Page 450 - Numerical Methods for Chemical Engineering

Fourier series and transforms in one dimension 439

property
+P
'
e −imπt/P inπt/P dt = 2Pδ mn (9.16)
e
−P
to obtain
+P
1 ' −inπt/P
c n = e f (t)dt (9.17)
2P
−P

The Fourier transform

The Fourier series representation assumes a known period 2P; however, often we wish to
analyze a function whose periodicity is unknown. From the exponential form of the Fourier
series, we obtain the Fourier transform by taking the limit P →∞. We begin by writing
the Fourier series as
+P
 
∞ 1 '
∞

˜ imπt/P −imπt /P imπt/P

f (t) = c m e =  e f (t )dt  e (9.18)
2P
m=−∞ m=−∞
−P
We now define
mπ π
ω m = ω = ω m+1 − ω m = (9.19)
P P
such that
1 1 $ π % 1
= = ω
2P 2π P 2π
and thus
+P
 
∞ '
1
˜ −iω m t iω m t
f (t) =  e f (t )dt  e ω (9.20)
2π
m=−∞
−P
In the limit P →∞, ω → 0, and the summation becomes an integral,
+∞
∞ '

F(ω m ) ω → F(ω)dω (9.21)
m=−∞
−∞
˜
Thus, assuming that f (t) = f (t), we obtain in the limit P →∞, the relation
 
+∞ ' +∞
1 −iωt iωt
'

f (t) =  e f (t )dt  e dω (9.22)
2π
−∞ −∞
This relation is satisfied by the Fourier transform pair
+∞ ' +∞
'
1 −iωt 1 iωt
F(ω) = √ f (t)e dt f (t) = √ F(ω)e dω (9.23)
2π 2π
−∞ −∞