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
ó
õ
0
f(t) · cos kwt · dt

bk  =  2/T · T
ó
õ
0
f(t) · sin kwt · dt
This can be written much more elegantly using complex numbers and functions as
f(t)  =  +¥
S
-¥
cn · einwt
The coefficients cn are obtained by
cn  =  T
ó
õ
0
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)  =  +¥
ó
õ
¥
g(w) · eiwt · dw
g(w)  =   (1/2p) ·  +¥
ó
õ
¥
f(t) · e– iwt  · dt
The simplicity, symmetry and elegance (not to mention their usefulness) of these Fourier integrals is just amazing!


With frame With frame as PDF

go to 3.3.2 Dipole Relaxation

go to Frequency Dependence of Orietaion Polarization

© H. Föll (Electronic Materials - Script)