Fourier TransformsFourier seriesSlide 3Slide 4Complex exponential notationEuler’s formulaComplex exponential formSlide 8Fourier transformSlide 10Slide 11Slide 12Slide 13Slide 14Convolution theoremUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don FussellFourier TransformsUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don FussellFourier seriesTo go from f( ) to f(t) substituteTo deal with the first basis vector being of length 2 instead of , rewrite asttT02)sin()cos()(000tnbtnatfnnn)sin()cos(2)(0010tnbtnaatfnnnUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don FussellFourier seriesThe coefficients becomedttktfTaTttk00)cos()(20dttktfTbTttk00)sin()(20University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don FussellFourier seriesAlternate formswhere))(cos(2))sin()tan()(cos(2))sin()(cos(2)(01000100010nnnnnnnnnntncatntnaatnabtnaatfnnnnnnabbac122tanandUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don FussellComplex exponential notationEuler’s formula)sin()cos( xixeixPhasor notation:xyiyxiyxzzyxzeziyxi122tanand))((whereUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don FussellEuler’s formulaTaylor series expansionsEven function ( f(x) = f(-x) )Odd function ( f(x) = -f(-x) )...!4!3!21432xxxxex...!8!6!4!21)cos(8642xxxxx...!9!7!5!3)sin(9753xxxxxx)sin()cos(...!7!6!5!4!3!21765432xixixxixxixxixeixUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don FussellComplex exponential formConsider the expressionSoSince an and bn are real, we can letand get)sin()()cos()()sin()cos()(000000tnFFitnFFtniFtnFeFtfnnnnnnnnntinn)(andnnnnnnFFibFFannFF 2)Im(and2)Re()Im(2and)Re(2nnnnnnnnbFaFFbFaUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don FussellComplex exponential formThusSo you could also writeninTtttinTttTttTttneFdtetfTdttnidttntfTdttntfidttntfTF000000000)(1))sin())(cos((1)sin()()cos()(10000ntninneFtf)(0)(University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don FussellFourier transformWe now haveLet’s not use just discrete frequencies, n0 , we’ll allow them to vary continuously tooWe’ll get there by setting t0=-T/2 and taking limits as T and n approach ntinneFtf0)(dtetfTFTtttinn000)(1University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don FussellFourier transformdtetfTedtetfTeeFtftTinTTntTintinTTntinntinn22/2/22/2/)(212)(1)(000dTT2limdnnlimdFeddtetfedtetfdetftititititi )(21)(2121)(21)(University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don FussellFourier transformSo we have (unitary form, angular frequency)Alternatives (Laplace form, angular frequency)deFtfFdtetfFtftiti)(21)())(()(21)())((1-FFdeFtfFdtetfFtftiti)(21)())(()()())((1-FFUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don FussellFourier transformOrdinary frequency2deFtfFdtetfFtftiti)()())(()()())((1-FFUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don FussellFourier transformSome sufficient conditions for applicationDirichlet conditions f(t) has finite maxima and minima within any finite interval f(t) has finite number of discontinuities within any finite intervalSquare integrable functions (L2 space)Tempered distributions, like Dirac deltadttf )(dttf2)]([21))(( tFUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don FussellFourier transformComplex form – orthonormal basis functions for space of tempered distributions)(222121dteetitiUniversity of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don FussellConvolution theoremTheoremProof (1))(*)()()()()*()(*)()()()()*(GFFGGFGFgffggfgf1-1-1-1-1-1-FFFFFFFFFFFF)()('')''(')'()'(')'(')'()'()*(''')'('gfdtetgdtetfdtettgdtetfdtdtettgtfgftitittititiFFF
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