Summary

Source code

This graphics has been realized with the help of the following Octave script:

#-------------------------------------------------------------------------------
# Specifications
#
order = 6;
pointNb = 100;
axesLength = 1.25;
textOffset = 0.05;
wLog = logspace(-1, 1, pointNb);
AdbMax = 10;
AdbMin = -40;

#===============================================================================
# Butterworth filter
#
[zeroes, poles, gain] = butter(order, 1, 's');

[num, den] = butter(order, 1, 's');
while ( length(num) < length(den) )
  num = [0, num];
endwhile;

#-------------------------------------------------------------------------------
# Poles plot
#
figure(1);
angle = linspace(pi/2, 3*pi/2, pointNb);
plot(cos(angle), sin(angle));

hold on;
plot([-axesLength, axesLength-1], [0, 0], 'k');
plot([0, 0], [-axesLength, axesLength], 'k');
plot(axesLength-1+textOffset*[0, 2, 0, 0], textOffset*3/4*[-1, 0, 1, -1], 'k');
plot(textOffset*3/4*[-1, 0, 1, -1], axesLength+textOffset*[0, 2, 0, 0], 'k');
plot(poles, 'xr');
hold off;

axis("equal", "off");
text(axesLength-1+textOffset, 1.5*textOffset, "re(s)");
text(textOffset, axesLength+1.5*textOffset, "im(s)");

for index = 1:order
  if imag(poles(index)) > 0
    text(real(poles(index))-textOffset, imag(poles(index))+textOffset, ...
    sprintf("p%i", index), "horizontalalignment", "right");
  endif
endfor

#-------------------------------------------------------------------------------
# Normalized filter, transfer functions plot
#
Adb = 20*log10(abs(freqs(num, den, wLog)));

figure(2);
semilogx(wLog, Adb);
hold on;
for index = 1:order
  if imag(poles(index)) > 0
    num = [0, 0, 1];
    den = [1, -2*real(poles(index)), abs(poles(index))^2];
    Adb = 20*log10(abs(freqs(num, den, wLog)));
    semilogx(wLog, Adb, 'g');
  endif
endfor
hold off;

axis([wLog(1), wLog(length(wLog)), AdbMin, AdbMax]);
xlabel('Frequency');
ylabel('Amplitude responses');
grid;

print -dsvg butterworth_amplitudes_6.svg

Licensing

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Category:CC-BY-SA-3.0#Butterworth%20amplitudes%206.svg