Fourier Transform Table
()
x
t ()
X
f ()X
ω
()t
δ
1 1
1 ()f
δ
2()
π
δω
0
()tt
δ
− 0
2
j
ft
e
π
− 0
j
t
e
ω
−
0
2
j
ft
e
π
0
()ff
δ
−
0
2( )
π
δω ω
−
0
cos(2 )ft
π
00
1()(
2ff ff
δδ
−+ +
)
00
()(
πδω ω δω ω
−+ +)
0
sin(2 )ft
π
00
1()(
2ff ff
j
δδ
−− +
)
00
()(j
πδω ω δω ω
−−−+)
()rect t sin ( )cf
sin 2
c
ω
π
sin ( )ct ()rect f
2
rect
ω
π
()tΛ 2
sin ( )cf 2
sin 2
c
ω
π
2
sin ( )ct ()f
Λ
2
ω
π
Λ
(), 0
t
eut
α
α
−> 1
2
j
f
α
π
+ 1
j
α
ω
+
(), 0
t
te u t
α
α
−>
()
2
1
2
j
f
απ
+
()
2
1
j
αω
+
||
,0
t
e
αα
−>
22
2
((2)f
α
απ
+
22
2
(()
α
α
ω
+
2
t
e
π
− 2
f
e
α
− 2
f
e
α
−
sgn( )t 1
j
f
π
2
j
ω
()ut 11
()
22
f
j
f
δ
π
+ 1
()
j
πδ ω
ω
+
()
dt
dt
δ
2
j
f
π
j
ω
0
()
n
tnT
δ
∞
=−∞
−
∑
00
1
n
n
f
TT
δ
∞
=−∞
−
∑
00
12
n
n
TT
π
δω
∞
=−∞
−
∑
