代码：
%% ------------------------------------------------------------------------%% Output Info about this m-filefprintf('\n***********************************************************\n');fprintf(' Problem 8.7 \n\n');banner();%% ------------------------------------------------------------------------% digital iir lowpass filterb = [1 1];a = [1 -0.9];figure('NumberTitle', 'off', 'Name', 'Problem 8.7 Pole-Zero Plot')set(gcf,'Color','white'); zplane(b,a); title(sprintf('Pole-Zero Plot'));%pzplotz(b,a);% corresponding system function Direct formK = 1; % gain parameterb = K*b; % denominator a = a; % numerator[db, mag, pha, grd, w] = freqz_m(b, a);% ---------------------------------------------------------------------% Choose the gain parameter of the filter, maximum gain is equal to 1 % ---------------------------------------------------------------------gain1=max(mag) % with polesK = 1/gain1[db, mag, pha, grd, w] = freqz_m(K*b, a);figure('NumberTitle', 'off', 'Name', 'Problem 8.7 IIR lowpass filter')set(gcf,'Color','white'); subplot(2,2,1); plot(w/pi, db); grid on; axis([0 2 -60 10]); set(gca,'YTickMode','manual','YTick',[-60,-30,0])set(gca,'YTickLabelMode','manual','YTickLabel',['60';'30';' 0']);set(gca,'XTickMode','manual','XTick',[0,0.25,0.5,1,1.5,1.75,2]);xlabel('frequency in \pi units'); ylabel('Decibels'); title('Magnitude Response in dB');subplot(2,2,3); plot(w/pi, mag); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Absolute'); title('Magnitude Response in absolute');set(gca,'XTickMode','manual','XTick',[0,0.25,0.5,1,1.5,1.75,2]);set(gca,'YTickMode','manual','YTick',[0,1.0]);subplot(2,2,2); plot(w/pi, pha); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Rad'); title('Phase Response in Radians');subplot(2,2,4); plot(w/pi, grd*pi/180); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Rad'); title('Group Delay');set(gca,'XTickMode','manual','XTick',[0,0.25,0.5,1,1.5,1.75,2]);%set(gca,'YTickMode','manual','YTick',[0,1.0]);% Impulse Responsefprintf('\n----------------------------------');fprintf('\nPartial fraction expansion method: \n');[R, p, c] = residuez(b,a)MR = (abs(R))' % Residue MagnitudeAR = (angle(R))'/pi % Residue angles in pi unitsMp = (abs(p))' % pole MagnitudeAp = (angle(p))'/pi % pole angles in pi units[delta, n] = impseq(0,0,50);h_chk = filter(b,a,delta); % check sequences% ------------------------------------------------------------------------------------------------% gain parameter K=0.05 % ------------------------------------------------------------------------------------------------h = ( 0.9.^n ) .* 0.1056 - 0.0556 * delta; % ------------------------------------------------------------------------------------------------figure('NumberTitle', 'off', 'Name', 'Problem 8.7 IIR lp filter, h(n) by filter and Inv-Z ')set(gcf,'Color','white'); subplot(2,1,1); stem(n, h_chk); grid on; %axis([0 2 -60 10]); xlabel('n'); ylabel('h\_chk'); title('Impulse Response sequences by filter');subplot(2,1,2); stem(n, h); grid on; %axis([0 1 -100 10]); xlabel('n'); ylabel('h'); title('Impulse Response sequences by Inv-Z');[db, mag, pha, grd, w] = freqz_m(h, [1]);figure('NumberTitle', 'off', 'Name', 'Problem 8.7 IIR filter, h(n) by Inv-Z ')set(gcf,'Color','white'); subplot(2,2,1); plot(w/pi, db); grid on; axis([0 2 -60 10]); set(gca,'YTickMode','manual','YTick',[-60,-30,0])set(gca,'YTickLabelMode','manual','YTickLabel',['60';'30';' 0']);set(gca,'XTickMode','manual','XTick',[0,0.25,1,1.75,2]);xlabel('frequency in \pi units'); ylabel('Decibels'); title('Magnitude Response in dB');subplot(2,2,3); plot(w/pi, mag); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Absolute'); title('Magnitude Response in absolute');set(gca,'XTickMode','manual','XTick',[0,0.25,1,1.75,2]);set(gca,'YTickMode','manual','YTick',[0,1.0]);subplot(2,2,2); plot(w/pi, pha); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Rad'); title('Phase Response in Radians');subplot(2,2,4); plot(w/pi, grd*pi/180); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Rad'); title('Group