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
The Fourier series of a saw-toothed wave is
The approximation improves as more oscillations are added.
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
"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
- 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
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
- , i. e. x(t) is absolutely integratable,
- variations of x(t) are limited in every finite time interval T and
- 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
firstname.lastname@example.org, July 15, 1996.
The original documentation and applets can be found at:
Modifications were made by Tom Huber,
email@example.com, September 27, 1996
This applet requires the
from Leigh Brookshaw to parse equations.
firstname.lastname@example.org, Revised 22-APR-97