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:

    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.