Fourier series

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Introduction

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 Introduction added on 10/06/05 by Jeff W