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Table of Fourier Transform PairsFunction, f(t)Definition of Inverse Fourier Transform1f (t ) 2p ò F (w )ejwtdwFourier Transform, F(w)Definition of Fourier Transform F (w ) - ò f (t )e- jwtdt- f (t - t 0 )F (w )e - jwt0f (t )e jw 0tF (w - w 0 )f (at )1wF( )aaF (t )2pf (-w )d n f (t )( jw ) n F (w )dt n(- jt ) n f (t )d n F (w)dw ntòf (t )dt- F (w ) pF (0)d (w )jwd (t )1e jw 0 t2pd (w - w 0 )sgn (t)2jwSignals & Systems - Reference Tables1

Fourier Transform TableUBC M267 Resources for 2005Fb(ω)F (t)Notes(0)Definition.(1)fb(ω)Inversion formula.(2)fb( t)2πf (ω)Duality property.(3)e at u(t)1a iωa constant, e(a) 0(4)2a ω2a constant, e(a) 0(5)Boxcar in time.(6)Boxcar in frequency.(7)Derivative in time.(8)Higher derivatives similar.(9)Z f (t)e iωt dtf (t)12πZ fb(ω)eiωt dω e a t β(t) 1,0,a2if t 1,if t 12 sinc(ω) 2sin(ω)ω1sinc(t)πβ(ω)f 0 (t)iω fb(ω)f 00 (t)(iω)2 fb(ω)di fb(ω)dωd2i2 2 fb(ω)dωbf (ω ω0 )tf (t)t2 f (t)eiω0 t f (t) t t0fkke iωt0 fb(kω)fb(ω)bg (ω)Derivative in frequency.(10)Higher derivatives similar.(11)Modulation property.(12)Time shift and squeeze.(13)Convolution in time.(14)(f g)(t) 0, if t 0u(t) 1, if t 01 πδ(ω)iωHeaviside step function.(15)δ(t t0 )f (t)e iωt0 f (t0 )Assumes f continuous at t0 .(16)eiω0 t2πδ(ω ω0 )Useful for sin(ω0 t), cos(ω0 t).(17)ZConvolution:(f g)(t) ZParseval: Zf (t u)g(u) du 1 f (t) dt 2π 2Z 2f (u)g(t u) du.fb(ω) dω.

jsgn(w )1ptu (t )pd (w ) å Fn e jnw 0t2ptrect ( )ttSa(BBtSa( )2p2wrect ( )Btri (t )wSa 2 ( )2n - A cos(ptt)rect ( )2t2t1jwå Fnd (w - nw 0 )n - wt)2Ap cos(wt )t (p ) 2 - w 22tcos(w 0 t )p [d (w - w 0 ) d (w w 0 )]sin(w 0 t )p[d (w - w 0 ) - d (w w 0 )]ju (t ) cos(w 0 t )p[d (w - w 0 ) d (w w 0 )] 2 jw 22w0 - wu (t ) sin(w 0 t )2p[d (w - w 0 ) - d (w w 0 )] 2w 22jw0 - wu (t )e -at cos(w 0 t )Signals & Systems - Reference Tables(a jw )w 02 (a jw ) 22

w0u (t )e -at sin(w 0 t )ew 02 (a jw ) 22a-a te -ta2 w22/( 2s 2 )s 2p e -s2w2 / 21a jwu (t )e -at1u (t )te -at(a jw ) 2Ø Trigonometric Fourier Series f (t ) a 0 å (a n cos(w 0 nt ) bn sin(w 0 nt ) )n 1where1a0 TTò02Tf (t )dt , a n ò f (t ) cos(w 0 nt )dt , andT02Tbn ò f (t ) sin(w 0 nt )dtT 0Ø Complex Exponential Fourier Seriesf (t ) å Fn ejwnt, wheren - Signals & Systems - Reference Tables1TFn ò f (t )e - jw 0 nt dtT 03

Some Useful Mathematical Relationshipse jx e - jxcos( x) 2e jx - e - jxsin( x) 2jcos( x y ) cos( x) cos( y ) m sin( x) sin( y )sin( x y ) sin( x) cos( y ) cos( x) sin( y )cos(2 x) cos 2 ( x) - sin 2 ( x)sin( 2 x) 2 sin( x) cos( x)2 cos2 ( x) 1 cos(2 x)2 sin 2 ( x) 1 - cos(2 x)cos 2 ( x) sin 2 ( x) 12 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 Tables4

Useful Integralsò cos( x)dxsin(x)ò sin( x)dx- cos(x)ò x cos( x)dxcos( x) x sin( x)ò x sin( x)dxsin( x) - x cos( x)òx2cos( x)dx2 x cos( x) ( x 2 - 2) sin( x)òx2sin( x)dx2 x sin( x) - ( x 2 - 2) cos( x)axdxe axaòeò xeòxaxdx2 axéx 1 ùe ax ê - 2 úëa a ûe dxé x 2 2x 2 ùe ax ê - 2 - 3 úa ûëa adx1ln a bxbò a bxdxò a 2 b 2x2Signals & Systems - Reference Tablesbx1tan -1 ( )aba5

Your continued donations keep Wikibooks running!Engineering Tables/Fourier Transform Table 2From Wikibooks, the open-content textbooks collection Engineering TablesJump to: navigation, searchSignalFourier transformunitary, angular frequencyFourier transformunitary, ordinary frequencyRemarks10The rectangular pulse and the normalized sinc function11Dual of rule 10. The rectangular function is an idealizedlow-pass filter, and the sinc function is the non-causalimpulse response of such a filter.12tri is the triangular function13Dual of rule 12.14Shows that the Gaussian function exp( - at2) is its ownFourier transform. For this to be integrable we must haveRe(a) 0.

common in opticsa 0the transform is the function itselfJ0 (t) is the Bessel function of first kind of order 0, rect isthe rectangular functionit's the generalization of the previous transform; Tn (t) is theChebyshev polynomial of the first kind.Un (t) is the Chebyshev polynomial of the second kindRetrieved from "http://en.wikibooks.org/wiki/Engineering Tables/Fourier Transform Table 2"Category: Engineering TablesViews

Fourier Transform Table UBC M267 Resources for 2005 F(t) Fb(!) Notes (0) f(t) Z1 1 f(t)e i!tdt De nition. (1) 1 2ˇ Z1 1 fb(!)ei!td! fb(!) Inversion formula. (2) fb( t) 2ˇf(!) Duality property. (3) e atu(t) 1 a i! aconstant, e(a) 0 (4) e ajtj 2a a2 !2 aconstant, e(a) 0 (5) (t) ˆ 1; if jtj 1, 0; if jtj 1 2sinc(!) 2