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Example: the Fourier Transform of a rectangle function: rect(t) 1/2 1/2 1/2 1/2 1 exp( ) [exp( )] 1 [exp( /2) exp(exp( /2) exp(2 sin(Fitdt it i ii i ii i ?? ? ? ?? ? ?? ? ? ? ? ? =?=? ? =? ?/2)] ? 1? ?/2) = ( /2) /2) = ( /2) ? F (sinc(??)= /2) Imaginary Component = 0 F(w)

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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

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