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132A_1_Midterm-W09sol

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

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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 πf t + Φ) , where Φ is a random variable uniformly distributed in [- π,π ) and A is a random vari- able that takes on the values 1 or- 1 with equal probability. Φ 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 πf t + Φ)] = E [ A ] E [ q ( t )cos(2 πf t + Φ)] because A and Φ 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( α )cos( β ) = cos( α + β ) + cos( α- β ) .) R X ( t 1 ,t 2 ) = E [ Aq ( t 1 )cos(2 πf t 1 + Φ) Aq ( t 2 )cos(2 πf t 2 + Φ)] = E [ A 2 q ( t 1 ) q ( t 2 )cos(2 πf t 1 + Φ)cos(2...
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132A_1_Midterm-W09sol - UCLA — Electrical Engineering...

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