Page 202 - Electromagnetics

Chapter 4

Temporal and spatial frequency domain
representation

4.1 Interpretation of the temporal transform

When a field is represented by a continuous superposition of elemental components, the
resulting decomposition can simplify computation and provide physical insight. Such rep-
resentation is usually accomplished through the use of an integral transform. Although
several different transforms are used in electromagnetics, we shall concentrate on the
powerful and efficient Fourier transform.
Let us consider the Fourier transform of the electromagnetic field. The field depends
on x, y, z, t, and we can transform with respect to any or all of these variables. However,
a consideration of units leads us to consider a transform over t separately. Let ψ(r, t)
represent any rectangular component of the electric or magnetic field. Then the temporal
˜
transform will be designated by ψ(r,ω):
˜
ψ(r, t) ↔ ψ(r,ω).
˜
Here ω is the transform variable. The transform field ψ is calculated using (A.1):
∞

˜ − jωt
ψ(r,ω) = ψ(r, t) e dt. (4.1)
−∞
The inverse transform is, by (A.2),
1 ∞ jωt
˜
ψ(r, t) = ψ(r,ω) e dω. (4.2)
2π −∞
˜
Since ψ is complex it may be written in amplitude–phase form:
ψ
˜
˜
ψ(r,ω) =|ψ(r,ω)|e jξ (r,ω) ,
ψ
where we take −π< ξ (r,ω) ≤ π.
Since ψ(r, t) must be real, (4.1) shows that
˜
˜
∗
ψ(r, −ω) = ψ (r,ω). (4.3)
Furthermore, the transform of the derivative of ψ may be found by differentiating (4.2).
We have
∂ 1 ∞ jωt
˜
ψ(r, t) = jωψ(r,ω) e dω,
∂t 2π
−∞

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