Infinite Impulse Response Digital Filter Design - Practical Analog and Digital Filter Design - page 151

to scale the magnitude as was the case for the impulse invariant design method. By
comparing this frequency response to that of Figure 6.1, we see a significant
difference. The reason for the difference is the different criteria placed on the
design. In the previous section, the emphasis was placed on matching an impulse
like input signal, while in this section the aim was to match a step like input
signal. As indicated in the figures, the different criteria produce filters with quite
different frequency responses. As in the previous section, this method of IIR filter
design is best suited to match low-frequency system responses.

6.3 BILINEAR TRANSFORM DESIGN

Both the impulse invariant and step invariant design methods provide good
approximations for lowpass and some bandpass analog filter responses. However,
they cannot provide good matching of high-frequency responses, which makes it
impossible to use them for highpass or bandstop filter design. In fact, they do not
provide the best methods for matching analog filter responses when a good match
is required throughout a wide range of frequencies. In addition, without careful
selection of the sampling frequency and strict band-limiting of the input signal,
distortion from aliasing can occur. Therefore, in this section we will discuss the
bilinear transformation that endeavors to make a reasonable match over the entire
filter frequency range. Of course, that provides a challenge since the analog
frequency range extends from zero to infinity and the digital frequency range
extends only from zero to π. However, a transformation from the analog s -domain
to the digital z -domain has been developed (as described in more detail in the
technical references provided at the end of the text). In this method, the
relationship between the s and z complex variables can be described by the
following equation, where T is the sampling period:

2 z − 1

s = ⋅ (6.14)

T z + 1

To better understand this relationship, we can represent the complex variable
z in the exponential form R ⋅ e j Ω

j Ω

2 R ⋅ e − 1 2 R ⋅ cos Ω + j ⋅ R ⋅ sin Ω − 1

s = ⋅ = ⋅

j Ω

T T R ⋅ cos Ω + j ⋅ R ⋅ sin Ω + 1

R ⋅ e + 1

(6.15)

This representation can be written in rectangular form as

2 [( R ⋅ cos Ω − 1 + j ⋅ R ⋅ sin Ω ] ⋅ [( R ⋅ cos Ω + 1 − j ⋅ R ⋅ sin Ω ]

s = ⋅ (6.16)

T [( R ⋅ cos Ω + 1 + j ⋅ R ⋅ sin Ω ] ⋅ [( R ⋅ cos Ω + 1 − j ⋅ R ⋅ sin Ω ]