Lectures 9-11: Fourier Transforms

Transcription

1 Lcurs 9-: ourr Transforms Rfrncs Jordan & Smh Ch7, Boas Ch5 scon 4, Kryszg Ch Wb s hp://wwwjhudu/sgnals/: go o Connuous Tm ourr Transform Proprs PHY6 Inroducon o ourr Transforms W hav sn ha any prodc funcon can b rprsnd by a ourr srs and ha any puls can b rprsnd by a half rang ourr srs In boh cass h shap s formd from summd sn and/or cosn rms of spcfc harmonc frquncs and ampluds Somms w may wsh o sudy hs dsrbuon of frquncs rahr han sar a h fnal summd ourr srs S blow from jhudu Two qusons may occur o us: Is hr anyhng analogous o a ourr srs for a funcon whch s no prodc? Th frquncs ncludd n h ourr sum s an nfn s n aks valus from o nfny bu by no mans ncluds all possbl frquncs s a dscr s of frquncs n s rsrcd o ngr valus, no a connuous spcrum Can w somhow modfy h srs o conan a connuous spcrum? Snc an ngral s h lm of a sum, you may no b surprsd o larn ha n h abov cass h ourr srs sum s rplacd by h ourr ransform whch dscrbs h frquncs prsn n h orgnal funcon ourr ransforms, somms calld ourr ngrals can b usd o rprsn - non-prodc funcons, g a sngl volag puls, a flash of lgh - a connuous spcrum of frquncs, g a connuous rang of colours of lgh or muscal pch Thy ar usd xnsvly n all aras of physcs and asronomy Dfnon of ourr Transforms In our sudy of ourr srs w focussd manly on h sn and cosn forms bu wh ourr ransforms s usually mor convnn o us a complx xponnal form Th formula ar: Smlarly, f d whr f d kx f x k dk whr k Phl Lghfoo 8/9 Lcurs 9- - Pag of 8 kx f x dx Essnally h formula for k dfns h ourr ransform of fx; lar w wll prov ha subsung hs no h ohr formula rurns h orgnal funcon fx rs w wll assum h formula ar ru, larn o us hm and obsrv som mporan proprs Th funcons fx and k smlarly f and ar calld a par of ourr ransforms or ngrals Th only dffrnc n h form of h ngrals s h sgn of h xponn, and n pracc s common o call hr funcon h ourr ransform of h ohr Nos and, or x and k, ar ofn calld conjuga varabls k s h wavnumbr, k /λ compar wh /T 3 Thr ar dffrn convnons abou h facor of Th convnon w ar usng, wh / apparng symmrcally, s h mos commonly usd by physcss

2 PHY6 pplcaons ourr ransforms ar usd n many dvrs aras of physcs and asronomy or xampl: In h opcs cours you wll fnd ha h nnsy of h raunhofr dffracon parn from an aprur s h modulus squard of h ourr ransform of h aprur Nx yar n nuclar physcs you wll fnd ha any wak scarng s found from h ourr ransform of h scarng ponal lhough nuclar physcss call h form facor nsad of h ourr ransform In quanum mchancs a localsd parcl has a sprad of momna Ths ar gvn by h ourr ransform of h wav pack 3 Exampls Exampl : rcangular op ha funcon nd h ourr ransform of h funcon f x p < x < p p > x and x > p k f x kx dx p p kx dx k kp kp [ ] Ths funcon occurs so ofn has a nam: s calld a snc funcon p sn kp kp snc x sn x x Or somms dfnd as sn x x ndng h valu a x s a ll rcky Th ass mhod s o us h srs xpanson of sn and look a h lm as x : x 3 x{ } x x sn x sn x x x[ ] Hnc lm lm 6 x lm{ } 3! 6 x x x x x 6 W s ha h majory of h funcon ls nsd h rgon < x < Ths s vn mor ru of h funcon snc x, whch s of nrs n opcs s lar sn x Th oal ara undr h curv can b valuad: dx x Exampl : Th Gaussan nd h ourr ransform of h Gaussan funcon f x a ax / No Th Gaussan s a funcon ncounrd frqunly n Quanum Mchancs and sascs Th consan a s rlad o h wdh: fx falls o / of s nal valu a x / a x ± / a Th facor of a / nsurs ha f x dx as rqurd for a probably dsrbuon Usng h formula abov, k ax a kx dx a ax kx dx Phl Lghfoo 8/9 Lcurs 9- - Pag of 8

