Fourier series

A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. All real signals can be generated using a fourier series.

Exponential form

$x(t)\sim {\sum }_{n=0}^{\infty }{c}_n\cdot e^{2\pi int/T}$

where: $c_n=\frac{1}{T}{\int }_{-P/2}^{P/2}x(t)e^{2\pi int/T}dt$ This can be proven to be the same as Komplekse indreproduktrom

Fourier series can be used to prove the formula for Fourier transform by putting by putting the formula for $c_n$ inside the exponential form formula, then using riemann sum which is explained in Integralet.

Sine-cosine form

$x(t)\sim A_0+{\sum }_{n=1}^{\infty }{A}_n\cos \left(\frac{2\pi nt}{T}\right)+B_{n}sin\left(\frac{2\pi nt}{T}\right)$ where: $A_0=\frac{1}{T}{\int }_{-P/2}^{P/2}x(t)dt$ $A_n=\frac{2}{T}{\int }_{-P/2}^{P/2}x(t)\cos \left(\frac{2\pi nt}{T}\right)dt$ $B_n=\frac{2}{T}{\int }_{-P/2}^{P/2}x(t)\sin \left(\frac{2\pi nt}{T}\right)dt$

This can be proven to be the same as Reelle indreproduktrom.