FinalExamF09

FinalExamF09 - Digital Signal Processing I Final Exam Fall...

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Unformatted text preview: Digital Signal Processing I Final Exam Fall 2009 ECE538 15 Dec. . 2009 Cover Sheet Test Duration: 120 minutes. Open Book but Closed Notes. Calculators NOT allowed. This test contains FIVE problems. All work should be done in the blue book provided. Do not return this test sheet, just return your blue book. 1 Problem 1. Consider a fnite-length sinewave oF the Form below where k o is an integer in the range 0 k o N-1. x [ n ] = e j 2 ko N n { u [ n ]-u [ n-N ] } (1) In addition, h [ n ] is a causal IR flter oF length L , where L < N . In this problem y [ n ] is the linear convolution oF the causal sinewave oF length N in Equation (1) with a causal IR flter oF length L , where L < N . y [ n ] = x [ n ] * h [ n ] (a) The region 0 n L-1 corresponds to partial overlap . The convolution sum can be written as: y [ n ] = ?? s k =?? h [ k ] x [ n-k ] partial overlap: n L-1 (2) Determine the upper and lower limits in the convolution sum above For 0 n L-1. (b) The region L n N-1 corresponds to full overlap. The convolution sum is: y [ n ] = ?? s k =?? h [ k ] x [ n-k ] full overlap: L n N-1 (3) (i) Determine the upper & lower limits in the convolution sum For L n N-1. (ii) Substituting x [ n ] in Eqn (1), show that For this range y [ n ] simplifes to: y [ n ] = H N ( k o ) e j 2 ko N n For L n N-1 (4) where H N ( k ) is the N-point DT oF h [ n ] evaluated at k = k o . To get the points, you must show all work and explain all details. (c) The region N n N + L-2 corresponds to partial overlap . The convolution sum: y [ n ] = ?? s k =?? h [ k ] x [ n-k ] partial overlap: N n N + L-2 (5) Determine the upper & lower limits in the convolution sum For N n N + L-2. (d) Add the two regions oF partial overlap at the beginning and end to Form: z [ n ] = y [ n ] + y [ n + N ] = ?? s k =?? h [ k ] x [ n-k ] For: 0 n L-1 (6) (i) Determine the upper and lower limits in the convolution sum above....
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FinalExamF09 - Digital Signal Processing I Final Exam Fall...

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