113_1_chapter10

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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, sayed@ee.ucla.edu. 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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 8

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online