Fourier series: Difference between revisions
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==Periodic Functions== |
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A continuous time signal <math>x(t)</math> is said to be periodic if there is a positive nonzero value of T such that |
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<math> s(t + T) = s(t)</math> for all <math>t</math> |
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==Dirichlet Conditions== |
==Dirichlet Conditions== |
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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 17:56, 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.
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see also:Orthogonal Functions
Principle author of this page: Aric Goe