用match-z方法，将模拟低通转换为数字低通
代码：
%% ------------------------------------------------------------------------
%% Output Info about this m-file
fprintf('
***********************************************************
');
fprintf(' <DSP using MATLAB> Problem 8.25
');
banner();
%% ------------------------------------------------------------------------
% -------------------------------
% ω = ΩT = 2πF/fs
% Digital Filter Specifications:
% -------------------------------
wp = 0.4*pi; % digital passband freq in rad/sec
ws = 0.6*pi; % digital stopband freq in rad/sec
Rp = 0.5; % passband ripple in dB
As = 50; % stopband attenuation in dB
Ripple = 10 ^ (-Rp/20) % passband ripple in absolute
Attn = 10 ^ (-As/20) % stopband attenuation in absolute
% Analog prototype specifications: Inverse Mapping for frequencies
T = 2; % set T = 1
Fs = 1/T;
OmegaP = wp/T; % prototype passband freq
OmegaS = ws/T; % prototype stopband freq
% Analog Butterworth Prototype Filter Calculation:
[cs, ds] = afd_butt(OmegaP, OmegaS, Rp, As);
% Calculation of second-order sections:
fprintf('
***** Cascade-form in s-plane: START *****
');
[CS, BS, AS] = sdir2cas(cs, ds)
fprintf('
***** Cascade-form in s-plane: END *****
');
% Calculation of Frequency Response:
[db_s, mag_s, pha_s, ww_s] = freqs_m(cs, ds, 0.5*pi);
% Calculation of Impulse Response:
%[ha, x, t] = impulse(cs, ds);
% Impulse Invariance Transformation:
%[b, a] = imp_invr(cs, ds, T);
% Calculation of Step Response:
[ha, x, t] = step(cs, ds);
% Step Invariance Transformation:
[b, a] = stp_invr(cs, ds, T); [C, B, A] = dir2par(b, a)
% Calculation of Frequency Response:
[db, mag, pha, grd, ww] = freqz_m(b, a);
%% -----------------------------------------------------------------
%% Plot
%% -----------------------------------------------------------------
figure('NumberTitle', 'off', 'Name', 'Problem 8.25 Analog Butterworth lowpass')
set(gcf,'Color','white');
M = 1; % Omega max
subplot(2,2,1); plot(ww_s, mag_s); grid on; axis([-M, M, 0, 1.2]);
xlabel(' Analog frequency in pi units'); ylabel('|H|'); title('Magnitude in Absolute');
set(gca, 'XTickMode', 'manual', 'XTick', [-0.3, -0.2, 0, 0.2, 0.3, 0.4, 0.6]);
set(gca, 'YTickMode', 'manual', 'YTick', [0, 0.0032, 0.5, 0.9441, 1]);
subplot(2,2,2); plot(ww_s, db_s); grid on; %axis([0, M, -50, 10]);
xlabel('Analog frequency in pi units'); ylabel('Decibels'); title('Magnitude in dB ');
set(gca, 'XTickMode', 'manual', 'XTick', [-0.3, -0.2, 0, 0.4, 0.6]);
set(gca, 'YTickMode', 'manual', 'YTick', [-65, -50, -1, 0]);
set(gca,'YTickLabelMode','manual','YTickLabel',['65';'50';' 1';' 0']);
subplot(2,2,3); plot(ww_s, pha_s/pi); grid on; axis([-M, M, -1.2, 1.2]);
xlabel('Analog frequency in pi nuits'); ylabel('radians'); title('Phase Response');
set(gca, 'XTickMode', 'manual', 'XTick', [-0.3, -0.2, 0, 0.4, 0.6]);
set(gca, 'YTickMode', 'manual', 'YTick', [-1:0.5:1]);
subplot(2,2,4); plot(t, ha); grid on; %axis([0, 30, -0.05, 0.25]);
xlabel('time in seconds'); ylabel('ha(t)'); title('Step Response');
figure('NumberTitle', 'off', 'Name', 'Problem 8.25 Digital Butterworth lowpass')
set(gcf,'Color','white');
M = 2; % Omega max
subplot(2,2,1); plot(ww/pi, mag); axis([0, M, 0, 1.2]); grid on;
xlabel(' frequency in pi units'); ylabel('|H|'); title('Magnitude Response');
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.4, 0.6, 1.0, M]);
set(gca, 'YTickMode', 'manual', 'YTick', [0, 0.0032, 0.5, 0.9441, 1]);
subplot(2,2,2); plot(ww/pi, pha/pi); axis([0, M, -1.1, 1.1]); grid on;
xlabel('frequency in pi nuits'); ylabel('radians in pi units'); title('Phase Response');
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.4, 0.6, 1.0, M]);
set(gca, 'YTickMode', 'manual', 'YTick', [-1:1:1]);
subplot(2,2,3); plot(ww/pi, db); axis([0, M, -100, 10]); grid on;
xlabel('frequency in pi units'); ylabel('Decibels'); title('Magnitude in dB ');
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.4, 0.6, 1.0, M]);
set(gca, 'YTickMode', 'manual', 'YTick', [-70, -50, -1, 0]);
set(gca,'YTickLabelMode','manual','YTickLabel',['70';'50';' 1';' 0']);
subplot(2,2,4); plot(ww/pi, grd); grid on; %axis([0, M, 0, 35]);
xlabel('frequency in pi units'); ylabel('Samples'); title('Group Delay');
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.4, 0.6, 1.0, M]);
%set(gca, 'YTickMode', 'manual', 'YTick', [0:5:35]);
figure('NumberTitle', 'off', 'Name', 'Problem 8.25 Pole-Zero Plot')
set(gcf,'Color','white');
zplane(b,a);
title(sprintf('Pole-Zero Plot'));
%pzplotz(b,a);
% ----------------------------------------------
% Calculation of Impulse Response
% ----------------------------------------------
figure('NumberTitle', 'off', 'Name', 'Problem 8.25 Imp & Freq Response')
set(gcf,'Color','white');
t = [0:0.01:80]; subplot(2,1,1); step(cs,ds,t); grid on; % Step response of the analog filter
axis([0,80,-0.2,1.5]);hold on
n = [0:1:80/T]; hn = filter(b,a,stepseq(0,0,80/T)); % Step response of the digital filter
stem(n*T,hn); xlabel('time in sec'); title ('Step Responses');
hold off
% Calculation of Frequency Response:
[dbs, mags, phas, wws] = freqs_m(cs, ds, 2*pi/T); % Analog frequency s-domain
[dbz, magz, phaz, grdz, wwz] = freqz_m(b, a); % Digital z-domain
%% -----------------------------------------------------------------
%% Plot
%% -----------------------------------------------------------------
subplot(2,1,2); plot(wws/(2*pi),mags,'b+', wwz/(2*pi)*Fs,magz,'r'); grid on;
xlabel('frequency in Hz'); title('Magnitude Responses'); ylabel('Magnitude');
text(-0.3,0.15,'Analog filter'); text(0.4,0.55,'Digital filter');
运行结果：
通带、阻带绝对指标
模拟原型Butterworth低通滤波器，直接形式系数
模拟原型Butterworth低通滤波器，串联形式系数
转换成数字低通后，并联形式系数
