Fourier Series. Topic 4 Fourier Series. sin. sin. Fourier Series. Fourier Series. Fourier Series. sin. b n. a n. sin

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1 Topic Fourier Series si Fourier Series Music is more th just pitch mplitue it s lso out timre. The richess o sou or ote prouce y musicl istrumet is escrie i terms o sum o umer o istict requecies clle hrmoics. The lowest requecy is clle the umetl requecy the pitch it prouces is use to me the ote. The lce o the mplitues o the ieret hrmoic requecies is resposile or the chrcteristic sou o ech istrumet. Ech istrumet elow is plyig sigle ote mile C Hz Toy Itrouctio to Fourier lysis Sigl geertor Clssicl guitr Pio Wht o you thik is est istrumet to emostrte timre? Bgpipes A sigle ote will coti ieret rctios o ech hrmoic st hrmoic hrmoic r hrmoic th hrmoic Hz Hz 78Hz Hz y B siω t y B siω t y B siω t y B siω t The summe output repets with the perio o the st hrmoic Fourier Series Fourier me the ol clim tht y repetig ptter coul thereore e represete y summe series o ie sie terms with term e to the sum to just the selie i eee. t ωt siωt ω Sie terms Summe hrmoics y totl B siωt B siωt... B siωt ω Cosie terms Fourier si Fourier Series t ωt siωt Fourier si t Fourier Series t t si T T Or i the ptter repets with perio where is istce The ptter will repet with perio o the st hrmoic requecy ω ω so sice ω ω ω T, the ω T we c write si Here perio o the shpe is t t t si T T where T is the perio o the repetig shpe

2 Fourier Series F F si For these cses is tke to e to simpliy epressios F si F si Nee to i, si F F si some ckgrou work require. Itegrtig over Itegrtig over si or ll Useul itegrls Try these itegrls usig the hits provie A B A B A B si Useul Itegrls summry or ll or ll Previous pge m m si { m } { m } si m m si m m m si si m or ll m Previous pge si or ll Previous pge si m or ll m Rememer o eve uctio

3 Fiig coeiciets o the Fourier Series Rememer how the Fourier series c e writte like this or perio si For simplicity let s mke so we c write si Tke this equtio itegrte oth sies over perio Clerly o the RHS the oly o-zero term is the term Hece we i si Fiig coeiciets o the Fourier Series Agi let so we c write si This time multiply oth sies y itegrte over perio Hece we i si O RHS, oly the term survives s it is oly term where Orthogolity Fiig coeiciets o the Fourier Series How to i ll coeiciets is cler. To i geerl epressio multiply oth sies o the Fourier series y m, the itegrte over perio: m m m O the RHS, oly the m term survives the itegrtio m m m m si m Coeiciets o the Fourier Series I similr wy, multiplyig oth sies o the Fourier series y sim, the itegrtig over perio we get: m si m The Fourier series c e writte with perio s The Fourier series coeiciets c e ou y:- si si m m Fiig the coeiciets o the Fourier Series Emple. - pge Fi Fourier series to represet this repet ptter. Step. Write ow the uctio i terms o. Wht is the perio?. Write ow the uctio i terms o. Wht is perio? Perio is Step. Use equtio to i? Step. Use equtio to i? Step. Use equtio to i? si. Use equtio to i? [ ]. Use equtio to i? si

4 Fiig coeiciets o the Fourier Series Fi Fourier series to represet this repet ptter.. Use equtio to i? si si si si Step. Write out vlues o or,,,,,. Step. Write out Fourier series With perio si, i the geeric orm replce with vlues or our emple Fiig coeiciets o the Fourier Series So wht oes the Fourier series look like i we oly use irst ew terms? Use FourierChecker t. si si si si.... Wht is whe? 8. Wht is I? I [ ]. Wht is whe?. Wht is whe?. Wht is whe 7? y y is. Wht is whe?. Wht is whe? 7. Wht is whe? is - - -

5 I 9. Tem A: Wht is? I [ ] y y is is step. Tem B: Descrie the ollowig step uctio i terms o? whe whe > y is is step. Tem A: Wht is I? I y is [ ] [ ] is step perios. Tem B: Descrie the ollowig step uctio over oe perio i terms o? whe > > whe > y is is - - step perios. Tem A: Wht is the itegrl o over oe perio? I [ ] [ ] y is is - -. Tem B: Descrie the ollowig step uctio over oe perio i terms o? whe > > whe > 7 y is is - step perio rise -

