132A_1_Midterm-W09sol

132A_1_Midterm-W09sol - UCLA Electrical Engineering Dept....

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Unformatted text preview: UCLA Electrical Engineering Dept. EE132A: Communication Systems Midterm Exam Solutions 1. Consider the random process X ( t ) = Aq ( t )cos(2 f t + ) , where is a random variable uniformly distributed in [- , ) and A is a random vari- able that takes on the values 1 or- 1 with equal probability. and A are statistically independent. The signal q ( t ) is a square wave with period T and is defined by q ( t ) = ( 1 , t < T/ 2 ,- 1 , T/ 2 t < T. (a) Compute the mean function, m X ( t ) = E [ X ( t )]. m X ( t ) = E [ Aq ( t )cos(2 f t + )] = E [ A ] E [ q ( t )cos(2 f t + )] because A and are independent. E [ A ] = (1)0 . 5 + (- 1)0 . 5 = 0, therefore m X ( t ) = 0. (b) Compute the autocorrelation function R X ( t 1 ,t 2 ) . (You may need the trigonomet- ric identity 2cos( )cos( ) = cos( + ) + cos( - ) .) R X ( t 1 ,t 2 ) = E [ Aq ( t 1 )cos(2 f t 1 + ) Aq ( t 2 )cos(2 f t 2 + )] = E [ A 2 q ( t 1 ) q ( t 2 )cos(2 f t 1 + )cos(2...
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This note was uploaded on 04/12/2010 for the course EE 132A taught by Professor Walker during the Spring '08 term at UCLA.

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132A_1_Midterm-W09sol - UCLA Electrical Engineering Dept....

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