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FinalExamF07

# FinalExamF07 - EE538 Final Exam Mon Dec 10 8-10 am RHPH 127...

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EE538 Final Exam Fall 2007 Mon, Dec 10, 8-10 am RHPH 127 Dec. 10, 2007 Cover Sheet Test Duration: 120 minutes. Open Book but Closed Notes. Calculators allowed!! This test contains five problems. Each of the five problems are equally weighted. All work should be done in the blue books provided. You must show all work for each problem to receive full credit. Do not return this test sheet, just return the blue books. 1

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EE538 Digital Signal Processing I Final Exam Fall 2007 Problem 1. This problem deals with the properties of deterministic autocorrelation. (a) Consider the symmetric sequence below which is one for 2 m 2 and zero for | m | > 2. Is this is a valid autocorrelation sequence? Justify your answer. r xx [ m ] = u [ m + 2] u [ m 3] (b) Consider the symmetric sequence below. Is this is a valid autocorrelation sequence? You need to explain your answer. r xx [ m ] = (3 − | m | )( u [ m + 2] u [ m 3]) (c) Let r xx [ m ] denote the autocorrelation sequence for the DT signal x [ n ]. Let y [ n ] = x [ n n o ], where n o is an integer. Let r yy [ m ] denote the autocorrelation sequence for the DT signal y [ n ]. Derive an expression relating r yy [ m ] and r xx [ m ]. That is, how is r yy [ m ] related to r xx [ m ]? (d) Let r xx [ m ] denote the autocorrelation sequence for the DT signal x [ n ]. Let y [ n ] = e j ( ω 0 n + θ ) x [ n ], where ω o is some frequency and θ is some phase value. Let r yy [ m ] denote the autocorrelation sequence for the DT signal y [ n ]. Derive an expression relating r yy [ m ] and r xx [ m ]. That is, how is r yy [ m ] related to r xx [ m ]? (e) Consider that the signal x [ n ] below, where a = 1 2 , is the input to the LTI system described by the difference equation in equation (2) below. x [ n ] = a n u [ n ] 1 a a n 1 u [ n 1] (1) y [ n ] = 1 4 y [ n 1] + x [ n ] (2) (i) Determine a closed-form analytical expression for the auto-correlation r xx [ m ] for x [ n ], when a = 1 2 . (Hint: examine the Z-Transform of x [ n ].) (ii) Determine a closed-form analytical expression for the cross-correlation r yx [ m ] be- tween the input and output.
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FinalExamF07 - EE538 Final Exam Mon Dec 10 8-10 am RHPH 127...

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