《DSP using MATLAB》Problem 8.7

代码：

%% ------------------------------------------------------------------------
%%            Output Info about this m-file
fprintf('\n***********************************************************\n');
fprintf('        <DSP using MATLAB> Problem 8.7 \n\n');
banner();
%% ------------------------------------------------------------------------

% digital iir lowpass filter
b = [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 form
K = 1;                                                                     % gain parameter
b = 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 poles
K = 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 Response
fprintf('\n----------------------------------');
fprintf('\nPartial fraction expansion method: \n');
[R, p, c] = residuez(b,a)
MR = (abs(R))'              % Residue  Magnitude
AR = (angle(R))'/pi         % Residue  angles in pi units
Mp = (abs(p))'              % pole  Magnitude
Ap = (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 Response
fprintf('\n----------------------------------');
fprintf('\nPartial fraction expansion method: \n');
[R, p, c] = residuez(b,a)
MR = (abs(R))'              % Residue  Magnitude
AR = (angle(R))'/pi         % Residue  angles in pi units
Mp = (abs(p))'              % pole  Magnitude
Ap = (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。