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University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier TransformsUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier seriesTo go from f(θ ) to f(t) substituteTo deal with the first basis vector being oflength 2π instead of π, rewrite asttT02!"#==)sin()cos()(000tnbtnatfnnn!!+="#=)sin()cos(2)(0010tnbtnaatfnnn!!++="#=University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier seriesThe coefficients becomedttktfTaTttk!+=00)cos()(20"dttktfTbTttk!+=00)sin()(20"University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier seriesAlternate formswhere! f (t) =a02+ ann=1"#(cos(n$0t) +bnansin(n$0t))=a02+ ann=1"#(cos(n$0t) % tan(&n)sin(n$0t))=a02+ cnn=1"#cos(n$0t +&n)!!"#$$%&'=+='nnnnnnabbac122tanand(University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellComplex exponential notationEuler’s formula)sin()cos( xixeix+=Phasor notation:!"#$%&='+==+==+'xyiyxiyxzzyxzeziyxi122tanand))((where((University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellEuler’s formulaTaylor series expansionsEven function ( f(x) = f(-x) )Odd function ( f(x) = -f(-x) )...!4!3!21432+++++=xxxxex...!8!6!4!21)cos(8642!+!+!=xxxxx...!9!7!5!3)sin(9753!+!+!=xxxxxx)sin()cos(...!7!6!5!4!3!21765432xixixxixxixxixeix+=+!!++!!+=University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellComplex exponential formConsider the expressionSoSince an and bn are real, we can letand get)sin()()cos()()sin()cos()(000000tnFFitnFFtniFtnFeFtfnnnnnnnnntinn!!!!!""#=#"#=#"#="++=+==$$$)(andnnnnnnFFibFFa!!!=+=nnFF =!2)Im(and2)Re()Im(2and)Re(2nnnnnnnnbFaFFbFa!==!==University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellComplex exponential formThusSo you could also writeninTtttinTttTttTttneFdtetfTdttnidttntfTdttntfidttntfTF!"""""==#=$$%&''()#=****+#+++000000000)(1))sin())(cos((1)sin()()cos()(10000!"#"=+=ntninneFtf)(0)($%University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier transformWe now haveLet’s not use just discrete frequencies, nω0 ,we’ll allow them to vary continuously tooWe’ll get there by setting t0=-T/2 and takinglimits as T and n approach ∞!"#"==ntinneFtf0)($dtetfTFTtttinn!+"=000)(1#University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier transformdtetfTedtetfTeeFtftTinTTntTintinTTntinntinn!!"""!!22/2/22/2/)(212)(1)(000#$#$=#$#$=$#$=%&%&&===!"dTT=#$%&'()*2lim!!="#dnnlim!!"!"""!!!!!!dFeddtetfedtetfdetftititititi## ###$$%$$%%$$%%$$%$$%=&'()*+==)(21)(2121)(21)(University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier transformSo we have (unitary form, angular frequency)Alternatives (Laplace form, angular frequency)!!"!"!!!deFtfFdtetfFtftiti##$$%%$$%====)(21)())(()(21)())((1-FF!!"!!!!deFtfFdtetfFtftiti##$$%%$$%====)(21)())(()()())((1-FFUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier transformOrdinary frequency! s ="2# ! F( f (t)) = F(s) = f (t)"##$e"i 2%stdtF-1(F(s)) = f (t) = F(&)ei 2%st"##$dsUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 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 finiteintervalSquare integrable functions (L2 space)Tempered distributions, like Dirac delta!<"!!#dttf )(!<"!!#dttf2)]([!"21))(( =tFUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier transformComplex form – orthonormal basis functions forspace of tempered


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UT CS 384G - Fourier Transforms

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