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FinalExamF08

# FinalExamF08 - Digital Signal Processing I ECE538 Final...

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Digital Signal Processing I Final Exam Fall 2008 ECE538 17 Dec.. 2008 Cover Sheet Test Duration: 120 minutes. Open Book but Closed Notes. Calculators NOT allowed. This test contains FIVE problems. All work should be done on blank 8.5” x 11” white sheets of paper (NOT provided). Do not return this test sheet, just return your answer sheets. 1

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Digital Signal Processing I Final Exam Fall 2008 ECE538 DSP I 17 Dec. 2008 Problem 1. [20 pts] You are given the matrix inversion lemma: bracketleftBigg A B C D bracketrightBigg - 1 = bracketleftBigg A - 1 + A - 1 B ( D - CA - 1 B ) - 1 CA - 1 - A - 1 B ( D - CA - 1 B ) - 1 - ( D - CA - 1 B ) - 1 CA - 1 ( D - CA - 1 B ) - 1 bracketrightBigg A symmetric-Toeplitz matrix is formed from the first three autocorrelation values of an AR process as r xx [0] = 1 / 3, r xx [1] = 1 / 6, and r xx [2] = 1 / 12 as R 3 = 1 / 3 1 / 6 1 / 12 1 / 6 1 / 3 1 / 6 1 / 12 1 / 6 1 / 3 (a) Use the matrix inversion lemma to compute the inverse of R 3 in terms of the inverse of R 2 = bracketleftBigg 1 / 3 1 / 6 1 / 6 1 / 3 bracketrightBigg R - 1 2 = bracketleftBigg 4 - 2 - 2 4 bracketrightBigg (b) Use the matrix inversion lemma to compute the solution a 3 = - R - 1 3 r 3 where: r 3 = 1 / 6 1 / 12 1 / 24 in terms of a 2 = - R - 1 2 r 2 = bracketleftBigg - 1 / 2 0 bracketrightBigg where: r 2 = bracketleftBigg 1 / 6 1 / 12 bracketrightBigg (c) What is the model order p of the AR process? 2
Problem 2. [20 points] A signal x [ n ] of length 12 is broken up into two overlapping blocks of length 8, denoted x 1 [ n ] and x 2 [ n ], respectively, for the purposes of filtering with h [ n ] = {- 1 , 2 , - 1 } of length M = 3 via the overlap-SAVE method. The last M - 1 = 2 points of x 1 [ n

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FinalExamF08 - Digital Signal Processing I ECE538 Final...

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