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434 Chapter 8 IIR FILTER DESIGN
is essentially band-limited to a lowpass or bandpass filter in which there
are no oscillations in the stopband.
8.4.3 BILINEAR TRANSFORMATION
This mapping is the best transformation method; it involves a well-known
function given by
s = 2 T
1 − z −1 1+sT/2
=⇒ z =
1+z−1 1 − sT/2
(8.65)
where T is a parameter. Another name for this transformation is the linear
fractional transformation because when cleared of fractions, we obtain
T
2 sz + T 2 s − z +1=0
which is linear in each variable if the other is fixed, or bilinear in s and z.
The complex plane mapping under (8.65) is shown in Figure 8.25, from
which we have the following observations:
1. Using s = σ + jΩ in(8.65), we obtain
(
z = 1+ σT 2 + j ΩT )/(
1 − σT 2
2 − j ΩT 2
)
(8.66)
Hence
1+ σT 2
σ0=⇒|z| =
+ j ΩT 2
∣1 − σT 2 − j ΩT 2
∣ < 1
=1
∣ > 1
jΩ
Im {z}
Unit Circle
σ
One-to-one
Transformation
1 + (sT/2)
= z
1 − (sT/2)
Re {z}
s-plane
z -plane
FIGURE 8.25 Complex-plane mapping in bilinear transformation
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