## 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:

The Fourier series of a square wave is

or

The Fourier series of a saw-toothed wave is

The approximation improves as more oscillations are added.

No Java, no applet! Sorry! But it would look like this:

A sample session would be as follows:
• To produce a saw-toothed wave, in the white box to the right of the word "Sin:" enter a formula such as 1/x or (-1^(x-1))/x. The variable "x" will be replaced by the term number, so the coefficients will have values of 1, 0.5, 0.3333,...
• IN ORDER FOR THE PROGRAM TO PARSE AN EXPRESSION, you must press the "Enter" key instead of leaving the box with the mouse or cursor keys.
• You can modify coefficients by using the formula box, the slider bars, or by entering an expression (such as 0.5 or -1/7) into the white box by each label.
• If your machine is capable of playing sounds, you should also hear a tone for the waveform you have produced. This may be turned off by pressing the "Audio Off" button.
• You may reset a coefficient to zero by clicking on the label button with the mouse, thus by clicking on the even numbered coefficients b2:, b4:, ..., you can produce a square wave.
• The applet can store up to 3 different waveforms (by clicking on Wave1, Wave2, Wave3) which is helpful for comparing different sequences or different numbers of terms.

Condition of Dirichlet:
The Fourier series of a periodic function x(t) exists, if
1. , 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.

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

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

This applet, gif images and HTML documentation were developed by Manfred Thole, thole@nst.ing.tu-bs.de, July 15, 1996. The original documentation and applets can be found at:

Modifications were made by Tom Huber, huber@gac.edu, September 27, 1996

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

Tom Huber, huber@gac.edu, Revised 24-Aug-2008

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