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# HW7 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department...

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Unformatted text preview: UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 310 Digital Signal Processing Problem Set 7 Fall 2003 Due: Wednesday, October 15, 2003 Problem 7.1 Compute the discrete-time Fourier transform X d ( ω ) of the following signals: 1. x ( n ) = α n sin( ω n ) u ( n ) for | α | < 1 2. x ( n ) = 2 n u (- n ) 3. x ( n ) = sin π 2 n πn sin π 4 n πn 4. x ( n ) = ( 1 2 ) | n | 5. x ( n ) = n ( 1 2 ) | n | 6. x ( n ) = (- 1) n Problem 7.2 Compute the signal x ( n ) for the following spectra 1. X d ( ω ) = cos 2 ( ω ) 2. X d ( ω ) = e- jω 1+ 1 6 e- jω- 1 6 e- j 2 ω Problem 7.3 Consider an LSI-system with impulse response h ( n ) = sin π 3 n πn Compute the output signal y ( n ) for the following input signals: 1. x ( n ) = ∑ ∞ k =-∞ δ ( n- 8 k ) 2. x ( n ) = δ ( n + 1) + δ ( n- 1) 3. x ( n ) = sin π 4 n πn 4. x ( n ) = sin π 4 n πn cos( 3 π 4 n ) Problem 7.4 Suppose X d ( ω ) = n 1 , | ω | ≤ π 6 , π 6 < | ω | ≤ π 1. Sketch X d ( ω ) 2. Find { x ( n ) } ∞ n =-∞ 3. If y ( n ) = e j π 3 n x ( n ), sketch Y d ( ω ) 4. If y ( n ) = cos( π 3 n ) x ( n ), sketch Y d ( ω ) Problem 7.5 Let x ( n ) = n 1 N , ≤ n ≤ N- 1 , else 1. Find closed-form expressions (no sums) for | X d ( ω ) | , and ∠ X d ( ω ) 2. For N = 6, plot | X d ( ω ) | . How will the shape of | X d ( ω ) | change if N increases?change if N increases?...
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HW7 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department...

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