IIR filter design - Continuous and Discrete Time Signals and Systems - page 733

where nums and denums specify the coefficients of the numerator and denom-
inator of the analog filter and fs is the sampling rate in samples/s. For
Example 16.4, the M ATLAB code is given by

>> fs = 0.5; % fs = 1/T = k/2 = 0.5
>> nums = [1.1897]; % numerator of the CT filter
>> denums = [1 1.5421 1.1897]; % denominator of CT filter
>> [numz,denumz] = bilinear (nums,denums,fs);
% coefficients of DT filter

which returns the values

numz = [0.3188 0.6376 0.3188];
denumz = [1.0000 0.1017 0.1735],

which are the same as the coefficients obtained in Example 16.4.

Filter design using M ATLAB Several additional functions are provided in
M ATLAB for directly determining the transfer function of the digital filters. The
buttord and butter functions, introduced in Chapter 7, can also be used to
compute IIR filters in the digital domain. The buttord function computes the
order N and cut-off frequency wn of the Butterworth filter, and the butter
function computes the coefficients of the numerator and denominator of the
z-transfer function of the Butterworth filter. For lowpass filters, the calling
syntaxes for the buttord and butter functions are given by

buttord function: [N, wn] = buttord(wp, ws, rp, rs) ;

butter function: [numz, denumz] = butter(N, wn) ,

where N is the order of the lowest-order digital Butterworth filter that loses no
more than rp dB in the pass band and has at least rs dB of attenuation in the
stop band. The frequencies wp and ws are the pass-band and stop-band edge
frequencies, normalized between zero and unity, where unity corresponds to
π radians/s. Similarly, wn is the normalized cut-off frequency for the Butter-
worth filter. The matrix numz contains the coefficients of the numerator, while
matrix denumz contains the coefficients of the denominator of the transfer
function of the Butterworth filter.
For Example 16.4, the M ATLAB code is given by

>> [N,wn] = buttord(0.25,0.75,20*log10(0.8),20*log10
(0.20));
>> [numz,denumz] = butter(N,wn);

which results in the following coefficients:

numz = [0.3188 0.6376 0.3188];
denumz = [1.0000 0.1017 0.1735],

which are identical to those obtained analytically in Example 16.4.