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File:Correlation function.svg

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Summary

Description
English: The blue curve is the cross-correlation of a square wave and a cosine function, as the phase lag of the cosine varies over one cycle. The amplitude and phase lag at the maximum value are the polar coordinates of one harmonic in the Fourier series expansion of the square wave. The corresponding Cartesian coordinates can be determined by evaluating the cross-correlation at just two phase lags separated by 90º.
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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SVG development
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The source code of this SVG is invalid due to 5 errors.
 
This W3C-invalid vector image was created with LibreOffice.
Gnu Octave source
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click to expand

This graphic was created with the help of the following Octave script:

  graphics_toolkit gnuplot
  pkg load signal
#{
The gnuplot graphics toolkit is not actively maintained and has a number
of limitations that are unlikely to be fixed.  Communication with gnuplot
uses a one-directional pipe and limited information is passed back to the
Octave interpreter so most changes made interactively in the plot window
will not be reflected in the graphics properties managed by Octave.  For
example, if the plot window is closed with a mouse click, Octave will not
be notified and will not update its internal list of open figure windows.
The qt toolkit is recommended instead.
#}
  set(0, "DefaultAxesTitleFontWeight", "bold")
  set(0, "DefaultAxesFontWeight",      "bold")
  set(0, "DefaultAxesFontSize",        14)

  P = 2000;                     % signal duration
  x = 0:P-1;			% sampling times

% Any function will do for the signal.  It needn't be periodic, but a square wave is convenient.
  period=233;			% square wave period
  s = square((x-period*.67)*2*pi/period);
  n = 4;			% a harmonic (any harmonic will do)

  num_angles=1000;		% eval the correlation function at 1000 phases
  correl = [];
  phi = (0:1:num_angles-1)/num_angles;
  for m=phi
% The right-hand side is equivalent to sum(s.*cos(2*pi*(n/P*x - m))
  correl(end+1) = real(sum(s.*exp(-i*2*pi*(n/P*x - m))));
  endfor

  figure("position",[1 1 700 700])
  plot(phi*360, correl, "color","b", "linewidth",3)
  xlim([0 360]);
  set(gca, 'xtick', [0 1 2 3 4]*90);

  [maxval,idx] = max(correl);
  cossin = sum(s.*exp(i*2*pi*n*x/P))
% The next two things are equal
  mod(atan2(imag(cossin), real(cossin))*360/(2*pi), 360)
  (idx-1)*360/num_angles
% The next two things are equal
  maxval
  abs(cossin)

  hold on
  stem((idx-1)*360/num_angles, correl(idx), "filled", "linewidth",4, "color",[1 0 1], "markerfacecolor",[1 0 1])
  stem( 0,correl( 0*num_angles/360+1),"filled", "linewidth",4, "color",[1 0 1], "markerfacecolor", [1 0 1])
  stem(90,correl(90*num_angles/360+1),"filled", "linewidth",4, "color",[1 0 1], "markerfacecolor", [1 0 1])

  title("Cross-correlation function", "fontsize",14);
  xlabel("phase lag (degrees)",       "fontsize",14)
  ylabel("correlation amplitude",     "fontsize",14)

Captions

Example of using two points of a correlation function to determine the location and height of its maximum

Items portrayed in this file

depicts

11 April 2022

image/svg+xml

09d1bd458779569d05dacbd6a5732edae1476426

70,305 byte

765 pixel

765 pixel

File history

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Date/TimeThumbnailDimensionsUserComment
current03:19, 12 April 2022Thumbnail for version as of 03:19, 12 April 2022765 × 765 (69 KB)Bob KUploaded own work with UploadWizard

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