Revision as of 04:12, 29 October 2006

Fourier series

The Fourier series is used to analyze arbitrary periodic functions by showing them as a composite of sines and cosines.

A function is considered periodic if $x(t)=x(t+T)\,$ for $T\neq 0$ .

The exponential form of the Fourier series is defined as $x(t)=\sum _{n=-\infty }^{\infty }\alpha _{n}e^{{j2\pi nt}/T}\,$

$e^{j\theta }=cos\theta +jsin\theta \,$

$\left\langle n\mid m\right\rangle =T\delta _{n,m}\,$

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	<math>e^{j \theta} = cos \theta + j sin \theta \, </math>		<math>e^{j \theta} = cos \theta + j sin \theta \, </math>

	<math> \left \langle n \mid m \right \rangle \,</math>		<math> \left \langle n \mid m \right \rangle = T \delta_{n,m} \,</math>