3 Ths ngral s pry rcky I can b shown ha So a ax k k a kx a a a k dx j nx jx 4n dx n Hr a n and j k PHY6 Hnc w hav found ha h ourr ransform of a Gaussan s a Gaussan! Th nvrs ransform can b prformd n a vry smlar way of cours gvng h nal Gaussan Rlvanc o Quanum mchancs k a ax L us look a h wdhs of h Gaussan / a f x and s ransform k W could ak any wdh w wand such as full wdh half maxmum, h man squard valu of x, or vn sgma so long as w wr conssn for boh Gaussans, bu o kp hngs smpl l s consdr whr ach funcon falls o / of s maxmum valu W can hrfor say h half wdh of fx s hn x / / a and for k s k / a So h full wdhs ar x / a 8 a and k a 8a W fnd h followng mporan rsul: h produc of h wdhs x k cons 8 Th produc of h wdhs s hrfor a consan, ndpndn of a, s xac valu drmnd by how h wdh s dfnd Ths s no jus ru for Gaussans I s ru for any funcon and s ourr ransform S uoral qusons! Th narrowr h funcon, h wdr h ransform, and vc vrsa Th broadr h funcon n ral spac x spac, h narrowr h ransform n k spac Or smlarly, workng wh m and frquncy, consan On can undrsand hs by hnkng abou wavpacks pur sn wav fx sn kx has unform nnsy hroughou all spac and comprss a sngl frquncy, x, k If w add oghr wo sn wavs of vry smlar k, gx sn kx sn k δkx, h sns add oghr consrucvly a h orgn bu bgn o cancl ach ohr ou nrfr dsrucvly furhr away s on adds oghr mor funcons wh a wdr rang of k s k ncrass, h wavs add consrucvly ovr an ncrasngly narrow rgon x dcrass, and nrfr dsrucvly vrywhr ls Th rsul abov s rlad o h uncrany rlaonshp n quanum mchancs Rmmbr ha momnum s rlad o k by p h k Thus p h k n dal fr parcl can b rprsnd by a wavfuncon kx ψ x, ψ, ha s has a dfn valu of k, a dfn momnum Corrspondngly hs wavfuncon xnds hrough all spac so w canno say whr h parcl s! k p, x parcl whch s localsd n spac has fn x mus b rprsnd by a wavpack wh a sprad of k, a sprad of momna W hav x k ~, hnc x p ~ ħ Ths s h uncrany rlaonshp bwn poson and momnum In h sam way, from ~, and h rlaonshp E h w hav E ~ ħ Ths s h uncrany rlaon bwn nrgy and m Phl Lghfoo 8/9 Lcurs 9- - Pag 3 of 8

4 PHY6 Phl Lghfoo 8/9 Lcurs 9- - Pag 4 of 8 Exampl 3: puls of radaon Consdr a puls of lgh gvn by < < ohrws f cos Th frquncy spcrum s gvn by d d d f cos d Groupng rms Bu rmmbr ha sn θ θ θ so w can wr sn sn sn sn Now l s mulply boh op and boom by sn sn Ofn >> so h scond rm s vry small and w nd only consdr h frs rm: sn Th frquncs ha ar prsn ar ssnally hos n h rang < < ± whr / So h full wdh of frquncs s / No ha f h puls s vry long h frquncy sprad s vry small ssnally h only frquncy obsrvd wll b Ths s as xpcd Bu for a shor puls hr wll b sgnfcan broadnng Ths rsul s rlvan o pulsd lasrs T-sapphr lasrs hav bn dvlopd whch gv vry shor pulss of lgh pulss lasng jus a fw fmosconds Such lgh pulss ar usd, for xampl, o prob rlaxaon phnomna n solds Howvr h frquncy of h lgh slf s only a ll grar han 5 Hz so hs mans ha w rally hav a shor cosn wav puls n m, and h frquncy s hrfor sprad Whl CW connuous wav lasrs can m lgh wh an xrmly narrow ln-wdh, pulsd lasr lgh mus, by s vry naur, hav a broadr ln-wdh nd h shorr h puls, h broadr h ln-wdh

