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Characteristics of Prototype Analog Filters 409
% db = Relative magnitude in db over [0 to wmax]
% mag = Absolute magnitude over [0 to wmax]
% pha = Phase response in radians over [0 to wmax]
% w = array of 500 frequency samples between [0 to wmax]
% b = Numerator polynomial coefficents of Ha(s)
% a = Denominator polynomial coefficents of Ha(s)
% wmax = Maximum frequency in rad/sec over which response is desired
%
w = [0:1:500]*wmax/500; H = freqs(b,a,w);
mag = abs(H); db = 20*log10((mag+eps)/max(mag)); pha = angle(H);
The impulse response h a (t) of the analog filter is computed using
MATLAB’s impulse function.
□ EXAMPLE 8.4 Design the analog Butterworth lowpass filter specified in Example 8.3 using
MATLAB.
Solution
MATLAB script:
>> Wp = 0.2*pi; Ws = 0.3*pi; Rp = 7; As = 16;
>> Ripple = 10 ^ (-Rp/20); Attn = 10 ^ (-As/20);
>> % Analog filter design:
>> [b,a] = afd_butt(Wp,Ws,Rp,As);
*** Butterworth Filter Order = 3
>> % Calculation of second-order sections:
>> [C,B,A] = sdir2cas(b,a)
C = 0.1238
B = 0 0 1
A = 1.0000 0.4985 0.2485
0 1.0000 0.4985
>> % Calculation of Frequency Response:
>> [db,mag,pha,w] = freqs_m(b,a,0.5*pi);
>> % Calculation of Impulse response:
>> [ha,x,t] = impulse(b,a);
The system function is given by
H a(s) =
0.1238
(s 2 +0.4985s +0.2485) (s +0.4985)
This H a(s) isslightly different from the one in Example 8.3 because in that
example we used Ω c = 0.5, while in the afd butt function Ω c is chosen to
satisfy the specifications at Ω p. The filter plots are shown in Figure 8.15. □
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