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**Unformatted text preview: **Chapter 9 THE Z-TRANSFORM 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. Our discussions so far in the book have been limited to the time-domain. In other words, we have been studying the properties of signals and systems, as well as determining the responses of systems, by focusing on their descriptions in the time domain. There is much more to be learned by studying signals and systems in the transform domain. Apart from simplifying many of the calculations we have been performing, such as solving difference equations and computing convolutions, the transform domain representation will allow us to get a deeper understanding of signals and systems by analysing them in the frequency domain. However, before plunging into the frequency-domain characterization of signals and sys- tems, we need to spend some time first introducing the concept of the z- transform, and later of the discrete-time Fourier transform. The z-Transform The (two-sided or bilateral) z- transform of a sequence x ( n ) is denoted by X ( z ) and is defined by the series X ( z ) = X n =- x ( n ) z- n . In other words, a (negative or positive) power of z- 1 is associated with each entry x ( n ) of the sequence and the result is summed to provide the function X ( z ). The independent variable z is in general complex-valued. For example, the z-transform of the finite-duration sequence x ( n ) = {- 2 , 4 , 3 } is simply X ( z ) =- 2 z + 4 + 3 z- 1 Observe that in this case the function X ( z ) is defined for all values of z in the complex plane, except at the points z = 0 and . We thus say that the region of convergence of X ( z ) is the entire complex plan except 0 and . 79 80 The z-Transform Chapter 9 Region of Convergence (ROC) The series defining the z-transform of a sequence may or may not converge for a specific value of the complex number z . The set of all values of z for which the series is absolutely summable is called the region of convergence ( ROC) of X ( z ). That is, ROC = ( z C such that X n =- | x ( n ) z- n | < ) 1. Finite-duration sequences . As we saw in the example above, the ROC of a finite- duration sequence x ( n ) is always the entire complex plane except possibly at the points z = 0 and/or z = : a) The point z = 0 is excluded when x ( n ) is nonzero for some positive n . b) The point z = is excluded when x ( n ) is nonzero for some negative n ....

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