Department of Electronics & Telecommunication Engineering C.V.Raman College of Engineering

Transcription

1 Lcur No Lcur-6-9 Ar rdig his lsso, you will lr ou Fourir sris xpsio rigoomric d xpoil Propris o Fourir Sris Rspos o lir sysm Normlizd powr i Fourir xpsio Powr spcrl dsiy Ec o rsr ucio o PSD. FOURIER SERIES EXPANSION: Frch Mhmici J.B.J. Fourir oud h y rirry priodic sigl c rprsd wih iii sris o siusoids wih udml rqucy d hrmoiclly rld rqucy ω =udml rqucy, ω = h hrmoic rqucy, whr =,, 3, 4 N. Fourir lysis is usd o lysis h sdy s rspos o work d rqucy lysis o sigls. Priodic Fucio: A ucio is sid o priodic wih im priod i i sisis h rlio ± =. A umrs o such priodic sigls r show i h ig. low. hus priodic ucio rps isl r ry scods. F F Fig. Fig. hr r wo orms o Fourir sris i rigoomric Fourir sris Expoil Fourir Sris Lcur 6-9 Sigl & Spcr Pg

2 rigoomric Fourir sris: h rigoomric Fourir sris or rirry priodic ucio is gi y [ cos si ] Whr s d s r kow s h Fourir sris coicis. Ky roos Formul si or ll ' ' os m, wh m Si. os m or ll m, Si. Si m or m os. os m or m Si. or os. Eluio o Fourir sris co-ici: his iols wo oprios i Eluio o coicis, d rucig o iii sris r ii umr o rms so h is rprsd wihi llowl rror. Lcur 6-9 Sigl & Spcr Pg

3 Eluio o : [ cos si] Igrig oh sids wihi o im priod. [ cos si ]. NB: Igrio o siusoidl ucio wihi limi -, whr is h im priod o h gi ucio is zro..... is kow s h rg lu o h ucio or D.. lu o h ucio. Eluio o Muliplyig cos i Equio d igrig oh sids or priod -, w will g [ cos si os. os os ] os os os. os o cos.cos 3 si.cos Eluio o Muliplyig Si i Equio d igrig oh sids or priod -, w will g Lcur 6-9 Sigl & Spcr Pg 3

4 [ cos si Si. Si Si ] Si Si Si yps o Symmry:. Si o 4 cos si si si i iii i E symmry Odd symmry Hl w symmry E symmry: I ucio sisis h rlio =- i is sid o symmry Exmpls: os θ is symmry ucio Odd symmry: I ucio sisis h rlio =-- h i is odd symmry ucio. Lcur 6-9 Sigl & Spcr Pg 4

5 Exmpls: - - Si θ is odd symmry ucio iii Hl w symmry: I ucio sisis h rlio =-+, h i is hl w symmry ucio Exmpl: h mpliud is k s A k = =4=A 4+=34=-A Similrly -=-4=-A - - So Si ucios r hl w symmry Fig: Exmpl os ohr hl w symmry Eluio o Fourir oicis i Symmry odiios i E symmry: Eluio o : Rplc y i h irs rm is symmry ucio, so =- Lcur 6-9 Sigl & Spcr Pg 5

6 Lcur 6-9 Sigl & Spcr Pg 6 5 Eluio o : cos cos cos cos w w w w Rplc wih i h irs rm cos cos w w is symmry ucio, so =- 6 cos 4 cos cos w w w Eluio o : si si si si w w w w Rplc wih i h irs rm si si w w is symmry ucio, so =- 7 si si w w Odd symmry: Eluio o : Rplc y i h irs rm

7 Lcur 6-9 Sigl & Spcr Pg 7 is odd symmry ucio, so =-- 8 Eluio o : cos cos cos cos w w w w Rplc wih i h irs rm cos cos w w is odd symmry ucio, so =-- 9 cos cos w w Eluio o : si si si si w w w w Rplc wih i h irs rm si si w w is odd symmry ucio, so =-- si 4 si si w w w

