Chapter 3 Fourier Series Representation of Period Signals

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1 ELG 3 Sigls d Sysms Chpr 3 Chpr 3 Fourir Sris Rprsio of Priod Sigls 3. Iroducio Sigls c b rprsd usig complx xpoils coiuous-im d discr-im Fourir sris d rsform. If h ipu o LI sysm is xprssd s lir combiio of priodic complx xpoils or siusoids, h oupu c lso b xprssd i his form. 3. A Hisoricl Prspciv By 87, Fourir hd compld wor h sris of hrmoiclly rld siusoids wr usful i rprsig mprur disribuio of body. H climd h y priodic sigl could b rprsd by such sris Fourir Sris. H lso obid rprsio for pridic sigls s wighd igrls of siusoids Fourir rsform. J Bpis Josph Fourir 3. h Rspos of LI Sysms o Complx Expoils I is dvgous i h sudy of LI sysms o rprs sigls s lir combiios of bsic sigls h possss h followig wo propris: h s of bsic sigls c b usd o cosruc brod d usful clss of sigls. /3 Yo

2 ELG 3 Sigls d Sysms Chpr 3 h rspos of LI sysm o ch sigl should b simpl ough i srucur o provid us wih covi rprsio for h rspos of h sysm o y sigl cosrucd s lir combiio of h bsic sigl. Boh of hs propris r providd by Fourir lysis. h imporc of complx xpoils i h sudy of LI sysm is h h rspos of LI sysm o complx xpoil ipu is h sm complx xpoil wih oly chg i mpliud; h is Coiuous im: s s H ( s, (3. Discr-im: z H ( z z, (3. whr h complx mpliud fcor H (s or H (z will b i grl b fucio of h complx vribl s or z. A sigl for which h sysm oupu is (possibl complx cos ims h ipu is rfrrd o s igfucio of h sysm, d h mpliud fcor is rfrrd o s h sysm s igvlu. Complx xpoils r igfucios. For ipu x ( pplid o LI sysm wih impuls rspos of h (, h oupu is y( h( τ τ dτ h( τ s( τ dτ s h( τ h( τ s( τ sτ dτ dτ, (3.3 whr w ssum h h igrl s h( τ τ dτ covrgs d is xprssd s s H ( s h( τ τ dτ, (3.4 h rspos o s is of h form y s ( H ( s, (3.5 I is show h complx xpoils r igfucios of LI sysms d H (s for spcific vlu of s is h h igvlus ssocid wih h igfucios. Complx xpoil squcs r igfucios of discr-im LI sysms. h is, suppos h LI sysm wih impuls rspos h [] hs s is ipu squc /3 Yo

3 ELG 3 Sigls d Sysms Chpr 3 x [ ] z, (3.6 whr z is complx umbr. h h oupu of h sysm c b drmid from h covoluio sum s [ ] h[ ] x[ ] h[ ] z z h[ z. (3.7 y ] Assumig h h summio o h righ-hd sid of Eq. (3.7 covrgs, h oupu is h sm complx xpoil muliplid by cos h dpds o h vlu of z. h is, y [ ] H ( z z, (3.8 whr ( z h[ z. (3.9 H ] I is show h complx xpoils r igfucios of LI sysms d H (z for spcific vlu of z is h h igvlus ssocid wih h igfucios z. h xmpl hr shows h usfulss of dcomposig grl sigls i rms of igfucios for LI sysm lysis: L 3 s s s 3, (3. from h igfucio propry, h rspos o ch sprly is s s H ( s s s H ( s s3 s3 3 3 H 3 ( s 3 d from h suprposiio propry h rspos o h sum is h sum of h rsposs, s s s3 y( H s H s ( ( 3H3( s3, (3. Grlly, if h ipu is lir combiio of complx xpoils, s x (, (3. h oupu will b 3/3 Yo

