Lightening Summary of Fourier Analysis

Transcription

1 Lghtnng Suary of Fourr Analyss D. Ad. Coponnts n Vctor Spacs You ar falar wth th fact that, n so vctor spac of your choosng, any vctor can b dcoposd nto coponnts along th drctons of so gvn bass: aˆ+ bj ˆ+ ckˆ avˆ (whr vˆ v ˆ, kˆ,tc) Although, n prncpl, th choc of bass s arbtrary, so chocs ar or convnnt than othrs. W usually choos th bass whr th bass vctors ar orthonoral (n vctor languag: ˆ ˆ jˆ jˆ kˆ kˆ, andˆ ˆj ˆ kˆ jˆ kˆ ). You can thnk of lots of good rasons for ths choc; on of partcular ntrst s that, gvn so, w can asly fnd th coponnt, or projcton, n a gvn drcton by sply dottng wth th bass vctor n that drcton. E.g. to fnd projcton along î as: ˆ aˆ ˆ+ bjˆ ˆ+ ckˆ ˆ a. Fourr Srs In th sa way that t s convnnt to dcopos vctors nto sus of or fundantal vctors, t s oftn convnnt to dcopos functons nto sus of or fundantal functons. So, just as w wrot avˆ for a vctor, w would lk to b abl to wrt a functon f() as f ( ) ag ( ),whr g ( ) ar so fundantal functons of our choosng. W can vsualz ths gotrcally by analogy wth th vctor apl abov. W thnk about so prodc functon, f ( ), as a vctor n a vctor spac whch s rally a functon spac. W pck a bass n th functon spac, g( ), to b a convnnt bunch of functons wth proprts that c, n so way, orthonoralty. W thn fnd th projcton of th vctor f ( ) along ach of th bass functons. Ths projctons ar thn th coffcnts a n th panson n trs of g( ). Whn th bass functons ar sn( )and os( ) c,,,,, ths dcoposton s calld a fourr srs. W wrt

2 f ( ) A cos( ) + B sn( ) th crcular functons ar a happy choc for th bass, bcaus thy hav th proprty of, (ovr whch thy cut a full cycl): bng orthogonal on th ntrval [ ] sn( )cos( n) anyn, sn( )sn( n) cos( )cos( n) n n n n W s that our bass vctors ar labld by th ndcs n and (and by sn-nss and cosn-nss ). Th procss of dottng vctors togthr corrsponds to ultplyng th approprat functons togthr and ntgratng ovr th ntrval. Our bass vctors ar orthogonal, snc dottng two dffrnt ons togthr producs. Not that dottng two of th sa vctors togthr gvs a. Thus to ak our bass orthonoral, w hav to rbr to dvd by whnvr w do on of ths ntgrals n a Fourr probl. To fnd th projcton of so f ( ) along a gvn bass vctor,.. th A n front of cos() for so, w sply follow th analogy wth th ral vctor cas, whr projcton along a gvn bass vctor s found by dottng wth that vctor. In th Fourr cas, w thn hav A f( )cos( ) d,,... B f( )sn( ) d,, Eapl: Fourr dcoposton of a squar wav Lts start wth an asy prodc functon, lk th squar wav: f ( ) + a whn < < a whn < < As s oftn th cas n Fourr analyss (and physcs n gnral), w can sav work by consdrng th sytry of th probl. Th functon gvn abov s odd on th ntrval; cosn s vn on th ntrval. Thus, th product f ( )cos( ) s odd, and th ntgral of ths product on th ntrval wll b. So all th A s. To fnd th B s:

