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HW4_ece220_2007_solution

HW4_ece220_2007_solution - Homework 4 Solution(ECE220 Fall...

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Unformatted text preview: Homework 4 Solution (ECE220 - Fall 2007) 1. (25 points) Convolution and power The auto correlation function is defined as: R x [ l ] , ∞ X k =-∞ x * [ k ] x [ k + l ] R x ( τ ) , Z + ∞-∞ x * ( t ) x ( t + τ ) dt. There was an error on the homework set, it should have been dt not dτ . (a) (10 points) Show that for any signal x [ n ], R x [0] = E x and that similarly, for analog signals x ( t ), R x (0) = E x . Solution: R x [0] = ∞ X k =-∞ x * [ k ] x [ k + 0] = ∞ X k =-∞ | x [ k ] | 2 = E x R x (0) = Z + ∞-∞ x * ( t ) x ( t + 0) dt = Z + ∞-∞ | x ( t ) | 2 dt = E x (b) (10 points) The following series of equations is the proof of the following lemma. Lemma Let a signal have x [ n ] autocorrelation R x [ l ] and an LTI system with impulse response h [ n ] have autocorrelation R h [ l ]. Then: R y [ l ] = R h [ l ] * R x [ l ] . (1) Proof: R y [ l ] = + ∞ X n =-∞ y * [ n ] y [ n + l ] (2) = + ∞ X n =-∞ ˆ + ∞ X k =-∞ h * [ k ] x * [ n- k ] !ˆ + ∞ X m =-∞ h [ m ] x [ n + l- m ] ! (3) ( m = k + p ) z}|{ = + ∞ X p =-∞ ˆ + ∞ X k =-∞ h * [ k ] h [ k + p ] !ˆ + ∞ X n =-∞ x * [ n- k ] x [ n- k + l- p ] ! (4) ( v = n- k ) z}|{ = ... (5) = R h [ l ] * R x [ l ] . (6) q.e.d. Say why you can go from (2) to (3). Say what property of products and sums is used, in addition to the change of variable m = k + p , to go from (3) to (4). Complete the right, to go from (3) to (4)....
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HW4_ece220_2007_solution - Homework 4 Solution(ECE220 Fall...

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