EEE 303: Signals and Linear Systems

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Cynthia Park
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1 33: Sigls d Lir Sysms Orhogoliy bw wo sigls L us pproim fucio f () by fucio () ovr irvl : f ( ) = c( ); h rror i pproimio is, () = f() c () h rgy of rror sigl ovr h irvl [, ] is, { }{ } = f () c () d = f () c () f () c () d f () d c () () c f d c f () d = f () d+ c () d c f () d () c f () d = + f () d c f () () d f () () d = + W w o miimiz h rror rgy by choosig suibl vlu of c I is s from h bov quio h h s d h ls rms r o h fucio of c W c miimiz by sig h d rm o zro his yilds, c = f() () d If f () d () r orhogol, c = hus if h wo sigls () d () r orhogol ovr irvl [, ], h, rgy of sum of orhogol sigls () = () + (); whr () d () r orhogol L, () () d = h, = () + () d = + + () () d + () d= + hus h rgy of h sum of wo orhogol sigls is qul o h sum of h rgis of h wo sigls Orhogol sigl spc W c pproim y sigl ovr h irvl [, ] by s of muully orhogol sigls (), (), L, ( ) s, rror, f () c() = () f() c() = h rgy of h rror sigl is miimizd if, f c d = c = δ δ = () () Sig = = δ ci δ ci f () d () () () i fd i ci = = () i i d = + = = f() () d ow, f () d c () d c f() () d + = cf i ( ) i( ) ci i ( ) d

2 Or, = () + = = f d c c c = () = f d c As,, d h rgy of f () is qul o h sum of h rgis of h orhogol compos f () = c (); = h bov quio is clld grlizd Fourir sris h s { () } h rgy of h sigl f () is, rigoomric Fourir sris L us cosidr sigl s: f c = is clld s of bsis fucios = his quio is clld Prsvl s horm {,cos ω,cos ω, L,cos ω, L; si ω, si ω, L,si ω, LL } A siusoidl wih frqucy ω is clld h h hrmoic of siusoidl frqucyω, whr is igr ω is clld h fudml compo m cos mωcos ωd = / m= m si mωsi ωd = / m= si mωcos ω d = for ll m d π hus h bov s is orhogol ovr irvl of durio, ω = So, w c prss f () by rigoomric Fourir sris s, f ( ) = + cosω + cos ω + L+ b siω + b si ω + L + ω ω = or, f ( ) = + ( cos + b si ) whr, = f() d, f()cos ωd = = f()cos ω d cos ω d, b = f()si d ω Compc form of FS b f() = c + ccos( ω+ ϕ) ; whr, c =, c = + b dϕ = = poil form of FS j ( ) = ω ; + ; = f d jω whr, d = c =, d d = f () d; d ( jb) = Rlioship bw c d d : c d = d = d d = ϕ, d = ϕ d d, = c =

3 mpl Drw h poil spcr of f( ) = 6 + cos(3 π / 4) + 8cos(6 π / ) + 4cos(9 π / 4) f j(3 /4) (3 /4) (6 /) (6 /) (9 /4) (9 /4) () 6 6[ π j π j j j j ] 4[ π π ] [ π = π ] or, f ( ) = h plo is show blow j π /4 j3 j π /4 j3 j π / j6 j π / j6 j π /4 j9 j π /4 j9 Fid h Fourir cofficis of h sigl f( ) = + siω+ cosω+ cos( ω+ π / 4) o jω o jω o jω o jω [s: + ( 656 ) + (656 ) + 5 (45 ) + 5 ( 45 ) ] 3 Fid h Fourir cofficis of h sigl () = si hus, d =, d =, d = d ll ohr d = Cosidr h priodic squr wv () s show i Figur blow Drmi h poil FS of () / jω jω A d = f () d = A d = jπ = m A hus d = A d d = [s] = m+ jπ [ ( ) ] Covrgc of Fourir sris A priodic sigl () hs FS rprsio if is sisfis h followig codiios: () d< () hs fii umbr of mim d miim

