Digital Signal Processing Systems, Basic Filtering Types, and Digital Filter Realizations - Digital Signal Processing: Fundamentals and Applications - page 167

Find the transfer function HðzÞ, the denominator polynomial AðzÞ, and the numerator polynomial BðzÞ.
Solution:
Taking the z-transform on both sides of the previous difference equation, we obtain
Y ðzÞ¼X ðzÞX ðzÞz 2 1:3Y ðzÞz 1 0:36Y ðzÞz 2

Moving the last two terms to the left side of the difference equation and factoring Y ðzÞ on the left side and X ðzÞ on
the right side, we obtain

Y ðzÞð1 þ 1:3z 1 þ 0:36z 2 Þ¼ð1 z 2 ÞX ðzÞ

Therefore, the transfer function, which is the ratio of Y ðzÞ over X ðzÞ, can be found to be

1 z 2
1 þ 1:3z 1 þ 0:36z 2

Y ð z Þ
X ðzÞ ¼

HðzÞ¼

From the derived transfer function HðzÞ, we can obtain the denominator polynomial and numerator polynomial as
AðzÞ¼1 þ 1:3z 1 þ 0:36z 2

and

BðzÞ¼1 z 2

The difference equation and its transfer function, as well as the stability issue of the linear time-invariant system,
will be discussed in the following sections

EXAMPLE 6.4
A digital system is described by the following difference equation:

yðnÞ¼xðnÞ0:5xðn 1Þþ0:36xðn 2Þ

Find the transfer function HðzÞ, the denominator polynomial AðzÞ, and the numerator polynomial BðzÞ.
Solution:
Taking the z-transform on both sides of the previous difference equation, we obtain

Y ðzÞ¼X ðzÞ0:5X ðzÞz 2 þ 0:36X ðzÞz 2

Therefore, the transfer function, that is the ratio of Y ðzÞ to X ðzÞ, can be found as

Y ð z Þ
X ðzÞ ¼ 1 0:5z 1 þ 0:36z 2

HðzÞ¼

From the derived transfer function HðzÞ, it follows that

AðzÞ¼1

BðzÞ¼1 0:5z 1 þ 0:36z 2