代码：
%% ------------------------------------------------------------------------
%% Output Info about this m-file
fprintf('
***********************************************************
');
fprintf(' <DSP using MATLAB> Problem 8.31
');
banner();
%% ------------------------------------------------------------------------
Fp = 3.2; % analog passband freq in kHz 6.4 kpi
Fs = 3.8; % analog stopband freq in kHz 7.6 kpi
fs = 8; % sampling rate in kHz 16.0 kpi
% -------------------------------
% Ω=(2/T)tan(ω/2)
% ω=2*[atan(ΩT/2)]
% Digital Filter Specifications:
% -------------------------------
wp = 2*pi*Fp/fs % digital passband freq in rad 0.8pi
%wp = Fp;
ws = 2*pi*Fs/fs % digital stopband freq in rad 0.95pi
%ws = Fs;
Rp = 0.5; % passband ripple in dB
As = 45; % 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 = 1/8000; % set T = 1
%fs = 1/T;
OmegaP = (2/T)*tan(wp/2) % prototype passband freq 1.9593pi 15675pi
OmegaS = (2/T)*tan(ws/2) % prototype stopband freq 8.089pi 64712pi
% Analog Chebyshev-1 Prototype Filter Calculation:
[cs, ds] = afd_chb1(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, 8*pi/T);
% --------------------------------------------------------------------
% find exact band-edge frequencies for the given dB specifications
% --------------------------------------------------------------------
[diff_to_45dB, ind] = min(abs(db_s+45))
db_s(ind-3 : ind+3) % magnitude response, dB
ww_s(ind)/(pi) % analog frequency in kpi units
%ww_s(ind)/(2*pi) % analog frequency in Hz units
[sA,index] = sort(abs(db_s+45));
AA_dB = db_s(index(1:8))
AB_rad = ww_s(index(1:8))/(pi)
AC_Hz = ww_s(index(1:8))/(2*pi)
% -------------------------------------------------------------------
% Calculation of Impulse Response:
[ha, x, t] = impulse(cs, ds);
% Impulse Invariance Transformation:
%[b, a] = imp_invr(cs, ds, T);
% Bilinear Transformation
[b, a] = bilinear(cs, ds, 1/T)
[C, B, A] = dir2cas(b, a)
% Calculation of Frequency Response:
[db, mag, pha, grd, ww] = freqz_m(b, a);
% --------------------------------------------------------------------
% find exact band-edge frequencies for the given dB specifications
% --------------------------------------------------------------------
[diff_to_45dB, ind] = min(abs(db+45))
db(ind-3 : ind+3) % magnitude response, dB
ww(ind)/(pi)
(2/T)*tan(ww(ind)/2)/pi
[sA,index] = sort(abs(db+45));
AA_dB = db(index(1:8))'
AB_rad = ww(index(1:8))'/pi
AC_Hz = (2/T)*tan(ww(index(1:8))'/2)/pi
% -------------------------------------------------------------------
%% -----------------------------------------------------------------
%% Plot
%% -----------------------------------------------------------------
figure('NumberTitle', 'off', 'Name', 'Problem 8.31 Analog Chebyshev-I lowpass')
set(gcf,'Color','white');
M = 1.0; % Omega max
subplot(2,2,1); plot(ww_s/pi, mag_s); grid on; %axis([-10, 10, 0, 1.2]);
xlabel(' Analog frequency in piHz units'); ylabel('|H|'); title('Magnitude in Absolute');
% set(gca, 'XTickMode', 'manual', 'XTick', [-8.089, -1.9593, 0, 1.9593, 8.089]); % T = 1
set(gca, 'XTickMode', 'manual', 'XTick', [-80000, -64712, -15675, 0, 15675, 64712, 80000]); % T = 1/8000
set(gca, 'YTickMode', 'manual', 'YTick', [0, 0.006, 0.94, 1.0, 1.5]);
subplot(2,2,2); plot(ww_s/pi, db_s); grid on; %axis([0, M, -50, 10]);
xlabel('Analog frequency in piHz units'); ylabel('Decibels'); title('Magnitude in dB ');
