113_1_chapter10

# 113_1_chapter10 - Chapter 10 PARTIAL FRACTIONS Copyright c...

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Chapter 10 PARTIAL FRACTIONS 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, [email protected] The z - transform of a sequence can be a rational function of z or not. In this chapter, we describe a useful method for inverse transforming rational z - transforms. The procedure is known as the partial fraction method. The method is best explained by examples. But let us Frst present some general facts. Partial Fractions Given a rational z - transform X ( z ), the method of partial fraction expansion expresses it as a sum of simpler transforms whose inversions are immediate. The simple transforms are usually terms of the forms: A, A z - p , A ( z - p ) 2 ,..., for some constants ( A,p ). These terms can also appear multiplied by powers of z or z - 1 . The main reason for choosing to express X ( z ) as a sum of such terms is that the inverse transforms of these terms are readily available. 1. ±or example, for a region of convergence of the form | z | > | p | we can invert every term as follows: A ←→ ( n ) A z - p ←→ Ap n - 1 u ( n - 1) A ( z - p ) 2 ←→ A · ( n - 1) p n - 2 u ( n - 1) Likewise, for a region of convergence of the form | z | < | p | we can invert every term as follows: A ←→ ( n ) A z - p ←→ - Ap n - 1 u ( - n ) A ( z - p ) 2 ←→ - A · ( n - 1) p n - 2 u ( - n ) 94

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95 2. Complex terms . Sometimes, a complex conjugate pair ( p,p * ) occurs that contributes with a sum of the form A z - p + A * z - p * . Its inverse z -transform for a region of convergence of the form | z | > | p | is [ Ap n - 1 + A * p * ( n - 1) ] u ( n - 1) . If we express { A,p } in polar forms, say A = | A | e , p = | p | e then the above result leads to the transform relation A z - p + A * z - p * ←→ 2 ·| A |·| p | n - 1 · cos[ ω ( n - 1) + α ] u ( n - 1) , | z | > | p | For a region of convergence of the form | z | < | p | , we obtain A z - p + A * z - p * ←→ - 2 ·| A |·| p | n - 1 · cos[ ω
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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_chapter10 - Chapter 10 PARTIAL FRACTIONS Copyright c...

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