File:Bessel-butterworth-filter.svg

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Original file ‎(SVG file, nominally 461 × 943 pixels, file size: 471 KB)

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Summary

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Description
Deutsch: Besselfilter
English: Comparison of a bessel filter, a Butterworth filter and 3 cascaded 1st order lowpass filters
Date
Source
from user Mik81
Author chris828

Licensing

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w:en:Creative Commons
attribution
This file is licensed under the Creative Commons Attribution 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 give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.

Matlab Code

[edit]
%% draw the plots

% Main program to calculate group delay of a butterworth bessel
% Author: mik81@de.wikipedia.org, chris828
% bessel especially butterworth with 3rd Order
 
clear all;
close all;
 
omega = (-2.:0.01:2.);
omega = 10..^omega;
time = (0.:0.01:20.);
step = zeros(length(time),1);
for i=200:1:length(step)
  step(i) = 1;
end
 
%
% Functions
%
 
besselThirdOrder = 1 ./ ...
   ( 1 + ( omega .*  1i ) ...
      + ( omega .*  1i).^2 .* (2 ./ 5) ...
      + ( omega .*  1i).^3 .* (1 ./ 15) ...
   );
besselFourthOrder = 1 ./ ...
  ( 1 + ( omega .* 1i ) ...
    + ( omega .* 1i).^2 .* (3 ./ 7) ...
    + ( omega .* 1i).^3 .* (2 ./ 21) ...
    + ( omega .* 1i).^4 .* (1 ./ 105) ...
  );
besselFifthOrder = 1 ./ ...
  ( 1 + ( omega .* 1i ) ...
    + ( omega .* 1i ).^2 .* (4 ./ 9) ...
    + ( omega .* 1i ).^3 .* (1 ./ 9) ...
    + ( omega .* 1i ).^4 .* (1 ./ 63) ...
    + ( omega .* 1i ).^5 .* (1 ./ 945) ...
  );
 
butterworthThirdOrder = 1 ./ ...
   ( (1 + (omega .* (1/1.8) .* 1i)) ...
     .* ( 1 + (omega .* (1/1.8) .* 1i) + (omega .* (1/1.8) .* 1i).^2 ));
% butterworthThirdOrderIIR_y_a = 1 ./ (1+ y_a_1 )
% butterworthThirdOrderIIR_y_b = ( 1 ./ ( 1 + y_b_1 + y_b_2^2 ))
threeFilterFirstOrder = 1 ./ ...
  ( (1 + ( 1/3.6 .* omega .* 1i)) ...
     .* (1 + ( 1/3.6 .* omega .* 1i)) ...
     .* (1 + ( 1/3.6 .* omega .* 1i)) ...
  );
 
%
% Calculate and plot gain
%
f1 = figure; 
subplot(3,1,1);
loglog( ...
    omega, abs(butterworthThirdOrder), 'r' ...
   , omega, abs(besselThirdOrder), 'g' ...
  , omega, abs(threeFilterFirstOrder), 'b' ...
   , [0.01, 100.], [1/sqrt(2), 1/sqrt(2)], 'm');
axis ([0.01, 100., 0.00001, 5.]);
legend('Butterworth 3. Ordnung',   'Besselfilter 3. Ordnung',  '3 Tiefpässe 1. Ordnung' , '-3dB Linie', 'Location', 'SouthWest');
set(gca, 'xtick', [0.01 0.1 1 10 100]);
set(gca, 'xticklabel', {'0.01', '0.1', '1', '10', '100'});
set(gca, 'ytick', [1e-5 1e-4 0.001 0.01 0.1 1]);
set(gca, 'yticklabel', {'1e-5', '1e-4', '0.001', '0.01', '0.1', '1'});
grid ('on');
xlabel('Normalized \omega / \omega_0');
ylabel('Gain');
%orient('landscape');
%print ('butterworthFrequency.png', '-dpng', '-landscape');
%print ('butterworthFrequency.svg', '-dsvg', '-landscape');
%print(f1, '-dpng', 'butterworthFrequency.png');
%print(f1, '-depsc2', 'butterworthFrequency.eps');
%
% Calculate and plot phase
%
 
% Butterworth
butterworthThirdOrderAngle = angle(butterworthThirdOrder);
% correct roll over of phase
shift = 0;
lastAngle = butterworthThirdOrderAngle(1);
 
for i=2:1:length(butterworthThirdOrderAngle)
  if ( lastAngle < 0 && butterworthThirdOrderAngle(i) > 0)
    shift = shift - 2*pi;
  end
  lastAngle = butterworthThirdOrderAngle(i);
  butterworthThirdOrderAngle(i) = butterworthThirdOrderAngle(i) + shift;
end
 
% Bessel
 
besselThirdOrderAngle = angle(besselThirdOrder);
 
% correct roll over of phase
shift = 0;
lastAngle = besselThirdOrderAngle(1);
for i=2:1:length(besselThirdOrderAngle)
  if ( lastAngle < 0 && besselThirdOrderAngle(i) > 0)
    shift = shift - 2*pi;
  end
  lastAngle = besselThirdOrderAngle(i);
  besselThirdOrderAngle(i) = besselThirdOrderAngle(i) + shift;
end
besselFourthOrderAngle = angle(besselFourthOrder);
% correct roll over of phase
shift = 0;
lastAngle = besselFourthOrderAngle(1);
 
for i=2:1:length(besselFourthOrderAngle)
  if ( lastAngle < 0 && besselFourthOrderAngle(i) > 0)
    shift = shift - 2*pi;
  end
  lastAngle = besselFourthOrderAngle(i);
  besselFourthOrderAngle(i) = besselFourthOrderAngle(i) + shift;
end
 
