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Quiz2sol_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 410 Digital Signal Processing Quiz 2 Thursday, September 18, 2008 Student Name: Prof. Bresler, Prof. Jones INSTRUCTIONS You may not use any calculators, cell phones, earphones (or other forms of electronic media) for this quiz. You may use one side of one sheet of handwritten notes. Show all your work to get full credit for your answers. When you are asked to “ calculate ”, “ determine or “ find ”, this means providing closed-form expressions, without summation or integral expressions. Problem 1 (20 points) Let X d ( ω ) = ( 0 | ω | ≤ ω c 1 ω c < | ω | < π where 0 < ω c < π . Determine x [ n ], the inverse DTFT of X d ( ω ). (Specify x [ n ] for all n Z and show your work step by step for complete credit.) Solution: x [ n ] = 1 2 π Z π - π X d ( ω ) · e jωn = 1 2 π •Z - ω c - π e jωn + Z π ω c e jωn Now for n = 0, we have x [0] = 2( π - ω c ) 2 π = 1 - ω c π (1) And, for n 6 = 0, we compute x [ n ] = 1 2 πjn e jωn fl fl - ω c - π + 1 2 πjn e jωn fl fl π ω c = 1 2 πjn h e - c n - » » » e - jπn + ' '' e jπn - e c n i = - ω c π sinc( ω c n ) , n 6 = 0 (2) Combining (1) and (2) we have x [ n ] = δ [ n ] - ω c π sinc( ω c n ) 1
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Alternative approach 1: Recall the modulating property of DTFT, using which we can show that x [ n ] = 2 y [ n ] · cos π + ω c 2 n where Y d ( ω ) = ( 1 , | ω | ≤ π - ω c 2 0 , otherwise y [ n ] = π - ω c 2 π sinc π - ω c 2 n
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