Table of Fourier Transform Pairs - Fermilab

Table of Fourier Transform Pairs

Function, f(t)

Fourier Transform, F(w)

Definition of Inverse Fourier Transform Definition of Fourier Transform

? f

(t)

=

1 2p

?

F (w )e jwt dw

-?

?

? F (w) = f (t)e - jwt dt -?

f (t - t0 )

F (w )e - jwt0

f (t)e jw0t

F (w - w 0 )

f (at)

1 F(w ) aa

F (t)

2pf (-w)

d n f (t) dt n

(- jt)n f (t)

t

? f (t )dt

-?

d (t)

e jw0t sgn (t)

( jw)n F (w)

d n F (w) dw n

F (w ) + pF (0)d (w) jw

1 2pd (w - w 0 ) 2 jw

Signals & Systems - Reference Tables

1

1 j pt u(t)

?

? Fn e jnw0t

n=-?

rect( t ) t

B 2p

Sa(

Bt 2

)

tri(t)

A

cos(

pt 2t

)rect(

t 2t

)

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

u(t) cos(w 0t)

u(t) sin(w 0t)

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

sgn(w )

pd (w) + 1 jw

?

2p ? Fnd (w - nw 0 ) n = -?

tSa(w2t )

rect(w ) B

Sa 2 (w2 )

Ap cos(wt ) t (p 2t ) 2 - w 2

p [d (w - w 0 ) + d (w + w 0 )]

p j

[d

(w

-

w0

)

-

d

(w

+

w0

)]

p 2

[d

(w

-

w0

)

+

d

(w

+

w0

)]

+

w

2 0

jw -w

2

p 2j

[d

(w

-w0)

-d

(w

+ w0 )] +

w2

w

2 0

-w2

(a + jw)

w

2 0

+

(a

+

jw ) 2

Signals & Systems - Reference Tables

2

u(t)e -at sin(w 0t) e -a t e -t 2 /(2s 2 ) u(t)e -at u(t)te -at

w0

w

2 0

+ (a

+

jw ) 2

2a a2 +w2

s 2p e -s 2w 2 / 2

1 a + jw

1 (a + jw) 2

? Trigonometric Fourier Series

?

f (t) = a0 + ? (an cos(w 0 nt) + bn sin(w 0 nt)) n =1

where

a0

=1 T

?T

0

f (t)dt

,

an

=

?2 T

T0

f

(t) cos(w 0nt)dt

, and

bn

=

2 T

T

?

0

f

(t) sin(w0 nt)dt

? Complex Exponential Fourier Series

?

? f (t) = Fne jwnt , where n=-?

? Fn

=

1 T

T 0

f

(t)e - jw0nt dt

Signals & Systems - Reference Tables

3

Some Useful Mathematical Relationships

cos(x)

=

e

jx

+ e - jx 2

sin( x)

=

e

jx

- e - jx 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(2x) = cos 2 (x) - sin 2 (x) sin(2x) = 2 sin(x) cos(x)

2cos2 (x) = 1 + cos(2x)

2 sin 2 (x) = 1 - cos(2x)

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

? cos(x)dx ? sin(x)dx ? x cos(x)dx ? x sin(x)dx ? x 2 cos(x)dx ? x 2 sin(x)dx ? eax dx

? xeax dx

? x 2eax dx

?

a

dx + bx

? dx

a2 + b 2x2

sin( x)

- cos(x)

cos(x) + x sin(x)

sin(x) - x cos(x)

2x cos(x) + (x 2 - 2) sin(x) 2x sin(x) - (x 2 - 2) cos(x)

eax a

eax

?x ?? a

-

1 a2

? ??

eax

? ? ?

x2 a

-

2x a2

-

2?

a

3

? ?

1 ln a + bx b

1 tan -1 ( bx )

ab

a

Signals & Systems - Reference Tables

5

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