EE562aFinSolF08

EE562aFinSolF08 - EE 562a 1. Distortion 101 Final Solution...

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EE 562a Final Solution December 11, 2008 1 1. Distortion 101 (a) Is the transformation of s ( u,t ) into d ( u,t ) a linear transformation? YES Is d ( u,t ) a wide-sense stationary random process? YES Is d ( u,t ) a strictly stationary random process? YES Is d ( u,t ) a Gaussian random process? YES (b) Use the fact that d ( u,t ) = A · r ( u,t ) - s ( u,t ) and make input exp( i 2 πft ) to determine the system function (e-value). G ( f ) = A · H ( f ) - 1 (c) Use S d ( f ) = | G ( f ) | 2 S s ( f ) and substitute for G ( f ) to get the result. S d ( f ) = | A · H ( f ) - 1 | 2 S s ( f ) (d) E {| d ( u,t ) | 2 } = R d (0) = Z -∞ S d ( f ) df = Z -∞ | A · H ( f ) - 1 | 2 S s ( f ) df = A 2 Z -∞ | H ( f ) | 2 S s ( f ) df | {z } a - 2 A Z -∞ Re { H ( f ) } S s ( f ) df | {z } b + Z -∞ S s ( f ) df | {z } c where Re {·} is the real-part operator. (Many of you were stuck at this point because you did not take the constant A outside the integral.) At this point one can complete the square or differentiate with respect to A and equate to zero, or complete the square with respect to A to get the answer. In completing the square, note that a · A 2 - 2 b · A + c = a ( A - b/a ) 2 + c - b 2 /a . This form gives the best choice of A and the minimum residual distortion. A min = b a = R -∞ Re { H ( f ) } S s ( f ) df R -∞ | H ( f ) | 2 S s ( f ) df (e) Evaluate c - b 2 /a in (d) to find the minimum mean-squared distortion. E {| d ( u,t ) | 2 } min = Z -∞ S s ( f ) df - h R -∞ Re { H ( f ) } S s ( f ) df i 2 R -∞ | H ( f ) | 2 S s ( f ) df Note : Invoking the general LMMSE theory to solve this problem seems like an unnecessary
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EE562aFinSolF08 - EE 562a 1. Distortion 101 Final Solution...

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