Page 520 - Electromagnetics

Appendix C

Some Fourier transform pairs

Note:

∞ 1 ∞
G(k) = g(x)e − jkx dx, g(x) = G(k)e jkx dk, g(x) ↔ G(k).
2π
−∞ −∞

rect(x) ↔ 2 sinc k (C.1)

k
2
(x) ↔ sinc (C.2)
2
2
sgn(x) ↔ (C.3)
jk

e jk 0 x ↔ 2πδ(k − k 0 ) (C.4)

δ(x) ↔ 1 (C.5)

1 ↔ 2πδ(k) (C.6)

n
d δ(x) n
↔ ( jk) (C.7)
dx n
n
d δ(k)
n
n
x ↔ 2π j (C.8)
dk n
1
U(x) ↔ πδ(k) + (C.9)
jk
∞ ∞
2π
δ t − n ↔ k 0 δ(k − nk 0 ) (C.10)
k 0
n=−∞ n=−∞
π k 2
e −ax 2 ↔ e − 4a (C.11)
a
1
−ax
e U(x) ↔ (C.12)
a + jk

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