Page 402 - Analog and Digital Filter Design

IIR Filter Design 399

The bilinear transform is a simple mathematical process. Starting with an analog
frequency response, H(s), bilinear transformation to produce H(z) is carried out
by substitution of s.

To see how this works, let's use a second-order analog (Butterworth) transfer
function:

1
H(s) =
s? +.JZs+l
Substituting for s gives:

This can be simplified by multiplying everything by the highest power denomi-
nator. which is (z + 1) squared, or (-1' i 2z + 1).
The equation then becomes:

(z'+2,+Ij
H(I) =
(z2 -22+1)+./2.(-.'-1)+(:'+2:+1)

Now z-' is a single clock cycle delay. which can be achieved easily in digital
systems. The equation can be restated in terms of delays by multiplying top and
bottom by giving:

Collecting terms on z-', -1-'. and so on, to give us coefficients for each deiay term,
this becomes:

(1 + 22-' + z-?)
H(z) =
3.4142 + 0.585786z-'

This equation can be compared to the equation for the biquad that follows:
Y(;) AO+Al.z-'+A2.z-'
H(z) = -
-
-
X(L) 1-B1.C' -m.z-'