13 Fast Fourier Transform (FFT)


 Barry Martin
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1 13 Fast Fourier Trasform FFT) The fast Fourier trasform FFT) is a algorithm for the efficiet implemetatio of the discrete Fourier trasform. We begi our discussio oce more with the cotiuous Fourier trasform Cotiuous ad Discrete Fourier Trasforms Revisited Let E k be the complex expoetial defied by ad cosider the complex ier product The ad for k l we have f, g 1 E k x) : e ikx fx)gx)dx. E k, E k 1 e } ikx {{ e ikx } dx 1, 1 E k, E l e ikx e ilx dx e ik l)x dx e ik l)x ik l) [e ik l) e ik l)] 1 ik l) 1 ik l) eik l) [ 1 e l k)i }{{} 1 As we just saw, the complex expoetials are orthoormal o the iterval [, ]. I fact, they are used as the basis fuctios for complex Fourier series with Fourier coefficiets c k 1 fx) k c k e ikx ft)e ikt dt f, E k ˆfk). As metioed earlier, the coefficiets c k ˆfk) are also kow as the Fourier trasform of f. Remark ote that the otatio used here is a bit differet from that used i Chapter 11. I particular, earlier we used the iterval, ). ]
2 I order to get to the discrete Fourier trasform we first trucate the Fourier series ad obtai P x) c k E k x) a trigoometric polyomial. This ca be verified by writig P x) c k e ikx c k e ix ) k so that P is a polyomial of degree i the complex expoetial e ix. We ow also cosider equally spaced data i the iterval [0, ), i.e., x j, fx j )) with x j j, j 0, 1,..., 1, 1) ad itroduce a discrete pseudo)ier product f, g 1 f ) j g ) j. 13) Remark This is oly a pseudoier product sice f, f 0 implies f 0 oly at the discrete odes j, j 0, 1,..., 1. Theorem 13.1 The set {E k } of complex expoetials is orthoormal with respect to the ier product, defied i 13). Proof First, E k, E k 1 1 ) ) j j E k E k e ikj/ e ikj/ }{{} 1 1. ext, for k l we have E k, E l ) ) j j E k E l e ikj/ e ilj/ ) e i k l) j. ow, sice k l, e i k l) 1 136
3 we ca apply the formula for the sum of a fiite geometric series to get Fially, sice E k, E l 1 e i k l) ) e i k l) 1 ) 1. e i k l) 1 the orthogoality result follows. This gives rise to a best approximatio result just as for stadard Fourier series): Corollary 13. Let data be give as i 1) ad cosider G spa{e k } with <. The the best least squares approximatio to the data from G with respect to the ier product 13)) is give by the discrete Fourier series with c k f, E k 1 gx) f c k E k x) ) j j ik e, k 0, 1,...,. Remark 1. The coefficiets c k of the discrete Fourier series are called the discrete Fourier trasform DFT) of f.. It is our goal to compute the DFT ad also evaluate g) via the fast Fourier trasform FFT). 3. ote that, eve if the data is real, ) the DFT will i geeral be complex. If, e.g., 4, ad the values f j, j 0, 1,, 3 are 4, 0, 3, 6 respectively, the the coefficiets c k are 13. The FFT Algorithm We start with the trigoometric polyomial c 0 13 c i c 1 c 3 1 6i. with DFT coefficiets P x) c k 1 f c k E k x) ) j j ik e 137
4 1 fx j )λ k ) j where x j j ad we set k i λ k e. Remark The values λ k or more aturally λ k ) are ofte referred to as the th roots of uity. This is illustrated i Figure 8 for l 4 l Re l 0 l Im l Figure 8: Roots of uity λ k, k 0, 1,..., 1, for 5. The computatio of each coefficiet c k correspods to evaluatio of a polyomial of degree 1 i λ k. Usig ested multiplicatio Horer s method) this ca be doe i O) operatios. Sice we have coefficiets, the polyomial P ca be costructed i O ) operatios. The major beefit of the fast Fourier trasform is that it reduces the amout of work to O log ) operatios. The FFT was discovered by Cooley ad Tukey i However, Gauss seemed to already be aware of similar ideas. Oe of the most popular moder refereces is the DFT Owers Maual by Briggs ad Heso published by SIAM i 1995). It is the dramatic reductio i computatioal complexity that eared the FFT a spot o the list of Top 10 Algorithms of the 0th Cetury see hadout). The FFT is based o the followig observatio: Theorem 13.3 Let data x k, fx k )), x k k, k 0, 1,..., 1, be give, ad assume that p ad q are expoetial polyomials of degree at most 1 which iterpolate 138
