Page 454 - Electromagnetics

(b) If f is real and causal, and f (0) is finite, then R(k) and X(k) are related by
the Hilbert transforms

1 ∞ R(k) 1 ∞ X(k)

X(k) =− P.V. dk , R(k) = P.V. dk .
π −∞ k − k π −∞ k − k
(c) If f is causal and has finite energy, it is not possible to have F(k) = 0 for
k 1 < k < k 2 . That is, the transform of a causal function cannot vanish over
an interval.
A causal function is completely determined by the real or imaginary part of its
spectrum. As with item 4, this is helpful when performing calculations or mea-
surements in the frequency domain. If the function is not band-limited however,
truncation of integrals will give erroneous results.
7. Time-limited vs. band-limited functions. Assume t 2 > t 1 .If f (t) = 0 for both t < t 1
and t > t 2 , then it is not possible to have F(k) = 0 for both k < k 1 and k > k 2
where k 2 > k 1 . That is, a time-limited signal cannot be band-limited. Similarly, a
band-limited signal cannot be time-limited.
8. Null function. If the forward or inverse transform of a function is identically zero,
then the function is identically zero. This important consequence of the Fourier
integral theorem is useful when solving homogeneous partial differential equations
in the frequency domain.
9. Space or time shift. For any fixed x 0 ,

f (x − x 0 ) ↔ F(k)e − jkx 0 . (A.3)

A temporal or spatial shift affects only the phase of the transform, not the magni-
tude.
10. Frequency shift. For any fixed k 0 ,

f (x)e jk 0 x ↔ F(k − k 0 ).

Note that if f ↔ F where f is real, then frequency-shifting F causes f to be-
come complex — again, this is important if F has been obtained experimentally or
through computation in the frequency domain.
11. Similarity. We have
1
k
f (αx) ↔ F ,
|α| α
where α is any real constant. “Reciprocal spreading” is exhibited by the Fourier
transform pair; dilation in space or time results in compression in frequency, and
vice versa.
12. Convolution. We have
∞

f 1 (x ) f 2 (x − x ) dx ↔ F 1 (k)F 2 (k)
−∞
and
1 ∞

f 1 (x) f 2 (x) ↔ F 1 (k )F 2 (k − k ) dk .
2π
−∞
© 2001 by CRC Press LLC