JOURNAL OF OBJECT TECHNOLOGY

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1 JOURNAL OF OBJECT TECHNOLOGY Publshed by ETH Zurch, Char of Software Engneerng JOT, 2010 Vol. 9, No. 2, March - Aprl 2010 The Dscrete Fourer Transform, Part 6: Cross-Correlaton By Douglas Lyon Abstract Ths paper s part 6 n a seres of papers about the Dscrete Fourer Transform (DFT) and the Inverse Dscrete Fourer Transform (IDFT). The focus of ths paper s on correlaton. The correlaton s performed n the tme doman (slow correlaton) and n the frequency doman usng a Short-Tme Fourer Transform (STFT). When the Fourer transform s an FFT, the correlaton s sad to be a fast correlaton. The approach requres that each tme segment be transformed nto the frequency doman after t s wndowed. Overlappng wndows temporally solate the sgnal by ampltude modulaton wth an apodzng functon. The selecton of overlap parameters s done on an ad-hoc bass, as s the apodzng functon selecton. Ths report s a part of project Fenestratus, from the skunk-works of DocJava, Inc. Fenestratus comes from the Latn and means, to furnsh wth wndows. 1 INTRODUCTION TO CROSS-CORRELATION Cross-Correlaton (also called cross-covarance) between two nput sgnals s a knd of template matchng. Cross-correlaton can be done n any number of dmensons. For the purpose of ths presentaton, we defne one-dmensonal normalzed crosscorrelaton between two nput sgnals as: r d = [(x[] x) (y[ d] y)] (x[] x) 2 (y[ d] y) 2 The coeffcent, r, s a measurement of the sze and drecton of the lnear relatonshp between varables x and y. If these varables move together, where they both rse at an dentcal rate, then r = +1. If the other varable does not budge, then r = 0. If the other varable falls at an dentcal rate, then r = -1. If r s greater than zero, we have postve correlaton. If r s less than zero, we have negatve correlaton. The sample non-normalzed cross-correlaton of two nput sgnals requres that r be computed by a sample-shft (tme-shftng) along one of the nput sgnals. For the numerator, ths s called a sldng dot product or sldng nner product. The dot product s gven by: (1) Douglas A. Lyon: The Dscrete Fourer Transform, Part 6: Cross-Correlaton, n Journal of Object Technology, vol. 9. no. 2, March - Aprl 2010 pp

2 THE DISCRETE FOURIER TRANSFORM, PART 6: CROSS-CORRELATION X Y = x y (2) When (1) s computed, for all delays, then the output s twce that of the nput. It s common to use the pentagon notaton when showng a cross correlaton: (x y) d = x * y +d (3) Where the astersk ndcates the complex conjugate (a negaton of the magnary part of the number). Input sgnals should ether have the same length, or there should be a polcy n place to make them the same (perhaps by zero paddng or data replcaton). If the nput sgnals are real-valued, then we can wrte: Comparng (4) wth the convoluton: (x y) d = x n * y n = x y +d (4) = x y n (5) Shows that Y s tme-reversed before shftng by n. In comparson, correlaton has shftng wthout the tme reversal. = 2 AN EXAMPLE, IN EXCEL Suppose you are gven two sgnals that have already had ther means subtracted. Correlate the sgnals, wthout dvdng by the standard devaton. The sgnals are: X=1,2,3,4 wth Y = 3, 2, 0, 1. x y y1 y2 y3 y4 y5 y6 y corr Fgure 1. Y s shfted 7 tmes Fgure 1 demonstrates that a movng cross correlaton requres that the kernel of the sgnal be shfted so that ts leadng edge appears and then s shfted untl only the tralng edge can be seen. After each shft, the rest of the sgnal s padded wth zeros. The row labeled corr contans the correlaton and results from the dot product of X wth Y. For example X Y1=12, X Y2=17, etc. 18 JOURNAL OF OBJECT TECHNOLOGY VOL. 9, NO. 2.

3 3 A SLOW CROSS CORRELATION We mplement the slow cross correlaton usng a sldng dot product: publc statc double[] shft(double d[], nt s) { double c[] = new double[d.length]; for (nt = 0; < c.length; ++) { f ((s + >= 0)&&(s+ < d.length)) c[] = d[s + ]; return c; publc statc vod man(fnal Strng[] args) { double x[] = {1,2,3,4; double y[] = {3,2,0,2; PrntUtls.prnt(x); PrntUtls.prnt(y); for (nt = -y.length+1; < y.length; ++){ double sldngy[] = shft(y, ); PrntUtls.prnt(sldngY); System.out.prntln("dot product:"+mat1.dot(x,sldngy)); Where the dot product s mplemented usng: statc publc double dot(fnal double[] a, fnal double[] b) { fnal nt alength = a.length; f (alength!= b.length) { System.out.prntln( "ERROR: Vectors must be of equal length n dot product."); return 0; double sum = 0; for (nt = 0; < alength; ++) { sum += a[] * b[]; return sum; The output matches that gven n Fgure 1: dot product: dot product: dot product: dot product: VOL. 9, NO. 2. JOURNAL OF OBJECT TECHNOLOGY 19