3 3 B f( )sn( d ) ( a)sn( ) d+ asn( ) d a [ cos( )],,3,... Now w can wrt th Fourr dcoposton of th squar wav: f ( ) B sn( ) a [ cos( )] sn( ) 4a sn sn3 sn Not that ths srs dos not trnat; n gnral you hav to add up an nfnt nubr of trs to rproduc f ( ) actly. In trs of th vctor spac analogy all that ans s that w nd a vctor spac wth an nfnt nubr of dnsons (th clbratd Hlbrt spac). But don t loos any slp ovr ths vctor spac coplcatons: th an pont s to s that what w ar dong s arrangng sns and cosns, wth th rght choc of frquncs and apltuds, so that thy ntrfr n such a way as to yld so gvn prodc functon. For our apl, th way th su looks aftr thr and fftn trs s shown blow: Fgur Fro A.B.Wood, "A Ttbook of Sound" You can try th graphcal approach yourslf wth th Fourr applt on th cours wbst.

4 4 4. Copl Eponntal Rprsntaton W know that th ultat forals for dalng wth sns and cosns s th copl ponntal. Usng th DMovr/Eulr rlaton θ cosθ + sn θ w rwrt th basc Fourr srs as f ( ) C Thn, th coffcnts nd up bng gvn by C f ( ) d,,,... ± ± and ar rlatd to th old coffcnts by C [ A + B] > C A C [ A B ] < (you should vrfy ths rlatons for yourslf). 5. Fourr Transfors Th prvous analyss s sngly applcabl only to functons prodc on [, ], snc sns and cosns hav th sa prodcty. W can gnralz thngs to work on any ntrval by th usual tchnqu of nsrtng a wavnubr: st kz, whr k / λ and λ s th dsrd wavlngth. Trgonortc functons wth argunt kz now hav prodcty of that wavlngth. Th Fourr achnry bcos f( ) C z λ λ z λ C f( zdz ) λ λ

5 5 Now, say w wshd to Fourr dcopos sothng lk a puls: f( z) b < z< a vrywhrls At frst, t looks lk troubl, ths thng has no prodcty at all. But, n fact, t dos: ts got an nfntly long wavlngth so long, that only on wavlngth fts on th ntr ral ln! To Fourr dcopos just us abov prssons n th lt λ. Now, carryng out th ltng procss for ths prssons s a athatcal dtal you can go look up n a book. For our purposs, suffc t to say that, for non-prodc functons a dscrt srs ovr a bunch of haroncs no longr works. W ust, nstad, suprpos wavs ovr a contnuous dstrbuton of frquncs: nstad of adjacnt trs 3 4 n th srs bng, say, sn(3 )andsn(4 ) or and, th adjacnt trs ar k k ' and, whr k and k' dffr only by an nfntsal dk. Our suaton thn turns nto an ntgral, kz f() z Fk ( ) dk kz whr Fk ( ) f() z dz kz s th thng that usd to b our st of bass vctors. (n.b. n th vctor spac/hlbrt kz spac pctur, stll s a bass. Th nubr of dnsons of th spac s now not only nfnt, but, an uncountabl nfnty, a contnuu). Th Fk ( ) s what usd to bc, th coffcnt of a gvn sn or cosn, or, quvalntly, th projcton of f ( ) along th as corrspondng to that sn or cosn. Instad of a dscrt st of apltuds, Fk ( ) s now a contnuous functon of k: Fk ( ) s th apltud of th sns and cosns that go wth bass vctor kz ), tc. kz (th projcton along It s ntrstng to not that, by a nor chang of dfnton, w can rwrt thngs as

6 6 f () z kz Fk ( ) dk Fk ( ) kz f ( z ) dz W s that th prssons ar dual to ach othr: w can thnk of Fk ( ) as th apltuds n th k-spac dcoposton of f ( ) or w can thnk of f ( ) as th apltuds n th z-spac dcoposton of Fk ( ). Functons f ( ) and Fk ( ) ar sad to b Fourr transfors of on anothr. 6. Fourr Transfor of a Gaussan Puls Start wth a snusodal wav wth wavnubr k ψ ( ) (, ) k ωt t A A constant For convnnc, w ar gong to say w ar lookng at a snapshot at a dfnt t t, and supprss all t varabls. Thn w hav just th wav n spac: ψ k ( ) A A constant Now w construct a puls fro ths wav by odulatng th apltud. A good choc s th Gaussan functon. Th wdth of th Gaussan (actually th half-wdth) s controlld by th factor whch s n th dnonator of th ponntal argunt: th bggr s, th wdr s ths curv. But t s a vry rgular wdth varaton: for all Gaussans 68% of th ara s btwn, and ths aks t nc to us. A ( ) constant So, now, posng ths odulaton, our faturlss sn wav s now a puls-lk wavfor: ψ ( ) k

