


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) » 
a_{0}/2 
+ a_{1} · cos wt + a_{2} · cos 2wt + ... + a_{n} · cos
nwt + .. 


+ b_{1} · sin wt + b_{2} · sin 2wt + ... + b_{n} · sin
nwt + .. 




and the Fourier coefficients
a_{k} and b_{k} (with the index
k = 0, 1, 2, ...) are determined by



a_{k} = 2/T ·

T
ó
õ
0 
f(t) · cos kwt · dt 
b_{k} = 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
¥ 
c_{n} · e^{inwt} 




The coefficients c_{n} are
obtained by 


c_{n} = 
T
ó
õ
0 
f(t) · e^{–
inwt} · dt 
= 
{

½(a_{n} –
ib_{n})


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 w_{0} = 2p/T. 


The coefficients c_{n}
define the spectrum of the
periodic function by giving the amplitudes of the harmonics that the function
contains. 


