Page 452 - Electromagnetics

Appendix A

Mathematical appendix

A.1 The Fourier transform

The Fourier transform permits us to decompose a complicated field structure into
elemental components. This can simplify the computation of fields and provide physical
insight into their spatiotemporal behavior. In this section we review the properties of
the transform and demonstrate its usefulness in solving field equations.

One-dimensional case
Let f be a function of a single variable x. The Fourier transform of f (x) is the function
F(k) defined by the integral

∞

F{ f (x)}= F(k) = f (x)e − jkx dx. (A.1)
−∞
Note that x and the corresponding transform variable k must have reciprocal units: if x
is time in seconds, then k is a temporal frequency in radians per second; if x is a length
in meters, then k is a spatial frequency in radians per meter. We sometimes refer to F(k)
as the frequency spectrum of f (x).
Not every function has a Fourier transform. The existence of (A.1) can be guaranteed
by a set of sufficient conditions such as the following:

∞
1. f is absolutely integrable: | f (x)| dx < ∞;
−∞
2. f has no infinite discontinuities;
3. f has at most finitely many discontinuities and finitely many extrema in any finite
interval (a, b).
While such rigor is certainly of mathematical value, it may be of less ultimate use to
the engineer than the following heuristic observation offered by Bracewell [22]: a good
mathematical model of a physical process should be Fourier transformable. That is, if the
Fourier transform of a mathematical model does not exist, the model cannot precisely
describe a physical process.
The usefulness of the transform hinges on our ability to recover f through the inverse
transform:

1 ∞
−1 jkx
F {F(k)}= f (x) = F(k) e dk. (A.2)
2π
−∞

© 2001 by CRC Press LLC