4 3 () mus hv fii umbr of discoiuiis wihi y fii irvl of A h poi of discoiuiy, h sum of h sris is h vrg of h lf d righ hd limis of () 4 Drmi h Fourir sris rprsio for () = si(π 3) + si(6 π) Propris of Fourir sris Liriy: FS FS (), y () b FS z() = A() + By() A + Bb im shifig: L, y () = ( ) h, FS (), FS jω y() [ ] 3 im rvrsl: FS FS (), ( ) 4 Cojugio d cojug symmry: FS FS (), () For rl d v (), = =, hus = his implis h h Fourir cofficis r rl d v For rl d odd (), = =, hus = his implis h h Fourir cofficis r purly imgiry d odd 5 Prsivl s rlio for priodic sigl: () j ω jω = () = = = Or, P = = + v = = his rlio is clld Prsivl s rlio for priodic sigl d d = ; j ω = = ffc of symmry 4 / v symmry: b =,, d = Odd symmry: Hlf wv odd symmry: ( /) = ( ) 4 / =, b, d b = j 4 / I his cs, =, = b = Oly odd hrmoics will b prs d h limi of igrio mpl Fid h Fourir sris of h ipu d oupu volg of idl full wv rcifir s show i Figur blow Fourir sris cofficis of v () / A jω =, = d / A jω jω Or, = jω ω i / jω A = { jω } / ( jω ) / / ja jω / jω / = + ω ; Formul: d = ( )

5 ja jπ jπ ja ja( ) Or, = cos π π + = = π π Fourir sris cofficis of v () A A / jω A jω / = A / =, = d d + / A jω j ( ω = ) d + 4A / A Or, = cos ω d = ( ) cos d cos si ( π ) = + For h bov problm, fid h vrg powr of ll h hrmoic compos of oupu ; Formul: ( ) Problm Fid h fudml priod, d h Fourir sris cofficis of h followig priodic sigl () Also fid h ol vrg powr of h scod d h hird hrmoic compos of h sigl j 3 [s] =, = ; P vg = π 8π 6 Covoluio of wo priodic sigls (wih sm priod): FS FS (), y() b FS z () = () y () c( = b ) z () hs h sm priod s () d y () 7 Igrio of priodic sigls: FS FS If (), h () d jω I will b priodic oly if = his propry idics h igror us high frqucy compos of h sigl 8 Diffriio of priodic sigls: d() If () FS FS, h ( jω) d hus h diffrior hcs h high frqucy compo of sigl mpl Drmi h poil Fourir sris of δ () jω π δ () = d ; ω = ; = / jω d = () d δ = / jω Hc, δ () = [s] = Formul: si d = ( si cos ) ; cos d ( cos si ) = + d = ( ) ;

6 h Gibb s Phomo If hr is jump discoiuiy i (), h rucd FS hibis oscillory bhvior roud h poi hr is ovrshoo of 9% i h viciiy of discoiuiy h firs p of oscillio, rgrdlss of h vlu of his bhvior is clld Gibbs phom Dcy r of mpliud spcrum If f () is smooh fucio, h mpliud spcrum dcys swifly wih frqucy I his cs fwr rms i FS provid good pproimio If f () hs jump discoiuiis, is syhsis rquir lrg umbr of high frqucy compos h mpliud spcrum dcys slowly s / For rigulr puls h spcrum dcys rpidly wih frqucy s / I grl, if h firs ( ) drivivs of () r coiuous d h driviv is discoiuous, h h mpliud spcrum dcys s / + LI sysm rspos o priodic ipus W hv s rlir h compl poil ipu o LI sysm grs oupu qul o h ipu j muliplid by h sysm frqucy rspos h is, h ipu () = ω rsuls i h oupu, y () = H( jω) jω j Usig liriy propry, if () = d ω = h, y () = dh( jω ) jω = mpl 5 h sigl () is ipu o LI sysm wih impuls rspos h ( ) = si( πu ) ( ) Fid h FS cofficis of h ipu d h oupu volg for 6% duy cycl d = si( ) si( ) d = ω ω ω = = sic π siπ DF: sic( ) : = π 3 3 6% duy cycl ms, = 6, h is, = 3; h, d = sic h frqucy rspos of h LI sysm is, ( ) jω H jω si( π) π = d = 5 ω + π + jω ' π h h FS coffici of h oupu sigl is, d = H( jω ) d= [s] 5 ( ω ) + π + jω

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

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