Signals and Systems 17 October 2012 Fall 2012 Quiz...

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Signals and Systems 17 October 2012 Fall 2012 Quiz: Wednesday 24 October 2012 Homework Assignment #9 1. Consider the DT, LTI system with the z -domain transfer function 2 2 ) 95 . 0 ( ) 6 / cos( ) 95 . 0 ( 2 ) ( + = z z z z H π (a) Plot the magnitude and phase of the frequency response of the system. (b) Program the corresponding difference equation, and iterate to determine the responses to the inputs ) ( ) 12 / cos( ) ( 1 k u k k x s = π , ) ( ) 6 / cos( ) ( 2 k u k k x s = π , ) ( ) 3 / cos( ) ( 3 k u k k x s = π . (c) Explain the amplitudes of the steady-state responses from part (b) in terms of the system frequency response. 2. Consider the sequence x 1 ( k ) = cos( k π /6). (a) Find the transfer function and the difference equation for a 2 nd -order FIR filter that has unity d.c. gain, but for which the given input x ( k ) produces zero forced response. (Your filter will have a “notch” at the given frequency, and H (1) = 1.) (b) Find the transfer function and the difference equation for a stable 2 nd -order filter that has unity d.c. gain, but for which the given input x ( k ) produces a large forced response.
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