Fourier Series for Real-valued Functions
vaThe Average value of the function f(t)
ω0The 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.
Fourier Series for Complex Valued Functions
Cn A complex valued coefficient
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
- 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.