Matlab/Octave

  

 

 

 

Filter Design Tutorial

 

 

 

Plotting Transfer Function

 

Example 1 :

 

 

    w = 0:0.01:2;

    s = j * w;

    Tf = 1 ./ (s.^2 + 0.6 .* s .+ 1);

    Tf_abs = abs(Tf);

    plot(w,Tf_abs);grid();

 

 

Example 2 :

 

 

    re = -2:.05:2;

    im = -2:.05:2;

     

    [a,b] = meshgrid(re,im);

    s = (a + b*j);

     

    Tf = 1 ./ (s.^2 + 0.6 .* s .+ 1);

    Tf_abs = abs(Tf);

     

    mesh(X,Y,Tf_abs);xlabel('a');ylabel('b');zlabel('|Tf(s)|');

 

    re = -2:.05:2;

    im = -2:.05:2;

     

    [a,b] = meshgrid(re,im);

    s = (a + b*j);

     

    Tf = 1 ./ (s.^2 + 0.6 .* s .+ 1);

    Tf_abs = abs(Tf);

     

    mesh(X,Y,Tf_abs);

    xlabel('a');ylabel('b');zlabel('|Tf(s)|');

    view(0,90);

    zmax = 20;

    zmin = -10;

    caxis([-zmax zmax]);

    colormap(hsv(20));

 

    re = -2:.025:2;

    im = -2:.025:2;

     

    [a,b] = meshgrid(re,im);

    s = (a + b*j);

     

    Tf = 1 ./ (s.^2 + 0.6 .* s .+ 1);

    Tf_abs = abs(Tf);

     

    contour(a,b,Tf_abs,200);

    xlabel('a');ylabel('b');

    xlim([-2 2]);ylim([-2 2]);zlim([0 10]);

 

 

Example 3 :

 

 

 

    download matlab/octable .m file

     

    re = -3:.05:3; im = -3:.05:3;

    a = 0.4;

    b = 1.0;

    z_List = [ 0 + 1.2*j 0 + 1.8*j 0 - 1.2*j, 0 - 1.8*j];

    p_List = [ a*cos(pi/2+pi/12)+b*sin(pi/2+pi/12)*j a*cos(pi/2+4*pi/12)+b*sin(pi/2+4*pi/12)*j a*cos(pi/2+pi/12)-b*sin(pi/2+pi/12)*j a*cos(pi/2+4*pi/12)-b*sin(pi/2+4*pi/12)*j];

     

     

    [X,Y] = meshgrid(re,im);

    s = (X + Y*j);

    s_jw = (0 + Y*j);

     

    num = ones(length(X),length(Y));

     

    for i=1:length(z_List)

       num = num .* (s .- z_List(i));   

    end;

     

     

    den = ones(length(X),length(Y));

     

    for i=1:length(z_List)

       den = den .* (s .- p_List(i));   

    end;

     

    Zw_num = ones(length(X),length(Y));

     

    for i=1:length(z_List)

       Zw_num = Zw_num .* (s_jw .- z_List(i));  

    end;

     

     

    Zw_den = ones(length(X),length(Y));

     

    for i=1:length(z_List)

       Zw_den = Zw_den .* (s_jw .- p_List(i));  

    end;

     

     

    Zabs = abs(num ./ den );

    Zabs = 10 * log10(Zabs);

    Zarg = arg(num ./ den);

    Zw = abs(Zw_num ./ Zw_den);

    Zw = 10 * log10(Zw);

    Zwarg = arg(Zw_num ./ Zw_den);

     

    zmax = 20;

    zmin = -10;

    subplot(2,2,1); mesh(X,Y,Zabs);axis([re(1) re(end) im(1) im(end) zmin zmax]);

            hold on;

            plot3(X.*0,Y,Zw, 'linewidth',3,'color','r');

            hold off;

            view(50,10);

        caxis([-zmax zmax]);

            colormap(hsv(20));

    subplot(2,2,2); mesh(X,Y,Zabs);axis([0 re(end) im(1) im(end) zmin zmax]);

            hold on;

            plot3(X.*0,Y,Zw, 'linewidth',3,'color','r');

            hold off;

            view(-90,10);

            caxis([-zmax zmax]);

            colormap(hsv(20));

    subplot(2,2,3); mesh(X,Y,Zabs);axis([re(1) re(end) im(1) im(end) zmin zmax]);

            hold on;

            plot3(X.*0,Y,Zw, 'linewidth',3,'color','r');

            hold off;

            view(0,90);

            caxis([-zmax zmax]);

            colormap(hsv(20));

    subplot(2,2,4); mesh(X,Y,Zarg);axis([0 re(end) im(1) im(end) -pi pi]);

            hold on;

            plot3(X.*0,Y,Zwarg, 'linewidth',3,'color','r');

            hold off;

            view(-90,10);

 

 

Impact of the location of pole from jw axis

 

 

 

Impact of Zero on jw axis

 

 

    re = -2:.05:2;

    im = -2:.05:2;

     

    [a,b] = meshgrid(re,im);

    s = (a + b*j);

     

    Tf = (s.^2 .+ 2) ./ (s.^2 + 0.6 .* s .+ 1); %Tf = 1 ./ (s.^2 + 0.6 .* s .+ 1);

    Tf_abs = abs(Tf);

    Tf_abs_log = 10 .* log(Tf_abs);

     

    subplot(1,2,1);

    mesh(X,Y,Tf_abs_log);xlabel('a');ylabel('b');zlabel('|Tf(s)| in dB');

     

    subplot(1,2,2);

    mesh(X,Y,Tf_abs_log);xlabel('a');ylabel('b');zlabel('|Tf(s)| in dB');

    view(0,90);

    zmax = 20;

    zmin = -10;

    caxis([-zmax zmax]);

    colormap(hsv(20));

 

 

    w = 0:0.01:2;

    s = j * w;

    Tf = (s.^2 .+ 2) ./ (s.^2 + 0.25 .* s .+ 1);  %Tf = 1 ./ (s.^2 + 0.6 .* s .+ 1);

    Tf_abs = abs(Tf);

    plot(w,Tf_abs); ylim([0 6]); grid();

 

 

Low Pass Filter - fir1

 

    Fs=8e3;

    Ts=1/Fs;

    Ns=512;

    t=[0:Ts:Ts*(Ns-1)];

    f1=500;

    f2=3200;

    x1=sin(2*pi*f1*t);

    x2=0.3*sin(2*pi*f2*t);

    x=x1+x2;

    a = 1;

    b = fir1(48,0.35,'low');

    y=filter(b,a,x);

    subplot(5,1,1); plot(x1);ylabel('x1');axis([0,length(x),-1.5,1.5]);

    subplot(5,1,2); plot(x2);ylabel('x2');axis([0,length(x),-1.5,1.5]);

    subplot(5,1,3); plot(x);ylabel('x=x1+x2');axis([0,length(x),-1.5,1.5]);

    subplot(5,1,4); plot(b);ylabel('b');axis([0,length(x),-0.5,0.5]);

    subplot(5,1,5); plot(y);ylabel('y=filter(b,a,x)');axis([0,length(x),-1.5,1.5]);