Delay');set(gca,'XTickMode','manual','XTick',[0,0.25,1,1.75,2]);%set(gca,'YTickMode','manual','YTick',[0,1.0]);% --------------------------------------------------% digital IIR comb filter% --------------------------------------------------b = K*[1 0 0 0 0 0 1];a = [1 0 0 0 0 0 -0.9];figure('NumberTitle', 'off', 'Name', 'Problem 8.7 Pole-Zero Plot')set(gcf,'Color','white'); zplane(b,a); title(sprintf('Pole-Zero Plot'));[db, mag, pha, grd, w] = freqz_m(b, a);figure('NumberTitle', 'off', 'Name', 'Problem 8.7 IIR comb filter')set(gcf,'Color','white'); subplot(2,2,1); plot(w/pi, db); grid on; axis([0 2 -60 10]); set(gca,'YTickMode','manual','YTick',[-60,-30,0])set(gca,'YTickLabelMode','manual','YTickLabel',['60';'30';' 0']);set(gca,'XTickMode','manual','XTick',[0,0.25,0.5,1,1.5,1.75,2]);xlabel('frequency in \pi units'); ylabel('Decibels'); title('Magnitude Response in dB');subplot(2,2,3); plot(w/pi, mag); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Absolute'); title('Magnitude Response in absolute');set(gca,'XTickMode','manual','XTick',[0,0.25,0.5,1,1.5,1.75,2]);set(gca,'YTickMode','manual','YTick',[0,1.0]);subplot(2,2,2); plot(w/pi, pha); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Rad'); title('Phase Response in Radians');subplot(2,2,4); plot(w/pi, grd*pi/180); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Rad'); title('Group Delay');set(gca,'XTickMode','manual','XTick',[0,0.25,0.5,1,1.5,1.75,2]);%set(gca,'YTickMode','manual','YTick',[0,1.0]);% Impulse Responsefprintf('\n----------------------------------');fprintf('\nPartial fraction expansion method: \n');[R, p, c] = residuez(b,a)MR = (abs(R))' % Residue MagnitudeAR = (angle(R))'/pi % Residue angles in pi unitsMp = (abs(p))' % pole MagnitudeAp = (angle(p))'/pi % pole angles in pi units[delta, n] = impseq(0,0,300);h_chk = filter(b,a,delta); % check sequences% ------------------------------------------------------------------------------------------------% gain parameter K=0.05 % ------------------------------------------------------------------------------------------------%h = 0.0211 * (( 0.9791.^n ) .* (2*cos(0.4*pi*n) + 2*cos(0.8*pi*n) + 1)) - 0.0556*delta; %L=5;h = 0.0176 * ( ( 0.9826.^n ) .* ( 2*cos(2*pi*n/3) + 2*cos(pi*n/3) + (-1).^n + 1) ) - 0.0556*delta; %L=6;% ------------------------------------------------------------------------------------------------figure('NumberTitle', 'off', 'Name', 'Problem 8.7 Comb filter, h(n) by filter and Inv-Z ')set(gcf,'Color','white'); subplot(2,1,1); stem(n, h_chk); grid on; %axis([0 2 -60 10]); xlabel('n'); ylabel('h\_chk'); title('Impulse Response sequences by filter');subplot(2,1,2); stem(n, h); grid on; %axis([0 1 -100 10]); xlabel('n'); ylabel('h'); title('Impulse Response sequences by Inv-Z');[db, mag, pha, grd, w] = freqz_m(h, [1]);figure('NumberTitle', 'off', 'Name', 'Problem 8.7 Comb filter, h(n) by Inv-Z ')set(gcf,'Color','white'); subplot(2,2,1); plot(w/pi, db); grid on; axis([0 2 -60 10]); set(gca,'YTickMode','manual','YTick',[-60,-30,0])set(gca,'YTickLabelMode','manual','YTickLabel',['60';'30';' 0']);set(gca,'XTickMode','manual','XTick',[0,0.25,1,1.75,2]);xlabel('frequency in \pi units'); ylabel('Decibels'); title('Magnitude Response in dB');subplot(2,2,3); plot(w/pi, mag); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Absolute'); title('Magnitude Response in absolute');set(gca,'XTickMode','manual','XTick',[0,0.25,1,1.75,2]);set(gca,'YTickMode','manual','YTick',[0,1.0]);subplot(2,2,2); plot(w/pi, pha); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Rad'); title('Phase Response in Radians');subplot(2,2,4); plot(w/pi, grd*pi/180); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Rad'); title('Group Delay');set(gca,'XTickMode','manual','XTick',[0,0.25,1,1.75,2]);%set(gca,'YTickMode','manual','YTick',[0,1.0]); 运行结果：
先计算单个IIR低通，
零极点
L=6阶梳状低通，系统函数部分分式展开如下
梳状低通滤波器零极点图
幅度谱、相位谱、群延迟
可以看出，在0到2π范围内，单个低通重复出现了6次，原来的谱压缩到六分之一。