% set(gca, 'XTickMode', 'manual', 'XTick', [-8.089, -1.9593, 0, 1.9593, 5.7, 8.089]); % T = 1
set(gca, 'XTickMode', 'manual', 'XTick', [-80000, -64712, -15675, 0, 15675, 45696, 64712, 80000]); % T = 1/8000
set(gca, 'YTickMode', 'manual', 'YTick', [-45, -1, 0]);
set(gca,'YTickLabelMode','manual','YTickLabel',['45';' 1';' 0']);
subplot(2,2,3); plot(ww_s/pi, pha_s/pi); grid on; %axis([-10, 10, -1.2, 1.2]);
xlabel('Analog frequency in piHz nuits'); ylabel('radians'); title('Phase Response');
% set(gca, 'XTickMode', 'manual', 'XTick', [-8.089, -1.9593, 0, 1.9593, 8.089]); % T = 1
set(gca, 'XTickMode', 'manual', 'XTick', [-80000, -64712, -15675, 0, 15675, 45696, 64712, 80000]); % T = 1/8000
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('Impulse Response');
figure('NumberTitle', 'off', 'Name', 'Problem 8.31 Digital Chebyshev-I 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(' Digital frequency in pi units'); ylabel('|H|'); title('Magnitude Response');
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.8, 0.95, M]);
set(gca, 'YTickMode', 'manual', 'YTick', [0, 0.0056, 0.9441, 1]);
subplot(2,2,2); plot(ww/pi, pha/pi); axis([0, M, -1.1, 1.1]); grid on;
xlabel('Digital frequency in pi nuits'); ylabel('radians in pi units'); title('Phase Response');
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.8, 0.95, M]);
set(gca, 'YTickMode', 'manual', 'YTick', [-1:1:1]);
subplot(2,2,3); plot(ww/pi, db); axis([0, M, -80, 10]); grid on;
xlabel('Digital frequency in pi units'); ylabel('Decibels'); title('Magnitude in dB ');
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.8, 0.93, 0.95, M]);
set(gca, 'YTickMode', 'manual', 'YTick', [-70, -45, -1, 0]);
set(gca,'YTickLabelMode','manual','YTickLabel',['70';'45';' 1';' 0']);
subplot(2,2,4); plot(ww/pi, grd); grid on; %axis([0, M, 0, 35]);
xlabel('Digital frequency in pi units'); ylabel('Samples'); title('Group Delay');
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.8, 0.95, M]);
%set(gca, 'YTickMode', 'manual', 'YTick', [0:5:35]);
figure('NumberTitle', 'off', 'Name', 'Problem 8.31 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 8.31 Imp & Freq Response')
set(gcf,'Color','white');
t = [0: 0.000005 : 8*0.0001]; subplot(2,1,1); impulse(cs,ds,t); grid on; % Impulse response of the analog filter
axis([0, 8*0.0001, -1.5*10000, 2.0*10000]);hold on
n = [0:1:7*0.0001/T]; hn = filter(b,a,impseq(0,0,7*0.0001/T)); % Impulse response of the digital filter
stem(n*T,hn); xlabel('time in sec'); title (sprintf('Impulse Responses T=%2d',T));
hold off
% Calculation of Frequency Response:
[dbs, mags, phas, wws] = freqs_m(cs, ds, 8*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/T, 'b+', wwz/(2*pi*T), magz, 'r'); grid on;
xlabel('frequency in Hz'); title('Magnitude Responses'); ylabel('Magnitude');
text(-0.8,0.15,'Analog filter', 'Color', 'b'); text(0.6,1.05,'Digital filter', 'Color', 'r');
%% -----------------------------------------------------------------------
%% MATLAB cheby1 function
%% -----------------------------------------------------------------------
% Analog Prototype Order Calculations:
ep = sqrt(10^(Rp/10)-1); % Passband Ripple Factor
A = 10^(As/20); % Stopband Attenuation Factor
OmegaC = OmegaP; % Analog Chebyshev-1 prototype cutoff freq
OmegaR = OmegaS/OmegaP; % Analog prototype Transition ratio
