This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 widesense stationary. Section 8.5: WideSense 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 widesense stationary. b. If Theta1 = Phi1 , show that X ( t ) and Y ( t ) are not jointly widesense stationary. c. If Theta1 = Phi1 , under what condition are X ( t ) and Y ( t ) jointly widesense stationary? 8.16 Explain why the following matrices can or cannot be valid autocorrelation matrices of a zeromean widesense 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 widesense 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 ) widesense stationary?...
View Full
Document
 Fall '07
 Carlton
 Correlation

Click to edit the document details