7 Although w startd fro a wav a dfnt frquncy, th Fourr transfor argunt tlls us to pct that ths puls s th suprposton of any wavs wth dffrnt k s. To s ths plctly, us th Fourr achnry: 7 ( ) k Fk ( ) dk Fk ( ) k ψ ( ) d Th apltud functon Fk ( ) contans th nforaton on th rang of k valus that ar ncludd n ψ ( ). So, w want to calculat k k Fk ( ) d + ( k k ) To do th ntgral, w coplt th squar n th ponntal argunt. To start, notc that th two trs n th argunt look vry slar to th squar of a bnoal d ( k k) + + k ( k) ( k k) (Ecpt for th ordr of k and k, w wll co to that) Thrfor f w start fro our ponntal functon + k ( k ) W can rvrs th ordr of k and k : k ( k) Thn add and subtract a tr suggstd by th prfct squar :

8 8 And rgroup: ( ) ( ) ( ) k k + k k k k + ( ) ( ) k k k k ( k k) Now rcall that ths s all a anpulaton of th ntgrand of F(k): Fk ( ) d + k ( k ) + ( k k) ( k k) ( k k) ( k k) ( k k) + d Mak th chang of varabls Thn ( k k) y +, d dy ( k k) ( ) y Fk dy ( k k ) Whr I hav lookd up th dfnt ntgral n a book, and also rvrsd th ordr of k and k agan, snc th squar s sytrc. (Th ntgral s not lntary, rqurs th thory of copl varabls.) Aftr ths sowhat long haul, w s a bautful thng: th dstrbuton of k s, tslf, turns out to b anothr Gaussan!

9 9 Fk ( ) ( k k ) Now, rbr Fk ( ) s th apltud of th wav wth wavnubr k. Whr F(k) s larg, that wavnubr s contrbutng a lot to th suprposton. W can thnk of F(k) as asurng th dstrbuton of wavlngths n th suprposton. Wth ths n nd, w ak th followng obsvatons ) Th wavlngths contrbutng to th puls ar n a Gaussan dstrbuton cntrd around k. Although th puls s suprposton of wavs wth dffrnt k, thy ar stll all clos to th orgnal k. Indd, th wav wth ks th on wth th largst apltud, w can thnk of t (sort of) as stll th fundantal wav n th bunch. ) Th half-wdth of dstrbuton of frquncs s, whch you vrfy by sng how th wdth ca out of th prsson for A(), and thn applyng to F(k). It s actly th rcprocal of th wdth of th puls (anothr bauty of th Gaussan functon.) If w call ths wdth k, thn, w can wrt k Ths s th statnt of th Fourr bandwdth thor. Th narrowr you ak puls n spac, th broadr th dstrbuton n wavnubr. (Ths aks sns n lght of Fourr thory: th or dscontnuous a functon you want to dcopos, th sallr th wavlngths rqurd, and thrfor th largr th k valus you hav to us. So, th spkr you ak your dstrbuton n, th wdr you hav to ak your dstrbuton n k. In gnral, th product of th wdths s a nubr of ordr. If w had workd n th t doan, w would hav gottn t ω Or, agan, th shortr th puls n t, th broadr th dstrbuton of frquncs n th Fourr dcoposton. Or th narrowr your puls, th or bandwdth s rqurd.

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