Page 410 - Analog and Digital Filter Design

IIR Filter Design 407

three steps: find the values of a and p, and then use in the frequency transfoor-
mation equation.

The first step is to find the value of a:

cos[2x(FL + FL)/2Fs]
a=
COS[~K(I;I~ FL),I'~Fs]
-
Next, find the value of @:

p = tan[2n(F,. - FL)/2Fs]

Now we can find the equation for H(z) by replacing s in the equation for H(s):

For example, suppose F,: = 3.0 kHz, FL = 0.5 kHz, and F5 = S kHz.

First find the values of a and p. a = 0.351 153302 and /3 = 1.496605763.

Hence, in this case, the substitution for s becomes:

I-?-'
s = 1.496605763
+
1 - 0.702306604~-~ 2-l
Consider a second-order Butterworth filter, with transfer function:

1
H(s) =
s2 +as+1
This has a s' term, and when the replacement for s is substituted factors of up
to z4 are produced. Therefore, as in the bandpass filter case, a simple digital
biquad stage is not sufficient for a second-order bandstop filter: two biquad
stages will be necessary.

Ill? Filter Stability

Stability is guaranteed in FIR filters because they have no feedback path. This
is not the case with IIR filters. Using a linear prototype, which is inherently
stable, will produce a stable IIR equivalent when processed by bilinear trans-
form. However, if the filter coefficients are rounded up, or down, it is possible