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??

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

[GIF, 60KB] The hardware applet can be viewed at the Institut für Netzwerktheorie und Schaltungstechnik, TU Braunschweig.

The source code (version 2000/03/25) is available according to the GNU Public License.

Other Versions of this Applet:
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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[ | Manfred Thole ]

Manfred Thole,
Last modified: Sat Jan 24 23:48:02 MET 2004