Page 461 - Numerical Methods for Chemical Engineering

450 9 Fourier analysis

∗
If f is a real signal, [F(ω)] = F(−ω), and thus
C g, f (ω) = G(ω)F(−ω) = G(ω)[F(ω)] ∗ for real functions (9.80)

Fourier transforms in multiple dimensions

The d-dimensional Fourier transform pair satisfies the relation
 
1 '  ' −i(q·r )  i(q·r)

f (r) = f (r )e dr e dq (9.81)
(2π) d  
d d

and, in agreement with the result for d = 1, (9.23) is defined as
1 ' −i(q·r) 1 ' i(q·r)
F(q) = f (r)e dr f (r) = F(q)e dq (9.82)
(2π) d/2 (2π) d/2
d d

As in one dimension, alternative definitions of the Fourier transform exist, but as long as
one is consistent and satisfies (9.81), the choice is rather arbitrary.

Convolution and correlation
d
The convolution of f (r), g(r), r ∈ ,is
1 '
∗ ∗
[g f ](s) = g(r) f (s − r)dr [G F](q) = G(q)F(q) (9.83)
(2π) d/2
d

The correlation of f (r) and g(r)is
1 '
C g, f (s) = g(r + s) f (r)dr C g, f (q) = G(q)F(−q) (9.84)
(2π) d/2
d

When f (r) and g(r) are real, C g, f (q) = G(q)F(−q) = G(q)[F(q)] .
∗

The discrete d-dimensional Fourier transform

The FFT algorithm is similar in multiple dimensions d > 2 to the case d = 1. Here, we
consider d = 2:
+∞ +∞
1 ' −i(q·r) 1 ' ' −i(q x x+q y y)
F(q) = f (r)e dr = f (x, y)e dxdy (9.85)
(2π) (2π)
2
−∞ −∞
We sample f (x, y) uniformly in 0 ≤ x < 2P x and 0 ≤ y < 2P y at
x j = ( j − 1)( x) x = (2P x )/N x j = 1, 2,..., N x
(9.86)
y k = (k − 1)( y) y = (2P y )/N y k = 1, 2,..., N y
N x = 2 and N y = 2 .
κ x
κ y