[PDF] [PDF] Fourier transform

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8 fév 2011 · Definition of Fourier Transform ◇ The forward and inverse Fourier Transform are defined for aperiodic signal as: A unit triangle function A(x):

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Review: Exponential Fourier Series (for Periodic Functions) { } 1 1 5 sinc(x) is the Fourier transform of a single rectangular pulse sin( ) Unit Triangle Pulse

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In electronics, Fourier analysis has been playing an important role, and a signal is often considered to be a superposition of many sine and cosine functions with  

Triangular Function Analysis* - ScienceDirect

In electronics, Fourier analysis has been playing an important role, and a signal is often considered to be a superposition of many sine and cosine functions with  

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Fourier Transform, F(w) Definition of Inverse Fourier Transform Р ¥ ¥- Fourier Transform Table UBC M267 tri is the triangular function 13 Dual of rule 12

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Let x(t) be a triangular pulse defined by x(t) = { 1 − t ; t < 1 0 ; else (a) By taking the derivative of x(t), use the derivative property to find the Fourier transform 

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For example, to find the Fourier series for a triangular wave as shown in Fig 2 we would calculate the coefficients as follows: 2See, for example, Boyce and 

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periodic with period 2L Any function with period 2L can be represented with a Fourier series -L L 2L 2L 0 Examples (triangle wave) (square wave) ( ) 1,0 1 ,

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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("quotesdbs_dbs9.pdfusesText_15