Digital Signal Processing Systems, Basic Filtering Types, and Digital Filter Realizations - Digital Signal Processing: Fundamentals and Applications - page 166

6.2 DIFFERENCE EQUATION AND TRANSFER FUNCTION
To proceed in this section, Equation (6.1) is rewritten as
yðnÞ¼b 0 xðnÞþb 1 xðn 1 Þþ/þ b M xðn MÞ
a 1 yðn 1 Þ/ a N yðn NÞ

With an assumption that all initial conditions of this system are zero, and with XðzÞ and YðzÞ denoting
the z-transforms of xðnÞ and yðnÞ , respectively, taking the z-transform of Equation (6.1) yields
YðzÞ¼b 0 XðzÞþb 1 XðzÞz 1

þ/þ b M XðzÞz M

(6.3)

a 1 YðzÞz 1 / a N YðzÞz N

Rearranging Equation (6.3) , we obtain
HðzÞ¼ YðzÞ

XðzÞ ¼ b 0 þ b 1 z 1

þ/þ b M z M

þ/þ a N z N ¼ BðzÞ

(6.4)

1 þ a 1 z 1 AðzÞ

where HðzÞ is defined as the transfer function with its numerator and denominator polynomials defined
below:

BðzÞ¼b 0 þ b 1 z 1 þ/þ b M z M (6.5)

AðzÞ¼ 1 þ a 1 z 1 þ/þ a N z N

(6.6)

Clearly the z-transfer function is defined as

z-transform of the output
z-transform of the input

ratio ¼

In DSP applications, given the difference equation, we can develop the z-transfer function and
represent the digital filter in the z-domain as shown in Figure 6.3 . Then the stability and frequency
response can be examined based on the developed transfer function.

z-transform input z-transform output
Digital filter transfer function

()

()

FIGURE 6.3

Digital filter transfer function.

EXAMPLE 6.3
A DSP system is described by the following difference equation:
yðnÞ¼xðnÞxðn 2Þ1:3yðn 1Þ0:36yðn 2Þ