# HW4sol_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 Homework 4 Solution Prof. Bresler, Prof. Jones September 24, 2008 Problem 1 (10 points) Complete the following signal ﬂow diagram for 8-point, radix-2 decimation-in-time FFT algorithm, and use it to compute the DFT of the 8-pt sequence x [ n ] = {- 1 , - 4 , - 2 , - 3 , 1 , 4 , 2 , 3 } by hand . Specify the value of the input sequence, output sequence, and the signal values after each stage, i.e. , at all the points marked by the solid circles in the diagram. Note: W 0 2 = 1; W 0 4 = 1; W 1 4 = - j ; W 0 8 = 1; W 1 8 = 1 2 - j 1 2 ; W 2 8 = - j ; W 3 8 = - 1 2 - j 1 2 Refer Figure 1. Problem 2 (10 points) The sequence x [ n ] = cos ± π 8 n ² , -∞ < n < was obtained by sampling a continuous-time signal x c ( t ) = cos(Ω 0 t ) , -∞ < t < at a sampling rate of 1024 samples/sec. What are three possible values of Ω 0 that can result in the same sequence of x [ n ]? Since w = Ω T and T = 1 1024 secs, the signal frequency is Ω 0 = ± π 8 + 2 πn ² 1 T = ± π 8 + 2 πn ² 1024 = 128 π + 2048 πn 1

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where n is any integer. Since cos function is symmetric, the following can be answers: Ω 0 = - 128 π + 2048 πn. Therefore, the possible three signal frequencies could be Ω 0 = 128 π, 2176 π, 4224 π, when n = 0 , 1 , 2. Problem 3 (20 points) The continuous-time signal x a ( t ) = cos(600 πt ) is sampled with a sampling period T to obtain a discrete-time signal x d [ n ] = x a ( nT ) . (a) Compute and sketch the magnitude of the continuous-time Fourier transform of x a ( t ) and the discrete-time Fourier transform of x d [ n ] for T = 1 ms . Continuous-time Fourier transform(CTFT) of
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HW4sol_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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