h5 - ECE 459 Handout 5 Fall 2000 HOMEWORK ASSIGNMENT 4...

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ECE 459 Fall 2000 Handout # 5 October 24, 2000 HOMEWORK ASSIGNMENT 4 Reading: Lecture notes (lectures 12-15), papers/books referenced in lecture notes. Due Date: Thursday, November 2, 2000 (in class) 1. ACF of Flat Fading Process { E ( t ) } . Consider the flat fading process { E ( t ) } for purely diffuse (Rayleigh) fading. We showed in class that for fixed t , E ( t ) is a zero mean, unit variance PCG ran- dom variable with a Rayleigh envelope. Since { E ( t ) } is Gaussian, all that is needed to characterize it completely (statistically) is its ACF. (a) Using equation (13.7) of the lecture notes, show that R E ( t + τ, t ) X n β 2 n e j 2 πf m τ cos θ n . This means that { E ( t ) } is (approximately) a stationary process. (b) Based on your answer to part (a), find the ACF’s and crosscorrelation functions of { E I ( t ) } and { E Q ( t ) } . Are { E I ( t ) } and { E Q ( t ) } independent processes? We can consider β 2 n to be the fraction of power gain of the channel corresponding to path n (or angle of arrival θ n ). Our goal now is to characterize the power gain in the multipath environment as a function of angle of arrival. Since the { β n } have been normalized so that n β 2 n = 1, we can define an angular power gain density p ( θ ) as p ( θ ) = X n β 2 n δ ( θ - θ n ) . (1) Then we can write R E ( τ ) = Z π - π p ( θ ) e j 2 πf m τ cos θ dθ . (2) Note that R E ( τ ) is strong function of angular power gain density.

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