Page 456 - Electromagnetics

for any continuous function f . With f (x) = 1 we obtain the familiar relation
∞

δ(x) dx = 1.
−∞
With f (x) = e − jkx we obtain

∞
δ(x)e − jkx dx = 1,
−∞
thus

δ(x) ↔ 1.
It follows that
1 ∞ jkx
e dk = δ(x). (A.4)
2π
−∞
Useful transform pairs. Some of the more common Fourier transforms that arise in
the study of electromagnetics are given in Appendix C. These often involve the simple
functions defined here:

1. Unit step function

1, x < 0,
U(x) = (A.5)
0, x > 0.
2. Signum function

−1, x < 0,
sgn(x) = (A.6)
1, x > 0.
3. Rectangular pulse function

1, |x| < 1,
rect(x) = (A.7)
0, |x| > 1.
4. Triangular pulse function

1 −|x|, |x| < 1,

(x) = (A.8)
0, |x| > 1.
5. Sinc function
sin x
sinc(x) = . (A.9)
x

Transforms of multi-variable functions
Fourier transformations can be performed over multiple variables by successive appli-
cations of (A.1). For example, the two-dimensional Fourier transform over x 1 and x 2 of
, x 3 ,..., x N ) given by
the function f (x 1 , x 2 , x 3 ,..., x N ) is the quantity F(k x 1 , k x 2
∞ ∞

x
x
f (x 1 , x 2 , x 3 ,..., x N ) e − jk x 1 1 dx 1 e − jk x 2 2 dx 2
−∞ −∞
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