5 part of the data accordig to px j ) fx j ), qx j ) fx j+1 ), j 0, 1,..., 1, the the expoetial polyomial of degree at most 1 which iterpolates all the give data is P x) e ix ) px) e ix ) q x ). 14) Proof Sice e ix e ix) has degree, ad p ad q have degree at most 1 it is clear that P has degree at most 1. We eed to verify the iterpolatio claim, i.e., show that By assumptio we have where P x k ) 1 P x k ) fx k ), k 0, 1,..., e ix k ) pxk ) + 1 e ix k e i k e i ) k 1) k. ) 1 e ix k q x k ) Therefore { px k ) if k eve, P x k ) q x k ) if k odd. Let k be eve, i.e., k j. The, by the assumptio o p P x j ) px j ) fx j ), j 0, 1,..., 1. O the other had, for k j + 1 odd), we have usig the assumptio o q) P x j+1 ) q x j+1 ) qx j ) fx j+1 ), j 0, 1,..., 1. This is true sice x j+1 j + 1) j x j. The secod useful fact tells us how to obtai the coefficiets of P from those of p ad q. Theorem 13.4 Let 1 1 p α j E j, q β j E j, ad P The, for j 0, 1,..., 1, 1 γ j E j. γ j 1 α j + 1 e ij/ β j γ j+ 1 α j 1 e ij/ β j. 139
6 Proof I order to use 14) we eed to rewrite q x ) 1 β j E j x ) 1 β j e ijx ) 1 β j e ijx e ij/. Therefore P x) e ix ) px) e ix ) q x ) e ix ) α j e ijx e ix ) β j e ijx e ij/ 1 1 [ α j + β j e ij/) e ijx + α j β j e ij/) e i+j)x]. However, the terms i paretheses together with the factor 1/) lead precisely to the formulæ for the coefficiets γ. Example We cosider the data x 0 y f0) f) ow we apply the DFT to fid the iterpolatig polyomial P. We have 1 ad P x) 1 γ j E j x) γ 0 e 0 + γ 1 e ix. Accordig to Theorem 13.4 the coefficiets are give by γ 0 1 α0 + β 0 e 0), γ 1 1 α0 β 0 e 0). Therefore, we still eed to determie α 0 ad β 0. They are foud via iterpolatio at oly oe poit cf. Theorem 13.3): px) qx) 0 α j E j x) α 0 such that px 0 ) fx 0 ) α 0 fx 0 ) 0 β j E j x) β 0 such that qx 0 ) fx 1 ) β 0 fx 1 ). With x 0 0 ad x 1 we obtai P x) 1 f0) + f)) + 1 f0) f)) eix.. 140
7 Example We ow refie the previous example, i.e., we cosider the data ow, ad P is of the form x 0 y f0) f ) P x) 3 f) 3 γ j E j x). f 3 Accordig to Theorem 13.4 the coefficiets are give by γ 0 1 α 0 + β 0 ), γ 1 1 ) α 1 + β 1 e i/ }{{}, i γ 1 α 0 β 0 ), γ 3 1 ) α 1 β 1 e i/ }{{}. i This leaves the coefficiets α 0, β 0 ad α 1, β 1 to be determied. They are foud via iterpolatio at two poits cf. Theorem 13.3): px) qx) ). 1 α j E j x) such that px 0 ) fx 0 ), px ) fx ), 1 β j E j x) such that qx 0 ) fx 1 ), qx ) fx 3 ). These are both problems of the form dealt with i the previous example, so a recursive strategy is suggested. A geeral recursive algorithm for the FFT is give i the Kicaid/Cheey textbook o pages 455 ad 456. I order to figure out the computatioal complexity of the FFT algorithm we assume ) m so that the recursive strategy for the FFT ca be directly applied. If is ot a power of, the the data ca be padded appropriately with zeros). I order to evaluate the iterpolatig expoetial polyomial P x) c k E k x) 1 γ j E j x) we ca use the FFT to compute the coefficiets c k or γ j ). Istead of givig a exact cout of the arithmetic operatios ivolved i this task, we estimate the miimum umber of multiplicatios required to compute the coefficiets of a expoetial polyomial with terms. 141