8 Lcur 6-9 Sigl & Spcr Pg 8 iii Hl w symmry: Eluio o : Rplc + y λ i h irs rm ;=λ- d =dλ d is hl w symmry ucio, so λ-=-λ Eluio o : cos cos cos cos w w w w Rplc + y λ i h irs rm ;=λ- d =dλ cos cos w d w is hl w symmry ucio, so λ-=-λ cos cos w d w w cos cos w d w odd wh w wh w w cos cos cos

9 or So 4 cos w or odd Eluio o : si w si w si w si w Rplc + y λ i h irs rm ;=λ- d =dλ si w d si w is hl w symmry ucio, so λ-=-λ si w w d si w si w d si w si w wh si w si w wh odd or So 4 3 si w or odd h Fourir sris xpsio or ucio hig hl w symmry cois oly h odd hrmoics rqucis. SUMMARY: i E Symmry = 4 cos w Odd Symmry Lcur 6-9 Sigl & Spcr Pg 9

10 iii 4 si w Hl w symmry, 4 4 si w or cos w or odd or odd I y ucio cois i E symmry d hl w symmry or cos w cos w or odd or odd Odd symmry d hl w symmry or si w si w Expoil Fourir sris: or odd or odd L is priodic sigl h ccordig o Fourir sris cos wo si w Lcur 6-9 Sigl & Spcr Pg

11 Lcur 6-9 Sigl & Spcr Pg jw jw jw jw jw jw j j j L j, d j Whr * is h complx cojug o, d =. h jw jw jw Whr is h xpoil Fourir sris oicis. jw w j w w j w j ] si [cos si cos W c rprs s cos Whr, d Φ r rld o,, s is lso kow s spcrl mpliud i. is h mpliud o h spcrl compo cos rqucy. Fourir sris Frqucy Spcrum: h plo o h mpliud o rqucy compo s h rqucy kow s h discr rqucy spcrum or li spcrum. h rqucy spcrum cosiss o discr lis. h lgh o h li rprss h mpliud o h corrspodig rqucy compo Phs Spcrum: h plo o h phs o h rqucy compo s h rqucy is kow s phs spcrum. Phs spcrum is odd ucio.

12 Smplig Fucio: h smplig ucio is did s is show i h ollowig igur. si x S x. h ucio x -π π π. FOURIER SERIES PROPOIES: I h Fourir sris rprsio o gi s propris r sisid y h sigl i im shi: Fourir sris rprsio o +τ, is iii i j j ' j Whr ' j im irsio: Fourir sris rprsio o - is gi y j j h h ollowig i.. h mgiud o rmis cos, phs is shid y 8. I rigoomric rprsio rmis cos u coms gi. im sclig: j j ' Whr =. i.. rmis cos u shis o w rqucy. I h sigl is comprssd i im domi > i is xpdd i rqucy domi; d i i is xpdd i im domi < h comprssd i rqucy domi. im drii: d j d j j j ' Lcur 6-9 Sigl & Spcr Pg

13 Whr =jπ Igrio: j j j j ' Whr j ' 3. APPLIAIONS OF FOURIER SERIES EXPANSION: i Rspos o Lir Sysm: Wh siusoidl xciio is pplid o lir sysm h rspos o h sysm is similrly siusoidl, i.., siusoidl worm prsrs h w shp. h rlioship o h rspos o h xciio is chrcrizd y h rlio o ipu - oupu mpliud d phs.l h ipu o h lir sysm h spcrl compo, w i j jw 3i- Filr i,w Hw V o,w h oupu o,w is rld o h ipu i,w y complx rsr ucio H j w w H w 3i- h oupu is j w jw j[ w w ], w H w i, w H w H w 3i-3 h physicl ipu ip is h sum o h spcrl compo d is complx cojug. i.. ip, w jw jw jw * jw jw R 3i-4 h corrspodig physicl oupu is op, w H w jw * H w jw 3i-5 Sic h oupu is + d rl h wo rms i 3i-5 mus complx cojug. Hc Hw =H * -w So Hw = H-w d θw =- θ-w. i.. Hw is ucio d θw is odd ucio. Hc h oupu o h sysm c xprssd s Lcur 6-9 Sigl & Spcr Pg 3

