stationary in the strict

sense. Show that ( ) ( )XY XYR Rτ τ− = .

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3. Let ( )Y t be a zero-mean, stationary, Gaussian random process and suppose that this process is the input to a square-law detector, i.e.:

(a) Show that the mean value of the detector output is (0){ ( )} CovYE Z t = . Note that ( ) { } { } { }CovY t t t tE Y Y E Y E Yτ ττ

+ += − .

(b) Show that the covariance function of the detector output is 2( ) ( )Cov 2CovYZ τ τ= . This shows that the detector output is stationary in

the wide sense. Hint: See Papoulis&Pillai’s book, 6-199 on Page 219. 4. Let ( )X t be a wide-sense stationary random process with

2 0

0

sin 22

( )XXff

R π τπ τ

τ σ= , where σ is a positive constant. Determine the spectral

density for this case.

( )Z t( )Y t yz

2. . ( ) ( )i e Z t Y t=

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5.

Let ( )X t be zero-mean, wide-sense stationary random process whose

spectral density ( )XS f is zero for values of frequency greater in magnitude than some frequency

0f .

Let 02p

pf fπ>>= .

Let Z be a random variable uniformly distributed between 0 and 2π . Let ( )X t and Z be independent.

(a) Show that the output process ( )Y t is stationary in the wide sense and that 1

2( ) ( )cosY XR R pτ τ τ= .

(b) Show that 1 14 4

( ) ( ) ( )Y X p X pS f S f f S f f= − + + .

( ) ( )cos( )Y t X t pt z= +( )X t ×

( ) cos( ) (modulator)W t pt z= +

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6. (Comb Filter) Let ( )X t be real wide-sense stationary R.P. with spectral density XS .

(a) Show that the system function of the given system has the squared magnitude:

2| ( )| 2(1 cos2 )H f fπ Δ= − Therefore, ( ) 2(1 cos2 ) ( )Y XS f f S fπ Δ= − . Such a system is sometimes called a comb filter.

(b) Show that for values of frequency which are small compared to 1Δ

we

obtain the approximate result 2 2 2( ) (4 ) ( )Y XS f f S fπ Δ≈ . Therefore, the system is acting like a differentiator for low- frequency inputs.

( )Y t( )X t +

delay unit invertortX Δ−−

tX Δ−Δ