5 PHY6 Exampl 4: Th on-sdd xponnal funcon Show ha h funcon f x k f x kx dx x < has ourr ransform λ x x > dx dx k λ k λ k [ ] λx kx x λ k λ k 4 Complxy, Symmry and h Cosn Transform Th formula for k nvolvs kx So n gnral, f fx s ral, k wll b complx In xampl 4, k s complx Howvr n xampls, and 3, k s ral Why? or ourr srs w found spcal rsuls for vn and odd funcons I s smlar for ourr ransforms W can wr k kx f x dx f x cos kxdx f xsn kxdx or an vn funcon, h scond ngral wll b zro h ngral of an odd funcon ovr a symmrc nrval Hnc f fx s ral and vn hn s ourr ransform s ral xampls,, and 3 Smlarly for an odd funcon h frs ngral wll b zro, so h ourr ransform s purly magnary In h gnral cas of a funcon wh no dfn symmry, h ourr ransform s complx xampl4 X X or vn funcons, also f x dx f x dx, so w can wr: k X f xcoskxdx Such ransforms k ar also vn, so h nvrs ransform can b smlarly smplfd So for an vn ral funcon w can wr f x kcoskxdx whr k f xcoskxdx k s hn calld h ourr cosn ransform of fx Th cosn ngral s somms bu no always! asr o valua Exampl 5: Rpa Exampl usng h ourr cosn ransform formula abov k f xcos kxdx p cos kx dx sn kx k p sn kp k 5 Us of ourr Transforms Gnral Commns Gvn ourr hory, w can undrsand why n opcs and many ohr branchs of physcs on sars by sayng, l us consdr a plan wav If w can solv o fnd h bhavour of a plan wav w can us ourr analyss o do anyhng by combnng plan wavs or xampl, n an lcroncs problm on mgh wan o know h rspons of a crcu o an lcrcal puls On can consdr h rspons of h crcu o a sngl frquncy and hn us ourr hory o wr h puls n rms of sngl frquncs Compuaonal Mhods vry ffcn mhod has bn dvsd for calculang h ourr ransform of a s of daa pons I s known as fas ourr Transforms or T I s usd vry wdly n many branchs of compuaonal physcs You may m hs n h fuur Phl Lghfoo 8/9 Lcurs 9- - Pag 5 of 8

