Signals & Systems - Reference Tables
1
Table of Fourier Transform Pairs
Function, f(t)
Fourier Transform, F(w)
Definition of Inverse Fourier Transform
ò
¥
¥-
= ww
p
w
deFtf
tj
)(
2
1
)(
Definition of Fourier Transform
ò
¥
¥-
-
= dtetfF
tjw
w )()(
)(
0
ttf -
0
)(

Fj
n
)()( tfjt
n
-
n
n
d
Fd
w
w)(
ò
¥-
t
df
tt
)(
)()0(
)(
wdp
w
w
F
j
F
+
)(t
d
1
tj

tjn
n
eF
0
w
å
¥
-¥=
-
n
n
nF
)(2
0
wwdp
)(
t
t
rect
)
2
(
w
t
tSa
)
2
(
2
Bt

w
t
p
w
t
t
p
-
A
)cos(
0
t
w
[]
)()(
00
wwdwwdp ++-
)sin(
0
t
w
[]
)()(
00
wwdwwd
p
+--
j
)cos()(
0

-
++--
j
)cos()(
0
tetu
t
w
a-
22
0
)(
)(
waw
wa
j
j
++
+
Signals & Systems - Reference Tables
3
)sin()(
0
tetu
t
w
a-
22
0
0

+
1
t
tetu
a-
)(
2
)(
1
wa
j
+
Ø Trigonometric Fourier Series
()
å
¥
=
++=
1
000
)sin()cos()(
n
nn
ntbntaatf
ww
where
ò
òò
=
==

-
¥
-¥=
==
T
ntj
n
n
ntj
n
dtetf
T
FeFtf
0
0
)(
1
where, )(
w
w
Signals & Systems - Reference Tables
4
Some Useful Mathematical Relationships
2
)cos(
jxjx
ee
x
-
+

=+
xx
)cos()cos()cos()cos(2 yxyxyx
++-=
)cos()cos()sin()sin(2 yxyxyx
+--=
)sin()sin()cos()sin(2 yxyxyx
++-=
Signals & Systems - Reference Tables
5
Useful Integrals
ò
dxx)cos(
)sin(x
ò
dxx)sin(
)cos(x
-
ò
dxxx )cos(
)sin()cos( xxx
+
ò
dxxx )sin(
)cos()sin( xxx
-
ò
dxxx )cos(
2
)sin()2()cos(2

a
x
e
x
a
ò
dxex
xa2
ú
û
ù
ê
ë
é
--
32
2
22
aa
x
a
x
e
x
a
ò
+
x
dx
ba