FinalExamF10 - Digital Signal Processing I Final Exam Fall...

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Unformatted text preview: Digital Signal Processing I Final Exam Fall 2010 ECE538 13 Dec.. 2010 Cover Sheet Test Duration: 120 minutes. Open Book but Closed Notes. Calculators NOT allowed. This test contains FOUR 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. (a) Let x [ n ] and y [ n ] be real-valued sequences both of which are even-symmetric: x [ n ] = x [- n ] and y [ n ] = y [- n ]. Under these conditions, prove that r xy [ ℓ ] = r yx [ ℓ ] for all ℓ . (b) Express the autocorrelation sequence r zz [ ℓ ] for the complex-valued signal z [ n ] = x [ n ]+ jy [ n ] where x [ n ] and y [ n ] are real-valued sequences, in terms of r xx [ ℓ ], r xy [ ℓ ], r yx [ ℓ ], and r yy [ ℓ ]. (c) Determine a closed-form expression for the autocorrelation sequence r xx [ ℓ ] for the signal x [ n ] below. x [ n ] = braceleftBigg sin( π 2 n ) πn bracerightBigg (1) (d) Determine a closed-form expression for the autocorrelation sequence r yy [ ℓ ] for the signal y [ n ] below. y [ n ] = (- 1) n x [ n ] = (- 1) n braceleftBigg sin( π 2 n ) πn bracerightBigg (2) (e) Determine a closed-form expression for the autocorrelation sequence r zz [ ℓ ] for the complex-valued signal z [ n ] formed with x [ n ] and y [ n ] defined above as the real and imaginary parts, respectively, as defined below. You must show all work and simplify as much as possible. z [ n ] = x [ n ] + jy [ n ] (3) (f) Plot r zz [ ℓ ]. 2 Problem 2. For all parts of this problem, the reconstructed spectrum is computed according to the equation below: Y r ( ω ) = N- 1 summationdisplay k =0 Y N ( k ) sin bracketleftBig N 2 parenleftBig ω- 2 πk N parenrightBigbracketrightBig N sin bracketleftBig 1 2 parenleftBig ω- 2 πk N parenrightBigbracketrightBig e- j N-...
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This note was uploaded on 02/19/2012 for the course ECE 538 taught by Professor Zoltolski during the Fall '08 term at Purdue University.

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FinalExamF10 - Digital Signal Processing I Final Exam Fall...

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