1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

380 CHAPTER 9 • Z·TRANSFORMS AND APPLICATIONS

We can factor X(z) and Y(z) out of the s ummations and write this in an equi-

valent form

p Q

Y(z)(l + L:>pz-P) = X(z) :~::)qz-q.

p = l q=O

From equation (9-33) we obtain

H(z) = Y(z) = L,~=O bqz-q

X(z) 1 + L,:=I apz-P'

which leads to the following important definition.

(9-33)

De finition 9 .4 (Transfer Function) The transfer function corresponding to the

Pth order djfference equation (9-29) is given by

H _ Y(z) _ bo + b1z-^1 + ~z-^2 + ... + bq-1z-Q+1 + bqz-Q

(z) - X(z) - 1 + a1z-^1 + a2z-^2 + ... + ap-1z-P+I + apz-P ·

(9-34)

Formula (9-34) is the transfer function for an infinite impulse response filter

(IIR filter). In t he special case when the denominator is unity it becomes the

transfer function for a finite impulse response filter (FIR filter).

I Definition 9 .5 (Unit-Sample Response) T he sequence h[nl = 3-^1 [H(z)I corre-

sponding to the transfer function H(z) = ~~:~ is called the unit-sample response.

Another important use of the transfer function is to st udy how a filter af-

fects various frequencies. In practice, a continuous-time signal is sampled at a