4 ELG 3 Sigls d Sysms Chpr 3 s y ( H( s, (3.3 Similrly for discr-im LI sysms, if h ipu is x [ ] z, (3.4 h oupu is y [ ] H( z z, ( Fourir Sris rprsio of Coiuous-im Priodic Sigls 3.3 Lir Combiios of hrmoiclly Rld Complx Expoils A priodic sigl wih priod of, x ( for ll, (3.6 W iroducd wo bsic priodic sigls i Chpr, h siusoidl sigl x ( cosω, (3.7 d h priodic complx xpoil ω, (3.8 Boh hs sigls r priodic wih fudml frqucy ω d fudml priod / ω. Associd wih h sigl i Eq. (3.8 is h s of hrmoiclly rld complx xpoils ω ( /,,,,... φ ( ± ± (3.9 Ech of hs sigls is priodic wih priod of (lhough for, h fudml priod of φ ( is frcio of. hus, lir combiio of hrmoiclly rld complx xpoils of h form 4/3 Yo

5 ELG 3 Sigls d Sysms Chpr 3 ω ( /, (3. is lso priodic wih priod of., x ( is cos. d, boh hv fudml frqucy qul o ω d r collcivly rfrrd o s h fudml compos or h firs hrmoic compos. d, h compos r rfrrd o s h scod hrmoic compos. d, h compos r rfrrd o s h h hrmoic compos. Eq. (3. c lso b xprssd s x *( * ω, (3. whr w ssum h x ( is rl, h is, x ( x *(. Rplcig by i h summio, w hv * ω, (3. which, by compriso wih Eq. (3., rquirs h *, or quivlly *. (3.3 o driv h lriv forms of h Fourir sris, w rwri h summio i Eq. (. s Subsiuig ω ( / [ ] * for, w hv ω ( / [ ] *. (3.4. (3.5 Sic h wo rms isid h summio r complx coug of ch ohr, his c b xprssd s R ω { }. (3.6 5/3 Yo

6 ELG 3 Sigls d Sysms Chpr 3 If is xprssd i polr from s θ A, h Eq. (3.6 bcoms h is R ( ω θ { A }. A cos( θ ω. (3.7 I is o commoly courd form for h Fourir sris of rl priodic sigls i coiuous im. Aohr form is obid by wriig i rcgulr form s B C h Eq. (3.6 bcoms [ B cos C ω ] ω si. (3.8 For rl priodic fucios, h Fourir sris i rms of complx xpoil hs h followig hr quivl forms: ω ( / A cos( ω θ [ B cos ω C ω ] si 6/3 Yo

7 ELG 3 Sigls d Sysms Chpr Drmiio of h Fourir Sris Rprsio of Coiuous-im Priodic Sigl Muliply boh sid of ω by ω, w obi ω ω ω, (3.9 Igrig boh sids from o / ω, w hv o h ω d ω ω d ( ω d, (3.3 ( ω, d, So Eq. (3.3 bcoms ω d, (3.3 h Fourir sris of priodic coiuous-im sigl ω ( / (3.3 ω ( / d d (3.33 Eq. (3.3 is rfrrd o s h Syhsis quio, d Eq. (3.33 is rfrrd o s lysis r of clld h Fourir sris cofficis of h quio. h s of coffici { } spcrl cofficis of x (. h coffici is h dc or cos compo d is giv wih, h is 7/3 Yo

8 ELG 3 Sigls d Sysms Chpr 3 d, (3.34 Exmpl: cosidr h sigl x ( siω. ω ω siω. Comprig h righ-hd sids of his quio d Eq. (3.3, w hv,, or Exmpl: h priodic squr wv, schd i h figur blow d dfi ovr o priod is, < x (, (3.35, < < / h sigl hs fudml priod d fudml frqucy ω /. o drmi h Fourir sris cofficis for x (, w us Eq. (3.33. Bcus of h symmry of x ( bou, w choos / / s h irvl ovr which h igrio is prformd, lhough y ohr irvl of lgh is vlid h hus ld o h sm rsul. For, d d, (3.36 For, w obi 8/3 Yo

