Fourier Series and Transforms

Fourier Series

Every (physically sensible) periodic function f(t) = f(t + T) with T = 1/n = 2p/w and n, w = frequency and angular frequency, respectively, may be written as a Fourier series as follows
f(t)  »   a0/2 + a1 · cos wt + a2 · cos 2wt + ... + an · cos nwt + ..
  + b1 · sin wt + b2 · sin 2wt + ... + bn · sin nwt + ..
and the Fourier coefficients ak and bk (with the index k = 0, 1, 2, ...) are determined by
ak  =  2/T · T
f(t) · cos kwt · dt

bk  =  2/T · T
f(t) · sin kwt · dt
This can be written much more elegantly using complex numbers and functions as
f(t)  =  +¥
cn · einwt
The coefficients cn are obtained by
cn  =  T
f(t) · e– inwt   · dt  =  {

½(an  –  ibn)
  for n > 0
½(a–n  +  ib–n)   for n < 0
The function f(t) is thus expressed as a sum of sin functions with the harmonic frequencies or simply harmonics w derived from the fundamental frequency w0 = 2p/T.
The coefficients cn define the spectrum of the periodic function by giving the amplitudes of the harmonics that the function contains.

Fourier Transforms

A nonperiodic function f(t) ("well-behaved"; we are not looking at some abominable functions only mathematicians can think of) can also be written as a Fourier series, but now the Fourier coefficients have some values for all frequencies w, not just for some harmonic frequencies.
Instead of a spectrum with defined lines at the harmonic frequencies, we now obtain a spectral density function g(w), defined by the following equations
f(t)  =  +¥
g(w) · eiwt · dw
g(w)  =   (1/2p) ·  +¥
f(t) · e– iwt  · dt
The simplicity, symmetry and elegance (not to mention their usefulness) of these Fourier integrals is just amazing!

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© H. Föll (Electronic Materials - Script)