An Intuitive but Not-All-That-Mathematically-Sound Explanation of the Fourier Transform

Transcription

1 An Inuv bu No-All-Tha-Mahmacally-Sound Explanaon of h Fourr Transform ) Inro by Dan Morrs Lk many folks ou hr, I hav a pry good da wha h Fourr Transform s. I blv dp n my har ha a m-doman sgnal can b rprsnd as a sum of snusods. I v wrn los of cod ha dpnds on compung Fourr Transforms and powr spcra, and rld on som lbrary o do h work (mmmmm Malab ff(x) ). Howvr, no bng a ral sgnal procssng yp, was for a long m a mysry why hs formula for h Fourr Transform ha I m salng from Wkpda: urns a sgnal no s frquncy componns. You do som suff wh complx numbrs and, hn vola, you hav a sgnal urnd no s frquncy componns. In alkng wh many ohr ngnrs who arn sgnal procssng yps, I ralzd ha many popl dar I say mos popl who us Fourr Transforms a las occasonally wr n h sam boa. If you askd hm o wr h formula for h Fourr Transform (whou Wkpda), hy would rmmbr ha hr mus b som and som, bu hy would b fuzzy on h dals. Thy may hav sn a drvaon hr or hr, bu ddn rally snk n. If ha s you, I hop I can shar wh you my basc nuon for h magc, a las nough o hlp you nllgnly undrsand how o compu a FT, n cas you r vr suck on a lonly sland wh no Malab and nd o ransform somhng no h frquncy doman. If you don hav any da wha h m and frquncy domans ar, you should look for a dffrn uoral. If you ar a mahmacally rgorous prson who s offndd by handwavng, you should look for a dffrn uoral. I wll us boh proof by xampl and proof by handwavng. If you wan o skp rgh o h mony and dcd whhr my casual nuon s sud o you, ry gong sragh o SECTION 4. Ohrws, hr w go l s sar wh h las dp and mos ncompl xplanaon vr proposd on wha h Fourr Transform dos. ) Las dp and mos ncompl xplanaon vr proposd on wha h Fourr Transform dos.

2 L s say I hav som sgnal x(: n hs cas I mad x( a bunch of snusods addd oghr, o mak hs mor clar. x( happns o b hr snusods wh frquncs of Hz, Hz, and Hz and ampluds 7,, and : 7sn( ) + sn( + sn( I opn my favor lbrary (my prsonal favor lbrary s Malab) and call ff(x) (acually I do a bunch of ohr suff o handl Malab covnons, c., s h cod a h nd of hs documn, whch ssnally appls h formula I wro up abov, and I g back a magc abl llng m how much of ach frquncy maks up my orgnal sgnal. Clarly n hs cas, I should g back h hr frquncs I pu n. Wha dos Malab show m (cod s a h nd of hs documn?

3 Looks good, I s h xac sam hr frquncs and ampluds ha I pu n. Gra so hs s a pry good dmonsraon of h FT s cor propry: lls m how much of ach frquncy maks up my orgnal sgnal. Now agan, w ask, how dos hs formula: do all ha magc? L s now jump no h lmd mahmacal background rqurd for my unsound FT nuon: h complx xponnal. ) Las dp and mos ncompl xplanaon vr proposd on h complx xponnal. Th rm ω appars all ovr h plac n sgnal procssng; w call hs h complx xponnal. Wha I m gong o do now s ry o convnc you ha hs rm ω s a cosn wav n whr ω drmns h frquncy,.. s mor or lss lk cos(ω. Whl hs s an awful smplfcaon ha hrows ou h complx par of our xponnal, s rally hlpful o jus sll your bran on hs concp, whch I m gong o ry o do now. If you alrady blv hs and wan o mov on o h Fourr Transform, skp o Scon 4. Frs of all hr s on hng w hav o ak on absolu fah, namly ha h mos magcal formula n all of mah Eulr s formula s ru: - Mos arg radrs of hs documn hav sn hs wzardry bfor; s drvaon s wll byond h scop of hs documn (and h scop of my had). Bu I v sn nough ms ha I blv, and hopfully you hav oo. Ths s h only hng w r rally gong o hav o ak on fah. Sw, so wha dos hs hav o do wh cosn wavs? Hr s whr h (hlpful) handwavng bgns. L s look a h complx xponnal ω agan, mor carfully, usng som xampl valus. Th prncpl hr s gong o b ha f hs funcon bhavs lk a cosn wav, hn s a cosn wav. I s a lap, sur, bu h pon hr s nuon, no rgor, so go wh m w r gong o plug n som valus now and s f ω looks mor or lss lk a cosn wav. I m gong o us xampl valus for h rm ω; w ll spara ω and lar. Th valus I m gong o pck for ω ar h frs fw mulpls of /, namly: 0*/, */, */, */, 4*/, and */ ohrws known as:

