Signals and systems/GF Fourier

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 $x(t)=\sum _{n=-\infty }^{\infty }\alpha _{n}e^{{j2\pi nt}/T}\,$

Determining the coefficient $\alpha _{n}\,$

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

$\int _{-T/2}^{T/2}x(t)\,dt=\sum _{n=-\infty }^{\infty }\alpha _{n}\int _{-T/2}^{T/2}e^{{j2\pi nt}/T}dt$

Identities

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

$\sin x={\frac {e^{jx}-e^{-jx}}{2j}}\,$

$\cos x={\frac {e^{jx}+e^{-jx}}{2}}\,$

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