Fourier Series and Applications
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1 9/7/9 Fourier Series d Applictios Fuctios epsio is doe to uderstd the better i powers o etc. My iportt probles ivolvig prtil dieretil equtios c be solved provided give uctio c be epressed s iiite su o sies d ies. I this sectio we will see how uctios c be epded hvig discotiuities lso. Applictios re i rottig chies Soud wves hert Bets. These trigooetric series re clled Fourier series d re soewht logous to Tylor series i tht both types o series provide es o epressig coplicted uctios i ters o certi ilir eleetry uctios. odiied by Peeyush Tewri Brod Use o Fourier Series Fourier series is used s es o solvig certi probles i prtil dieretil equtios. However Fourier series hve uch wider pplictio i sciece d egieerig d i geerl re vluble tools i the ivestigtios o periodic pheoe. For eple bsic proble i spectrl lysis is to resolve icoig sigl ito its hroic copoets which outs to tructig ti its Fourier series represettio. ti I soe requecy rges the seprte ters correspod to dieret colors or to dieret udible toes. The gitude o the coeiciet deteries the plitude o ech copoet. odiied by Peeyush Tewri
2 9/7/9 Iportt oruls A t rtio o *9±θ ±se rtio o θ whe is evethe sig or is to be decided ro the qudrt i which the gle *9±θ lies. E: si 57si69 - si-/. A t rtio o *9±θ ±co rtio o rtio o θ whe is odd. The sig or is to be decided ro the qudrt i which the gle *9±θ lies. T5t 95-cot5- ' '' ''' '''' uvd uv u v u v u v u v... v5 Where dsh deotes dieretitio d suies itegrtio w r to. odiied by Peeyush Tewri Iportt Foruls SiCosy Si y Si y CosSiy CosCosy SiSiy Si Cos Cos 5 Si si si it y Si y Cos y Cos si si 5 y y y 7 si... odiied by Peeyush Tewri
3 9/7/9 Iportt Foruls si si si d d d d odiied by Peeyush Tewri 5 si si Iportt Foruls i d d d d i i i si si si odiied by Peeyush Tewri 6 d d si si si
4 9/7/9 Iportt Foruls si d si si si si si d odiied by Peeyush Tewri 7 Fourier Series Represettio o Fuctios We begi with series o the or b si O the set o poits where this series coverges it deies uctio whose vlue t ech poit is the su o the series or tht vlue o. I this cse the series is sid to be the Fourier series o. Our iedite gols re to deterie wht uctios c be represeted s su o Fourier series d to id soe es o coputig the coeiciets i the series correspodig to give uctio. odiied by Peeyush Tewri 8
5 9/7/9 Periodic Fuctios We irst develop properties o si/ d / where is positive iteger. The irst property is their periodic chrcter. A uctio is periodic with period T > i the doi o cotis T wheever it cotis d i T or ll. See the grph s below. odiied by Peeyush Tewri 9 Periodicity o the Sie d Cosie Fuctios For periodic uctio o period T T or ll. Si d Cos re periodic with period /. Also T is lso period d so is y ultiple o T. The sllest vlue o T or which is periodic is clled the udetl period o. I d g re two periodic uctios with coo period T the g d c c g re lso periodic with period T. I prticulr si / d / re periodic with period T /. si si odiied by Peeyush Tewri 5
6 9/7/9 Discotiuities odiied by Peeyush Tewri Orthogolity The stdrd ier product u v o two rel-vlued uctios u d v o the itervl β is deied by u v β u v d The uctios u d v re orthogol o β i their ier product u v is zero: u v β u v d A set o uctios is utully orthogol i ech distict pir o uctios i the set is orthogol. odiied by Peeyush Tewri 6
7 9/7/9 Orthogolity o Sie d Cosie The uctios si / d / or utully orthogol set o uctios o - with - / / d δ - / si / d - si / si / d δ where δ i d δ i These results c be obtied by direct itegrtio; odiied by Peeyush Tewri Fourier Epsio - Suppose the series coverges d cll its su : si b Coeiciets c be oud s ollows. d By orthogolity b d d si d d d odiied by Peeyush Tewri 7
8 9/7/9 Coeiciet Foruls Thus ro the previous slide we hve d K To id the coeiciet we hve d d d b si d Thus the coeiciets re give by d K Siilrly the coeiciets b re give by b si d K odiied by Peeyush Tewri 5 The Euler-Fourier Forul- Thus the coeiciets re give by the equtios d K b si d K which kow s the Euler-Fourier oruls. Note tht these oruls deped oly o the vlues o i the itervl -. Sice ech ter o the Fourier series b si is periodic with period pi the series coverges or ll whe it coverges i - d is deteried or ll by its vlues i -. odiied by Peeyush Tewri 6 8
9 9/7/9 9 Fid the F S. to represet - ro-pi to pi. si b d d odiied by Peeyush Tewri 7 d d si [] s d b si odiied by Peeyush Tewri 8 si
10 9/7/9 Usig coe. Just obtied We get we get b si si odiied by Peeyush Tewri 9 Obti the Fourier epsio o e - i the itervl -. EX As e e e e d e sih sih d e { si } odiied by Peeyush Tewri
11 9/7/9 b b e si d e { si} sih odiied by Peeyush Tewri Hece the F. S. Epsio sih sih sih si sih sih odiied by Peeyush Tewri
12 9/7/9 Eple uctio -<< is eve so ll b d d d d odiied by Peeyush Tewri si si Eple cotiues Hece the Fourier series is give by g y... odiied by Peeyush Tewri...
