《DSP using MATLAB》Problem 8.33

代码：

%% ------------------------------------------------------------------------
%%            Output Info about this m-file
fprintf('\n***********************************************************\n');
fprintf('        <DSP using MATLAB> Problem 8.33.2 \n\n');

banner();
%% ------------------------------------------------------------------------

% Digital Filter Specifications:
wp = 0.45*pi;                 % digital passband freq in rad
ws = 0.50*pi;                 % digital stopband freq in rad
Rp = 0.5;                     % passband ripple in dB
As = 60;                      % 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;                           % set T = 1
Fs = 1/T;
OmegaP = (2/T)*tan(wp/2);        % Prewarp(Cutoff) prototype passband freq
OmegaS = (2/T)*tan(ws/2);        % Prewarp(cutoff) prototype stopband freq

OmegaP/pi
OmegaS/pi

% ---------------------------------------------------------------
%      method 1:   afd_chb1 function
% ---------------------------------------------------------------
% 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, pi/T);

delta_w = 2*pi/1000;
Rp_s = -(min(db_s(501:1:floor(OmegaP/delta_w)+501)));                       % Actual Passband Ripple

fprintf('\nActual Passband Ripple is %.4f dB.\n', Rp_s);

As_s = -round(max(db_s(floor(OmegaS/delta_w)+501:1:1000)));                % Min Stopband attenuation
fprintf('\nMin Stopband attenuation is %.4f dB.\n', As_s);

% 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);

delta_w = 2*pi/1000;
Rp = -(min(db(1:1:ceil(wp/delta_w)+1)));                       % Actual Passband Ripple

fprintf('\nActual Passband Ripple is %.4f dB.\n', Rp);

As = -round(max(db(ceil(ws/delta_w)+1:1:501)));                % Min Stopband attenuation
fprintf('\nMin Stopband attenuation is %.4f dB.\n', As);

%% -----------------------------------------------------------------
%%                             Plot
%% -----------------------------------------------------------------
figure('NumberTitle', 'off', 'Name', 'Problem 8.33.2 Analog Chebyshev-1 lowpass')
set(gcf,'Color','white');
M = 1;                          % Omega max

subplot(2,2,1); plot(ww_s/pi, 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', [-0.6366, -0.5437, 0, 0.5437, 0.6366]);
set(gca, 'YTickMode', 'manual', 'YTick', [0, 0.001, 0.5, 0.9441, 1]);

subplot(2,2,2); plot(ww_s/pi, 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', [-0.6366, -0.5437, 0, 0.5437, 0.6366]);
set(gca, 'YTickMode', 'manual', 'YTick', [-65, -60, -1, 0]);
set(gca,'YTickLabelMode','manual','YTickLabel',[ '65'; '60';'1 ';' 0']);

subplot(2,2,3); plot(ww_s/pi, 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', [-0.6366, -0.5437, 0, 0.5437, 0.6366]);
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 8.33.2 Digital Chebyshev-1 lowpass by afd_chb1 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, 0.45, 0.50, 1.0, 1.5, 1.55, M]);
set(gca, 'YTickMode', 'manual', 'YTick', [0, 0.001, 0.5, 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, 0.45, 0.50, 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('Digital frequency in \pi units'); ylabel('Decibels'); title('Magnitude in dB ');
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.45, 0.50, 1.0, M]);
set(gca, 'YTickMode', 'manual', 'YTick', [-65, -60, -1, 0]);
set(gca,'YTickLabelMode','manual','YTickLabel',['65'; '60';' 1';' 0']);

subplot(2,2,4); plot(ww/pi, grd); grid on; axis([0, M, 0, 200]);
xlabel('Digital frequency in \pi units'); ylabel('Samples'); title('Group Delay');
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.45, 0.50, 1.0, M]);
%set(gca, 'YTickMode', 'manual', 'YTick', [0:20:100]);

figure('NumberTitle', 'off', 'Name', 'Problem 8.33.2 Pole-Zero Plot')
set(gcf,'Color','white');
zplane(b,a);
title(sprintf('Pole-Zero Plot'));
%pzplotz(b,a);

% ---------------------------------------------------------------
%           method 2:  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);

delta_w = 2*pi/1000;
Rp = -(min(db(1:1:ceil(wp/delta_w)+1)));                       % Actual Passband Ripple

fprintf('\nActual Passband Ripple is %.4f dB.\n', Rp);

As = -round(max(db(ceil(ws/delta_w)+1:1:501)));                % Min Stopband attenuation
fprintf('\nMin Stopband attenuation is %.4f dB.\n', As);

%% -----------------------------------------------------------------
%%                             Plot
%% -----------------------------------------------------------------  

figure('NumberTitle', 'off', 'Name', 'Problem 8.33.2 Digital Chebyshev-1 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, 0.45, 0.50, 1, M]);
set(gca, 'YTickMode', 'manual', 'YTick', [0, 0.001, 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, 0.45, 0.50, 1, 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, 0.45, 0.50, 1, M]);
set(gca, 'YTickMode', 'manual', 'YTick', [-70, -60, -15, -1, 0]);
set(gca,'YTickLabelMode','manual','YTickLabel',['70'; '60';'15';' 1';' 0']);

subplot(2,2,4); plot(ww/pi, grd); axis([0, M, 0, 150]); grid on;
xlabel('Digital frequency in \pi units'); ylabel('Samples'); title('Group Delay');
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.45, 0.50, 1, M]);
set(gca, 'YTickMode', 'manual', 'YTick', [0:20:100]);

% ----------------------------------------------
%       Calculation of Impulse Response
% ----------------------------------------------
figure('NumberTitle', 'off', 'Name', 'Problem 8.33.2 Imp & Freq Response')
set(gcf,'Color','white');
t = [0:0.5:160]; subplot(2,1,1); impulse(cs,ds,t); grid on;   % Impulse response of the analog filter
axis([0,160,-0.4,0.5]);hold on

n = [0:1:160/T]; hn = filter(b,a,impseq(0,0,160/T));           % Impulse response of the digital filter
stem(n*T,hn); xlabel('time in sec'); title (sprintf('Impulse Responses, T=%f',T));
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*Fs, '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', 'Color', 'b'); text(0.4,0.55,'Digital filter', 'Color', 'r');

　　运行结果：

这里只放chebyshev-1型，第2小题

通带、阻带绝对指标，以及模拟滤波器频带截止频率，

模拟chebyshev-1型低通滤波器，幅度谱、相位谱和脉冲响应

采用afd_chb1函数（双线性变换法），得到数字chebyshev-1低通滤波器，其幅度谱、相位谱和群延迟响应

采用MATLAB自带cheby1函数，得到数字低通，幅度谱、相位谱和群延迟响应

以下三图是模拟低通、两函数得到数字低通，各自最低阻带衰减对比，可见，MATLAB自带的cheby1函数设计的数字低通可以达到70dB。

至于其它的代码这里不放了，步骤类似，最后给出设计滤波器阶数和阻带衰减对比结果