Fourier Series and FFT

by AppliedMathematics.info

The decomposition of a periodic function as a fourier series is of great importance in many areas of physics such as vibration, acoustics and electromagnetics. It is also a fundamental technique of digital signal processing.

Fourier Series

A periodic function f(t) of period T can be expanded in the form

f(t) = a0
2
+

n = 1 
an cos( 2 n p
T
)+

n = 1 
bn cos( 2 n p
T
)
where the an and bn are real numbers defined by

a0 = 2
T

T

0 
f(t) dt ,

an = 2
T

T

0 
f(t) cos( 2n pt
T
) dt ,

bn = 2
T

T

0 
f(t) sin( 2n pt
T
) dt .
Note that the domain of integration may be taken to be [-T/2, T/2].

Alternatively, we may use the complex form of the series,

f(t) = Re



n = 0 
(an - i bn) exp( i 2 n p
T
t )

with

an - i bn = 2
T

T

0 
f(t) exp(- i 2n p
T
t ) dt .

Fourier Transform of Discretely Sampled Data

Usually the periodic function f(t) will be realised only at discrete sample points. Suppose that f1, f2, ... fN is a sequence of discrete values of f(t); fk = f( [(k-1 )/N] T). By substitution of these values into the Fourier transform equation above and using the rectangle integration rule,

an - i bn cn = 1
N
N

k = 0 
fk e-2 i n pk/N

The Fast Fourier Transform is an efficient algorithm for obtaining the cn from the fk.


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