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Signals & Systems - Reference Tables

1

Table of Fourier Transform Pairs

Function, f(t)Fourier Transform, F(")

ÂJ

Z""

deFtf tj )(21)(

Definition of Fourier Transform

ÂJJ

ZdtetfF

tj"")()( 0 ttfJ 0 tj eF J tj et f0 0 ""JF )(tf~ )(1 ~F)(tF)(2"Jf nn dttfd)( )()(""Fj n )()(tfjt n J nn dFd ÂJ tdf'')( 1tj e 0 )(2 0 (t)sgn "j 2

Fourier Transform Table

UBC M267 Resources for 2005

F(t) bF(!)

Notes(0)

f(t) Z 1 -1 f(t)e -i!t dtDenition.(1) 1 2Z 1 -1 bf(!)e i!t d! bf(!)

Inversion formula.

(2)bf(-t)

2f(!)Duality property.(3)

e -at u(t) 1 a+i! aconstant,0(4) e -ajtj 2a a 2 2 aconstant,0(5) (t)=1;ifjtj<1,

0;ifjtj>12sinc(!)=2sin(!)

Boxcar in time.(6)

1 sinc(t) (!)Boxcar in frequency. (7)f 0 (t)i!bf(!)Derivative in time.(8) f 00 (t)(i!) 2 bf(!)

Higher derivatives similar.(9)

tf(t)id d!bf(!)

Derivative in frequency.(10)

t 2 f(t)i 2 d 2 d! 2 bf(!)

Higher derivatives similar.(11)

e i! 0 t f(t) bf(!-! 0 )Modulation property.(12) ft-t 0 k ke -i!t

0bf(k!)

Time shift and squeeze.(13)

(fg)(t) bf(!)bg(!)

Convolution in time.(14)

u(t)=0;ift<0

1;ift>0

1 i!+(!)

Heaviside step function.(15)

(t-t 0 )f(t)e -i!t 0 f(t 0 )Assumesfcontinuous att 0 .(16) e i! 0 t 2(!-! 0 )Useful for sin(! 0 t), cos(! 0 t).(17)

Convolution:(fg)(t)=Z

1 -1 f(t-u)g(u)du=Z 1 -1 f(u)g(t-u)du.

Parseval:

Z 1 -1 jf(t)j 2 dt=1 2Z 1 -1bf(!) 2 d!.

Signals & Systems - Reference Tables

2 tj 1 )sgn(" )(tu 1)( H

JÂZntjn

n eF 0

JÂZ

J nn nF)(2 0 trect )2(" 'Sa )2(2BtSaB )(Brect" )(ttri )2( 2 "Sa )2()2cos(

trecttA

22
)2()cos(" J A )cos( 0 t"xz)()( 00 )sin( 0 t" xz)()( 00 j )cos()( 0 ttu" xz 22
000 )()(2 JHHHJ j )sin()( 0 ttu" xz 22
02 00 )()(2 JHHJJ j )cos()( 0 tetu t ~J 22
0 )()("~""~jjHHH

Signals & Systems - Reference Tables

3 )sin()( 0 tetu t ~J 22
00 jHH t e ~J 22
2 H )2/( 22
t e J2/ 22
2 J e t etu ~J "~jH 1 t tetu ~J 2 )(1"~jH

õ Trigonometric Fourier Series

EF Z HHZ 1000
)sin()cos()( nnn ntbntaatf"" where ZZZ T nT T n dtnttfTbdtnttfTadttfTa 000 000 )sin()(2 and, )cos()(2 , )(1

õ Complex Exponential Fourier Series

JÂ

JÂZ

ZZ T ntj n nntj n dtetfTFeFtf 0 0 )(1 where, )(

Signals & Systems - Reference Tables

4

Some Useful Mathematical Relationships

2)cos(

jxjx eex J HZ jeex jxjx

2)sin(

J JZ )sin()sin()cos()cos()cos(yxyxyxŠZÎ )sin()cos()cos()sin()sin(yxyxyxÎZÎ )(sin)(cos)2cos( 22
xxxJZ )cos()sin(2)2sin(xxxZ )2cos(1)(cos2 2 xxHZ )2cos(1)(sin2 2 xxJZ

1)(sin)(cos

22
ZHxx )cos()cos()cos()cos(2yxyxyxHHJZ )cos()cos()sin()sin(2yxyxyxHJJZ )sin()sin()cos()sin(2yxyxyxHHJZ

Signals & Systems - Reference Tables

5

Useful Integrals

dxx)cos( )sin(x dxx)sin( )cos(xJ dxxx)cos( )sin()cos(xxxH dxxx)sin( )cos()sin(xxxJ dxxx)cos( 2 )sin()2()cos(2 2 xxxxJH dxxx)sin( 2 )cos()2()sin(2 2 xxxxJJ dxe x~ ae x~ dxxe x~

ėĘĖćĈĆJ2

1 a axe x~ dxex x~2 JJ 322
22
aax axe x~ Hxdx~ x~Hln1 H 222
xdx~ )(tan1 1 x J

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Engineering Tables/Fourier Transform Table 2

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Signal Fourier transform

unitary, angular frequency Fourier transform unitary, ordinary frequency Remarks 10 The rectangular pulse and the normalized sinc function 11 Dual of rule 10. The rectangular function is an idealized 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.

14 Shows that the Gaussian function exp( - at2) is its own Fourier transform. For this to be integrable we must have

Re(a) > 0.

common in optics a>0 the transform is the function itself J

0(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.

U n (t) is the Chebyshev polynomial of the second kind Retrieved from "http://en.wikibooks.org/wiki/Engineering_Tables/Fourier_Transform_Table_2"

Category: Engineering Tables

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