Table of Fourier Transform Pairs
Function, f(t)
Definition of Inverse Fourier Transform
1
f (t ) =
2p
¥
ò F (w )e
jwt
dw
Fourier Transform, F(w)
Definition of Fourier Transform
¥
F (w ) =
-¥
ò f (t )e
- jwt
dt
-¥
f (t - t 0 )
F (w )e - jwt0
f (t )e jw 0t
F (w - w 0 )
f (at )
1
w
F( )
a
a
F (t )
2pf (-w )
d n f (t )
( jw ) n F (w )
dt n
(- jt ) n f (t )
d n F (w)
dw n
t
ò
f (t )dt
-¥
F (w )
+ pF (0)d (w )
jw
d (t )
1
e jw 0 t
2pd (w - w 0 )
sgn (t)
2
jw
Signals & Systems - Reference Tables
1
j
sgn(w )
1
pt
u (t )
pd (w ) +
¥
¥
å Fn e jnw 0t
2p
t
rect ( )
t
tSa(
B
Bt
Sa( )
2p
2
w
rect ( )
B
tri (t )
w
Sa 2 ( )
2
n = -¥
A cos(
pt
t
)rect ( )
2t
2t
1
jw
å Fnd (w - nw 0 )
n = -¥
wt
)
2
Ap cos(wt )
t (p ) 2 - w 2
2t
cos(w 0 t )
p [d (w - w 0 ) + d (w + w 0 )]
sin(w 0 t )
p
[d (w - w 0 ) - d (w + w 0 )]
j
u (t ) cos(w 0 t )
p
[d (w - w 0 ) + d (w + w 0 )] + 2 jw 2
2
w0 - w
u (t ) sin(w 0 t )
2
p
[d (w - w 0 ) - d (w + w 0 )] + 2w 2
2j
w0 - w
u (t )e -at cos(w 0 t )
Signals & Systems - Reference Tables
(a + jw )
w 02 + (a + jw ) 2
2
w0
u (t )e -at sin(w 0 t )
e
w 02 + (a + jw ) 2
2a
-a t
e -t
a2 +w2
2
/( 2s 2 )
s 2p e -s
2
w2 / 2
1
a + jw
u (t )e -at
1
u (t )te -at
(a + jw ) 2
Ø Trigonometric Fourier Series
¥
f (t ) = a 0 + å (a n cos(w 0 nt ) + bn sin(w 0 nt ) )
n =1
where
1
a0 =
T
T
ò0
2T
f (t )dt , a n = ò f (t ) cos(w 0 nt )dt , and
T0
2T
bn = ò f (t ) sin(w 0 nt )dt
T 0
Ø Complex Exponential Fourier Series
f (t ) =
¥
å Fn e
jwnt
, where
n = -¥
Signals & Systems - Reference Tables
1T
Fn = ò f (t )e - jw 0 nt dt
T 0
3
Some Useful Mathematical Relationships
e jx + e - jx
cos( x) =
2
e jx - e - jx
sin( x) =
2j
cos( x ± y ) = cos( x) cos( y ) m sin( x) sin( y )
sin( x ± y ) = sin( x) cos( y ) ± cos( x) sin( y )
cos(2 x) = cos 2 ( x) - sin 2 ( x)
sin( 2 x) = 2 sin( x) cos( x)
2 cos2 ( x) = 1 + cos(2 x)
2 sin 2 ( x) = 1 - cos(2 x)
cos 2 ( x) + sin 2 ( x) = 1
2 cos( x) cos( y ) = cos( x - y ) + cos( x + y )
2 sin( x) sin( y ) = cos( x - y ) - cos( x + y )
2 sin( x) cos( y ) = sin( x - y ) + sin( x + y )
Signals & Systems - Reference Tables
4
Useful Integrals
ò cos( x)dx
sin(x)
ò sin( x)dx
- cos(x)
ò x cos( x)dx
cos( x) + x sin( x)
ò x sin( x)dx
sin( x) - x cos( x)
òx
2
cos( x)dx
2 x cos( x) + ( x 2 - 2) sin( x)
òx
2
sin( x)dx
2 x sin( x) - ( x 2 - 2) cos( x)
ax
dx
e ax
a
òe
ò xe
òx
ax
dx
2 ax
éx 1 ù
e ax ê - 2 ú
ëa a û
e dx
é x 2 2x 2 ù
e ax ê - 2 - 3 ú
a û
ëa a
dx
1
ln a + bx
b
ò a + bx
dx
ò a 2 + b 2x2
Signals & Systems - Reference Tables
bx
1
tan -1 ( )
ab
a
5