Lab de fourier

Table of Fourier Transform Pairs
Function, f(t)
Definition of Inverse Fourier Transform

Fourier Transform, F(w)
Definition of Fourier Transform

1 f (t ) = 2p
f (t - t 0 )

¥

-¥

ò F(w )e

jwt

dw

F (w ) =

¥

-¥

ò f (t )e

- jwt

dt

F (w )e - jwt0 F (w - w 0 )

f (t )e jw 0t

f (at )

1 w F( ) a a 2pf (-w ) ( jw ) n F (w )

F (t ) d n f (t ) dt n (-jt ) n f (t )

d n F (w) dw n

-¥

ò

t

f (t )dt

F (w ) + pF (0)d (w ) jw
1 2pd (w - w 0 ) 2 jw

d (t )
e jw 0 t
sgn (t)

Signals & Systems - Reference Tables

1

j

1 ptsgn(w )

u (t )

pd (w ) +
¥

1 jw

n = -¥

å Fn e jnw 0t

¥

2p

n = -¥

å Fnd (w - nw 0 )
wt ) 2

t rect ( ) t

tSa(

B Bt Sa( ) 2p 2 tri (t )

w rect ( ) B w Sa 2 ( )2

A cos(

pt t )rect ( ) 2t 2t

Ap cos(wt ) t (p ) 2 - w 2 2t
p [d (w - w 0 ) + d (w + w 0 )]

cos(w 0 t ) sin(w 0 t )

p [d (w - w 0 ) - d (w + w 0 )] j
p [d (w - w 0 ) + d (w + w 0 )] +2 jw 2 2 w0 - w
2 p [d (w - w 0 ) - d (w + w 0 )] + 2w 2 2j w0 - w

u (t ) cos(w 0 t )

u (t ) sin(w 0 t )

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

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

Signals & Systems -Reference Tables

2

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

w0
2 w 0 + (a + jw ) 2

e

-a t

2a a2 +w2

e -t

2

/( 2s 2 )

s 2p e -s

2

w2 / 2

u (t )e -at

1 a + jw 1 (a + jw ) 2

u(t )te -at

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

ò0

T

2T f (t )dt , a n = ò f (t ) cos(w 0 nt )dt , and T02T bn = ò f (t ) sin(w 0 nt )dt T 0

Ø Complex Exponential Fourier Series
f (t ) =

n = -¥

å Fn e

¥

jwnt

, where

1T Fn = ò f (t )e - jw 0 nt dt T 0

Signals & Systems - ReferenceTables

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(...