% three filter first order
 
threeFilterFirstOrderAngle = angle(threeFilterFirstOrder);
 
% correct roll over of phase
shift = 0;
lastAngle = threeFilterFirstOrderAngle(1);
for i=2:1:length(threeFilterFirstOrderAngle)
  if ( lastAngle < 0 && threeFilterFirstOrderAngle(i) > 0)
    shift = shift - 2*pi;
  end
  lastAngle = threeFilterFirstOrderAngle(i);
  threeFilterFirstOrderAngle(i) = threeFilterFirstOrderAngle(i) + shift;
end
 
subplot(3,1,2);
semilogx( ...
   omega, butterworthThirdOrderAngle, 'r'...
   , omega, besselThirdOrderAngle, 'g'...
   , omega, threeFilterFirstOrderAngle, 'b');
legend('Butterworthfilter 3. Ordnung', 'Besselfilter 3. Ordnung', '3 Tiefpässe 1. Ordnung', 'Location', 'SouthWest');
axis ([0.01, 100., -pi*9/4, pi/4]);
set(gca, 'xtick', [0.01 0.1 1 10 100]);
set(gca, 'xticklabel', {'0.01', '0.1', '1', '10', '100'});
grid ('on');
xlabel('Normalized \omega / \omega_0');
ylabel('phase in rad');
%
% Calculate and plot group delay with phase
%
 
%  -deltaAngle/deltaOmega
 
% butterworth
lastAngle = butterworthThirdOrderAngle(1);
lastOmega = omega(1);
butterworthThirdOrderGroupDelay = zeros(length(butterworthThirdOrder),1);
for i=2:1:length(butterworthThirdOrderAngle)
  butterworthThirdOrderGroupDelay(i) = ...
     -1. * (butterworthThirdOrderAngle(i)-lastAngle)/(omega(i)-lastOmega);
  lastOmega = omega(i);
  lastAngle = butterworthThirdOrderAngle(i);
end
butterworthThirdOrderGroupDelay(1) = butterworthThirdOrderGroupDelay(2);
 
% bessel
lastAngle = besselThirdOrderAngle(1);
lastOmega = omega(1);
besselThirdOrderGroupDelay = zeros(length(besselThirdOrder),1);
for i=2:1:length(besselThirdOrderAngle)
  besselThirdOrderGroupDelay(i) = ...
     -1. * (besselThirdOrderAngle(i)-lastAngle)/(omega(i)-lastOmega);
  lastOmega = omega(i);
  lastAngle = besselThirdOrderAngle(i);
end
besselThirdOrderGroupDelay(1) = besselThirdOrderGroupDelay(2);
 
lastAngle = besselFourthOrderAngle(1);
lastOmega = omega(1);
besselFourthOrderGroupDelay = zeros(length(besselFourthOrder),1);
for i=2:1:length(besselFourthOrderAngle)
  besselFourthOrderGroupDelay(i) = ...
     -1. * (besselFourthOrderAngle(i)-lastAngle)/(omega(i)-lastOmega);
  lastOmega = omega(i);
  lastAngle = besselFourthOrderAngle(i);
end
besselFourthOrderGroupDelay(1) = besselFourthOrderGroupDelay(2);
 
% three first order
lastAngle = threeFilterFirstOrderAngle(1);
lastOmega = omega(1);
threeFilterFirstOrderGroupDelay = zeros(length(threeFilterFirstOrder),1);
for i=2:1:length(threeFilterFirstOrderAngle)
  threeFilterFirstOrderGroupDelay(i) = ...
     -1. * (threeFilterFirstOrderAngle(i)-lastAngle)/(omega(i)-lastOmega);
  lastOmega = omega(i);
  lastAngle = threeFilterFirstOrderAngle(i);
end
threeFilterFirstOrderGroupDelay(1) = threeFilterFirstOrderGroupDelay(2);
 
subplot(3,1,3);
semilogx( ...
    omega, butterworthThirdOrderGroupDelay, 'r'...
   , omega, besselThirdOrderGroupDelay, 'g'...
  , omega, threeFilterFirstOrderGroupDelay, 'b');
axis ('auto');
grid ('on');
xlabel('Normalized \omega / \omega_0');
ylabel('Groupdelay');
legend('Butterworth 3. Ordnung', 'Bessel 3. Ordnung', '3 Tiefpässe 1. Ordnung', 'Location', 'SouthWest');
set(gca, 'xtick', [0.01 0.1 1 10 100]);
set(gca, 'xticklabel', {'0.01', '0.1', '1', '10', '100'});

%% save plots
print(f1, '-dpng', 'butterworth.png');
print(f1, '-depsc', 'butterworth.eps');

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Date/TimeThumbnailDimensionsUserComment
current18:03, 4 April 2015Thumbnail for version as of 18:03, 4 April 2015461 × 943 (471 KB)Chris828 (talk | contribs)text as path
17:58, 4 April 2015Thumbnail for version as of 17:58, 4 April 2015461 × 943 (273 KB)Chris828 (talk | contribs)User created page with UploadWizard

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