# Fourier Series

#### Fourier Series for Real-valued Functions

• va
The Average value of the function f(t)
• ω0
The fundamental frequency of f(t) in radians per second
• an
The magnitude of the even component
• bn
The magnitude of the odd component
• Each nω0 is a harmonic of the fundamental frequency. $f ⁡ t = v a + Σ n = 1 ∞ ⁡ a n ⁢ cos ⁡ n ⁢ ω 0 ⁢ t + b n ⁢ sin ⁡ n ⁢ ω 0 ⁢ t$

Where: $ω 0 = 2 ⁢ π T = τ T$ $v a = 1 T ⁢ ∫ t 0 t 0 + T ⁡ f ⁡ t ⁢ d t$ $a n = 2 T ⁢ ∫ t 0 t 0 + T ⁡ f ⁡ t ⁢ cos ⁡ n ⁢ ω 0 ⁢ t ⁢ d t$ $b n = 2 T ⁢ ∫ t 0 t 0 + T ⁡ f ⁡ t ⁢ sin ⁡ n ⁢ ω 0 ⁢ t ⁢ d t$

#### Fourier Series for Complex Valued Functions

• Cn
A complex valued coefficient
• $f ⁡ t = Σ n = - ∞ ∞ ⁡ C n ⁢ e i ⁢ n ⁢ ω 0 ⁢ t$

Where: $C n = 1 T ⁢ ∫ t 0 t 0 + T ⁡ f ⁡ t ⁢ e - i ⁢ n ⁢ ω 0 ⁢ t ⁢ d t$

A Fourier Series is a representation of a periodic function using an infinite summation of sines and cosines. If a finite summation is used the resulting expression will approximate the function. The approximation can be made to any necessary precision by carrying out a larger summation.

The necessary conditions for a function to have a Fourier transform are not known, however Dirichlet's Conditions are a set of sufficient conditions for the existence of the Fourier series:

• f(t) has only one value for any t
• f(t) has a finite number of discontinuities in its period
• f(t) has a finite number of minima and maxima in its period
• $\underset{{t}_{0}}{\overset{{t}_{0}+T}{\int }}\left|f\left(t\right)\right|dt$ is finite

If the function f(t) is even then the odd portion of the Fourier series is 0 such that b=0. Likewise if f(t) is odd then a=0 and av=0. Because of superposition, if the function can be made odd by shifting it along the dependent axis then a=0 still but av must still be calculated. There are also simplifications if f(t) has half- or quarter-wave symmetry.