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Unformatted text preview: UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 410 Digital Signal Processing Homework 2 Due Wednesday, September 10, 2008 Prof. Bresler / Prof. Jones 1. Find a closed-form expression (no sums) for the DTFT of the following sequences: (a) (b) (c) (d) (e) 8 , where | | 1 5 , where | | 1 10 4 , where | | 1 0.6 cos 2. The DTFT of x[n] is as given in the following figure. Determine x[n] for each case. (a)
Xd(ω) 2 1 ‐π − 3π − 4 π
2 ω 0
π 2 3π 4 π (b) Xd(ω) 2 1 ω ‐π 0 π 3. Find a closed-form expression for the inverse DTFT of the following sequence: 4 2 sin 2 10cos 6 . Determine in terms 4. Let [n] be an arbitrary signal, not necessarily real-valued, with DTFT of for the following : (a) (b) (c) 5. Let , where * denotes complex conjugate 3 be a signal with DTFT as shown in the following figure. Determine and sketch the DTFT of cos . 6. Consider the complex sequence . | for the case of large N. and sketch | (a) Find a closed-form expression (no sums) for Explain how does the plot change as N decreases. (b) Find a closed-form expression (no sums) for the N-point DFT of the finite length sequence , 0,1, … , 1. Sketch | | for the case of large N. (c) Find the DFT of for the case of , where is an integer. Sketch | |. 7. Find a closed-form expression (no sums) for the DFT of the following finite-length sequence of length : 0, 1, You can assume that is even. ,0 ,0 1 1 ...
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