Fourier Transform Pairs PDF

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Summary

A Table showing a function and it's Fourier Transform...

Description

Table of Fourier Transform Pairs

Function, f(t) Definition of Inverse Fourier Transform

f (t ) =

1 2p

Fourier Transform, F(w w) Definition of Fourier Transform

¥ jwt ò F (w )e dw

¥

ò

F (w ) =

-¥

f ( t) e - jwt dt

-¥

f (t - t 0 )

F (w )e

f (t )e j w0 t

F (w - w 0 )

f (at )

1 w F( ) a a

F (t )

2pf ( -w )

d n f (t )

( jw ) n F (w )

- jw t 0

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

ej

2pd (w - w 0 )

w 0t

sgn (t)

Signals & Systems - Reference Tables

2 jw

1

j

sgn(w )

1 pt

u (t )

pd (w) + ¥

¥

å Fn e

jn w 0t

å Fn d (w - nw 0 )

2p

n= -¥

n = -¥

t rect ( ) t

tSa (

B Bt Sa( ) 2p 2

w rect( ) B

tri(t )

2 w Sa ( ) 2

A cos(

1 jw

pt t ) rect( ) 2t 2t

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 )

p w2 [d (w - w 0 ) - d (w + w 0 )] + 2 2 2j w 0 -w

u(t ) e-a t cos(w 0 t ) w02

Signals & Systems - Reference Tables

(a + j w) + (a + jw) 2

2

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

w0 2 w0

e

2

2

2a a 2 +w 2

-a t

e -t

+ (a + jw )

/( 2s 2 )

s 2p e

u (t )e -at

- s 2w 2 / 2

1 a + jw 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 a0 =

1 T

T

ò0

f (t ) dt , a n =

2T f (t ) cos(w 0 nt) dt , and T 0ò

2T b n = ò f (t ) sin(w 0 nt )dt T0

Ø Complex Exponential Fourier Series ¥

f (t ) =

å Fn e

j wnt

, where

n = -¥

Signals & Systems - Reference Tables

Fn =

1 T

T

ò f (t )e

- jw 0 nt

dt

0

3

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

- jx

e jx - e sin( x) = 2j

- jx

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

éx 1 ù e ax ê - 2 ú ëa a û

ò xe

ax

òx

e dx

éx 2 2 x 2 ù e ê - 2 - 3ú a û ëa a

dx

1 ln a + bx b

dx

2 ax

ò a + bx dx

òa 2 + b 2 x2

Signals & Systems - Reference Tables

ax

1 - bx tan 1 ( ) ab a

5...