6 . Tem A: Wht is the itegrl o over oe perio? I 7 [ ] [ 7] y is is - step perio rise - Fourier series. Tem B: I we were to represet the uctio elow s Fourier series wht coul you sy out the vlue o? is selie shiter. Hl wy etwee 7 is. So o 9 y is si is - step perio perio rise 7 [ ] [7 ] perio Fourier series si Fourier series si 7. Tem A: I we were to represet the uctio elow s Fourier series wht coul you sy out the vlues o the terms? y is is - step perio rise - o uctio so ll terms re zero 8. Tem B: I we were to represet the uctio elow s Fourier series wht coul you sy out the sig o the term? y is is - step perio rise - Fourier series si ecture 7 Fourier Series 8. Tem B: I we were to represet the uctio elow s Fourier series wht coul you sy out the vlue o the term? It woul hve egtive mplitue y is 8 si si - step perio - rise - is - st sie hrmoic umetl - Toy More emples o Fourier series Descriig pulses with Fourier series

7 Fiig coeiciets o the Fourier Series Fiig coeiciets o the Fourier Series Fi Fourier series to represet this repet ptter. Steps to clculte coeiciets o Fourier series. Write ow the uctio i terms o. Wht is perio?. Use equtio to i? ] [ Perio is. Use equtio to i? si. Use equtio to i? si si si si Fiig coeiciets o the Fourier Series Fiig coeiciets o the Fourier Series. Write out vlues o or,,,,,.. Use equtio to i?. Write out Fourier series with perio,, i the geeric orm replce with vlues or our emple si... si si si si So wht oes this Fourier series look like i we oly use irst ew terms? Use Fourier_checker o wesite... si si si Fourier series terms oly or sie ie egrees totl mplitue Fiig coeiciets o the Fourier Series Fiig coeiciets o the Fourier Series Fi Fourier series to represet this repet ptter. Steps to clculte coeiciets o Fourier series. Write ow the uctio i terms o. Wht is perio?. Use equtio to i? et sie - i coeiciets Right sie - i coeiciets Perio is Fiig coeiciets o the Fourier Series Fiig coeiciets o the Fourier Series et sie - i coeiciets Right sie - i coeiciets si si First terms to Attece sheet Coeiciets Perio is vu uv uv v v si si si Itegrte y prts si si u u

8 Perio is Perio is si Coeiciets si Coeiciets Itegrte y prts si si uv uv vu u u v si v si si si si Coeiciets Perio is si si 9 si 9 si si si si... Check our Fourier series usig Fourier_checker.ls si si si si si... 9 Fourier Series o eve o repetig uctios

9 Fourier Series o eve o repetig uctios Fourier Series pplie to pulses or or or > Oly sie terms require to eie o uctio Oly ie terms require to eie eve uctio Oly eve uctio c hve oset. Or spce ser light pulse Iitil isplcemet o guitr strig Electroic wveuctio o molecule C Fourier series e use to represet sigle pulse evet rther th iiitely repetig ptter? Yes. It s ie i you hve sigle pulse occurrig etwee to prete tht it is prt o iiitely repetig ptter. All you hve to o the is work out the Fourier series or the iiite ptter sy tht the pulse is represete y tht uctio just etwee There is etter wy. I the oly coitio is tht the prete uctio e perioic, sice we kow tht eve uctios coti oly ie terms o uctios oly sie terms, why o t we rw it either like this or this? O uctio oly sie terms ecomes Eve uctio oly ie terms ut oly look etwee Wht is perio o repetig ptter ow? This pproch is ie ut it les to lot o work i the itegrtio stge. Hl-rge sie series where Hl-rge sie series We sw erlier tht or uctio with perio the Fourier series is:- si where si si si I the hl rge cse we hve uctio o perio which is o so cotis oly sie terms, so the ormule ecome:- si si si where si Rememer, this is ll to simpliy the Fourier series. We re still oly llowe to look t the uctio etwee ODD ODD ODD EVEN si ODD EVEN EVEN si

