In this section we will study periodic signals in terms of their frequency f t is said to be periodic if (4.1)

Transcription

1 Fourier Series Iroducio I his secio we will sudy periodic sigals i ers o heir requecy is said o be periodic i coe Reid ha a sigal ( ) ( ) ( ) () or every, where is a uber Fro his deiiio i ollows ha ( ) ( ) () or every ieger, posiive or egaive For exaple i he ( ) (( ) ) ( ) ( ) Siilarly, or he equaio ( ) ( ' ) ( ' ) ( ) holds he salles posiive uber or which () holds is called a period Le us ider he well kow siusoidal ucio Asi ( φ) where A is he apliude, is he agular requecy, ad φ is he phase Sice ( ( ) φ) A ( φ ) Asi si

2 6 ad is he salles agle ha saisies he equaio he ( α ) si α si Le us ider wo periodic sigals ( ) ad ( ) ( ) ad ( ) respecively he coo period o posiive uber saisyig () ( ) ( ) ad ( ) ( ) wih periods ad,, i i exiss, is he salles or every Observe ha ay, such ha ( ) ( ) ( ) ( or all, ca be expressed i he or where is a ieger Likewise ay, such ha ) or all, ca be expressed i he or, where is a ieger Hece, he codiio uder which he ucios ( ) ad ( ) have a coo period is he above equaio saes ha he raio us be a raioal uber he coo period is he lowes coo uliple (over he posiive iegers ield) o ad I he raio is a irraioal uber, he () ad () do o have a coo period I such a case iegers ad such ha do o exis Le h ( x) be a arbirary ucio I ( ) is a periodic sigal wih period he he ucio deied by he equaio ( ) h( ()) g

3 7 is periodic because ( ) h( )) h( ( ) ) g( ) g ( () Hece, he period o g is or less ha (see Exaple ) Exaple Le us ider he siusoidal sigals he periods o hese sigals are: ( ) ( 6 ) 5 ( ) si, Cosequely, we obai hus, he raio is a raioal uber hereore, he wo sigals have a coo period give by Now we ider he sigals: he periods o hese sigals are: ( ) ( 5 ) g 5 ( ) si g, ad he raio

4 8 is a irraioal uber Hece, he sigals do o have ay coo period Exaple Le us ider he periodic sigal wih period ad le h ( x) x he ucio () h( ()) g is ( ) ( ) g I is a periodic ucio wih period (see Fig) (a) () (b) g() Fig Plos o sigals ( ) ad g( ) hus, he period o g () is oly hal ha o ( )

5 9 Fourier series Deiiio We ider a periodic sigal ( ) wih he period I soe codiios, () as a which will be discussed laer, are saisied, we ca express he ucio series o sie ad ie ucios or () a a a b si b si si () a ( a b ) () where Expressio o he righ had side o () is called he rigooeric Fourier series he su is called Fourier series ad is ers are called he haroics he -h haroic is he er We label w () a b si a b a a b a b b si c a b (5) θ a a b b siθ (6) a b obaiig w () c ( θ siθ si ) c ( θ ) (7) hus, he -h haroic is a siusoidal ucio wih he apliude θ ad he agular requecy he irs haroic c, he phase

6 w () a b si c ( θ ) is called he udaeal ad is requecy is called he udaeal requecy he a er Fourier series a is he -h haroic Hece, a equivale or o he () a c ( θ ) c ( θ ) c c ( θ ) (8) wih c a ollows he plo o as a ucio o is called he apliude specru ad he plo c θ ( ) o as a ucio o is called he phase specru o he sigal hey are called he requecy specra Exaple Le us ider a periodic sigal ( ) icludig he ollowig haroics: rad where s c w 5 ( ) ( ) ( ) ( 5 ) ( ) 5 w w5 5 Figure shows hese haroics ad illusraes how he sigal ( ) ro is haroics ogeher is buil up (a) c

7 (b) w () (c) w () (d) w 5 () (e) () Fig Cosrucio o he sigal ( ) isig o our haroics

8 Evaluaio o Fourier series coeicies Our obecive is o evaluae he a ad b coeicies i he Fourier series () he coeicies are called he Fourier coeicies o he sigal I order o deerie a Le us ider he iegral ( ) we iegrae boh sides o equaio () ro o () d ad ad b d si d (9) he ucio whereas has he period Hece, holds Sice iegral o over is period is zero he d d d () Siilarly, we have si d ()

9 Subsiuig () ad () io (9) we obai or () d ad a a () d () Forula () ca be odiied by addig a arbirary value upper lii o iegraio o he lower ad a () d () Equaio () shows ha a is he average value o ( ) over he period o copue we uliply boh sides o he Fourier series (9) by a ad he iegrae ro o obaiig () d a d ( a b si ) d () As explaied above, he irs er o he righ had side o () is zero o rearrage he oher ers we use he ollowig orulas: si d (5) i d (6) i Hece, we have () d a d a

