IIR filter design - Continuous and Discrete Time Signals and Systems - page 734

16.4 Designing hig hpass, bandpass, and bandstop IIR filters

In the following examples, we design the highpass, bandpass, and bandstop IIR
filters.

Example 16.5
Example 15.5 designed a highpass FIR filter for the following specifications:

(i) pass-band edge frequency Ω p = 0 . 5 π radians/s;

(ii) stop-band edge frequency Ω s = 0 . 125 π radians/s;

(iii) pass-band ripple ≤ 0.01 dB;
(iv) stop-band attenuation ≥ 60 dB.

Design an IIR filter with the same specifications.

Solution
Choosing k = 1 (sampling interval T = 2), step 1 transforms the pass-band and
stop-band corner frequencies into the CT frequency domain:

pass-band corner frequency ω p = tan(0 . 5 Ω p ) = tan(0 . 25 π ) = 1 radian/s;

stop-band corner frequency ω s = tan(0 . 5 Ω s ) = tan(0 . 0625 π ) = 0 . 1989 radians/s .

Step 2 designs the analog filter based on the transformed specifications. In
Chapter 7, we presented the design methodology for deriving the transfer
function of the analog highpass filter analytically. Here, we use M ATLAB
to calculate the analog elliptic filter based on the above specifications:

>> wp = 1; ws = 0.1989; Rp = 0.01; Rs = 60 ;
>> [N,wn] = ellipord (wp,ws,Rp,Rs, 's');
% Order and cut off frequency
% of the analog elliptic filter
>> [nums,denums]=ellip (N,Rp,Rs,wn,'high','s');
% Tx function of the analog
% elliptic filter

which yields the following transfer function for the analog filter:
H ( s ) = 0 . 9988 s 4 + 0 . 0542 s 2 + 0 . 000373

s 4 + 1 . 872 s 3 + 1 . 824 s 2 + 1 . 04 s + 0 . 3732 .
Step 3 derives the z-transfer function of the digital filter using the bilinear trans-
formation. This is achieved by using the bilinear function in M ATLAB .

>> [numz,denumz] = bilinear(nums,denums,0.5) % DT Filter

The resulting filter is given by
H ( z ) = 0 . 1725 z 4 − 0 . 6539 z 3 + 0 . 9638 z 2 − 0 . 6539 z + 0 . 1725
z 4 − 0 . 6829 z 3 + 0 . 7518 z 2 − 0 . 138 z + 0 . 0468

.