4 THE DISCRETE FOURIER TRANSFORM, PART 6: CROSS-CORRELATION dot product: dot product: The mplementaton s clearly not optmzed, but t s correct and serves to llustrate the sldng dot product nature of the cross correlaton. 4 A FAST CORRELATION A slow mplementaton of the movng cross correlaton algorthm, as shown n Fgure 1, wll take O(N**2) tme. Further, the unoptmzed mplementaton, shown n secton 3, has hgh constant tme overhead. Usng the FFT and the correlaton theorem, we accelerate the correlaton computaton. The correlaton theorem says that multplyng the Fourer transform of one functon by the complex conjugate of the Fourer transform of the other gves the Fourer transform of ther correlaton. That s, take both sgnals nto the frequency doman, form the complex conjugate of one of the sgnals, multply, then take the nverse Fourer transform. Ths s expressed by: f (x) g(x) F *(u)g(u) (6) We now compare the slow correlaton wth the fast correlaton: publc statc vod testcorrelaton() { fnal double x[] = {1,2,3,4; fnal double y[] = {3,2,0,2; PrntUtls.prnt(x); PrntUtls.prnt(y); System.out.prntln("cross cor (slow):"); PrntUtls.prnt(Mat1.slowCorrelaton(x, y)); System.out.prntln("Fast xcor:"); PrntUtls.prnt(SgProc.correl(x, y, 0)); The output follows: cross cor (slow): Fast xcor: The fast correlaton makes use of the FFT, and gets dentcal results to the slow correlaton. So how much faster s the FFT than the slow correlaton for small szed arrays? The testng code follows: 20 JOURNAL OF OBJECT TECHNOLOGY VOL. 9, NO. 2.

5 publc statc vod man(strng[] args) { for (nt = 1; < 5; ++) { System.out.prntln("sze:" + *256); speedtestcorrelaton(256 * ); //man2(args); publc statc vod speedtestcorrelaton(nt n) { fnal double x[] = new double[n]; fnal double y[] = new double[n]; StopWatch sw = new StopWatch(); sw.start(); Mat1.slowCorrelaton(x, y); sw.stop(); sw.prnt("slow correlaton s done"); sw.start(); SgProc.correl(x, y, 0); sw.stop(); sw.prnt("fast correlaton s done"); Even for modest arrays, we see substantal speedup (between 2.5 and 12 tmes faster, for small array szes): sze:256 slow correlaton s done 0.01 seconds fast correlaton s done seconds sze:512 slow correlaton s done seconds fast correlaton s done seconds sze:768 slow correlaton s done seconds fast correlaton s done seconds sze:1024 slow correlaton s done seconds fast correlaton s done seconds 5 SUMMARY Ths paper shows how the FFT can be used to speed up cross correlaton. Further, t shows that even for small array szes, substantal speed up can be obtaned by usng the fast cross correlaton. Arguments for usng the FFT to accelerate the cross correlaton are often not supported wth specfc data on computaton tme (a stuaton, whch ths paper remedes) [Lyon 97]. The cross correlaton has uses n many felds of scentfc endeavor (musc, dentfcaton of blood flow, astronomcal event processng, speech processng, pattern recognton, fnancal engneerng, etc.). One of the basc problems wth the term normalzaton when appled to the crosscorrelaton s that t s defned n dfferent places dfferently. For example, Pratt suggests that the number of elements n the normalzaton (an even a square root) s VOL. 9, NO. 2. JOURNAL OF OBJECT TECHNOLOGY 21

6 THE DISCRETE FOURIER TRANSFORM, PART 6: CROSS-CORRELATION not needed [Pratt]. Lews suggests usng both the square root and the average [Lews]. Therefore the queston of whch normalzaton to use s applcaton-specfc. REFERENCES [Lews] Fast Normalzed Cross-Correlaton by J.P. Lews, last accessed 8/24/09. [Lyon 97] Java Dgtal Sgnal Processng, Douglas A. Lyon and H. Rao, M&T Press (an mprnt of Henry Holt). November [Pratt] W. Pratt, Dgtal Image Processng, John Wley, New York, About the author Douglas A. Lyon (M'89-SM'00) receved the Ph.D., M.S. and B.S. degrees n computer and systems engneerng from Rensselaer Polytechnc Insttute (1991, 1985 and 1983). Dr. Lyon has worked at AT&T Bell Laboratores at Murray Hll, NJ and the Jet Propulson Laboratory at the Calforna Insttute of Technology, Pasadena, CA. He s currently the co-drector of the Electrcal and Computer Engneerng program at Farfeld Unversty, n Farfeld CT, a senor member of the IEEE and Presdent of DocJava, Inc., a consultng frm n Connectcut. Dr. Lyon has authored or co-authored three books (Java, Dgtal Sgnal Processng, Image Processng n Java and Java for Programmers). He has authored over 40 journal publcatons. Emal: lyon@docjava.com. Web: 22 JOURNAL OF OBJECT TECHNOLOGY VOL. 9, NO. 2.