homework11

# homework11 - COMER ECE 600 Homework 11 1 Show that if is a...

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C OMER ECE 600 Homework 11 1. Show that if ° is a random variable with ± ( ² ) = E[e i ° ° ] and ± (1) = ± (2) = 0, then the process X( t ) = cos( ³ t + ° ) is wide-sense stationary. 2. Consider a zero-mean strict-sense stationary random process with ) 30 cos( 18 ) 20 cos( 50 ) ( R X πτ πτ τ + = as input to the following system: a. Find the variance of X( t ). b. What value of A will minimize the mean square value of Y( t )? What is the mean square value of Y( t ) for this value of A? 3. White noise X( t ) with spectral density 1 V 2 /Hz is input to a linear time-invariant system with impulse response h( t ) = u( t ) – u( t -1). If the output of the system is Y( t ), then a. Determine R XY ( t 1 , t 2 ). b. Determine R YY ( t 1 , t 2 ). 4. Let X 1 , …, X n be jointly Gaussian random variables with the same mean ´ and with covariance function where | ± | < 1. ° ° ± ° ° ² ³ = - = = , , 0 1 | | , , ) X , Cov(X 2 2 otherwise j i j i j i ρσ σ a. Find the mean and variance of . X X S 1 n n + + = ° b. Find the characteristic function of S n .

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