《DSP using MATLAB》Problem 8.31

代码：

%% ------------------------------------------------------------------------
%%            Output Info about this m-file
fprintf('\n***********************************************************\n');
fprintf('        <DSP using MATLAB> Problem 8.31 \n\n');

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('\n***** Cascade-form in s-plane: START *****\n');
[CS, BS, AS] = sdir2cas(cs, ds)
fprintf('\n***** Cascade-form in s-plane: END *****\n');

% 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('\n\n ********** Chebyshev-I Filter Order = %3.0f  \n', 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。