Chapter 3 Fourier Series Representation of Periodic Signals
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1 Chptr Fourir Sris Rprsttio of Priodic Sigls If ritrry sigl x(t or x[] is xprssd s lir comitio of som sic sigls th rspos of LI systm coms th sum of th idividul rsposs of thos sic sigls Such sic sigl must: cpl of rprstig lrg umr of sigls hv systm rsposs simpl ough for computtiol covic Complx xpotils giv low r such sic sigls/fuctios: st for C systms z for D systms whr s z r complx umrs Both hv th proprty tht: Rspos to LI systm hs th sm form s th iput with chg i th mplitud oly A fuctio with this proprty is clld Eig fuctio d th corrspodig mplitud rtio is clld Eig vlu st H(s st z H(s H[z] x(t y(t x[] H[z] z y[]
2 st d z Eig fuctios H(s d H[z] Eig vlus Cosidr iput x(t st i C systm h corrspodig output: y(t x(t h(t st H(s Similrly for D systm: Iput: x[] z Output: h( τ h( τ z sτ sτ h( τ x(t τ dτ dτ H(s dτ y[] x[] h[] h[]z st H[z] z h( τ whr ( h[]x[ ] s(tτ dτ h[]z
3 whr H[z] h[] z ( C systms: LI systm hus with iput x(t th output : s t y(t H(s H(s H(s s t I grl: If x(t ( th ( y(t D systms: If x[] z (5 s t th (6 y[] H(s st H[z s t If th iput to LI systm is lir comitio of complx xpotils its output is lso th lir comitio of th sm xpotils ] z s t s t s t
4 I Fourir sris such complx xpotils r: s for C systms z for D systms Fourir Sris of C Priodic Sigls: A sigl x(t with fudmtl frqucy of is xprssd s lir comitio of complx xpotil d t its hrmoics ± t ± t s: x(t t fudmtl frqucy scod hrmoics third hrmoics so o (7 RHS Is clld th Fourir sris d is clld th -th hrmoic compot Fourir Sris Cofficit : From q (7 w my writ: x(t t t t ( t
5 Itgrtig ovr itrvl to ( is th fudmtl priod: t ( t ( t x(t dt dt ( dt ( Usig Eulr s formul: ( t dt cos( t dt si( for for t dt hus ( t dt for for (9 From q (: x(t t dt 5
6 or t x(t dt ( As x(t is priodic q (9 is pplicl to y tim itrvl Hc t x(t dt ( Eq (7 is clld Fourir sythsis qutio d q ( s Fourir lysis qutio If x(t is rl priodic fuctio th x( t x ( t cosqutly x(t t Comprig q (7 with q (: t x (t t ( t 6
7 Prolm p 5 of txt Fid fudmtl frqucy d Fourir co-fficit of: 5 x(t cos ( t si ( t Solutio: Usig Eulr s formul: x(t t t 5 t 5 t - t Hc th fudmtl frqucy is / It cosists of th fudmtl d d 5 th hrmoics olyhus x(t Prolm p 56 of txt d Fid th Fourir sris of: t - t 5 5 5t -5 x(t 5 5t givig - - t
8 Solutio: x(t t Hr (/ x(t [ dt (cos ( Prolm 5 p 9 of txt [Vry importt rsult]: t dt [ dt] δ(t t [ dt ] ] δ(t [ ( - t ] dt] Fid th Fourir sris of: x(t Solutio: - -/ - / t t dt t si( si( [ ]
9 Spcil cs: Lim Covrgc of th Fourir Sris: For fuctio to hv Fourir sris th sris is to covrgt h covrgc coditios for x(t is giv y Dirichlt coditios s follows: Ovr y priod x(t: i must solutly itgrl ovr y priod i Exmpl: x( t dt < x(t h fuctio hs discotiuitis d hc ot covrgt ii dos ot hv mor th fiit umr of discotiuitis ovr y priod t iii discotiuitis my hv t most fiit umr of fiit 9
10 Gis Phomo: As th umr of trms is icrsd Fourir sris rprsts th fuctio mor ccurtly At discotiuitis th sris hs vlu ½ th sum of vlus ust for d ust ftr th discotiuity h sris shows rippls t discotiuity As icrss th rippls gt comprssd towrd th discotiuity ut its p mplitud rmid uchgd Proprtis of Fourir Sris: Lirity: If x(t im Shift: h y(t A x(t B y(t x(t x(t t A B t
11 im Rvrsl: x(t x( t im Sclig: x(t x( αt Multiplictio: ( α t x(t y(t x(ty(t Cougt: x(t x (t l l l Cougt Symmtry: For rl x(t:
12 Drivtiv: dx(t dt Prsvl s Rltio: x(t dt Cosult l p 6 of txt Go through xmpl p of txt Exmpls: Pro 6 p5 of txt Which of th followig r rl vlud? Which r v? x (t x (t x (t ( t 5 cos( si( t 5 t 5
