10月1日，新中国70周岁生日，上午观看了盛大的庆祝仪式，整齐的方阵，先进的武器，尊敬的先辈英雄，欢乐的人们，愿我们的
国家越来越好，人民生活越来越好。
接着做题。
代码：
%% ------------------------------------------------------------------------
%% Output Info about this m-file
fprintf('
***********************************************************
');
fprintf(' <DSP using MATLAB> Problem 8.30
');
banner();
%% ------------------------------------------------------------------------
% -----------------------------------
% Ω=(2/T)tan(ω/2)
% ω=2*[atan(ΩT/2)]
% Digital Filter Specifications:
% -----------------------------------
wp = 0.4*pi; % digital passband freq in rad
ws = 0.6*pi; % digital stopband freq in rad
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 = (2/T)*tan(wp/2); % prototype passband freq
OmegaS = (2/T)*tan(ws/2); % 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, 2*pi/T);
% --------------------------------------------------------------------
% find exact band-edge frequencies for the given dB specifications
% --------------------------------------------------------------------
%ind = find( abs(ceil(db_s))-50 == 0 )
[diff_to_50dB, ind] = min(abs(db_s+50))
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+50));
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, Fs);
[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
% --------------------------------------------------------------------
%ind = find( abs(ceil(db))-50 == 0 )
[diff_to_80dB, ind] = min(abs(db+50))
db(ind-3 : ind+3) % magnitude response, dB
ww(ind)/(pi)
%ww(ind)*Fs/(2*pi)
(2/T)*tan(ww(ind)/2)/pi
[sA,index] = sort(abs(db+50));
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 8.30 Analog Butterworth 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', [-0.876, -0.463, 0, 0.463, 0.876]); % T=1
set(gca, 'XTickMode', 'manual', 'XTick', [-0.44, -0.23, 0, 0.23, 0.44]); % T=2
set(gca, 'YTickMode', 'manual', 'YTick', [0, 0.0032, 0.5, 0.9441, 1]);
subplot(2,2,2); plot(ww_s/pi, db_s); grid on; axis([-M, M, -100, 10]);
xlabel('Analog frequency in pi units'); ylabel('Decibels'); title('Magnitude in dB ');
%set(gca, 'XTickMode', 'manual', 'XTick', [-0.876, -0.463, 0, 0.463, 0.8591, 0.876]); % T=1
set(gca, 'XTickMode', 'manual', 'XTick', [-0.44, -0.23, 0, 0.23, 0.4295, 0.44]); % T=2
set(gca, 'YTickMode', 'manual', 'YTick', [-90, -50, -1, 0]);
set(gca,'YTickLabelMode','manual','YTickLabel',['90';'50';' 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', [-OmegaS, -OmegaP, 0, OmegaP, OmegaS]/pi);
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 8.30 Digital Butterworth lowpass by afd_butt 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, 0.4, 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('Digital 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('Digital frequency in pi units'); ylabel('Decibels'); title('Magnitude in dB ');
%set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.4, 0.594, 0.6, 1.0, M]); % T=1
set(gca, 'XTickMode', 'manual', 'XTick', [0, 0.4, 0.594, 0.6, 1.0, M]); % T=2
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('Digital 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 8.30 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 8.30 Imp & Freq Response')
set(gcf,'Color','white');
t = [0:0.5:60]; subplot(2,1,1); impulse(cs,ds,t); grid on; % Impulse response of the analog filter
axis([0,60,-0.3,0.5]);hold on
n = [0:1:60/T]; hn = filter(b,a,impseq(0,0,60/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');
%% +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
%% MATLAB butter function
%% +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
% Analog Prototype Order Calculations:
N = ceil((log10((10^(Rp/10)-1)/(10^(As/10)-1)))/(2*log10(OmegaP/OmegaS)));
fprintf('
********** Butterworth Filter Order = %3.0f
', N)
OmegaC = OmegaP/((10^(Rp/10)-1)^(1/(2*N))); % Analog BW prototype cutoff freq
wn = 2*atan((OmegaC*T)/2); % Digital BW cutoff freq
% Digital Butterworth Filter Design:
wn = wn/pi; % Digital Butterworth cutoff freq in pi units
[b, a] = butter(N, wn); [C, B, A] = dir2cas(b, a)
% Calculation of Frequency Response:
[db, mag, pha, grd, ww] = freqz_m(b, a);
figure('NumberTitle', 'off', 'Name', 'Problem 8.30 Digital Butterworth lowpass by butter 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(' 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 8.30 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 8.30 Imp & Freq Response')
set(gcf,'Color','white');
t = [0:0.5:60]; subplot(2,1,1); impulse(cs,ds,t); grid on; % Impulse response of the analog filter
axis([0,60,-0.3,0.5]);hold on
n = [0:1:60/T]; hn = filter(b,a,impseq(0,0,60/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：
非归一化Butterworth模拟原型低通滤波器，直接形式的系数，
模拟低通串联形式的系数：
用双线性变换法，转换成数字Butterworth低通，直接形式的系数如下
数字低通串联形式系数
模拟Butterworth低通原型滤波器的幅度谱、相位谱和脉冲响应
双线性变换法，得到的数字Butterworth低通滤波器，起幅度谱、相位谱和群延迟响应
数字低通系统函数的零极点图
下图的上半部分，模拟低通和数字低通的脉冲响应对比，可以看出形态不一致。
采用MATLAB自带的butter函数求取数字低通，其幅度谱、相位谱和群延迟。
与上面afd_butt函数所得结果相比，相位谱和群延迟稍有不同。
零极点图，也稍有不同，零点部分靠的更紧密。
脉冲响应，与上一种方法对比，结果一样，看不出区别。
