Page 447 - Numerical Methods for Chemical Engineering

9 Fourier analysis

Fourier analysis treats the representation of periodic functions as linear combinations of sine
and cosine basis functions. In chemical engineering, Fourier analysis is applied to study
time-dependent signals in spectroscopy and to analyze the spatial structure of materials
from scattering experiments. Here, the basic foundation of Fourier analysis is presented,
with an emphasis upon implementation in MATLAB.

Fourier series and transforms in one dimension

We begin our discussion of Fourier analysis by considering the representation of a periodic
function f (t) with a period of 2P, f (t + 2P) = f (t). If f (t) has a finite number of local
extrema and a finite number of times t j ∈ [0, 2P] at which it is discontinuous, Dirichlet’s
theorem states that it may be represented as the Fourier series
∞
1 mπt mπt
˜
f (t) = a 0 + a m cos + b m sin (9.1)
2 P P
m=1
such that at all t where f (t) is continuous, f (t ) = f (t ), and at all points t j where f (t)is

˜
˜
discontinuous, f (t j ) is the average of the right-and left-hand limits:
f + (t j ) + f − (t j )
˜
f (t j ) = f + (t j ) = lim f (t j + ε) f − (t j ) = lim f (t j − ε) (9.2)
2 ε→0 ε→0
a 0 , {a 1 , a 2 ,... }, and {b 1 , b 2 ,... } are calculated using the orthogonality properties of sine
and cosine functions. First, to compute a 0 , we integrate f (t) over the period [0, 2P], and do
˜
the same for f (t):
2P 2P
' '
˜
f (t)dt = f (t)dt
0 0
2P 2P
∞ '
'
a 0 mπt mπt
= dt + a m cos + b m sin dt (9.3)
2 P P
0 m=1 0
As the cosine and sine terms integrate to 0,
2P
1 '
a 0 = f (t) dt (9.4)
P
0
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