Tài liệu Table of Fourier Transform Pairs ppt

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
)(

tj
eF
w
w
-
tj
etf
0
)(
w
)(
0
ww -
F
)( tf a
)(
1
a
w
a
F
)(tF )(2
wp -
f
n
n
dt
tfd )(
)()(
ww

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

e
0
w
)(2
0
wwpd -
(t)
sgn
w
j
2
Signals & Systems - Reference Tables
2
t
j
p
1
)sgn(
w
)(
tu
w
wpd
j
1
)(
+
å
¥
-¥=n

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

Sa
B
p
)(
B
rect
w
)(ttri
)
2
(
2
w
Sa
)
2
()
2
cos(
tt
p
t
rect
t
A
22
)
2
(
)cos(

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

ttu
w
[]
22
0
00
)()(
2
ww
w
wwdwwd
p
-
+++-
j
)sin()(
0
ttu
w
[]
22
0
2
00
)()(
2
ww
w
wwdwwd
p

-
++--
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

)( waw
w
j++
t
e
a-
22
2
wa
a
+
)2/(
22
s
t
e
-
2/
22
2
ws
ps
-
e
t
etu
a-
)(
wa
j

+
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
n
T
T
n
dtnttf
T
b
dtnttf
T
adttf
T
a
0
0
0
0
0
0
)sin()(
2
and, )cos()(
2
, )(
1
w
w
Ø Complex Exponential Fourier Series
ò
å

-
¥
-¥=
==
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
-
+

=
j
ee
x
jxjx
2
)sin(
-
-
=
)sin()sin()cos()cos()cos( yxyxyx
m=±
)sin()cos()cos()sin()sin( yxyxyx
±=±
)(sin)(cos)2cos(
22
xxx
-=
)cos()sin(2)2sin( xxx
=
)2cos(1)(cos2
2
xx
+=
)2cos(1)(sin2
2
xx
-=
1)(sin)(cos
22

=+
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

2
xxxx
-+
ò
dxxx )sin(
2
)cos()2()sin(2
2
xxxx
--
ò
dxe
xa
a
e
x
a
ò
dxxe
xa
ú
û
ù
ê
ë
é
-
2
1
a

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

x
ba
b
+
ln
1
ò
+
222
x
dx
ba
)(tan
1
1
a
b
ab
x
-