4 0, /,, /,, and /. If you don wan o rad any mah and you rus m no o cha, you can skp unl h </borng algbra> and I ll jus show you h rsuls hr. I m jus dong algbra hr, bu hlps m blv ha hs all maks sns. <borng algbra> Wha s h valu of h complx xponnal whn ω 0? Wll w don nd fancy mah for hs on, snc anyhng 0 : ω 0 0. Okay, wha s h valu of h complx xponnal whn ω /? Wll, usng Eulr s formula: ( ) ( ) ) ( ω Okay, wha s h valu of h complx xponnal whn ω? Wll, usng Eulr s formula: ω Okay, wha s h valu of hs xprsson whn ω /? Wll, usng Eulr s formula: ( ) ( ) ( ) ( ) ( ) * * * ) ( ω Okay, wha s h valu of hs xprsson whn ω? Wll, usng Eulr s formula: ( ) ( ) ω Okay, wha s h valu of h complx xponnal whn ω /? Wll, usng Eulr s formula: ( ) ( ) ( ) ( ) ( ) * * * * * ) ( ω </ borng algbra >

5 Now l s mak a abl of hos rsuls, and a nc pry plo: Valu of ω Rsul (valu of ω ) Ral par of rsul Imagnary par of rsul 0/ 0 / 0 * / / - 0 -* 4/ 0 / 0 * So l s plo h ral par of our rsul (bcaus magnary numbrs ar oo much o dal wh rgh now): Valus of h complx xponnal. 0. -jw x Dos ha plo rmnd you of anyhng? Eh? Eh? Mayb rmnds you of cos(x):

6 . 0. cos(x) x In fac f you pluggd all h valus you wand no h complx xponnal ( ω ), h ral par of your rsuls would look jus lk a cosn wav. Th complx par, urns ou, would look jus lk a sn wav a h sam frquncy! How bloody sw s ha? So wha s up wh ω and, whch I d oghr? Hopfully you s whr I m gong wh hs rally ω s usually usd o rprsn h frquncy of my complx xponnal, and s h npu varabl. So f s h cosn wav w jus walkd hrough, s a cosn wav wh wc h frquncy. I ll lav as an xrcs for you o plug n valus and convnc yourslf of ha f you so dsr. So wha w hav, from all hs, wha I wan you o blv, s ha ω s sor of lk a cosn wav wh frquncy ω whch, by h wll-known handwavng horm, ranslas o ω s a cosn wav wh frquncy ω Now w r rady o mov on o h Fourr Transform slf. 4) Las dp and mos ncompl xplanaon vr proposd why h Fourr Transform works. Okay, now quppd wh our dp blf ha ω s a cosn wav wh frquncy ω (f you skppd hr from h bgnnng, I hop you blv ha, l s prnd w r ohrws hgh school algbra sudns, and ngag n a ll dalog wh a random dud.

7 Random dud: I nd o know how many 7 s hr ar n 49. Hgh school algbra sudn: Duh, jus dvd 49 by 7. Thr ar svn 7 s n 49. Random dud: I nd o know how many x s hr ar n y. Hgh school algbra sudn: Duh, jus dvd y by x. Thr ar y/x x s n y. Random dud: I nd o know how much of h frquncy Hz hr s n my sgnal x(. Hgh school algbra sudn: Duh, jus dvd x( by Hz. Thr s x(/hz amoun of Hz n your sgnal. Now hs sudn may no vn know wha hrz mans, bu hs s a gra nuon sor of how would w dvd x( by a sgnal rprsnng a cran frquncy ω (n hs cas Hz)? Wll, now ha w blv ha ω s a cosn wav wh frquncy ω, I b w can us ha how abou hs: x( amoun of sgnal wh frquncy ω n x( x( ω Ths s a nonsns mahmacal xprsson, bu s rally gng clos o h formula for h Fourr ransform unforunaly, hs s an xprsson ha looks a a sngl valu of. And ha dosn mak sns; a sngl valu of x( s jus a numbr, and dosn hav frquncy componns. Wha I rally wan o know s wha h amoun of sgnal wh frquncy ω hr s n my nr x(. So wha f w ngra hs ovr all valus of? ω amoun of sgnal x( wh frquncy ω n x( x( ω ω d Is could b s ha h Fourr Transform? I damn sur looks lk h formula a h op of hs documn ha I sol from wkpda. L s jus fnsh hs up by rplacng amoun of sgnal wh frquncy ω n x( by h mor convnonal noaon X(ω): X ( ω ) x( ω d So how dos hs rlad o h ff(x) funcon ha I lk o call all h m n Malab or som ohr lbrary? Ths formula aks h sgnal x( and lls m how much of on frquncy hr s