10 or a () d () d,, (7) I a siilar aer we id b () si d () si d,, (8) Exaple Le us ider he periodic sigal wih period havig he waveor show i Fig () - Fig Sigal ( ) or Exaple We wish o express his sigal by a Fourier series he waveor o ( ) described as ollows is o id a, we apply () () or or < < < < a d d Nex, we calculae he Fourier coeicies a give by ():

11 5 si si d d a Sice we obai,, a o deerie we use (8): b ( ) ( ) b d si d si d si d si Hece, i ollows,, b odd is eve is i i 6 hus, he coeicies are b, b, b, b, b, b, b, b ad he Fourier series has he or () 7 si 7 6 si5 5 6 si si 6

12 6 Sice c a b b ad or odd: a b θ siθ a b a b he θ or odd he apliude ad phase specra o he sigal are show i Fig (a) 5 c (b) θ Fig Specra o he sigal or Exaple Properies o Fourier series I his secio we will discuss soe waveor properies o periodic ucios

13 7 A ucio ( ) is said o be eve i, or every, i saisies he codiio ( ) ( ) Exaples o he eve ucios are:, si, si Noe ha i a ucio is eve, is graph is syeric i he verical axis (see Fig5) () A ucio Fig 5 A eve ucio ( ) is said o be odd i, or every, i saisies he codiio ( ) ( ) Exaples o odd ucios: si si Noe ha i a ucio is odd, is graph is syeric i he origi (see Fig6) () Fig 6 A odd ucio

14 8 () A periodic sigal wih period is said o possess odd hal-wave syery i, or every, i saisies he codiio Fucio () si has his propery; aoher exaple is show i Fig7 () Eve ucios () Fig 7 A ucio possessig odd hal-wave syery Le be a periodic eve ucio he or ay he equaio holds his equaio iplies () d () d a () d () d (9) I a ucio is eve, he is Fourier series does o coai he ers

15 9 b si,, because hey violae he relaio ( ) ( ) ( ) b b si si As a aer o ac he expressio or a ca be reduced as ollows: Sice boh () a ad d () d () are eve, he Cosequely, we have ( ) ( ) ( ) a () d () d,, () Suarizig, we sae ha he Fourier series o a eve periodic sigal coais oly ad ie ers a Odd ucios () For a odd ucio ad ay he equaio () d holds; hece, we coclude ha a I a ucio is odd he is Fourier series does o coai he ers a ad a,, because hey violae he relaio ( ) ( )

16 Siilarly as previously, i ca be proved ha he orula or sipliied ad has he or b ca be b () si d () hus, he Fourier series o a odd periodic sigal coais oly sie ers Odd hal-wave syery ucios I he case o odd hal-wave syery he a er haroics cao exis because hey violae he codiio a ad all eve (), ie a a b or eve he coeicies a ad b or odd are give by: a () d () b () si d () he above orulas ca be derived usig he ollowig relaioships or odd: () ( ) () si () si( ) () si Usig hese expressios we ca reduce he ierval o iegraio wice ad obai orulas () ad ()

17 he advaages o he syery properies o periodic ucios are as ollows: (i) We kow i advace wha coeicies will be zero; equely, we do o wase ie o copue he (ii) he oher coeicies ca be obaied usig sipliied expressios 5 Expoeial Fourier series We ider he rigooeric Fourier series si () a ( a b ) () ad subsiue: ( e ) e ( e ) si e As a resul, we obai a b () a ( e e ) ( e e ) a a b a b e e (5) Le a (6) ( a b ) (7) he we have ( a b ) Sice a ad b b (see (7) ad (8)) a

18 ( a b ) (8) Hece, we coclude ha ad or a pair o coplex cougae ubers Iserig (6), (7) ad (8) io (5) yields () c~ e e (9) he hird er o he righ had side o (9) ca be rearraged as ollows e e Usig () we rewrie (9) i he copac or () () e c~ () called he expoeial Fourier series he coeicies are give by (7) where a ad b are speciied by (7) ad (8) repeaed below: Hece, i ollows a b () () d si d () d () ()( si ) d si d

19 Usig Euler s orula α e α siα, we obai () e d,, () Usually we se or, he is () e d () e d,, () he coeicies are geerally coplex ad ca be expressed i he polar or e φ,, () Usig (8) we obai he relaioships: c~ φ φ ad he plo o versus is called he apliude specru ad he plo o versus is called he phase specru Noe ha or φ Sice () d a (5) b a b a a a b e he, usig (5) ad (6), we ca express i ers o c ad c θ e θ (6) Assuig a arbirary ieger ˆ i he expoeial Fourier series, we prove ha he su o wo ers correspodig o ˆ ad ˆ gives he -h haroic:

