Page 457 - Electromagnetics

∞ ∞

x
x
e
= f (x 1 , x 2 , x 3 ,..., x N ) e − jk x 1 1 − jk x 2 2 dx 1 dx 2 .
−∞ −∞
The two-dimensional inverse transform is computed by multiple application of (A.2),
recovering f (x 1 , x 2 , x 3 ,..., x N ) through the operation
1 ∞ ∞ x x
, x 3 ,..., x N ) e jk x 1 1 jk x 2 2 dk x 1 dk x 2 .
e
, k x 2
F(k x 1
(2π) 2 −∞ −∞
Higher-dimensional transforms and inversions are done analogously.
Transforms of separable functions. If we are able to write
f (x 1 , x 2 , x 3 ,..., x N ) = f 1 (x 1 , x 3 ,..., x N ) f 2 (x 2 , x 3 ,..., x N ),

then successive transforms on the variables x 1 and x 2 result in
, x 3 ,..., x N ).
f (x 1 , x 2 , x 3 ,..., x N ) ↔ F 1 (k x 1 , x 3 ,..., x N )F 2 (k x 2
In this case a multi-variable transform can be obtained with the help of a table of one-
dimensional transforms. If, for instance,

f (x, y, z) = δ(x − x )δ(y − y )δ(z − z ),
then we obtain
F(k x , k y , k z ) = e − jk x x e − jk y y e − jk z z
by three applications of (A.1).
A more compact notation for multi-dimensional functions and transforms makes use
of the vector notation k = ˆ xk x + ˆ yk y + ˆ zk z and r = ˆ xx + ˆ yy + ˆ zz where r is the position
vector. In the example above, for instance, we could have written

δ(x − x )δ(y − y )δ(z − z ) = δ(r − r ),
and

∞ ∞ ∞

F(k) = δ(r − r )e − jk·r dx dy dz = e − jk·r .
−∞ −∞ −∞
Fourier–Bessel transform. If x 1 and x 2 have the same dimensions, it may be con-
venient to recast the two-dimensional Fourier transform in polar coordinates. Let x 1 =
= p sin θ, where p and ρ are defined on (0, ∞)
ρ cos φ, k x 1 = p cos θ, x 2 = ρ sin φ, and k x 2
and φ and θ are defined on (−π, π). Then
π ∞
F(p,θ, x 3 ,..., x N ) = f (ρ, φ, x 3 ,..., x N ) e − jpρ cos(φ−θ) ρ dρ dφ. (A.10)
−π 0
If f is independent of φ (due to rotational symmetry about an axis transverse to x 1 and
x 2 ), then the φ integral can be computed using the identity

1 π − jx cos(φ−θ)
J 0 (x) = e dφ.
2π −π
Thus (A.10) becomes

∞
F(p, x 3 ,..., x N ) = 2π f (ρ, x 3 ,..., x N )J 0 (ρp)ρ dρ, (A.11)
0

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