# hw4 - Homework 4 ECE 220 Due 1 A LTI system is described...

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Homework 4 ECE 220 Due: March 29, 2005 1. A LTI system is described the the difference equation y [ n ] = 1 3 ( x [ n ] + x [ n - 1] + x [ n - 2]) (a) Determine the system function H ( z ) for this system. (b) Plot the poles and zeros of H ( z ) in the z-plane. (c) From H ( z ) obtain an expression for H ( e j ˆ ω ), the frequency response of the system. (d) Sketch the frequency response (magnitude and phase) as a function of frequency for - π ˆ ω π . (e) What is the output of the system if the input is x [ n ] = 2 + cos( π 4 n ) - 2 cos( π 3 n ) 2. Figure 1 depicts a cascade connection of two LTI systems as described in Section 7-5.1 of your text. (a) Use z -transforms to show that the system function for the overall system (from x [ n ] to y [ n ]) is H ( z ) = H 1 ( z ) H 2 ( z ), where Y ( z ) = H ( z ) X ( z ). (b) Use the result of (2a) to show that the order of the system is not important. That is, show that for the same input x [ n ] to the systems, the overall outputs are the same so that w [ n ] = y [ n ]. (c) Suppose that both systems above are 3-point averages described by the difference equation in Problem 1. Determine the system function H ( z ) = H 1 ( z ) H 2 ( z ) for the overall system.

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