1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

340 CHAPTER 9 • Z· TRANSFORMS AND APPLICATIONS

z-transform, provided that it is a. rational function; that is,

X(z) = P(z) = bo + b1z^1 + b2z~ + ... + bp_ 1 zP-l + bvz" ,

Q(z) a1l + a1z^1 + a>iz^2 + ... + a 9 -1zq- L + a 9 z9

(9-3)

where P(z) and Q(z) a.re polynomials of degree p and q, respectively.

From our knowledge of rational functions, we see that an admissible z-

tra.nsform is defined everywhere in the complex plane except at a finite number of

isolated singularities that are poles and occur at the points where Q(z) = O. The

Laurent series expansion in (9-1) can be obtained by a partial fraction manipu-

lation and followed by geometric series expansions in powers of ~. However, the

important feature of formula (9-3) is the calculation of the inverse z-tra.nsform

via residues. For convenience we restate this concept.

t Corollary 9.1 (Inverse z-transform) Let X(z) be the z-transform of the se-

quence {x,.}. Then by Theorems 9.1 and 9.2 x,. is given by the formula

k

Xn = x[n] = 3-^1 [X(z)] = l:Res[X(z)z"-^1 ,zj]

j=l

where z1, z2, .. ., zk a.re the poles of f(z) = X(z)z"-^1.

t Corollary 9.2 (Inverse z-transform) Let X(z) be the z-transform of t he se-

quence. If X(z) has simple poles at the points z1, z2, ... , Zk then Xn is given by

the formula.

k

Xn = x[n] = 3 -^1 [X(z)] = L( Jim (z - z;)X(z)z"-^1 ).

z-zi

i=l

• EXAMPLE 9.1 Find the z-transform of the unit pulse or impulse sequence

x = o[n] = { 1 for n = 0

n 0 otherwise.