113_1_chapter11 - Chapter 11 TRANSFER FUNCTIONS Copyright c...

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Unformatted text preview: Chapter 11 TRANSFER FUNCTIONS Copyright c 1996 by Ali H. Sayed. All rights reserved. These notes are distributed only to the students attending the undergraduate DSP course EE113 in the Electrical Engineering Department at UCLA. The notes cannot be reproduced without written consent from the instructor: Prof. A. H. Sayed, Electrical Engineering Department, UCLA, CA 90095, sayed@ee.ucla.edu. The z- transform is a very important tool in the study of linear time-invariant systems, and especially in the study of systems that are described by constant-coefficient linear dif- ference equations . It allows us to tackle several questions very directly, as we are going to illustrate in this chapter. Transfer Function The transfer function of an LTI system is defined as the z-transform of its impulse response sequence h ( n ). We shall denote it by H ( z ) so that H ( z ) = k =- h ( k ) z- k We must always associate a region of convergence ( ROC) with a transfer function. For example, the transfer function of an LTI system with impulse response sequence n u ( n ) is H ( z ) = z z- for | z | > | | . In other words, given h ( n ) we can determine H ( z ) by simply evaluating its z- transform. Determining H ( z ) from Difference Equations Actually, we do not need to know the impulse response sequence of an LTI system in order to determine its transfer function. For instance, when an LTI system is described in terms of a relaxed constant-coefficient difference equation, we can determine its transfer function more directly as illustrated by the following example. Consider a relaxed and causal system that is described by the difference equation y ( n )- 1 2 y ( n- 1) = x ( n ) . Since the system is relaxed, and since this is a constant-coefficient difference equation, we know that it describes an LTI system. Now taking z- transforms of both sides of the 102 103 equation, and using the properties of z- transforms, we obtain the algebraic equation Y ( z )- 1 2 z- 1 Y ( z ) = X ( z ) . Here Y ( z ) denotes the z- transform of { y ( n ) } and X ( z ) denotes the z- transform of { x ( n ) } . The { x ( n ) ,y ( n ) } denote an arbitrary input-output pair. The above equation can be solved to yield Y ( z ) X ( z ) = 1 1- 1 2 z- 1 . We claim that the ratio Y ( z ) X ( z ) (of output to input z- transforms) is the transfer function H ( z ) of the system. To see this, assume x ( n ) = ( n ) then y ( n ) = h ( n ). Hence, if X ( z ) = 1 we obtain Y ( z ) = H ( z ), which shows that we must have H ( z ) = 1 1- 1 2 z- 1 . Since the system is assumed to be causal, the impulse response sequence is necessarily right- sided and the ROC has to be | z | > 1 / 2....
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This note was uploaded on 11/07/2009 for the course EE 113 taught by Professor Walker during the Spring '08 term at UCLA.

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113_1_chapter11 - Chapter 11 TRANSFER FUNCTIONS Copyright c...

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