Fourier Transform Pairs

Downloaded from http://uspas.fnal.gov/materials/11ODU/FourierTransformPairs.pdf. Uses j instead of i for imaginary unit.

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 )

Heaviside function

¥

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

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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

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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

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