File:Bessel-butterworth-filter 2.png
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Summary[edit]
| Description |
English: Comparison of a bessel filter, a Butterworth filter and 3 cascaded 1st order lowpass filters
Deutsch: de:Besselfilter |
| Source | Own work |
| Author | Own work |
This plot was created with Gnuplot.
Licensing[edit]
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.
GNU Octave[edit]
% Main program to calculate group delay of a butterworth bessel
% Author: mik81@de.wikipedia.org
% bessel especially butterworth with 3rd Order
clear 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
%
xlabel("Normalized omega / omega_0");
ylabel("Gain");
loglog( \
omega, abs(butterworthThirdOrder), ";Butterworth 3. Ordnung;" \
, omega, abs(besselThirdOrder), ";Besselfilter 3. Ordnung;" \
% , omega, abs(besselFourthOrder), ";Besselfilter 4. Ordnung;" \
, omega, abs(threeFilterFirstOrder), ";3 Tiefpaesse 1. Ordnung;" \
, [0.01, 100.], [1/sqrt(2), 1/sqrt(2)], ";-3dB;");
axis ([0.01, 100., 0.00001, 5.]);
grid ("on");
print ("butterworthFrequency.png", "-dpng", "-landscape");
%
% 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
xlabel("Normalized omega / omega_0");
ylabel("phase in rad");
% !!! unavoidable !!!
__gnuplot_set__ ytics pi/2;
semilogx( \
omega, butterworthThirdOrderAngle, ";Butterworthfilter 3. Ordnung;"
, omega, besselThirdOrderAngle, ";Besselfilter 3. Ordnung;"
% , omega, besselFourthOrderAngle, ";Besselfilter 4. Ordnung;" );
, omega, threeFilterFirstOrderAngle, ";3 Tiefpaesse 1. Ordnung;");
% legend (, );
axis ([0.01, 100., -pi*9/4, pi/4]);
grid ("on");
print ("butterworthAngle.png", "-dpng", "-landscape");
%
% 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);
xlabel("Normalized omega / omega_0");
ylabel("Groupdelay");
% !!! unavoidable !!!
__gnuplot_set__ ytics auto;
semilogx( \
omega, butterworthThirdOrderGroupDelay, ";Butterworth 3. Ordnung;"
, omega, besselThirdOrderGroupDelay, ";Bessel 3. Ordnung;"
% , omega, besselFourthOrderGroupDelay, ";Besselfilter 4. Ordnung;");
, omega, threeFilterFirstOrderGroupDelay, ";3 Tiefpaesse 1. Ordnung;");
axis ("auto");
grid ("on");
print ("butterworthGroupDelay.png", "-dpng", "-landscape");
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 20:03, 16 January 2008 | | 640 × 1,440 (11 KB) | Mik81 (talk | contribs) | == Summary == {{Information |Description= de:Besselfilter |Source= selfmade |Date= |Author= |Permission= |other_versions= }} Category:Linear filters == Licensing == {{cc-by-3.0}} |
| 20:01, 16 January 2008 |  | 640 × 1,440 (14 KB) | Mik81 (talk | contribs) | {{Information |Description= de:Besselfilter |Source= selfmade |Date= |Author= |Permission= |other_versions= }} Category:Linear filters |
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