Page 406 - Analog and Digital Filter Design

IIR Filter Design 403

Now by substituting into the original Butterworth transfer function

1
H(s) =
s2 +&+I

Using oD = 2n(3.4/8) = 2.6703538, the function cot(oJ2) = cot(1.33517688) =
Iftan( 1.33517688) = 0.2400788:

Comparing to design method 1 and 1, it can be seen that the equation for H(z)
using design method 2 is identical (within the error limits of my calculator). The
only difference is that the second method is a single step.

Highpass Frequency Scaling
Design a second-order IIR highpass filter with the same characteristics as the
lowpass design produced in the previous section. The filter should have a
passband edge at 3.4kHz, a sampling clock of 8 kHz, and should be based
on the lowpass analog (Butterworth) transfer function:

As in the case of lowpass design, there are several ways to produce a highpass
IIR filter design. First I will follow the step-by-step procedure outlined previ-
ously: pre-warp the analog equation and then use the bilinear transform to
produce the coefficients. The second method is more complex. but produces the
design in a single step.

Design Method 1
The analog frequency response must be pre-warped using the following
equation:

s = ..t(oD/2).[=J