homework10

homework10 - 2 2 1 sin ( T / 2) E[S ] S ( ) 2 sin ( T / 2)...

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C OMER ECE 600 Homework 10 1. Let X( t ) and Y( t ) be jointly wide-sense stationary random processes, and Z( t ) be derived as shown in the figure below. Given that R XY ( ) = 10e -2 u( ), find E[Z( t )]. 2. Show that if X( t ) is a strict-sense stationary process and ± is a random variable independent of X( t ), then the process Y( t ) = X( t - ± ) is strict-sense stationary. 3. Let X( t ) and V( t ) be continuous parameter random processes. Find the mean and variance for T=5 and for T=100 of the random variable T T T 1 N X( )d 2T t t - = , where X( t ) = 10 + V( t ). Assume that E[V( t )] = 0 and R V ( ) = 2 ² ( ). 4. Show that | R XY ( )| ³ ½[R XX (0) + R YY (0)]. 5. Show that if X( t ) is a wide-sense stationary random process and 1 1 S X( T), n k k n = = ± then 2 2 X
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Unformatted text preview: 2 2 1 sin ( T / 2) E[S ] S ( ) 2 sin ( T / 2) n d n - = & (multiplier) - + X( t ) Gain 2 1 Second Delay Y( t ) Gain 2 Z( t ) 6. Consider a random process X( t ) that assumes the values 1 or -1. Suppose that X(0) = 1 with probability , and that X( t ) changes polarity with each occurrence of an event in a Poisson process of rate & . Find ( i ) the pmf of X( t ), ( ii ) the mean and variance of X( t ), and ( iii ) the autocovariance function of X( t ). 7. Show that the random process X( t ) of Problem 6 is stationary. Then show that X( t ) approaches stationary behavior as t even if P(X(0) = 1) ....
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