Today is Children's Day, not his family
Code:
%% ------------------------------------------------------------------------
%% Output Info about this m-file
fprintf('\n***********************************************************\n');
fprintf(' <DSP using MATLAB> Problem 8.4 \n\n');
banner();
%% ------------------------------------------------------------------------
% digital Notch filter
r = 0.7
%r = 0.9
%r = 0.99
omega0 = pi/2;
% corresponding system function Direct form
b0 = 1.0; % gain parameter
b = b0*[1 -2*cos(omega0) 1]; % numerator with poles
a = [1 -2*r*cos(omega0) r*r]; % denominator
% precise resonant frequency and 3dB bandwidth
omega_r = acos((1+r*r)*cos(omega0)/(2*r));
delta_omega = 2*(1-r);
fprintf('\nNotch Freq is : %.4fpi unit, 3dB bandwidth is %.4f \n', omega_r/pi,delta_omega);
%
[db, mag, pha, grd, w] = freqz_m(b, a);
[db_b, mag_b, pha_b, grd_b, w] = freqz_m(b, 1);
% ---------------------------------------------------------------------
% Choose the gain parameter of the filter, maximum gain is equal to 1
% ---------------------------------------------------------------------
gain1=max(mag) % with poles
gain2=max(mag_b) % without poles
[db, mag, pha, grd, w] = freqz_m(b/gain1, a);
[db_b, mag_b, pha_b, grd_b, w] = freqz_m(b/gain2, 1);
figure('NumberTitle', 'off', 'Name', 'Problem 8.4 Notch filter with poles')
set(gcf,'Color','white');
subplot(2,2,1); plot(w/pi, db); grid on; axis([0 2 -60 10]);
set(gca,'YTickMode','manual','YTick',[-60,-30,0])
set(gca,'YTickLabelMode','manual','YTickLabel',['60';'30';' 0']);
set(gca,'XTickMode','manual','XTick',[0,0.5,1,1.5,2]);
xlabel('frequency in \pi units'); ylabel('Decibels'); title('Magnitude Response in dB');
subplot(2,2,3); plot(w/pi, mag); grid on; %axis([0 1 -100 10]);
xlabel('frequency in \pi units'); ylabel('Absolute'); title('Magnitude Response in absolute');
set(gca,'XTickMode','manual','XTick',[0,0.5,1,1.5,2]);
set(gca,'YTickMode','manual','YTick',[0,1.0]);
subplot(2,2,2); plot(w/pi, pha); grid on; %axis([0 1 -100 10]);
xlabel('frequency in \pi units'); ylabel('Rad'); title('Phase Response in Radians');
subplot(2,2,4); plot(w/pi, grd*pi/180); grid on; %axis([0 1 -100 10]);
xlabel('frequency in \pi units'); ylabel('Rad'); title('Group Delay');
set(gca,'XTickMode','manual','XTick',[0,0.5,1,1.5,2]);
%set(gca,'YTickMode','manual','YTick',[0,1.0]);
figure('NumberTitle', 'off', 'Name', 'Problem 8.4 Notch filter without poles')
set(gcf,'Color','white');
subplot(2,2,1); plot(w/pi, db_b); grid on; axis([0 2 -60 10]);
set(gca,'YTickMode','manual','YTick',[-60,-30,0])
set(gca,'YTickLabelMode','manual','YTickLabel',['60';'30';' 0']);
set(gca,'XTickMode','manual','XTick',[0,0.25,0.5,1,1.5,1.75]);
xlabel('frequency in \pi units'); ylabel('Decibels'); title('Magnitude Response in dB');
subplot(2,2,3); plot(w/pi, mag_b); grid on; %axis([0 1 -100 10]);
xlabel('frequency in \pi units'); ylabel('Absolute'); title('Magnitude Response in absolute');
set(gca,'XTickMode','manual','XTick',[0,0.5,1,1.5,2]);
set(gca,'YTickMode','manual','YTick',[0,1.0]);
subplot(2,2,2); plot(w/pi, pha_b); grid on; %axis([0 1 -100 10]);
xlabel('frequency in \pi units'); ylabel('Rad'); title('Phase Response in Radians');
xlabel('frequency in \pi units'); ylabel('Decibels'); title('Magnitude Response in dB');
subplot(2,1,2); plot(w/pi, pha, 'r--'); grid on; hold on;%axis([0 1 -100 10]);
plot(w/pi, pha_b); hold off;
xlabel('frequency in \pi units'); ylabel('Rad'); title('Phase Response in Radians');
figure('NumberTitle', 'off', 'Name', 'Problem 8.4 Pole-Zero Plot')
set(gcf,'Color','white');
zplane(b,a);
title(sprintf('Pole-Zero Plot, r=%.2f \\omega=%.2f\\pi',r,omega0/pi));
%pzplotz(b,a);
figure('NumberTitle', 'off', 'Name', 'Problem 8.4 Pole-Zero Plot')
