wss prob1

# wss prob1 - ECE302 Spring 2006 HW12 Solutions 1 Solutions...

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Unformatted text preview: ECE302 Spring 2006 HW12 Solutions April 27, 2006 1 Solutions to HW12 Note: These solutions are D. J. Goodman, the authors of our textbook. I have annotated and corrected them as necessary. Text in italics is mine. Problem 10.10.2 • Let A be a nonnegative random variable that is independent of any collection of samples X ( t 1 ) ,... ,X ( t k ) of a wide sense stationary random process X ( t ). Is Y ( t ) = A + X ( t ) a wide sense stationary process? Problem 10.10.2 Solution To show that Y ( t ) is wide-sense stationary we must show that it meets the two requirements of Definition 10.15, namely that its expected value and autocorelation function must be independent of t . Since Y ( t ) = A + X ( t ), the mean of Y ( t ) is E [ Y ( t )] = E [ A ] + E [ X ( t )] = E [ A ] + μ X (1) The autocorrelation of Y ( t ) is R Y ( t,τ ) = E [( A + X ( t )) ( A + X ( t + τ ))] (2) = E bracketleftbig A 2 bracketrightbig + E [ A ] E [ X ( t )] + AE [ X ( t + τ )] + E [ X ( t ) X ( t + τ )] (3) = E bracketleftbig A 2 bracketrightbig + 2 E [ A ] μ X + R X ( τ ) , (4) where the last equality is justified by the fact that we are given that X ( t ) is wide sense stationary. We see that neither E [ Y ( t )] nor R Y ( t,τ ) depend on t . Thus Y ( t ) is a wide sense stationary process. Problem 10.11.1 • X ( t ) and Y ( t ) are independent wide sense stationary processes with expected values μ X and μ Y and autocorrelation functions R X ( τ ) and R Y ( τ ) respectively. Let W ( t ) = X ( t ) Y ( t ). (a) Find μ W and R W ( t,τ ) and show that W ( t ) is wide sense stationary. (b) Are W ( t ) and X ( t ) jointly wide sense stationary? Problem 10.11.1 Solution (a) Since X ( t ) and Y ( t ) are independent processes, E [ W ( t )] = E [ X ( t ) Y ( t )] = E [ X ( t )] E [ Y ( t )] = μ X μ Y . (1) In addition, R W ( t,τ ) = E [ W ( t ) W ( t + τ )] (2) = E [ X ( t ) Y ( t ) X ( t + τ ) Y ( t + τ )] (3) = E [ X ( t ) X ( t + τ )] E [ Y ( t ) Y ( t + τ )] (4) = R X ( τ ) R Y ( τ ) (5) We can conclude that W ( t ) is wide sense stationary. ECE302 Spring 2006 HW12 Solutions April 27, 2006 2 (b) To examine whether X ( t ) and W ( t ) are jointly wide sense stationary, we calculate R WX ( t,τ ) = E [ W ( t ) X ( t + τ )] = E [ X ( t ) Y ( t ) X ( t + τ )] . (6) By independence of X ( t ) and Y ( t ), R WX ( t,τ ) = E [ X ( t ) X ( t + τ )] E [ Y ( t )] = μ Y R X ( τ ) . (7) Since W ( t ) and X ( t ) are both wide sense stationary and since R WX ( t,τ ) depends only on the time difference τ , we can conclude from Definition 10.18 that W ( t ) and X ( t ) are jointly wide sense stationary....
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wss prob1 - ECE302 Spring 2006 HW12 Solutions 1 Solutions...

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