Digital Signal Processing Systems, Basic Filtering Types, and Digital Filter Realizations - Digital Signal Processing: Fundamentals and Applications - page 169

The transfer function HðzÞ can be factored into the pole-zero form:

b 0 ð z z 1 Þð z z 2 Þ/ð z z M Þ
ðz p 1 Þðz p 2 Þ/ðz p N Þ

HðzÞ¼ (6.7)

where the zeros z i can be found by solving roots of the numerator polynomial, while the poles p i can be solved for
the roots of the denominator polynomial.

EXAMPLE 6.6
Consider the following transfer functions:

1 z 2
1 þ 1:3z 1 þ 0:36z 2

HðzÞ¼

Convert it into the pole-zero form.
Solution:
We first multiply the numerator and denominator polynomials by z 2 to achieve the advanced form in which both
numerator and denominator polynomials have positive powers of z, that is,

ð1 z 2 Þz 2
ð1 þ 1:3z 1 þ 0:36z 2 Þz 2 ¼

z 2 1
z 2 þ 1:3z þ 0:36

HðzÞ¼

Letting z 2 1 ¼ 0, we get z ¼ 1andz ¼1. Setting z 2 þ 1:3z þ 0:36 ¼ 0leadstoz ¼0:4andz ¼0:9.
We then can write numerator and denominator polynomials in the factored form to obtain the pole-zero form:
HðzÞ¼ ð z 1Þð z þ 1Þ
ðz þ 0:4Þðz þ 0:9Þ

6.2.1 Impulse Response, Step Response, and System Response
The impulse response hðnÞ of the DSP system HðzÞ can be obtained by solving its difference equation
using a unit impulse input dðnÞ . With the help of the z-transform and noticing that XðzÞ¼Zfdð n Þg 1,
we yield

hðnÞ¼Z 1 fHðzÞXðzÞg ¼ Z 1

fHðzÞg

(6.8)

Similarly, for a step input, we can determine step response assuming zero initial conditions. Letting

z
z 1

XðzÞ¼Z½uðnÞ ¼

the step response can be found as

yðnÞ¼Z 1 HðzÞ z

z 1

(6.9)

Furthermore, the z-transform of the general system response is given by
YðzÞ¼HðzÞXðzÞ

(6.10)