13 9/7/9 etsdeie S N N S S K S odiied by Peeyush Tewri 5 odiied by Peeyush Tewri 6
14 9/7/9 odiied by Peeyush Tewri 7 Now coe. i Fourier Epsio i - Suppose the series coverges d cll its su : The coeiciets c be oud s ollows. si b d b d d d odiied by Peeyush Tewri 8 By orthogolity d b si d d
15 9/7/9 Coeiciet Foruls Thus ro the previous slide we hve d K To id the coeiciet we hve d d d b si d Thus the coeiciets re give by d d Siilrly the coeiciets b re give by b si d K K odiied by Peeyush Tewri 9 The Euler-Fourier Foruls Thus the coeiciets re give by the equtios d K b si d K which kow s the Euler-Fourier oruls. Note tht these oruls deped oly o the vlues o i the itervl -. Sice ech ter o the Fourier series b si is periodic with period the series coverges or ll whe it coverges i - d is deteried or ll by its vlues i -. odiied by Peeyush Tewri 5
16 9/7/9 Eple : Trigulr Wve o Cosider the uctio below. < < This uctio represets trigulr wve d is periodic with period T. See grph o below. I this cse. Assuig tht hs Fourier series represettio id the coeiciets d b. odiied by Peeyush Tewri Eple : Coeiciets o First we id : d d The or we hve 8/ d d where we hve used itegrtio by prts. Siilrly it c be show tht b odd eve odiied by Peeyush Tewri 6
17 9/7/9 Eple : Fourier Epsio o Thus b d 8/ odd eve The b si 8 5 K 5 8 / / odiied by Peeyush Tewri Eple : Fuctio o Cosider the uctio below. < < < < 6 < < This uctio is periodic with period T 6. I this cse. Assuig tht hs Fourier series represettio id the coeiciets d b. odiied by Peeyush Tewri 7
18 9/7/9 8 Eple : Coeiciets o First we id : Usig the Euler-Fourier oruls we obti d d si si K d odiied by Peeyush Tewri 5 si K d b Eple : Fourier Epsio o Thus b d The i si b K si odiied by Peeyush Tewri 6 K 5 5 si
19 9/7/9 Eple : Trigulr Wve Cosider gi the uctio ro Eple < < s grphed below d its Fourier series represettio 8 / We ow eie speed o covergece by idig the uber o ters eeded so tht the error is less th. or ll. odiied by Peeyush Tewri 7 Eple : Prtil Sus The th prtil su i the Fourier series is 8 s / d c be used to pproite the uctio. The coeiciets diiish s - so the series coverges irly rpidly. This is see below i the grph o s s d. odiied by Peeyush Tewri 8 9
20 9/7/9 Eple : Errors To ivestigte the covergece i ore detil we ider the error uctio e - s. Give below is grph o e 6 o. Note tht the error is gretest t d where the grph o hs corers. Siilr grphs re obtied or other vlues o. odiied by Peeyush Tewri 9 Eple : Uior Boud Sice the iu error occurs t or we obti uior error boud or ech by evlutig e t oe o these poits. For eple e 6.7 d hece e 6 <. o d equetly or ll. odiied by Peeyush Tewri
21 9/7/9 Eple : Speed o Covergece The tble below shows vlues o e or other vlues o d these dt poits re plotted below lso. Fro this iortio we c begi to estite the uber o ters tht re eeded to chieve give level o ccurcy. To gurtee tht e. we eed to choose. e_ odiied by Peeyush Tewri At st Thks. odiied by Peeyush Tewri
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