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 n·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!

To index

3.3.2 Dipole Relaxation

Frequency Dependence of Orietaion Polarization

© H. Föll (Electronic Materials - Script)