Transformative Application

fst Frr trnsfrm (FFT) s n lgrthm tht cmpts th Dscrt Frr Trnsfrm (DFT) f sqnc, r ts nvrs (DFT). Frr nlyss cnvrts sgnl frm ts rgnl dmn (ftn tm r spc) t rprsnttn n th frqncy dmn nd vc vrs. Th DFT s btnd by dcmpsng sqnc f vls nt cmpnnts f dffrnt frqncs. Ths prtn s sfl n mny flds, bt cmptng t drctly frm th dfntn s ftn t slw t b prctcl. n FFT rpdly cmpts sch trnsfrmtns by fctrzng th DFT mtrx nt prdct f sprs (mstly zr) fctrs. s rslt, t mngs t rdc th cmplxty f cmptng th DFT frm ( n 2 ) {\txtstyl (n^{2})} , whch rss f n smply ppls th dfntn f DFT, t ( n lg ⁡ n ) {\txtstyl (n\lg n)} , whr n s th dt sz. Th dffrnc n spd cn b nrms, spclly fr lng dt sts whr n my b n th thsnds r mllns. n th prsnc f rnd-ff rrr, mny FFT lgrthms r mch mr ccrt thn vltng th DFT dfntn drctly r ndrctly. Thr r mny dffrnt FFT lgrthms bsd n wd rng f pblshd thrs, frm smpl cmplx-nmbr rthmtc t grp thry nd nmbr thry. Fst Frr trnsfrms r wdly sd fr pplctns n ngnrng, msc, scnc, nd mthmtcs. Th bsc ds wr pplrzd n 1965, bt sm lgrthms hd bn drvd s rly s 1805. n 1994, Glbrt Strng dscrbd th FFT s "th mst mprtnt nmrcl lgrthm f r lftm", nd t ws ncldd n Tp 10 lgrthms f 20th Cntry by th mgzn Cmptng n Scnc & ngnrng. Th bst-knwn FFT lgrthms dpnd pn th fctrztn f n, bt thr r FFTs wth ( n lg ⁡ n ) {\dsplystyl (n\lg n)} cmplxty fr ll, vn prm, n. Mny FFT lgrthms dpnd nly n th fct tht − 2 π / n {\txtstyl ^{-2\p /n}} s n n'th prmtv rt f nty, nd ths cn b ppld t nlgs trnsfrms vr ny fnt fld, sch s nmbr-thrtc trnsfrms. Snc th nvrs DFT s th sm s th DFT, bt wth th ppst sgn n th xpnnt nd 1/n fctr, ny FFT lgrthm cn sly b dptd fr t.