If the signal of interest is periodic with period TT, then it has a Fourier Series:
x
(
t
)
=
∑
n
=
-
∞
∞
c
n
e
j
Ω
0
n
t
x
(
t
)
=
∑
n
=
-
∞
∞
c
n
e
j
Ω
0
n
t
(1)
Using the linearity of the Fourier Transform, we have
X
(
j
Ω
)
=
∑
n
=
-
∞
∞
c
n
F
e
j
Ω
0
n
t
=
2
π
∑
n
=
-
∞
∞
c
n
δ
(
Ω
-
n
Ω
0
)
X
(
j
Ω
)
=
∑
n
=
-
∞
∞
c
n
F
e
j
Ω
0
n
t
=
2
π
∑
n
=
-
∞
∞
c
n
δ
(
Ω
-
n
Ω
0
)
(2)
where FF corresponds to the Fourier Transform of the signal within the brackets.