13 Solutio: x (t is ot rl-vlud x t (t x ls ( ls ( (t x rl is (t x ls cos( ls cos( (t : x is rl (t x ls si ls si ls si (t : x
14 Prolm ( p55 of txt Fid th Fourir sris of x(t: x(t Solutio: x(t t for t ; d x(t t t - - t - hc whr t t dt [ cos( { t ] ( t [ t ] dt }
15 Fourir Sris Rprsttio of D Sigls D Fourir Sris: As with C sigls D sigl c xprssd s sris comitio of complx xpotils h sris howvr is fiit Lt x[] priodic with fudmtl priod i x[]x[] Cosidr complx fuctio For itgr r or φ φ [ ] r [] ( ( ( r( φ [ ] φ r[] r ( ( φ [] rpts itslf with priod of hus st of vlus of φ [] is ough to rprst priodic D sigl ( r( φ [] 5
16 hrfor priodic D sigl my xprssd s ( whr <> ms y ritrry coscutiv xpotils Eq ( is th D Fourir sris is clld Fourir cofficit Dtrmitio of : Cosidr ow: x[] φ[] x[] < > With r < > < > r (r < > x[] < > if ls - r r (r (r < > < > < > < > (r( < > < > x[] r r ± ± 6
17 Rplcig r y x[] < > ( Eq ( is clld sythsis qutio d q ( th lysis qutio As i itrvl rpts itslf i y othr priods ± ± Sic th sris is fiit hvig trms it is lwys covrgt Exmpl: Exmpl p of txt Fid th Fourir sris of: x[] - Hr (m m m si si m 7
18 For ( ( With 9 thr r 9 trms g d If th RHS Of q ( is tructd to lss th 9 trms distortio occurs S Fig p of txt ot tht Gis phomo dos ot occur i D sigl pproximtio Prolm 9 p 5 of txt Fid th Fourir sris of Solutio: x[] m {δ[ - m] δ[ -- m]} x[]
19 Hr givig < > [ x[] ] {δ[] δ[ ]} [ ; tc ] Prolm p 5 of txt 6 Fid th Fourir cofficits of x[] y[] d z[] giv y: x[] cos( 6 y[] si( 6 z[] x[] y[] Solutio: 6 for x[] y[] d hc for z[] 6 9
20 Hc (c ( ( 6 cos( x[] > < - givig ] [ si( si( y[] - - ( ( c c c c c c si c givig c d d to covr Choosi g c x[]y[] l l - - l l l > < > <
21 Fourir Sris d LI Systms For systm with impuls rspos H( or H[ ] : x(t x[] h(t h[] y(t y[] τ H( H( y(t y[] For C: For D: H( H( h( τ τ h[] C: With Fourir sris of x(t: dτ (5 (5 t x(t D: y(t y[] < > H( t o fid th systm output y(t or y[]: Fid H( or H( w usig q 5 or 5 Fid F S of x(t or x[] Fid y(t or y[] usig q 6 or q7 H( (6 (7
22 Pro p 6 of txt Giv h(t - t fid y(t with ( x(t δ(t (c x(t giv s: Solutio: ( H( Hr h( τ - x(t - -/ / τ dτ t x(t dt τ τ dτ -τ τ dτ x(t - - t δ(t dt δ(t t dt t
23 Hc (c whr Pro p 6: Giv: ] [ H( H( y(t t t t t t t t ] [ si( y(t si( si( (t x odd for ] [ si( for v for δ ] [ x[] ls h[]
24 Fid th Fourir cofficits of output y[] Whr h[] (- h[] H( [] x[] hr x[] δ > < > < ] [ H( y[] ] [ ] [ x[]
25 Frqucy-Shpig or Frqucy-Slctiv Filtrs: Idl Filtrs: Low-pss: H( < > c c H( - c c High-pss: Bd-pss: H( < > c c H( - c c H( < < ls H( Bd-stop: - H( < < ls H( o-idl Filtrs: C Filtr: R - x(t v i (t C y(t v (t 5
26 Systm diffrtil qutio: or dy(t RC y(t x(t dt x(t t y(t H( ( t ( RCH( givig Y( H( X( RC H( dy(t dt H( -t - H( RC t h rspos plot: H( H( - his is low-pss filtr 6
27 D Filtrs Cosidr first-ordr filtr giv y th diffrc qutio: y[] y[-] x[] < With x[] H( H( H( [ H( ] ( givig h mgitud H( d th phs gl ( for 6 d -6 r show i Fig p 6 of th txt or H For < < it is low-pss filtr For > > - it is high-pss filtr 7
28 Prolm 6 p 5 of txt With th giv impuls rspos H( fid th filtr output with th followig iputs: Solutio: Hr d y[] x o is pssd x [] ( x[] si( [] ( ( H( (si c H( ( H( givig H( 5/ / / 5/ x [] - - -
29 Hc is locd / is pssd 9 othrs ( ( ] [ [] x 6 6 si( [] x ( ( si( ] [ H( H( H( H( H( H( y[] ( ( > <
[ ] Review. For a discrete-time periodic signal xn with period N, the Fourier series representation is
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