8 Lemma 13.5 Let R) be the miimum umber of multiplicatios required to compute the coefficiets of a expoetial polyomial with terms. The R) R) +. Proof There are coefficiets γ j computed recursively via the formulas γ j 1 α j + 1 e ij/ β j γ j+ 1 α j 1 e ij/ β j listed i Theorem If the factors 1 e ij/, 0, 1,..., 1, are precomputed, the each of these formulas ivolves multiplicatios for a total of multiplicatios. Moreover, the coefficiets α j ad β j are determied recursively usig R) multiplicatios each. Theorem 13.6 Uder the same assumptios as i Lemma 13.5 we have R m ) m m. Proof We use iductio o m. For m 0 there are o multiplicatios sice P x) c 0. ow, assume the iequality holds for m. The, by Lemma 13.5 with m ) R m+1) R m ) R m ) + m. By the iductio hypothesis this is less tha or equal to m m + m m + 1) m+1. Settig m m log i Theorem 13.6 implies R) log, i.e., the fast Fourier trasform reduces the amout of computatio required to evaluate a expoetial polyomial from O ) if doe directly) to O log ). For 1000 this meas roughly 10 4 istead of 10 6 operatios. Remark 1. Applicatio of the classical fast Fourier trasform is limited to the case where the data is equally spaced, periodic, ad of legth m. The ofte suggested strategy of paddig with zeros to get a data vector of legth m ) or the applicatio of the FFT to operiodic data may produce uwated oise.. Various versio of the FFT for oequally spaced data have also bee proposed i recet years e.g., FFT by Daiel Potts, Gabriele Steidl ad Mafred Tasche). 14
9 Remark Give a vector a [a 0, a 1,..., a ] T of discrete data, Corollary 13. tells us that the discrete Fourier trasform â of a is computed compoetwise as â k 1 j ik a j e, k 0, 1,..., 1. Of course, the FFT ca be used to compute these coefficiets. There is a aalogous formula for the iverse DFT, i.e., a j j ik â k e, j 0, 1,..., 1. We close with a few applicatios of the FFT. Use of the FFT for may applicatios beefits from the fact that covolutio i the fuctio domai correspods to multiplicatio i the trasform domai, i.e., f g ˆfĝ so that we have f g ˆfĝ )ˇ. Some applicatios are ow Filters i sigal or image processig. Compressio of audio of video sigals. E.g., the MP3 file format makes use of the discrete cosie trasform. Deoisig of data cotamiated with measuremet error see, e.g., the Maple worksheet FFT.mws). Multiplicatio with cyclic matrices. A cyclic matrix is of the form a 0 a 1... a 1 a a a 0 a 1... a 1 C a 1 a a 0 a a 1... a 1 a a 0 ad multiplicatio by C correspods to a discrete covolutio, i.e., Cy) i C ij y j a j i y j, i 0,...,, where the subscript o a is to be iterpreted cyclically, i.e., a +1+i a i or a i a +1 i. Thus, a coupled system of liear equatios with cyclic system matrix ca be decoupled ad thus trivially solved) by applyig the discrete Fourier trasform. 143
10 High precisio iteger arithmetic also makes use of the FFT. umerical solutio of differetial equatios. Remark Aother trasform techique that has some similarities to but also some sigificat differeces from) the FFT is the fast wavelet trasform. Usig socalled scalig fuctios ad associated wavelets as orthogoal basis fuctios oe ca also perform best approximatio of fuctios or data). A advatage of wavelets over the trigoometric basis used for the FFT is the fact that ot oly do we have localizatio i the trasform domai, but also i the fuctio domai. 144
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