6 PHY6 Physcs/sronomy Exampls 5 Opcs: Dffracon of Lgh rom school physcs lssons, you may b famlar wh h dffracon of lgh a small aprurs Many of you wll sudy hs n mor dal nx smsr PHY7 Consdr a small sl llumnad unformly by lgh of wavlngh λ h sl, h lgh amplud fx, and hus also h lgh nnsy fx, wll b smlar o h op ha funcon xampl abov Obsrvng h dffracon parn on a dsan scrn, h nnsy a any pon s gvn by h ourr ransform squard: k snθ whr k and θ s λ h angular poson on a dsan scrn rom xampl, h nnsy a h scrn hrfor has a snc dsrbuon w obsrv a brgh cnral frng and rgularly spacd sd frngs of dcrasng nnsy Smlarly n all cass of scarng, h nnsy of h scard lgh s gvn by h squar of h T of 7 h objc ha dos h scarng Snc lgh has a small wavlngh, λ ~3 m w only g a 6 rasonabl rang of θ valus for a small objc g d m In vryday lf, you may b abl o s a dffracon parn by lookng a a sodum srlgh hrough an umbrlla or hrough a ms of randrops ourr ransforms go boh ways, so also from lookng a a dffracon parn w can dduc h naur of h objc causng h scarng or xampl, crysal lacs can scar X-rays and from h dffracon parns h crysal srucur can b dducd 5 Nuclar Physcs: scarng of lcrons Consdr a bam of lcrons scard by proons ull analyss rqurs rlavsc quanum mchancs Bu w xpc h sam faurs as n ohr scarng: h scarng wll b largs for kd < and vry small for kd >> Th dpndnc of h scarng on k s known as h form facor Som dcads ago, proons wr hough o b lmnary parcls, so w would xpc o fnd d o b of h ordr of h sz of small nucl Howvr h daa dos no f h prdcons!! Rmmbr ha a broadr k ndcas a narrowr fx Ths daa was h frs vdnc for quarks and gluons! 53 Tlcommuncaons: bandwdh lmaons s w hav sn n xampl 3, h shorr h puls, h broadr s h frquncy dsrbuon n h ourr srs rqurd o dscrb Th lcommuncaons ndusry consanly ry o mprov daa ransfr ra along cabls Typcally h daa aks h form of a dgal sgnal and mprovd spd s achvd by shornng h lnghs of h s and s Ths xnds h frquncy dsrbuon of h ourr srs ha dscrbs If h bandwdh lm of a lphon cabl s MHz hn only frquncs blow MHz can pass, ffcvly clppng h hgh frquncy nd of h daa sgnal and dformng h shap of h logc puls squar wav a ll som pon hs wll lm daa ransfr Phl Lghfoo 8/9 Lcurs 9- - Pag 6 of 8

7 6 Dla uncons PHY6 Th Drac dla funcon δx s vry usful n many aras of physcs I s no an ordnary funcon, n fac proprly spakng can only lv nsd an ngral Hr w dfn, xplor s proprs, hn us o prov h ourr ngral horm Essnally h dla funcon s an nfnly narrow spk ha has un ara δx s a spk cnrd a x, δx x s a spk cnrd a x x Th produc of h dla funcon δx x wh any funcon fx s zro xcp whr x ~ x ormally, for any funcon fx f x δ x x dx f and δ x x dx x b f In fac hs ngrals can also b wrn x δ x x dx f x and a b δ x x dx a whr a < x < b, snc δx x s also dfnd o b zro vrywhr xcp a x x Thr ar svral ways n whch w pcur δx x Th smpls s as h lm as h of a rcangl of hgh h, wdh /h, and hus ara h/h h /h Exampls x x Gvn ha f x δ x x dx f, a fnd x sn x δ x x dx, b fnd x δ x a dx, c fnd h T of fx δx a a b c Phl Lghfoo 8/9 Lcurs 9- - Pag 7 of 8

8 7 Proof of h ourr Transform Thorm PHY6 Th formula for ourr ransforms can b drvd from h formula for ourr srs by consdrng a prodc funcon hn lng h prod nd o nfny You can fnd hs yp of drvaon n Kryszg scon 7, Jordan & Smh scon 7 no ha h auhors work n rms of f no and show h proof for sn srs / ngrals and Boas scon 54 uss h complx form bu no hr dffrn convnon rgardng facors of L Consdr h ngral I x kx dk L L kx kx L Lx Lx sn Lx L sn Lx W hav I x dk [ ] L cf xampl L x x x Lx Ths snc funcon nds o L/ as x and away from hr bcoms small sn x sn Lx W sad arlr ha dx Smlarly x dx x So w s ha akng h lm as L of Ix w hav h proprs w wan n a dla funcon L kx So w can dfn a dla funcon as δ x Lm L dk L Hnc kx δ x dk or k xx δ x x dk W wll us hs dfnon o prov h ourr ngral horm Earlr w sad kx f x k dk whr k kx f x dx Gvn a funcon fx, h scond formula dfns k Wha nds o b provd s ha subsung hs k no h frs formula dos yld h orgnal funcon fx Pu h scond ngral no h frs: kx kx kx f x k dk f x dx dk kx kx k x x f x dx dk f x dx dk f x δ x x dx f x Hnc, usng our dfnon of a dla funcon, w hav provd h horm Phl Lghfoo 8/9 Lcurs 9- - Pag 8 of 8