Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 111

b o = 1 b o = 1

x ( n ) 2

z −1

y ( n )

z −1 z −1

− 0.3

5

− 0.5

− 1

Pole Zero

z −1

− 0.1

6

Poles Zeros

Implementation of 2 ð z 1 Þð z þ 2 Þð z þ 3 Þ=½ z ð z þ 0 : 5 Þð z 2 þ 0 : 3z þ 0 : 1 Þ Fig. 2.33

2.6.8 Design of IIR Digital Filters

Often IIR digital filters are designed by converting analog prototype filters (such as
Butterworth, Chebychev, and Elliptic filters) into digital filters via suitable
transformation methods. The basic prototype filter type is an analog low-pass filter,
and filter transformations are necessary to obtain LP, HPF, BPF, or BSF digital
filters.
Typically the required digital filter specifications are initially transformed into
their analog counterparts to help in designing the proper analog prototype. For this
to be effective the transformation should:

1. preserve stability and
2. map the jx axis into the circumference of the unit circle e jX :

There are two approaches for transforming digital IIR filters into analog
prototypes:

1. Time-domain matching using the impulse response,
2. Frequency-domain matching using the frequency response.

2.6.8.1 Time-Domain Design: Impulse Response Matching

In the impulse response matching method (a.k.a. the impulse invariance method),
the impulse response of the digital IIR filter is simply taken to be a sampled
version of the impulse response of the analog prototype filter. That is,

h ð n Þ¼ h a ð nT s Þ; n ¼ 0 ; 1 ; 2 ; ... :

If the transfer function of the analog prototype filter H a (f) is bandlimited to
(- B, B) and f s /2 C B, then there would be no aliasing. This in turn would imply
that the digital transfer function H ð e jX Þ would be a scaled and repeated copy of
H a (f). With this understanding it becomes apparent that the impulse response
matching method is not suitable to design a digital HPF, due to aliasing problems.