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Unformatted text preview: C OMER ECE 600 Homework 11 1. Show that if & is a random variable with ยก ( ยข ) = E[e i & & ] and ยก (1) = ยก (2) = 0, then the process X( t ) = cos( ยฃ t + & ) is wide-sense stationary. 2. Consider a zero-mean strict-sense stationary random process with ) 30 cos( 18 ) 20 cos( 50 ) ( R X ฯฯ ฯฯ ฯ + = 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 ) โ 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 , โฆ, X n be jointly Gaussian random variables with the same mean ยค and with covariance function where | ยก | < 1. & & ยก & & ยข ยฃ =- = = , , 1 | | , , ) X , Cov(X 2 2 otherwise j i j i j i ฯฯ ฯ a. Find the mean and variance of...
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This note was uploaded on 10/12/2011 for the course ECE 600 taught by Professor Staff during the Spring '08 term at Purdue.
- Spring '08