October 1, the new China 70 birthday, in the morning to watch the grand ceremony, neat square, advanced weapons, respect for ancestors hero, happy people, we would like to
National getting better, getting better and better people's lives.
Then do problems.
Code:
%% ------------------------------------------------------------------------
%% Output Info about this m-file
fprintf('\n***********************************************************\n');
fprintf(' <DSP using MATLAB> Problem 8.30 \n\n');
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('\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, 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('\n\n ********** Butterworth Filter Order = %3.0f \n', 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'));
% ----------------------------------------------
% 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);% Of Digital-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');
operation result:
Non-normalized analog prototype Butterworth low pass filter of direct form coefficients,
Analog lowpass coefficient series of the form:
By bilinear transform, into a digital Butterworth low-pass, direct form coefficients as follows
Digital low pass coefficients tandem
Amplitude of the analog Butterworth lowpass prototype filter spectrum, the phase spectrum and the impulse response
Bilinear transform, to obtain a digital Butterworth low-pass filter, from the amplitude spectrum and phase spectrum, group delay response
Digital low pass pole-zero function of the system of FIG.
FIG upper portion of the lower half, the analog low-pass and low-pass impulse response digital comparison can be seen inconsistent form.
Butter comes with MATLAB function for determining the digital low-pass, its amplitude spectrum and phase spectrum, group delay.
Compared with the result of the function afd_butt obtained above, phase and group delay spectrum is slightly different.
Pole-zero plot, is also slightly different, in part by zero closer.
Impulse response, can not see the difference.