8 n ha whol sgnal. Th usual mplmnaon of a Fourr Transform dos hs for los and los of frquncs (ofn you jus ll how many you wan, and rurns a abl of X(ω) valus for dffrn frquncs. Ths s xacly wha h plo s n scon. Hopfully ha a las hlps you rmmbr h basc concps of h Fourr Transform; I m gong o ouch brfly on h nvrs Fourr Transform. ) Las dp and mos ncompl xplanaon vr proposd on why h nvrs Fourr Transform works. Acually, scrach ha, ha s oo much, l s do hs ) A good way o rmmbr wha h nvrs Fourr Transform looks lk f you bough no Scon 4. Th nvrs Fourr Transform s h oppos of h Fourr Transform. If I m h magc IFT oracl, you would gv m a bunch of frquncy/amplud pars n ohr words, you r sayng: hr s how much of ach frquncy lv n som sgnal. I would hn ll you wha h orgnal sgnal was. So hs s h oppos of h Fourr Transform. Wh no rgards o mahmacal dph, wha s h oppos of hs? X ( ω ) x( ω d How abou w jus swch all h s and ω s, and urn h nsd of h ngral upsd-down: x( ω ω X ( ω) ω dω Ths s vry vry clos o h convnonal rprsnaon for h IFT, whch ncluds a normalzaon facor and looks lk hs: x( ω ω X ( ω) ω dω I hav bn vry carlss wh absolu valus hr, so you can jus rus ha h / s ncssary jus o mak sur ha f I pu x( no a Fourr Transform and ak h IFT of h rsul, I g x( back. 6) Wha dd w lav ou?

9 L s s asd from all mahmacal dph and jusfcaon, h mos mporan omsson hr s ha my logc whch s rally mor lk a pnumonc dvc complly gnors phas and complly gnors h complx par of h FT. Ths s okay for wo rasons () hs s jus a basc nuon and a way of rmmbrng wha h FT looks lk, and () many folks who ar lkly o rad hs documn us h FT ofn jus for compung powr spcra (whch s dfnd as h magnud of h ral par of h FFT of a sgnal). Phas? Wha phas. W don nd no snkng phas. Complx par? Wha complx par. W don nd no snkng complx par. Also, o b clar, hroughou hs documn I usd noaon conssn wh h connuous Fourr Transform, bu as far as undrsandng, rcognzng, and rmmbrng h basc formula, hs all appls jus fn o h dscr Fourr Transform (DFT and IDFT) and h dscr-m Fourr Transform (DTFT and IDTFT), whos ypcal formulas look vry vry much lk h FT and IFT I v prsnd hr. Appndx A: Malab cod % Quck dmonsraon of usng h Fourr ransform n Malab % o masur h frquncy conn of a sgnal. % % Dan Morrs, 006 % hp://cs.sanford.du/~dmorrs % Numbr of pons w'll hav n our sgnal N 04; % Th m of our "rcordng nrval", abrarly 0 sconds T 0; % Dfn m a m axs [0:N-]/N; % Convr m o sconds *T; % L's add oghr hr sn wavs wh dffrn frquncs and % ampluds f ; a 7; f ; a ; f ; a ; % Dfn our s funcon f a*sn(*p*f* + a*sn(*p*f* + a * sn(*p*f*; % Plo our s funcon and mak look pry

10 fgur(); plo(,f); xl xlabl('tm (s)'); s(xl,'fonsz',); yl ylabl('sgnal (mad-up uns)'); s(yl,'fonsz',); l l('mad-up sgnal for dmonsrang ff'); s(l,'fonsz',); s(gcf,'mnubar','non'); %% % Tak h fourr ransform of our sgnal, hn ak s magnud, % snc for oday, w'r no nrsd n phas. To convr o h % ampluds w provdd n our orgnal sgnal, dvd by N/; ha's jus % h Malab convnon for h ff funcon. p abs(ff(f))/(n/); % W'r only nrsd n posv frquncs for hs dmonsraon, so % only us h scond half of h suff w go back from ff(). Th frs % half conand ngav frquncs. p p(:n/); % Fnd h corrspondng frquncy n Hz (w g frquncs back from FFT % n radans) frq [0:N/-]/T; % Plo our Fourr componns and mak hm look pry fgur(); % plo h powr plo(frq,p); xl xlabl('tm (s)'); s(xl,'fonsz',); yl ylabl('sgnal (mad-up uns)'); s(yl,'fonsz',); l l('fourr componns of our sgnal'); s(l,'fonsz',); s(gcf,'mnubar','non'); % Zoom n on an nrsng rang of h graph xlm([0,0]);