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20092ee132A_1_hwk3 - R Z 2 Sum of two modulated Gaussians...

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EE132A, Spring 2009 Communication Systems Prof. John Villasenor Handout # 8 TA: Pooya Monajemi and Erica Han Homework 3 Assigned: Monday, April 13, 2009 Due: Monday, April 20, 2009 Reading Assignment: Chapter 5 and Chapter 7 (section 7.1). 1. Constants and Carriers in Autocorrelations. (a) Let A be a deterministic constant, and let Z(t) be a zero-mean stationary process. Find R X ( τ ) for X(t) = A + Z(t) . You may express your answer in terms of R Z ( τ ) . (b) Let f c be a deterministic constant, θ be uniformly distributed between ± π , and Z(t) as above. Assume Z(t) and θ are statistically independent. Find R X ( τ ) for X(t) = cos(2 π f c t + θ ) + Z(t) . You may once again express your answer in terms of
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Unformatted text preview: R Z ( ) . 2. Sum of two modulated Gaussians. Let X and Y be statistically independent Gaussian-distributed random variables, each with zero mean and unit variance. Define the random process Z(t) = X cos(2 t) + Y sin(2 t) . (a) Compute the mean function μ Z (t) . (b) Find the probability density function of Z(t) at a specified time t 1 . (c) Compute the autocorrelation function R Z (t 1, t 2 ) . (d) Is Z(t) wide sense stationary? (Justify your answer). 3. Problem 5.40 in Proakis and Salehi. 4. Problem 5.44 in Proakis and Salehi. 5. Problem 5.50 in Proakis and Salehi....
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