Page 453 - Electromagnetics

When this is possible we write

f (x) ↔ F(k)
and say that f (x) and F(k) form a Fourier transform pair. The Fourier integral theorem
states that

−1 −1
FF { f (x)}= F F{ f (x)}= f (x),
except at points of discontinuity of f . At a jump discontinuity the inversion formula
+
−
returns the average value of the one-sided limits f (x ) and f (x ) of f (x). At points of
continuity the forward and inverse transforms are unique.
Transform theorems and properties. We now review some basic facts pertaining
to the Fourier transform. Let f (x) ↔ F(k) = R(k) + jX(k), and g(x) ↔ G(k).
1. Linearity. αf (x) + βg(x) ↔ αF(k) + βG(k) if α and β are arbitrary constants.
This follows directly from the linearity of the transform integral, and makes the
transform useful for solving linear differential equations (e.g., Maxwell’s equations).
2. Symmetry. The property F(x) ↔ 2π f (−k) is helpful when interpreting transform
tables in which transforms are listed only in the forward direction.
∗
3. Conjugate function. We have f (x) ↔ F (−k).
∗
4. Real function. If f is real, then F(−k) = F (k). Also,
∗
∞ ∞

R(k) = f (x) cos kx dx, X(k) =− f (x) sin kx dx,
−∞ −∞
and
1 ∞ jkx
f (x) = Re F(k)e dk.
π 0
A real function is completely determined by its positive frequency spectrum. It is
obviously advantageous to know this when planning to collect spectral data.
5. Real function with reflection symmetry. If f is real and even, then X(k) ≡ 0 and

∞ 1 ∞

R(k) = 2 f (x) cos kx dx, f (x) = R(k) cos kx dk.
0 π 0
If f is real and odd, then R(k) ≡ 0 and

∞ 1 ∞
X(k) =−2 f (x) sin kx dx, f (x) =− X(k) sin kx dk.
0 π 0
(Recall that f is even if f (−x) = f (x) for all x. Similarly f is odd if f (−x) =− f (x)
for all x.)
6. Causal function. Recall that f is causal if f (x) = 0 for x < 0.
(a) If f is real and causal, then

2 ∞ ∞

X(k) =− R(k ) cos k x sin kx dk dx,
π 0 0
2 ∞ ∞

R(k) =− X(k ) sin k x cos kx dk dx.
π 0 0
© 2001 by CRC Press LLC