6.3 Design of IIR Filters
In this section, we discuss the design of digital filters that have an infinite impulse
response. In designing IIR filters, the usual starting point will be an analog filter transfer
function H(s). Because analog filter design is a mature and well-developed field, it is
not surprising that we begin the design of digital IIR filters in the analog domain and
then convert the design into the digital domain. The problem is to determine a digital
filter H(z) which will approximate the performance of the desired analog filter H(s).
There are two methods, the impulse-invariant method and the bilinear transform, for
designing digital IIR filters based on existing analog IIR filters. Instead of designing
the digital IIR filter directly, these methods map the digital filter into an equivalent
analog filter, which can be designed by one of the well-developed analog filter
design methods. The designed analog filter is then mapped back into the desired digital
filter.
The impulse-invariant method preserves the impulse response of the original analog
filter by digitizing the impulse response of analog filter, but not its frequency (magni-
tude) response. Because of inherent aliasing, this method is inappropriate for highpass
or bandstop filters. The bilinear-transform method yields very efficient filters, and is
well suited for the design of frequency selective filters. Digital filters resulting from the
bilinear transform will preserve the magnitude response characteristics of the analog
filters, but not the time domain properties. In general, the impulse-invariant method is
good for simulating analog filters, but the bilinear-transform method is better for
designing frequency selective IIR filters.
6.3.1 Review of IIR Filters
As discussed in Chapters 3 and 4, an IIR filter can be specified by its impulse response
fhn, n 0, 1, . . . , Ig, I/O difference equation, or transfer function. The general
form of the IIR filter transfer function is defined in (4.3.10) as
Hz
LÀ1
l0
b
l
z
Àl
1
M
m1
a
m
z
Àm
:
6:3:1
The design problem is to find the coefficients b
l
and a
m
so that H(z) satisfies the given
specifications. This IIR filter can be realized by the I/O difference equation
yn
LÀ1
l0
b
l
xn À l À
M
m1
a
m
yn À m:
6:3:2
The impulse response h(n) of the IIR filter is the output that results when the input is
the unit impulse response defined in (3.1.1). Given the impulse response, the filter
output y(n) can also be obtained by linear convolution expressed as
DESIGN OF IIR FILTERS
255