Fourier Synthesis

A periodic signal can be described by a Fourier decomposition as a Fourier series, i. e. as a sum of sinusoidal and cosinusoidal oscillations. By reversing this procedure a periodic signal can be generated by superimposing sinusoidal and cosinusoidal waves. The general function is:

x(t)=a0/2+a1*cos(1*w0*t)+b1*sin(1*w0*t)+a2*cos(2*w0*t)+b2*sin(2*w0*t)+ ..

The Fourier series of a square wave is

x(t)=sin(w0*t)+1/3*sin(3*w0*t)+1/5*sin(5*w0*t)+ ...


x(t)=cos(w0*t)-1/3*cos(3*w0*t)+1/5*cos(5*w0*t)- ...

The Fourier series of a saw-toothed wave is

x(t)=sin(w0*t)+1/2*sin(2*w0*t)+1/3*sin(3*w0*t)+ ...

The approximation improves as more oscillations are added.

No Java, no applet! Sorry! But it would look like this:
Ugh! Even no images??

A sample session would be as follows:
Condition of Dirichlet:
The Fourier series of a periodic function x(t) exists, if
  1. \int_T0 |x(t)|dt < oo, i. e. x(t) is absolutely integratable,
  2. variations of x(t) are limited in every finite time interval T and
  3. there is only a finite set of discontinuities in T.

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The source code (version 96/09/27) is available according to the GNU Public License

This applet uses the package. HotJava users should set Class access to Unrestricted.

This applet, gif images and HTML documentation were developed by Manfred Thole,, July 15, 1996. The original documentation and applets can be found at:



Modifications were made by Tom Huber,, September 27, 1996

This applet requires the graph2d package from Leigh Brookshaw to parse equations.

Tom Huber,, Revised 24-Aug-2008
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