11.4 IIR Filter Design 657
FIGURE 11.11
Relation betweenandω
forK= 1.
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
− 10
− 8
− 6
− 4
− 2
0
2
4
6
8
10
Ω
(rad/sec)
ω/π
which when plotted displays a linear relation around the low frequencies but it warps as we get into large
frequencies (see Figure 11.11).
The relation between the frequencies is obtained by lettingσ=0 in the second equation in Equa-
tion (11.23). The linear relationship at low frequencies can be seen using the expansion of the tan(.)
function
=K
[
ω
2
+
ω^3
24
+···
]
≈
ω
Ts
for small values ofωorω≈Ts. As frequency increases the effect of the terms beyond the first one
makes the relation nonlinear. See Figure 11.11.
To compensate for the nonlinear relation between the frequencies, or the warping effect, the following
steps to design a discrete filter are followed:
- Using the frequency warping relation (Eq. 11.24) the specified discrete frequenciesωpandωstare
transformed into specified analog frequenciespandst. The magnitude specifications remain
the same in the different bands—only the frequency is being transformed.
- Using the specified analog frequencies and the discrete magnitude specifications, an analog filter
HN(s)that satisfies these specifications is designed.
- Applying the bilinear transformation to the designed filterHN(s), the discrete filterHN(z)that
satisfies the discrete specifications is obtained.
