Fourier TransformsFourier seriesSlide 3Slide 4Complex exponential notationEuler’s formulaComplex exponential formSlide 8Fourier transformSlide 10Slide 11Slide 12Slide 13Slide 14University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don FussellFourier TransformsUniversity of Texas at Austin CS384G - Computer Graphics Fall 2008 Don FussellFourier seriesTo go from f( ) to f(t) substituteTo deal with the first basis vector being of length 2 instead of , rewrite asttT02ωπθ ==)sin()cos()(000tnbtnatfnnnωω +=∑∞=)sin()cos(2)(0010tnbtnaatfnnnωω ++=∑∞=University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don FussellFourier seriesThe coefficients becomedttktfTaTttk∫+=00)cos()(20ωdttktfTbTttk∫+=00)sin()(20ωUniversity of Texas at Austin CS384G - Computer Graphics Fall 2008 Don FussellFourier seriesAlternate formswhere€ f (t) =a02+ ann=1∞∑(cos(nω0t) +bnansin(nω0t))=a02+ ann=1∞∑(cos(nω0t) − tan(ϕn)sin(nω0t))=a02+ cnn=1∞∑cos(nω0t + ϕn)⎟⎟⎠⎞⎜⎜⎝⎛−=+=−nnnnnnabbac122tanand ϕUniversity of Texas at Austin CS384G - Computer Graphics Fall 2008 Don FussellComplex exponential notationEuler’s formula)sin()cos( xixeix+=Phasor notation:⎟⎠⎞⎜⎝⎛=−+==+==+−xyiyxiyxzzyxzeziyxi122tanand))((whereϕϕUniversity of Texas at Austin CS384G - Computer Graphics Fall 2008 Don FussellEuler’s formulaTaylor series expansionsEven function ( f(x) = f(-x) )Odd function ( f(x) = -f(-x) )...!4!3!21432+++++=xxxxex...!8!6!4!21)cos(8642−+−+−=xxxxx...!9!7!5!3)sin(9753−+−+−=xxxxxx)sin()cos(...!7!6!5!4!3!21765432xixixxixxixxixeix+=+−−++−−+=University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don FussellComplex exponential formConsider the expressionSoSince an and bn are real, we can letand get)sin()()cos()()sin()cos()(000000tnFFitnFFtniFtnFeFtfnnnnnnnnntinnωωωωω−−∞=∞−∞=∞−∞=−++=+==∑∑∑)(andnnnnnnFFibFFa−−−=+=nnFF =−2)Im(and2)Re()Im(2and)Re(2nnnnnnnnbFaFFbFa−==−==University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don FussellComplex exponential formThusSo you could also writeninTtttinTttTttTttneFdtetfTdttnidttntfTdttntfidttntfTFϕωωωωω==−=⎟⎟⎠⎞⎜⎜⎝⎛−=∫∫∫∫+−+++000000000)(1))sin())(cos((1)sin()()cos()(10000∑∞−∞=+=ntninneFtf)(0)(ϕωUniversity of Texas at Austin CS384G - Computer Graphics Fall 2008 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)(ωdtetfTFTtttinn∫+−=000)(1ωUniversity of Texas at Austin CS384G - Computer Graphics Fall 2008 Don FussellFourier transformdtetfTedtetfTeeFtftTinTTntTintinTTntinntinnππωωωππ22/2/22/2/)(212)(1)(000−∞−∞=−∞−∞=∞−∞=∫∑∫∑∑===ωπdTT=⎟⎠⎞⎜⎝⎛∞→2limωω =∞→dnnlimωωπωπππωωωωωωdFeddtetfedtetfdetftititititi∫∫ ∫∫∫∞∞−∞∞−−∞∞−−∞∞−∞∞−=⎥⎦⎤⎢⎣⎡==)(21)(2121)(21)(University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don FussellFourier transformSo we have (unitary form, angular frequency)Alternatives (Laplace form, angular frequency)ωωπωπωωωdeFtfFdtetfFtftiti∫∫∞∞−−∞∞−====)(21)())(()(21)())((1-FFωωπωωωωdeFtfFdtetfFtftiti∫∫∞∞−−∞∞−====)(21)())(()()())((1-FFUniversity of Texas at Austin CS384G - Computer Graphics Fall 2008 Don FussellFourier transformOrdinary frequency€ s =ω2π € F( f (t)) = F(s) = f (t)−∞∞∫e−i 2π stdtF-1(F(s)) = f (t) = F(φ)ei 2π st−∞∞∫dsUniversity of Texas at Austin CS384G - Computer Graphics Fall 2008 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 delta∞<∫∞∞−dttf )(∞<∫∞∞−dttf2)]([πδ21))(( =tFUniversity of Texas at Austin CS384G - Computer Graphics Fall 2008 Don FussellFourier transformComplex form – orthonormal basis functions for space of tempered
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