IIR filter design - Continuous and Discrete Time Signals and Systems - page 727

where num and denum specify the coefficients of the numerator and denomi-
nator of the analog filter and fs is the sampling rate in samples/s. For Example
16.3, the M ATLAB code is given by

>>fs=1; %fs=1/T=1
>> nums = [1.1332]; % numerator of CT filter
>> denums = [1 1.5055 1.1332]; % denominator of CT filter
>> [numz,denumz] = impinvar (nums,denums,fs);
% coefficients of the DT
% filter

which returns the following values:

numz = 0.4848 and denumz = [1.0000 -0.6876 0.2219].

The transfer function of the second-order IIR filter is given by

0 . 4848 z
z 2 − 0 . 6875 z + 0 . 2219 ,

H ( z ) =

which yields the same expression as the one derived in Example 16.3.
For the third-order Butterworth filter, the M ATLAB code for the impulse
invariance transformation is given by

>>fs=1; %fs=1/T=1
>> nums = [2.6702]; % numerator of the CT filter
>> denums = [1 2.7747 3.8494 2.6702];
% denominator of the CT filter
>> [numz,denumz] = impinvar (nums,denums,fs);
% coeffs of the DT filter

which returns the following values:

numz = [0 0.4695 0.1907] and denumz = [1.0000 -0.6106
0.3398 -0.0624].

The transfer function of the third-order IIR filter is given by

0 . 4695 z 2 + 0 . 1907 z
z 3 − 0 . 6106 z 2 + 0 . 3398 z − 0 . 0624 .

H ( z ) =