14 Lcur 6-9 Sigl & Spcr Pg 4 cos j w w H H w H Normlizd Powr i Fourir Expsio: osidr wo rms o h Fourir sris xpsio Fudml d h irs hrmoics 4 cos cos ' h ormlizd powr S o is ] ' [ ' S By xsio h ormlizd powr ssocid wih h ir Fourir sris is S N.B: h powr d ormlizd powr r ssocid wih h rl worms o wih h complx worms. For xpoil Fourir sris h ormlizd powr is du o h produc rms * j j ol ormlizd powr is S * I complx rprsio, h powr ssocid wih priculr rqucy = is o ssocid wih h spcrl compo d, rhr h comiio o h spcrl compo hus h powr is * * = *

15 V - V S V - V V A wo sidd powr spcrum iii Powr Spcrl Dsiy PSD: S h sum S o h ormlizd powr i ll spcrl compos rom =- o Normlizd powr ds h rqucy i rg d is ds ds d d dsd is clld h ormlizd powr spcrl dsiy G. h powr i h rg d is Gd. h powr i h rg - is S G d Ad powr i h rg o - is S G d Lcur 6-9 Sigl & Spcr Pg 5

16 h quiis i o wo quios h o physicl sigiicc u h ol powrs i h rl rqucy rg - h physicl sigiicc, d h powr is gi s S G d G d o id h powr spcrl dsiy, diri S. Bu i w h hrmoics G=. So hrmoics G gis impuls o srgh qul o h jump i S. Hc G i Ec o rsr ucio o PSD: L i is h ipu o ilr hig psd G i. I i is h spcrl mpliud o h ipu sigl h i i G Whr i i j L h oupu is o hig spcrl mpliud o, h h corrspodig psd is o G Ad o o j I H is h rsr ucio o h ilr h h ipu d oupu spcrl mpliuds r rld s o =H i ; Hc o = H i Susiuig i h quio or G o w h G =G i H. Assigms:. Fid h Fourir sris xpsio or h ollowig w orms i - Lcur 6-9 Sigl & Spcr Pg 6

17 iii A - - -A i Drmi h Fourir xpsio or h ollowig sigls. x or. x cos cos. 5 c. cos x d. cos cos x 3. Show h or rl x - ull w rciir oupu hl w rciir oupu Lcur 6-9 Sigl & Spcr Pg 7

18 x cos o, d x si Whr x d x o do h d odd prs o x x x x x x d xo 4. L x d y wo priodic sigls wih priod, d x d y dos h Fourir sris coicis o hs wo sigls. Show h * x y x y * 5. Show h or ll priodic physicl sigl h h ii powr, h coicis o h Fourir sris xpsio x d o zro s. 6. A priodic rigulr worm is did y or d ±= lcul h rcio o h ormlizd powr o his worm which is coid i is irs hr hrmoics. 7. Fid G or h ollowig olgs. A impuls ri o srgh I d priod. A puls ri o mpliud A, durio τ=ia, d priod 8. Plo G or olg sourc rprsd y impuls ri o srgh I d priod or =,,, iiiy. omm o his limiig rsul 9. G i is h powr spcrl dsiy o squr w olg o pk-o-pk mpliud d priod. h squr-w is ilrd y low-pss R ilr wih 3dB rqucy. h oupu is k cross h cpcior. lcul G i. Fid G o. A symmricl squr-w o zro m lu, pk-o-pk olg ol, d priod sc is pplid o idl low-pss ilr. h ilr hs rsr ucio H = i h rqucy rg Hz, d H= lswhr. Plo h powr spcrl dsiy o h ilr oupu Wh is h ormlizd powr o h ipu squr w? Wh is h ormlizd powr o h ilr oupu? Lcur 6-9 Sigl & Spcr Pg 8