Transcription of Table of Fourier Transform Pairs
1 Signals & Systems - Reference Tables1 Table of Fourier Transform PairsFunction, f(t) Fourier Transform , F( )Definition of Inverse Fourier Transform deFtftj)(21)(Definition of Fourier Transform dtetfFtj )()()(0ttf 0)(tjeF tjetf0)( )(0 F)(tf )(1 F)(tF)(2 fnndttfd)()()( Fjn)()(tfjtn nndFd )( tdf )()()0()( FjF )(t 1tje0 )(20 (t)sgn j2 Fourier Transform TableUBC M267 Resources for 2005F(t)bF(!)Notes(0)f(t)Z1 1f(t)e i!tdtDe nition.(1)12 Z1 1bf(!)ei!td!bf(!)Inversion formula.(2)bf( t)2 f(!)Duality property.(3)e atu(t)1a+i!
2 Aconstant,<e(a)>0(4)e ajtj2aa2+!2aconstant,<e(a)>0(5) (t)= 1;ifjtj<1,0;ifjtj>12sinc(!)=2sin(!)!Boxcar in time.(6)1 sinc(t) (!)Boxcar in frequency.(7)f0(t)i!bf(!)Derivative in time.(8)f00(t)(i!)2bf(!)Higher derivatives similar.(9)tf(t)idd!bf(!)Derivative in frequency.(10)t2f(t)i2d2d!2bf(!)Higher derivatives similar.(11)ei!0tf(t)bf(! !0)Modulation property.(12)f t t0k ke i!t0bf(k!)Time shift and squeeze.(13)(f g)(t)bf(!)bg(!)Convolution in time.(14)u(t)= 0;ift<01;ift>01i!+ (!)Heaviside step function.(15) (t t0)f(t)e i!
3 T0f(t0)Assumesfcontinuous att0.(16)ei!0t2 (! !0)Useful for sin(!0t), cos(!0t).(17)Convolution:(f g)(t)=Z1 1f(t u)g(u)du=Z1 1f(u)g(t u) :Z1 1jf(t)j2dt=12 Z1 1 bf(!) 2d!.Signals & Systems - Reference Tables2tj 1)sgn( )(tu j1)( ntjnneF0 nnnF)(20 )( trect)2( Sa)2(2 BtSaB )(Brect )(ttri)2(2 Sa)2()2cos( trecttA22)2()cos( A)cos(0t )()(00 )sin(0t )()(00 j)cos()(0ttu 22000)()(2 j)sin()(0ttu 220200)()(2 j)cos()(0tetut 220)()( jj Signals & Systems - Reference Tables3)sin()(0tetut 2200)( j te 222 )2/(22 te 2/22 2 etetu )( j 1ttetu )(2)(1 j Trigonometric Fourier Series 1000)sin()cos()(nnnntbntaatf where TnTTndtnttfTbdtnttfTadttfTa000000)sin()( 2 and, )cos()(2 , )
4 (1 Complex Exponential Fourier Series TntjnnntjndtetfTFeFtf00)(1 where, )( Signals & Systems - Reference Tables4 Some Useful Mathematical Relationships2)cos(jxjxeex jeexjxjx2)sin( )sin()sin()cos()cos()cos(yxyxyx )sin()cos()cos()sin()sin(yxyxyx )(sin)(cos)2cos(22xxx )cos()sin(2)2sin(xxx )2cos(1)(cos22xx )2cos(1)(sin22xx 1)(sin)(cos22 xx)cos()cos()cos()cos(2yxyxyx )cos()cos()sin()sin(2yxyxyx )sin()sin()cos()sin(2yxyxyx Signals & Systems - Reference Tables5 Useful Integrals dxx)cos()sin(x dxx)sin()cos(x dxxx)cos()sin()cos(xxx dxxx)sin()cos()sin(xxx dxxx)cos(2)sin()2()cos(22xxxx dxxx)sin(2)cos()2()sin(22xxxx dxex aex dxxex 21aaxex dxexx 2 32222aaxaxex xdx x ln1 222xdx )(tan11 x Your continued donations keep Wikibooks running!)
5 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 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.
6 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 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 Retrieved from " " Category: Engineering Tables Views
