Table of Fourier Transform Pairs

Sa (x) = sin(x) / x sinc function

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

sgn(w )

j

pt

u(t)

pd (w) + 1

jw

?

? Fn e jnw0t

n=-?

?

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

rect( t ) t

tSa(w2t )

B 2p

Sa(

Bt 2

)

Sa (x) = sin(x) / x sinc function

rect(w ) B

tri(t)

tri(t) = (1-|t|)rect(t/2) triangle function = rect(t)*rect(t)

A

cos(

pt 2t

)rect(

t 2t

)

Sa 2 (w2 )

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

Sa (x) = sin(x) / x sinc function

cos(w 0t)

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

sin(w 0t)

p j

[d

(w

-

w0

)

-

d

(w

+

w0

)]

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

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

Your continued donations keep Wikibooks running!

Engineering Tables/Fourier Transform Table 2

From Wikibooks, the open-content textbooks collection

< Engineering Tables Jump to: navigation, search

Signal

Fourier transform unitary, angular frequency

Fourier transform unitary, ordinary frequency

Remarks

10

The rectangular pulse and the normalized sinc function

Dual of rule 10. The rectangular function is an idealized

11

low-pass filter, and the sinc function is the non-causal

impulse response of such a filter.

12

tri is the triangular function

13

Dual of rule 12.

Shows that the Gaussian function exp( - at2) is its own

14

Fourier transform. For this to be integrable we must have

Re(a) > 0.

common in optics

Retrieved from "" Category: Engineering Tables

Views

a>0

the transform is the function itself J0(t) is the Bessel function of first kind of order 0, rect is the rectangular function it's the generalization of the previous transform; Tn (t) is the Chebyshev polynomial of the first kind.

Un (t) is the Chebyshev polynomial of the second kind

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download