set(gcf,'Color','white');
zplane(b,1);
title(sprintf('Pole-Zero Plot, r=%.2f \\omega=%.2f\\pi',r,omega0/pi));
%pzplotz(b,a);
% Impulse Response
fprintf('\n----------------------------------');
fprintf('\nPartial fraction expansion method: \n');
b = b/gain1;
[R, p, c] = residuez(b , a)
MR = (abs(R))' % Residue Magnitude
AR = (angle(R))'/pi % Residue angles in pi units
Mp = (abs(p))' % pole Magnitude
Ap = (angle(p))'/pi % pole angles in pi units
[delta, n] = impseq(0,0,50);
h_chk = filter(b , a , delta); % check sequences
% ------------------------------------------------------------------------
% gain parameter b0=1
% ------------------------------------------------------------------------
h = -0.5204*( 0.7.^n ) .* (2*cos(pi*n/2) ) + 2.0408 * delta; % r=0.7
%h = -0.1173*( 0.9.^n ) .* (2*cos(pi*n/2) ) + 1.2346 * delta; % r=0.9
%h = -0.0102*( 0.99.^n ) .* (2*cos(pi*n/2) ) + 1.0203 * delta; % r=0.99
% ------------------------------------------------------------------------
% ------------------------------------------------------------------------
% gain parameter b0 = equation
% ------------------------------------------------------------------------
%h = -0.3877*( 0.7.^n ) .* (2*cos(pi*n/2) ) + 1.5204 * delta; % r=0.7
%h = -0.1173*( 0.9.^n ) .* (2*cos(pi*n/2) ) + 1.2346 * delta; % r=0.9
%h = -0.0102*( 0.99.^n ) .* (2*cos(pi*n/2) ) + 1.0203 * delta; % r=0.99
% ------------------------------------------------------------------------
figure('NumberTitle', 'off', 'Name', 'Problem 8.4 Notch filter, h(n) by filter and Inv-Z ')
set(gcf,'Color','white');
subplot(2,1,1); stem(n, h_chk); grid on; %axis([0 2 -60 10]);
xlabel('n'); ylabel('h\_chk'); title('Impulse Response sequences by filter');
subplot(2,1,2); stem(n, h/gain1); grid on; %axis([0 1 -100 10]);
xlabel('n'); ylabel('h'); title('Impulse Response sequences by Inv-Z');
[db, mag, pha, grd, w] = freqz_m(h/gain1, [1]);
figure('NumberTitle', 'off', 'Name', 'Problem 8.4 Notch filter, h(n) by Inv-Z ')
set(gcf,'Color','white');
subplot(2,2,1); plot(w/pi, db); grid on; axis([0 2 -60 10]);
set(gca,'YTickMode','manual','YTick',[-60,-30,0])
set(gca,'YTickLabelMode','manual','YTickLabel',['60';'30';' 0']);
set(gca,'XTickMode','manual','XTick',[0,0.5,1,1.5,2]);
xlabel('frequency in \pi units'); ylabel('Decibels'); title('Magnitude Response in dB');
subplot(2,2,3); plot(w/pi, mag); grid on; %axis([0 1 -100 10]);
xlabel('frequency in \pi units'); ylabel('Absolute'); title('Magnitude Response in absolute');
set(gca,'XTickMode','manual','XTick',[0,0.5,1,1.5,2]);
set(gca,'YTickMode','manual','YTick',[0,1.0]);
subplot(2,2,2); plot(w/pi, pha); grid on; %axis([0 1 -100 10]);
xlabel('frequency in \pi units'); ylabel('Rad'); title('Phase Response in Radians');
subplot(2,2,4); plot(w/pi, grd*pi/180); grid on; %axis([0 1 -100 10]);
xlabel('frequency in \pi units'); ylabel('Rad'); title('Group Delay');
set(gca,'XTickMode','manual','XTick',[0,0.5,1,1.5,2]);
%set(gca,'YTickMode','manual','YTick',[0,1.0]);
% Given resonat frequency and 3dB bandwidth
delta_omega = 0.04;
omega_r = pi*0.5;
r = 1 - delta_omega / 2
operation result:
Notch filter, ω0 = 0.5π, the introduction of the poles r = 0.7
System functions partial fraction expansion
The system as in FIG pole-zero
The magnitude spectrum, the phase spectrum, group delay
Zero is located at the home position, corresponding to remove zero, as follows
After the removal of zero, the amplitude spectrum of a notch filter, phase and group delay spectrum
With the introduction of zero, near the notch frequency band narrower (red), blue is the case where no zero. As said the book, the notch frequency ω0
Both little difference.
Results pole mold r = 0.9 and 0.99, the hold here.