% Chapter 8: Example 8.11%            Impulse Invariance Transformation%                  Butterworth Lowpass Filter Design%% Digital Filter Specifications:wp = 0.2*pi;                         % digital Passband freq in Hzws = 0.3*pi;                         % digital Stopband freq in HzRp = 1;                              % Passband ripple in dBAs = 15;                             % Stopband attenuation in dB% Analog Prototype Specifications: Inverse mapping for frequenciesT = 1;                               % Set T=1OmegaP = wp * T;                     % Prototype Passband freqOmegaS = ws * T;                     % Prototype Stopband freqep = sqrt(10^(Rp/10)-1);             % Passband Ripple parameterRipple = sqrt(1/(1+ep*ep));          % Passband RippleAttn = 1/(10^(As/20));               % Stopband Attenuation% Analog Butterworth Prototype Filter Calculation:[cs,ds] = afd_butt(OmegaP,OmegaS,Rp,As);%%*** Butterworth Filter Order =  6 % Impulse Invariance transformation:[b,a] = imp_invr(cs,ds,T);[C,B,A] = dir2par(b,a)%%C =  []%%B = 1.8557   -0.6304%%   -2.1428    1.1454%%    0.2871   -0.4466%%A = 1.0000   -0.9973    0.2570%%    1.0000   -1.0691    0.3699%%    1.0000   -1.2972    0.6949% Plottingfigure(1); subplot(1,1,1)[db,mag,pha,grd,w] = freqz_m(b,a);subplot(2,2,1); plot(w/pi,mag); title('Magnitude Response')xlabel('frequency in pi units'); ylabel('|H|'); axis([0,1,0,1.1])set(gca,'XTickMode','manual','XTick',[0,0.2,0.3,1]);set(gca,'YTickmode','manual','YTick',[0,Attn,Ripple,1]); gridsubplot(2,2,3); plot(w/pi,db); title('Magnitude in dB');xlabel('frequency in pi units'); ylabel('decibels'); axis([0,1,-40,5]);set(gca,'XTickMode','manual','XTick',[0,0.2,0.3,1]);set(gca,'YTickmode','manual','YTick',[-50,-15,-1,0]); gridset(gca,'YTickLabelMode','manual','YTickLabels',['50';'15';' 1';' 0'])subplot(2,2,2); plot(w/pi,pha/pi); title('Phase Response')xlabel('frequency in pi units'); ylabel('pi units'); axis([0,1,-1,1]);set(gca,'XTickMode','manual','XTick',[0,0.2,0.3,1]);set(gca,'YTickmode','manual','YTick',[-1,0,1]); gridsubplot(2,2,4); plot(w/pi,grd); title('Group Delay')xlabel('frequency in pi units'); ylabel('Samples'); axis([0,1,0,10])set(gca,'XTickMode','manual','XTick',[0,0.2,0.3,1]);set(gca,'YTickmode','manual','YTick',[0:2:10]); grid