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20072ee132A_1_hwk3 - R Z 2 Sum of two modulated Gaussians...

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EE132A, Spring 2007 Communication Systems Prof. John Villasenor Handout # 8 TA: Choo Chin (Jeffrey) Tan Homework 3 Assigned: Monday, April 16, 2007 Due: Monday, April 23, 2007 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. 6. Problem 7.2 in Proakis and Salehi....
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