File:Bessel-butterworth-filter 2.png
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Summary
[edit]DescriptionBessel-butterworth-filter 2.png |
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 |
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This plot was created with Gnuplot by n.
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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");
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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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