wss prob1

wss prob1 - ECE302 Spring 2006 HW12 Solutions April 27,...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ] + X (1) The autocorrelation of Y ( t ) is R Y ( t, ) = E [( A + X ( t )) ( A + X ( t + ))] (2) = E bracketleftbig A 2 bracketrightbig + E [ A ] E [ X ( t )] + AE [ X ( t + )] + E [ X ( t ) X ( t + )] (3) = E bracketleftbig A 2 bracketrightbig + 2 E [ A ] X + R X ( ) , (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, ) 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 X and Y and autocorrelation functions R X ( ) and R Y ( ) respectively. Let W ( t ) = X ( t ) Y ( t ). (a) Find W and R W ( t, ) 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 )] = X Y . (1) In addition, R W ( t, ) = E [ W ( t ) W ( t + )] (2) = E [ X ( t ) Y ( t ) X ( t + ) Y ( t + )] (3) = E [ X ( t ) X ( t + )] E [ Y ( t ) Y ( t + )] (4) = R X ( ) R Y ( ) (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, ) = E [ W ( t ) X ( t + )] = E [ X ( t ) Y ( t ) X ( t + )] . (6) By independence of X ( t ) and Y ( t ), R WX ( t, ) = E [ X ( t ) X ( t + )] E [ Y ( t )] = Y R X ( ) . (7) Since W ( t ) and X ( t ) are both wide sense stationary and since R WX ( t, ) depends only on the time difference , we can conclude from Definition 10.18 that W ( t ) and X ( t ) are jointly wide sense stationary....
View Full Document

Page1 / 8

wss prob1 - ECE302 Spring 2006 HW12 Solutions April 27,...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online