9 ELG 3 Sigls d Sysms Chpr 3 ω d ω ω ω ω ω (3.37 si( ω ω si( ω h bov figur is br grph of h Fourir sris cofficis for fixd d svrl vlus of. For his xmpl, h cofficis r rl, so hy c b dpicd wih sigl grph. For complx cofficis, wo grphs corrspodig o h rl d imgiry prs or mpliud d phs of ch coffici, would b rquird. 3.4 Covrgc of h Fourir Sris If priodic sigl x ( is pproximd by lir combiio of fii umbr of hrmoiclly rld complx xpoils x ( ω. (3.38 9/3 Yo

10 ELG 3 Sigls d Sysms Chpr 3 L ( do h pproximio rror, ( x ( ω. (3.39 h cririo usd o msur quiivly h pproximio rror is h rgy i h rror ovr o priod: E ( d. (3.4 I is show (problm 3.66 h h priculr choic for h cofficis h miimiz h rgy i h rror is ω d. (3.4 I c b s h Eq. (3.4 is idicl o h xprssio usd o drmi h Fourir sris cofficis. hus, if x ( hs Fourir sris rprsio, h bs pproximio usig oly fii umbr of hrmoiclly rld complx xpoils is obid by rucig h Fourir sris o h dsird umbr of rms. h limi of E s is zro. O clss of priodic sigls h r rprsbl hrough Fourir sris is hos sigls which hv fii rgy ovr priod, d <, (3.4 Wh his codiio is sisfid, w c gur h h cofficis obid from Eq. (3.33 r fii. W dfi ( ω, (3.43 h ( d, (3.44 /3 Yo

11 ELG 3 Sigls d Sysms Chpr 3 h covrgc gurd wh x ( hs fii rgy ovr priod is vry usful. I his cs, w my sy h x ( d is Fourir sris rprsio r idisiguishbl. Alriv s of codiios dvlopd by Dirichl h gurs h quivlc of h sigl d is Fourir sris rprsio: Codiio : Ovr y priod, x ( mus b bsoluly igrbl, h is d <, (3.45 his gurs ch coffici will b fii, sic ω d d <. (3.46 A priodic fucio h viols h firs Dirichl codiio is, < <. Codiio : I y fii irvl of im, x ( is of boudd vriio; h is, hr r o mor h fii umbr of mxim d miim durig sigl priod of h sigl. A xmpl of fucio h ms Codiio bu o Codiio : si, <, (3.47 Codiio 3: I y fii irvl of im, hr r oly fii umbr of discoiuiis. Furhrmor, ch of hs discoiuiis is fii. A xmpl h viols his codiio is fucio dfid s x (, < 4, x ( /, 4 < 6, x ( / 4, 6 < 7, x ( / 8, 7 < 7. 5, c. h bov hr xmpls r show i h figur blow. /3 Yo

12 ELG 3 Sigls d Sysms Chpr 3 h bov r grlly phologicl i ur d cosquly do o ypiclly ris i prcicl coxs. Summry: For priodic sigl h hs o discoiuiis, h Fourir sris rprsio covrgs d quls o h origil sigl ll h vlus of. For priodic sigl wih fii umbr of discoiuiis i ch priod, h Fourir sris rprsio quls o h origil sigl ll h vlus of xcp h isold pois of discoiuiy. Gibbs Phomo: r poi, whr x ( hs ump discoiuiy, h pril sums x ( of Fourir sris xhibi subsil ovrshoo r hs dpois, d icrs i will o dimiish h mpliud of h ovrshoo, lhough wih icrsig h ovrshoo occurs ovr smllr d smllr irvls. his phomo is clld Gibbs phomo. /3 Yo

13 ELG 3 Sigls d Sysms Chpr 3 A lrg ough vlu of should b chos so s o gur h h ol rgy i hs rippls is isigific. 3.5 Propris of h Coiuous-im Fourir Sris oio: suppos x ( is priodic sigl wih priod d fudml frqucy ω. h if h Fourir sris cofficis of x ( r dod by, w us h oio x FS (, o sigify h pirig of priodic sigl wih is Fourir sris cofficis. 3/3 Yo

