Experiment 6: Fourier Series

Transcription

1 Fourier Series

2 Experime 6: Fourier Series Theory A Fourier series is ifiie sum of hrmoic fucios (sies d cosies) wih every erm i he series hvig frequecy which is iegrl muliple of some pricipl frequecy d mpliude h vries iversely wih is frequecy. The usefuless of such series is h y periodic fucio f wih period T c be wrie s Fourier series i he followig wy: f ( ) cos( k ) b si( k ) k k k where is he fudmel frequecy of he fucio; h is, 2 T Where T is he fudmel period of he fucio. The Fourier coefficies, he sd b s i he equio, my be compued by he followig se of iegrls: T T f ( ) d k 2 T T f ( )cos( k ) d b k 2 T T f ( )si( k ) d Now h we hve wy o represe he fucio, we c use he Fourier series jus s if i were he fucio iself d ivesige is behvior i elecroic sysems.

3 Why sie d cosie????! Vecors We re ow goig o discuss some formlism of hree-dimesiol vecors expressed i Cresi coordies, for he purpose of mkig comprisos o Fourier series. I Cresi spce, y vecor c be wrie s lier combiio of he muully perpediculr bsis vecors xˆ, yˆ, z ˆ i he followig wy: where he coefficies re give by V xˆ yˆ zˆ x y z V. xˆ, V. yˆ, V. zˆ x y z The bove expressios for he coefficies c be esily derived from he followig perpediculriy (or, more geerlly, orhogoliy) relios: xˆ. yˆ yˆ. zˆ zˆ. xˆ xˆ. xˆ yˆ. yˆ zˆ. zˆ Ay vecor c be decomposed io se of ppropriely weighed orhoorml bsis vecors Exmple: rˆ xˆ yˆ x x y??,?? y

4 Perfume ier produc wih he bsis vecors rˆ xˆ yˆ x rˆ xˆ, rˆ yˆ x x y y y The orhogoliy codiios simply express h he bsis vecors xˆ, yˆ, zˆ re lierly idepede. Alhough his secio bou vecors is elemery d my pper uecessry, we will see presely ( d you my hve figured i ou by ow) here re similriies bewee his vecor formlism d h of he Fourier series Compriso bewee Vecors d Fourier Series The wo previous secios were wrie suggesively, o mke comprisos bewee he formlism for he Fourier series d for vecors. The similriies bewee he wo c provide us wih some isigh bou Fourier series(for hose wih kowledge of lier lgebr,hese similriies rise sice we c cree ier produc sources for boh hree-dimesio vecors d for periodic fucios of give period).is should be cler h he hrmoic fucios mkig up periodic fucio re logous o he ui vecors mkig up vecor,d he coefficies, b i Fourier series re logous o he compoes Vi of vecor: cos,si xˆ, yˆ, zˆ, b,, x y z So, he hrmoic fucios re he elemes h go io mkig ceri periodic fucio (hey will be he sme for ll fucios wih he sme period), d he coefficies re he mou of ech hrmoic we eed o mke he priculr fucio. This wy of hikig bou Fourier series is exremely powerful d will serve you well if you ler i ow. So, if you hve y doubs h you fully udersd he ide, rered he previous secio d lk bou i wih ohers uil you do udersd.

5 Compre Fourier Series o vecor decomposiio: Vecor Decomp. Fourier Series B si cos A C D

6 LAB work : b w 2 si x ( ).5 cos cos3 cos5 cos =lispce(-2*pi,2*pi,); x=squre(+pi/2,5) plo(,x) xis([-2*pi,2*pi,-.2,.2]); grid o sumerms=zeros(6,legh()); sumerms(,:)=.5; for =:size(sumerms,)- sumerms(+,:)=(2/(pi*)*si(pi*/2))*cos(*); ed x_n=cumsum(sumerms); id=; for N=[,:2:size(sumerms,)-] id=id+; subplo(3,3,id) plo(,x_n(n+,:),'k',,x,'r--') xis([-2*pi,2*pi,-.2,.2]); xlbel('') ylbel(['x_{',um2sr(n),'}()']) ed

7 x ().5 x ().5 x 3 () x 5 ().5 x 7 ().5 x 9 () x ().5 x 3 ().5 x 5 () The compc rigoomeric Fourier series f C C cos( w ) d so he coefficies give by o o C is he mpliude or coefficie is he hrmoic θ is he phse w is he rdi frequecy T he period C b,,,, C 2 2 o o b FREQUENCY SPECTRA Ampliude specrum - The plo of mpliude C versus he (rdi) frequecy? (use he fucio C b 2 2 Phse specrum - The plo of he phse b versus he (rdi) frequecy?. These wo plos covey ll of he iformio h he plo of f() s fucio of does. The frequecy specr of sigl cosiue he frequecy-domi descripio of f() wheres f() s fucio of is he ime-domi descripio Bdwidh The differece (i rdis) of he highes d lowes frequecies i he phse specrum is hebdwidh.

8 Time Vs Frequecy Domi

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10 Fourier Series wih GUI Gibbs Pheome Explio We begi his discussio by kig sigl wih fiie umber of discoiuiies (like squre pulse) d fidig is Fourier Series represeio. We he emp o recosruc i from hese Fourier coefficies. Wh we fid is h he more coefficies we use, he more he sigl begis o resemble he origil. However, roud he discoiuiies, we observe ripplig h does o seem o subside. As we cosider eve more coefficies, we oice h he ripples rrow, bu do o shore. As we pproch ifiie umber of coefficies, his ripplig sill does o go wy. This is whe we pply he ide of lmos everywhere. While hese ripples remi (ever droppig below 9% of he pulse heigh), he re iside hem eds o zero, meig h he eergy of his ripple goes o zero. This mes h heir widh is pprochig zero d we c sser h he recosrucio is excly he origil excep he pois of discoiuiy. Below we will use he squre wve, log wih is Fourier Series represeio, d show severl figures h revel his pheomeo more mhemiclly.

11 Exmple Figure shows severl Fourier series pproximios of he squre wve usig vried umber of erms, deoed by K: Whe comprig he squre wve o is Fourier series represeio i Figure, i is o cler h he wo re equl. The fc h he squre wve's Fourier series requires more erms for give represeio ccurcy is o impor. However, close ispecio of Figure does revel poeil issue: Does he Fourier series relly equl he squre wve ll vlues of? I priculr, ech sep-chge i he squre wve, he Fourier series exhibis pek followed by rpid oscillios. As more erms re dded o he series, he oscillios seem o become more rpid d smller, bu he peks re o decresig. Cosider his mhemicl quesio iuiively: C discoiuous fucio, like he squre wve, be expressed s sum, eve ifiie oe, of coiuous oes? Oe should les be suspicious, d i fc, i c' be hus

12 expressed. This issue brough Fourier much criicism from he Frech Acdemy of Sciece (Lplce, Legedre, d Lgrge comprised he review commiee) for severl yers fer is preseio o 87. I ws o resolved for lso ceury, d is resoluio is ieresig d impor o udersd from prcicl viewpoi. The exreous peks i he squre wve's Fourier series ever dispper; hey re ermed Gibb's pheomeo fer he Americ physicis Josih Willrd Gibbs. They occur wheever he sigl is discoiuous, d will lwys be prese wheever he sigl hs jumps.

13 Exercise -Compue d plo he Fourier coefficies (he specrum) for he followig periodic sigl. 2-Use Fourier Series Demo i demosre h he rigle wve eed less umber of Fourier coefficie h he squre wve.