File:Circular convolution example.svg
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![File:Circular convolution example.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Circular_convolution_example.svg/462px-Circular_convolution_example.svg.png)
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Summary
DescriptionCircular convolution example.svg |
English: Circular convolution can be expedited by the FFT algorithm, so it is often used with an FIR filter to efficiently compute linear convolutions. These graphs illustrate how that is possible. Note that a larger FFT size (N) would prevent the overlap that causes graph #6 to not quite match all of #3. |
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Date | ||||
Source | Own work | |||
Author | Bob K | |||
Permission (Reusing this file) |
I, the copyright holder of this work, hereby publish it under the following license:
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Other versions |
This file was derived from: Circular convolution example.png |
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SVG development InfoField | ![]() This W3C-invalid vector image was created with LibreOffice. |
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Gnu Octave source InfoField | click to expand
This graphic was created with the help of the following Octave script: % Options
frame_background_gray = true;
if frame_background_gray
graphics_toolkit("qt") % has "insert text" option
% graphics_toolkit("fltk") % has cursor coordinate readout
frame_background = .94*[1 1 1];
d = 2; % amount to add to text sizes
else
graphics_toolkit("gnuplot") % background will be white regardless of value below
frame_background = .94*[1 1 1];
d=0;
endif
% (https://octave.org/doc/v4.2.1/Graphics-Object-Properties.html#Graphics-Object-Properties)
% Speed things up when using Gnuplot
set(0, "DefaultFigureColor",frame_background)
set(0, "DefaultAxesFontsize",10+d) % size of numeric tick labels
set(0, "DefaultTextFontsize",12+d)
set(0, "DefaultAxesXtick",[])
set(0, "DefaultAxesYtick",[])
set(0, "DefaultLineLinewidth",1)
xmax = 3000;
%=======================================================
hfig = figure("position",[100 100 488 512], "color",frame_background);
x1 = .02; % left margin
x2 = .02; % right margin
y1 = .08; % bottom margin for annotation
y2 = .08; % top margin for title
dy = .04; % vertical space between rows
width = 1-x1-x2;
height= (1-y1-y2-5*dy)/6; % space allocated for each of 6 rows
x_origin = x1;
y_origin = 1; % start at top of graph area
%=======================================================
y_origin = y_origin -y2 -height; % position of top row
% subplot() undoes all the "color" attempts above. (gnuplot bug)
subplot("position",[x_origin y_origin width height])
L = 100;
f = ones(1,L)/L;
plot(-100:200-1, [zeros(1,100) f*L zeros(1,100)], "linewidth",2, "color","magenta")
xlim([-100 xmax]); ylim([0 2])
title("Circular convolution example", "fontsize",16)
text(100, 1.6, "h[n]")
%text(xmax/2, 0.4, '\leftarrow n \rightarrow')
text(2500, 0.330, '\leftarrow n \rightarrow')
y_origin = y_origin -dy -height;
subplot("position",[x_origin y_origin width height])
a = [zeros(1,20) ones(1,L) zeros(1,300) 0.5*ones(1,100) zeros(1,1000-L-20-400)];
b = [zeros(1,1000-L-20) ones(1,L) zeros(1,20)];
a1 = [zeros(1,1000) a zeros(1,1000)];
b1 = [zeros(1,1000) b zeros(1,1000)];
plot(1:length(a1), a1, "color","blue", 1:length(a1), b1, "color","red")
xlim([0 xmax]); ylim([0 2])
text(200, 1.6, "X[n]")
y_origin = y_origin -dy -height;
subplot("position",[x_origin y_origin width height])
a1 = conv(a1,f);
b1 = conv(b1,f);
plot(1:length(a1), a1+b1, "color","green", "linewidth",2)
xlim([0 xmax]); ylim([0 2*max(a1)])
text(200, 1.6, "X[n] * h[n]")
%text(200, 1.6, "X[n] ∗ h[n]", "interpreter","none") % requires PERL post-processor
y_origin = y_origin -dy -height;
subplot("position",[x_origin y_origin width height])
a = [a a a];
b = [b b b];
L = 1:length(a);
plot(L, a, "color","blue", L, b, "color","red")
xlim([0 xmax]); ylim([0 2.5])
set(gca,"xtick", [1000 2000]);
%set(gca,"xticklabel",["N" "2N"])
set(gca,"xticklabel",[]); text(981,-.5, "N"); text(1955,-.5, "2N")
text(200, 2.0, 'X_N[n]')
y_origin = y_origin -dy -height;
subplot("position",[x_origin y_origin width height])
a1 = conv(a,f);
b1 = conv(b,f);
b1(1:90) = b1(3000+[1:90]);
L = 1:length(a1);
plot(L,a1,"color","blue", L,b1, "color","red")
xlim([0 xmax]); ylim([0 2*max(a1)])
text(200, 1.6, 'components of X_N[n] * h[n]') % can't use "interpreter","none" here
y_origin = y_origin -dy -height;
subplot("position",[x_origin y_origin width height])
c = a1+b1;
L = length(c);
k=1100;
plot(1:k, c(1:k), "color","red", k+(1:900), c(k+(1:900)), "color","green",...
"linewidth",2, (k+900+1):xmax, c((k+900+1):xmax), "color","red")
xlim([0 xmax]); ylim([0 2*max(a1+b1)])
text(200, 1.6, 'X_N[n] * h[n]') % can't use "interpreter","none" here
text(1263, -.6, "X[n] * h[n]", "fontsize",16)
%text(1274, -.6, "X[n] ∗ h[n]", "interpreter","none", "fontsize",16) % requires PERL post-processor
% After a call to annotation(), the cursor coordinates change to the units used below.
annotation("line", [.367 .367], [.113 .022])
annotation("line", [.664 .664], [.113 .022])
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Captions
Example of using circular convolution to produce linear convolution
Items portrayed in this file
depicts
4 June 2019
image/svg+xml
bb8b6d187e4336614bc012e902caef51f08aa9cc
40,476 byte
486 pixel
462 pixel
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 00:17, 29 January 2020 | ![]() | 462 × 486 (40 KB) | commons>Bob K | replace white figure background with gray |
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Width | 130.3mm |
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Height | 137.16mm |
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