Fourier

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

- jw t

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 ) +

¥

1
jw

¥

å Fn e jnw 0t

2p

t
rect ( )
t

tSa(

B
Bt
Sa( )
2p
2

w
rect ( )
B

tri (t )

wSa 2 ( )
2

n = -¥

A cos(

pt
t
)rect ( )
2t
2t

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 0t )

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 - wu (t )e -at cos(w 0 t )

Signals & Systems - Reference Tables

(a + jw )
2
w 0 + (a + jw ) 2

2

w0

u (t )e -at sin(w 0 t )

e

2
w 0 + (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 0nt ) )
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
T0

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
T0

3

Some Useful Mathematical Relationships
e jx + e - jx
cos( x) =
2
e jx...