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Unformatted text preview: 8.10 Problems 297 c. Find the crosscorrelation function R XY ( t , t + ) , and show that X ( t ) and Y ( t ) are jointly wide-sense stationary. Section 8.5: Wide-Sense Stationary Processes 8.15 Two random processes X ( t ) and Y ( t ) are defined as follows: X ( t ) = A cos ( w 1 t + Theta1) Y ( t ) = B sin ( w 2 t + Phi1) where w 1 , w 2 , A , and B are constants, and Theta1 and Phi1 are statistically inde- pendent random variables, each of which is uniformly distributed between 0 and 2 . a. Find the crosscorrelation function R XY ( t , t + ) , and show that X ( t ) and Y ( t ) are jointly wide-sense stationary. b. If Theta1 = Phi1 , show that X ( t ) and Y ( t ) are not jointly wide-sense stationary. c. If Theta1 = Phi1 , under what condition are X ( t ) and Y ( t ) jointly wide-sense stationary? 8.16 Explain why the following matrices can or cannot be valid autocorrelation matrices of a zero-mean wide-sense stationary random process X ( t ) . a. G = 1 1 . 2 . 4 1 1 . 2 1 . 6 . 9 . 4 . 6 1 1 . 3 1 . 9 1 . 3 1 b. H = 2 1 . 2 . 4 1 1 . 2 2 . 6 . 9 . 4 . 6 2 1 . 3 1 . 9 1 . 3 2 c. K = 1 . 7 . 4 . 8 . 5 1 . 6 . 9 . 4 . 6 1 . 3 . 1 . 9 . 3 1 298 Chapter 8 Introduction to Random Processes 8.17 Two jointly stationary random processes X ( t ) and Y ( t ) are defined as fol- lows: X ( t ) = 2cos ( 5 t + ) Y ( t ) = 10sin ( 5 t + ) where is a random variable that is uniformly distributed between 0 and 2 . Find the crosscorrelation functions R XY ( ) and R YX ( ) . 8.18 State why each of the functions, F ( ) , G ( ) , and H ( ) , shown in Figure 8.7, can or cannot be a valid autocorrelation function of a wide-sense stationary process. 8.19 A random process Y ( t ) is given by Y ( t ) = A cos ( wt + ) where A , w , and are independent random variables. Assume that A has a mean of 3 and a variance of 9, is uniformly distributed between and , and w is uniformly distributed between 6 and 6. Determine if the process is stationary in the wide sense. 8.20 A random process X ( t ) is given by X ( t ) = A cos ( t ) + ( B + 1 ) sin ( t ) < t < where A and B are independent random variables with E [ A ] = E [ B ] = and E [ A 2 ] = E [ B 2 ] = 1. Is X ( t ) wide-sense stationary?...
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- Fall '07