g = sqrt(A*A-1)/ep; % Analog prototype Intermediate cal
N = ceil(log10(g+sqrt(g*g-1))/log10(OmegaR+sqrt(OmegaR*OmegaR-1)));
fprintf('
********** Chebyshev-I Filter Order = %3.0f
', N)
% Digital Chebyshev-1 Filter Design:
wn = wp/pi; % Digital Chebyshev-1 cutoff freq in pi units
[b, a] = cheby1(N, Rp, wn)
[C, B, A] = dir2cas(b, a)
% Calculation of Frequency Response:
[db, mag, pha, grd, ww] = freqz_m(b, a);
% --------------------------------------------------------------------
% find exact band-edge frequencies for the given dB specifications
% --------------------------------------------------------------------
[diff_to_45dB, ind] = min(abs(db+45))
db(ind-3 : ind+3) % magnitude response, dB
ww(ind)/(pi)
(2/T)*tan(ww(ind)/2)/pi
[sA,index] = sort(abs(db+45));
AA_dB = db(index(1:8))'
AB_rad = ww(index(1:8))'/pi
AC_Hz = (2/T)*tan(ww(index(1:8))'/2)/pi
% -------------------------------------------------------------------
%% -----------------------------------------------------------------
%% Plot
%% -----------------------------------------------------------------
figure('NumberTitle', 'off', 'Name', 'Problem 8.31 Digital Chebyshev-I lowpass by cheby1 function')
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('Digital frequency in pi units'); ylabel('|H|'); title('Magnitude Response');
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.8, 0.95, M]);
set(gca, 'YTickMode', 'manual', 'YTick', [0, 0.0056, 0.9441, 1]);
subplot(2,2,2); plot(ww/pi, pha/pi); axis([0, M, -1.1, 1.1]); grid on;
xlabel('Digital frequency in pi nuits'); ylabel('radians in pi units'); title('Phase Response');
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.8, 0.95, 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('Digital frequency in pi units'); ylabel('Decibels'); title('Magnitude in dB ');
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.8, 0.93, 0.95, M]);
set(gca, 'YTickMode', 'manual', 'YTick', [-60, -45, -1, 0]);
set(gca,'YTickLabelMode','manual','YTickLabel',['60';'45';' 1';' 0']);
subplot(2,2,4); plot(ww/pi, grd); grid on; %axis([0, M, 0, 35]);
xlabel('Digital frequency in pi units'); ylabel('Samples'); title('Group Delay');
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.8, 0.95, M]);
%set(gca, 'YTickMode', 'manual', 'YTick', [0:5:35]);
figure('NumberTitle', 'off', 'Name', 'Problem 8.31 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 8.31 Imp & Freq Response')
set(gcf,'Color','white');
t = [0: 0.000005 : 8*0.0001]; subplot(2,1,1); impulse(cs,ds,t); grid on; % Impulse response of the analog filter
axis([0, 8*0.0001, -1.5*10000, 2.0*10000]);hold on
n = [0:1:7*0.0001/T]; hn = filter(b,a,impseq(0,0,7*0.0001/T)); % Impulse response of the digital filter
stem(n*T,hn); xlabel('time in sec'); title (sprintf('Impulse Responses T=%2d',T));
hold off
% Calculation of Frequency Response:
[dbs, mags, phas, wws] = freqs_m(cs, ds, 8*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/T, 'b+', wwz/(2*pi*T), magz, 'r'); grid on;
xlabel('frequency in Hz'); title('Magnitude Responses'); ylabel('Magnitude');
text(-0.8,0.15,'Analog filter', 'Color', 'b'); text(0.6,1.05,'Digital filter', 'Color', 'r');
运行结果：
这里放上T=1/8000sec的结果。
模拟chebyshev-1型低通，幅度谱、相位谱和脉冲响应
采用双线性变换法，得到数字chebyshev-1型低通滤波器，幅度谱、相位谱和群延迟响应
采用MATLAB自带cheby1函数得到的数字低通，其幅度谱、相位谱和群延迟
cheby1函数得到的数字低通，和相应的模拟原型的脉冲响应，二者形态不同。
