Page 451 - Numerical Methods for Chemical Engineering

440 9 Fourier analysis

One can propose alternative definitions that also satisfy (9.22), such as
+∞ +∞
1 −iωt
' '
iωt
F(ω) = f (t)e dt f (t) = F(ω)e dω (9.24)
2π
−∞ −∞
As long as one is consistent, the convention used for the Fourier transform is arbi-
trary. Here, we use the definition consistent with MATLAB, (9.23). Note that as F(0) =
√ - +∞
1/ 2π f (t)dt, we assume that f (t) integrates to zero.
−∞
The discrete Fourier transform

We next consider computing the Fourier transform from the N sampled data
{ f 1 , f 2 ,..., f N } , f k = f (t k ), taken at the N uniform times t k = (k − 1) t.Ifwehave
samples taken at nonuniform times, we use interpolation to generate the values of f (t)at
time values that are uniformly-spaced before computing the Fourier transform. We have no
data outside of the time period [0, (N − 1) t], so let us assume that f (t) is periodic outside
of this range. As generally f 1 = f N , the period is
2P = N( t) (9.25)

We wish to compute the Fourier transform
+∞
1 ' −iωt
F(ω) = √ f (t)e dt (9.26)
2π
−∞
butaswehaveonly N pieces of information {f 1 , f 2 ,..., f N }, we can determine F(ω)
independently at only N different frequencies ω n . What are the frequencies for which we
compute F(ω)? First, if f (t + 2P) = f (t), we can represent f (t) as the Fourier series
∞
iω m t mπt
f (t) = c m e ω m = (9.27)
P
m=−∞
Thus, F(ω) takes the form
∞

F(ω) = A F(ω m )δ(ω − ω m ) (9.28)
m=−∞
as is shown by substitution into the inverse Fourier transform:
+∞ +∞
∞
1 ' iωt 1 ' iωt
f (t) = √ F(ω)e dω = √ A F(ω m )δ(ω − ω m ) e dω
2π 2π m=−∞
−∞ −∞
+∞
∞ ' ∞
A iωt A iω m t
f (t) = √ F(ω m ) δ(ω − ω m )e dω = √ F(ω m ) e
2π 2π
m=−∞ m=−∞
−∞
(9.29)
Thus, for a function with f (t + 2P) = f (t), the Fourier transform is nonzero only at the