Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 108

2.6.7 IIR Digital Filters

2.6.7.1 Structure and Implementation of IIR Digital Filters

An IIR digital filter is a system whose impulse response h(n) has an infinite
number of non-zero samples. To implement IIR filters in practice, recursion is
often used. That is, to create the output, not only are previous values of the input
used, but also previous values of the output. This kind of recursive implementation
is essentially a feedback system. The general formula for the I/O relations of a
recursive IIR filter are:

y ð n Þ¼ b o x ð n Þþ b 1 x ð n 1 Þþþ b M x ð n M Þ
a 1 y ð n 1 Þ a N y ð n N Þ:

: ð 2 : 30 Þ

The above equation is often referred to as a difference equation. Taking the
z-transform of both sides of the equation yields:

Y ð n Þ¼ b o X ð z Þþ b 1 z 1 X ð z Þþþ b M z M X ð z Þ
a 1 z 1 Y ð z Þ a N z N Y ð z Þ:
)H ð z Þ¼ Y ð z Þ

ð 2 : 31 Þ

P i ¼ 0 b i z i
1 þ P k ¼ 1 a k z k :

X ð z Þ ¼ b o þ b 1 z 1 þþ b M z M 1 þ a 1 z 1 a N z N ¼

Hence, unlike FIR filters, IIR filters can have poles at locations other than z = 0.
In fact, the above equations can also represent a FIR filter if one puts
a 1 = a 2 = ... = 0. Normally, the number of poles N is larger than the number of
zeros M, and the order of the filter is decided by the number of its poles. For the
IIR filter to be stable, all poles should be inside the unit circle.

2.6.7.2 IIR versus FIR Filters

IIR filters usually have lower orders than FIR filters with similar performance
with respect to sharpness of cutoff, passband ripple, etc. Because of the lower
orders IIR filters tend to require fewer delay elements and digital multipliers for
hard-ware implementation, and they require fewer computations in software
implementations.
One of the key disadvantages of IIR filters is that because of their inherent
use of feedback, they can have stability problems. Quantization effects are also
much more serious in IIR filters, again due to their use of feedback. Addition-
ally, it is generally not possible to have a linear phase transfer function with IIR
filters.