File:Butterworth poles 3.svg

Summary[edit]

DescriptionButterworth poles 3.svg	English: pole location for a 3rd order Butterworth lowpass filter
Date	7 June 2012
Source	Own work
Author	Fcorthay
SVG development	The source code of this SVG is valid. This vector image was created with Inkscape.

Source code[edit]

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

#-------------------------------------------------------------------------------
# Specifications
#
order = 3;
fc = 1;
pointNb = 100;
axesLength = 1.25;
textOffset = 0.05;
wLog = logspace(-2, 2, pointNb);
AdbMax = 10;
AdbMin = -60;

makeGraphics = 1;

figureIndex = 0;

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

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

#-------------------------------------------------------------------------------
# Poles plot
#
figureIndex = figureIndex+1;
figure(figureIndex);
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");
  elseif imag(poles(index)) == 0
    text(real(poles(index))-textOffset, imag(poles(index))+textOffset, ...
    "pr", "horizontalalignment", "right");
  endif
endfor

if (makeGraphics != 0)
  print -dsvg butterworth_poles_3.svg
endif

#-------------------------------------------------------------------------------
# Amplitude plot
#
Adb = 20*log10(abs(freqs(num, den, wLog)));

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

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

if (makeGraphics != 0)
  print -dsvg butterworth_amplitudes_3.svg
endif

#-------------------------------------------------------------------------------
# Phase plot
#
phase = arg(freqs(num, den, wLog));

figureIndex = figureIndex+1;
figure(figureIndex);
semilogx(wLog, phase);
hold on;
for index = 1:order
  if imag(poles(index)) > 0
    numP = [0, 0, 1];
    denP = [1, -2*real(poles(index)), abs(poles(index))^2];
    phase = arg(freqs(numP, denP, wLog));
    semilogx(wLog, phase, 'g');
  elseif imag(poles(index)) == 0
    numP = [0, 1];
    denP = [1, -real(poles(index))];
    phase = arg(freqs(numP, denP, wLog));
    semilogx(wLog, phase, 'g');
  endif
endfor
semilogx([wLog(1), wLog(length(wLog))], -pi/2*[1, 1], 'c');
semilogx([wLog(1), wLog(length(wLog))],   0  *[1, 1], 'c');
semilogx([wLog(1), wLog(length(wLog))],  pi/2*[1, 1], 'c');
hold off;

axis([wLog(1), wLog(length(wLog)), -pi, pi]);
xlabel('Frequency');
ylabel('Phase responses');

if (makeGraphics != 0)
  print -dsvg butterworth_phases_3.svg
endif

Licensing[edit]

I, the copyright holder of this work, hereby publish it under the following licenses: This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. You are free: to share – to copy, distribute and transmit the work to remix – to adapt the work Under the following conditions: attribution – You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). share alike – If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one. http://creativecommons.org/licenses/by-sa/3.0 CC BY-SA 3.0 Creative Commons Attribution-Share Alike 3.0 truetrue Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue You may select the license of your choice.