**Unformatted text preview: **UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 410 Digital Signal Processing Homework 8 Due Wednesday, October 22, 2008 Prof. Bresler / Prof. Jones 1. (25 points) Consider the system shown in the diagram below: where 2 1 and . a) Find the impulse response, of the overall system. (Hint: z-transform) b) Determine the difference equation representation of the overall system that relates the output to the input. c) Is the overall system causal? Is it BIBO stable? Explain your reasoning for full credit. d) Determine the frequency response, , of the overall system. e) Plot the magnitude and phase response of the overall system for 0 using MATLAB. 2. (15 points) Determine the frequency response function their impulse response. Plot the magnitude response | interval ,. a) b) c) 0.9| | 0.4 0.5 || 0.5 cos 0.1 for each of the LSI systems described by | and the phase response over the 3. (30 points) A LSI system can be described by the following difference equation: a) Create a function in MATLAB that takes inputs vectors , and , where is the vector of the coefficients , is the vector of the coefficients and is a vector of the frequencies on the interval , , that outputs the frequency response, , in a vector. The frequency response, , should only be evaluated at the frequencies that the vector provides. your function should have a header line like the following: function [H] = diffeq2freqresp(b,a,w) b) Use the function developed in part (a) to plot the magnitude and phase for each of the following systems: i) ii) ∑ 2 0.95 1 0.9025 2 4. (15 points) A LSI system is described by the following difference equation: 2 0.81 2 Determine the steady-state response of this system to the following input signals: a) x n b) x n c) x n 5 ∑ 10 k 1 3 cos 1 cos 2 sin 5. (15 points) A first–order feedback system is shown below with a constant feedback gain, K. x[n] + - p[n] y[n] K The plant’s impulse response is given by 2 . For what values of K is the system stable? NOTE: Please hand in any relevant plots and your MATLAB functions or scripts. ...

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