Fourier series: Difference between revisions

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==Periodic Functions==
A continuous time signal <math>x(t)</math> is said to be periodic if there is a positive nonzero value of T such that
<center>
<math> s(t + T) = s(t)</math> for all <math>t</math>
</center>
==Dirichlet Conditions==
==Dirichlet Conditions==
The conditions for a periodic function <math>f</math> with period 2L to have a convergent Fourier series.
The conditions for a periodic function <math>f</math> with period 2L to have a convergent Fourier series.

Revision as of 16:56, 6 December 2004

Periodic Functions

A continuous time signal is said to be periodic if there is a positive nonzero value of T such that

for all

Dirichlet Conditions

The conditions for a periodic function with period 2L to have a convergent Fourier series.

Theorem:

Let be a piecewise regular real-valued function defined on some interval [-L,L], such that has only a finite number of discontinuities and extrema in [-L,L]. Then the Fourier series of this function converges to when is continuous and to the arithmetic mean of the left-handed and right-handed limit of at a point where it is discontinuous.

The Fourier Series

A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions.

.


see also:Orthogonal Functions

Principle author of this page: Aric Goe