20 e ˆ ˆ ˆ e ˆ c~ e ˆ ˆ c~ ˆ ( ) ( ˆ e Re e ) ( ˆ θˆ ) Re cˆ e cˆ ( ˆ θˆ ) o derive he above relaio, equaios (8) ad (6) have bee applied Exaple 5 Le us ider a periodic ucio ( ) equaio si ˆ () A or ˆ, show i Fig8, speciied by he where For his ucio we id he coeicies () A Fig 8 Periodic ucio or Exaple 5 o deerie we use (7) Sice ( ) is eve he b ad orula (7) reduces o a (7) where a is speciied by () repeaed below or coveiece () d,, a

21 5 Hece, we have,, A a d si ad apply he rigooeric ideiy ( ) ) si( ) si( si β α β α β α idig A A A a d ) si( ) si( Sice ad, he A A a

22 6 Nex we use (7) ad calculae: A c~ a a A 5 A a 5 o id a we apply equaio () Exaple 6 A A a () d d si A Le us ider a recagular-wave sigal ( ) idicaed i Fig9 () - Fig 9 Recagular-wave sigal ( ) or Exaple 6 o id he coeicies, we apply () repeaed below () e d where

23 7 () < < < < or or ad Hece, we have e e d e d e Usig relaioship we obai aer siple rearragees ( ),,, e e ± ± (8) Noe ha he sigal is odd ad has odd hal-wave syery propery; hece, as well as () a a ad or every b or a eve Cosequely or a eve akig io accou he above saees ad relaioships (8) ad (8) we obai: c~ c~ e 5 5 e e e 5 5 e e

24 8 hus, he expoeial Fourier series expasio o ( ) is 5 5 () e e e e e e 5 5 he apliude ad phase specra are idicaed i Figs ad c~ Fig Apliude specru o he sigal or Exaple 6 φ Fig Phase specru o he sigal or Exaple 6 Sice, he apliude specru is syeric i he verical axis Siilarly, sice φ φ, he phase specru is syeric i he origi 6 Covergece o Fourier series No every periodic sigal ca be represeed by a Fourier series ad he quesio arises uder wha codiios he Fourier series exiss his quesio is aswered by he Dirichle codiios as ollows:

25 9 (i) Over ay period () us be absoluely iegrable, ie () d < () ( ) (ii) I ay period is o bouded variaio, ie has a os a iie uber o axia ad iia (iii) I ay period () has a os a iie uber o discoiuiies ad each o hese discoiuiies is iie Uder hese codiios he Fourier series o ( ) coverges o ( ) where () is coiuous ad he series coverges o [ ( ) ( )] i i a all pois a ay poi i o discoiuiy ) Noe ha he codiio ( i guaraees ha each coeicie is iie, because () e d () e d () d < A periodic ucio ha violaes his codiio is depiced i Fig () - Fig Fucio ha violaes he irs Dirichle codiio A exaple o a ucio ha does o saisy codiio (ii) is

26 wih as depiced i Fig e si () < () Fig Fucio ha violaes he secod Dirichle codiio A exaple o a ucio ha violaes codiio (iii) is idicaed i Fig () Fig Fucio ha violaes he hird Dirichle codiio Dirichle codiios do o require ha a ucio ( ) be coiuous i order o possess a Fourier expasio hus, is waveor ay iss o a uber o disoied arcs as depiced i Fig5

27 () c b a Fig 5 Sigal wih discoiuiy pois he sigal show i Fig5 has wo pois o discoiuiy a ad over he period Seig io he Fourier series o his sigal we obai Siilarly, seig we obai ( a c) b I ca be show ha Fourier coeicies ed o zero as approaches iiiy hereore he Fourier series ca be rucaed aer a iie uber o ers he rucaed series icludig N ers will be called he parial su ad labeled s N () Le us ider a periodic sigal ( ) havig he waveor depiced i Fig6 his sigal saisies he Dirichle codiios () Fig 6 Recagular pulse rai sigal

28 A ± he pois o discoiuiy occur Figure 7 shows he parial sus s N () o his sigal or N 7, N, N 5 ad N 5 Noe ha he parial sus exhibi ripples ad oscillaios Furherore, he peak value o hese ripples (a) s 7 () (b) s ()

29 (c) s 5 () (d) s 5 () Fig 7 A illusraio o he Gibbs pheoeo i he eighborhood o he discoiuiy pois does o decrease as N icreases ad reais approxiaely 9% o he heigh o he sigal or ay iie N hus, he deviaios i he viciiy o he discoiuiies are raher large his behavior o is kow as he Gibbs pheoeo s N ()