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20092ee132A_1_hwk3_sol

20092ee132A_1_hwk3_sol - EE132A Spring 2009 Communication...

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Unformatted text preview: EE132A, Spring 2009 Communication Systems Prof. John Villasenor Handout# 10 TA: Pooya Monajemi and Erica Han Homework 3 Solutions 1. Constants and Carriers in Autocorrelations . (a) ( )( ) [ ] 1 2 1 2 1 2 2 1 2 1 2 2 1 2 2 1 2 2 1 2 ( ) [ ( ) ( )] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , where X Z Z R t t E X t X t E A Z t A Z t E A Z t A Z t A Z t Z t A A A E Z t Z t A R t t A R t t τ τ , = = + + = + + + = + ⋅ + ⋅ + = + − = + = − (b) Now, we have ( ) cos(2 ) ( ) c X t f t Z t π θ = + + ( ) [ ( ) ( )] [(cos(2 ) ( ))(cos(2 ( ) ) ( ))] [cos(2 )cos(2 ( ) )] [ ( )cos(2 ( ) )] [cos(2 ) ( )] [ ( ) ( )] X c c c c c c R E X t X t E f t Z t f t Z t E f t f t E Z t f t E f t Z t E Z t Z t τ τ π θ π τ θ τ π θ π τ θ π τ θ π θ τ τ = + = + + + + + + = + + + + + + + + + + + Now since θ and ( ) Z t are independent and since ( ) Z t is zero mean, the second and third expectations of the previous equations are equal to zero. ( ) [cos(2 )cos(2 ( ) )] [ ( ) ( )] 1 [cos(4 2 2 ) cos(2 )] ( ) 2 X c c c c c Z R E f t f t E Z t Z t E f t f f R τ π θ π τ θ τ π π τ θ π τ τ = + + + + + = + + + + Since θ has a uniform distribution between π ± , the PDF of θ is: 1 ( ) 2 f θ θ π θ π π = , − ≤ ≤ and [cos(4 2 2 )] ( )cos(4 2 2 ) 1 cos(4 2...
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20092ee132A_1_hwk3_sol - EE132A Spring 2009 Communication...

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