10 Hl-rge ie series Fourier Series pplie to pulses Hl-rge sie ie series The Fourier series or pulse such s Agi, or uctio with perio the Fourier series is:- where si Agi we hve uctio o perio ut this time it is eve so cotis oly ie terms, so the ormule ecome:- where si EVEN EVEN EVEN EVEN c e writte s either hl rge sie or ie series. However the series is oly vli etwee Hl rge sie series Hl rge ie series si where where et s o emple to emostrte this si Hl Rge Fourier Series Fi Hl Rge Sie Series which represets the isplcemet, etwee, o the pulse show to the right The pulse is eie s or with legth So si si Itegrte y prts uv uv vu so set u si v v si s u si si si Hl rge sie series Hl Rge Fourier Series Fi Hl Rge Sie Series which represets the isplcemet, etwee, o the pulse show to the right si si si si si si... where C we check this o Fourier_checker.ls si Hl Rge Fourier Series si si si si si... Fourier Series pplie to pulses Why is this useul? I Qutum you hve see tht there eist speciic solutios to the wve equtio withi potetil well suject to the give oury coitios. Ψ whe Ψ Ψ B si Give complicte geerl solutio we c ecovolve ito hrmoic terms

11 ecture 7 Summry Prctice questios olie t Norml series Eve o uctios Pulses Repet perio o other ritrry repet perios ecture 8 Summry Prctice questios olie t Comple Fourier series Prsevl s theorum Revisio & Prctice si si si si Comple Fourier Series I my res o physics, especilly Qutum mechics, it is more coveiet to ier wves writte i their comple orm Rememer:- e i e i si si e i i The comple orm o the Fourier series c e erive y ssumig solutio o the orm i c e the multiplyig oth sies y e im itegrtig over perio: im i im i m e c e e c e i e i For m itegrl vishes. For m itegrl. So c e Comple Fourier Series The comple orm o the Fourier series c e erive y ssumig solutio o the orm i c e the multiplyig oth sies y e im itegrtig over perio: im i im i m e c e e c e i m e m i si m i For m itegrl vishes. For m itegrl. So c e Comple Fourier Series Thereore or perio o the comple Fourier series is give s:- c e i where c e i The more geerl epressio or uctio with perio c e epresse s:- i i c e where c e et s try emple rom the otes.. Comple Fourier Series emple Emple. Fi the comple Fourier series or i the rge - i the repet perio is. i i c e perio is, so we write c e i Itegrtio y prts u v uv v u with u v e c e i C e i i i e i e i i e i e i i So u v e i i i e i i e i e i i i i e i i i i e e e e

12 Comple Fourier Series emple Emple Fi the comple Fourier series or i the rge - i the repet perio is. Prsevl s Theorem pplie to Fourier Series The eergy i virtig strig or electricl sigl is proportiol to the squre o the mplitue o the wve i i i i i Sice the C e e e e C i i i i We wt to i ctul vlues or C so it woul e helpul to covert epressio or C ito sie ie terms usig the str epressios:- So we c write i i i i e e si e e i i i i i i si so C sice i c e i so i e Prsevl s Theorem pplie to Fourier Series Cosier gi the str Fourier series with perio tke or simplicity s. si Squre oth sies the itegrte over perio: [ ] si The RHS will give oth squre terms cross term. Whe we itegrte, ll the cross terms will vish. All the squres o the ies sies itegrte to give hl the perio. Hece:- [ ] [ ] Prctice revisio Try olie questios & 7 Hece Prsevl s theorem tells us tht the totl eergy i virtig system is equl to the sum o the eergies i the iiviul moes. Fourier Series i movies!!!! I music i the umetl requecy o ote is Hz The the hrmoics re t Hz, Hz, Hz, Hz, Hz But the octves re t Hz, Hz, 8 Hz, Hz, Hz I you wte to epli the Fourier series to lie you proly pick otes tht showe you uerstoo this So s greetig why ot try G, A, F, F octve lower, C 9Hz Hz 9Hz 7Hz Hz Why re these otes specil...we ll tke the st hrmoic s F ottom.8hz 8 th hrmoic th hrmoic th octve 8 th hrmoic th hrmoic th hrmoic t totl B si t B si t B si t B si t B si t