Fiω=∫−∞∞fte−(iωt)d
t
F
ω
t
f
t
ω
t

(1)
ft=12π∫−∞∞Fiωeiωtd
ω
f
t
1
2
ω
F
ω
ω
t

(2)
Both formulas are

*symmetrical* except for the

±±
sign and

12π
1
2
. Useful! Say

ft
↔
Fiω
f
t
↔
F
ω
through CTFT, and

Giω=fiω
G
ω
f
ω
, that is: the same form as some time-domain signal

f
f
. Then what is

gt
g
t
?

gt=12π∫−∞∞Giωeiωtd
ω
=12π∫−∞∞fiωeiωtd
ω
g
t
1
2
ω
G
ω
ω
t
1
2
ω
f
ω
ω
t

(3)
Let

τ=ω
τ
ω
, and

i
in

fiω
f
ω
doesn't matter, then:

gt=12π∫−∞∞fτeiτtd
τ
=∫−∞∞fτe−(iτ(−t))d
τ
=12πF−(it)
g
t
1
2
τ
f
τ
τ
t
τ
f
τ
τ
t
1
2
F
t

(4)
In other words:

gt
g
t
takes the form of FT

F
F
(reversed and scaled,

Figure 1).

CTFT of
∀
t
:gt=1
t
g
t
1
(Figure 2).

Giω=∫−∞∞1e−(iωt)d
t
G
ω
t
1
ω
t

(5)
We know Figure 3.

So by duality we get

Figure 4.

Uiω=πδω+1iω
U
ω
δ
ω
1
ω
.

See Lathi pp. 250-1 for derivation.