14 ELG 3 Sigls d Sysms Chpr Liriy L x ( d y ( do wo priodic sigls wih priod d which hv Fourir sris cofficis dod by d b, h is x FS FS ( d y b h w hv (, FS z A By( c A ( Bb. ( im Shifig Wh im shif o priodic sigl x (, h priod of h sigl is prsrvd. If x FS (, h w hv FS ω. (3.49 h mgiuds of is Fourir sris cofficis rmi uchgd im Rvrsl If x FS (, h x FS (. (3.5 im rvrsl pplid o coiuous-im sigl rsuls i im rvrsl of h corrspodig squc of Fourir sris cofficis. If x ( is v, h is, h Fourir sris cofficis r lso v,. Similrly, if x ( is odd, h is, h Fourir sris cofficis r lso odd, im Sclig If x ( hs h Fourir sris rprsio ( rprsio of h im-scld sigl α is x ω, h h Fourir sris 4/3 Yo

15 ELG 3 Sigls d Sysms Chpr 3 ( αω α. (3.5 h Fourir sris cofficis hv o chgs, h Fourir sris rprsio hs chgd bcus of h chg i h fudml frqucy Muliplicio Suppos x ( d y ( r wo priodic sigls wih priod d h x FS (, y FS ( b. Sic h produc x ( y( is lso priodic wih priod, is Fourir sris cofficis h is l l FS x ( y( h b. (3.5 l h sum o h righ-hd sid of Eq. (3.5 my b irprd s h discr-im covoluio of h squc rprsig h Fourir cofficis of x ( d h squc rprsig h Fourir cofficis of y ( Coug d Coug Symmry ig h complx coug of priodic sigl x ( hs h ffc of complx cougio d im rvrsl o h corrspodig Fourir sris cofficis. h is, if x FS (, h x *(. (3.53 FS * If x ( is rl, h is, x ( x *(, h Fourir sris cofficis will b coug symmric, h is *. (3.54 5/3 Yo

16 ELG 3 Sigls d Sysms Chpr 3 From his xprssio, w my g vrious symmry propris for h mgiud, phs, rl prs d imgiry prs of h Fourir sris cofficis of rl sigls. For xmpl: From Eq. (3.54, w s h if x ( is rl, is rl d. If x ( is rl d v, w hv, from Eq. (3.54 *, so * h Fourir sris cofficis r rl d v. If x ( is rl d odd, h Fourir sris cofficis r rl d odd Prsvl s Rlio for Coiuous-im priodic Sigls Prsvl s Rlio for Coiuous-im priodic Sigls is d, (3.55 Sic ω d d, so h is h vrg powr i h h hrmoic compo. hus, Prsvl s Rlio ss h h ol vrg powr i priodic sigl quls h sum of h vrg powrs i ll of is hrmoic compos. 6/3 Yo

17 ELG 3 Sigls d Sysms Chpr Summry of Propris of h Coiuous-im Fourir Sris Propry Priodic Sigl Fourir Sris Cofficis Priodicwih priod d y( fudm l frqucy ω Liriy Ax ( By( A Bb im Shifig ω Frqucy shifig Mω M Cougio x *( * im Rvrsl im Sclig α, α > (Priodic wih priod / α Priodic Covoluio x τ y( τ dτ Muliplicio x ( y( Diffriio Igrio Coug Symmry for Rl Sigls Rl d Ev Sigls Rl d Odd Sigls Ev-Odd Dcomposiio of Rl Sigls / ( b b l lb dx ( ω d x ( d (fii vlud d priodic oly if ω ( / x ( Ev x ( Od x ( rl x ( rl d v x ( rl d odd { } [ rl] { } [ rl] Prsvl s Rlio for Priodic Sigls d l * R{ } R{ } Im{ } Im{ } rl d v purly imgiry d odd R { } Im { } 7/3 Yo

18 ELG 3 Sigls d Sysms Chpr 3 Exmpl: Cosidr h sigl g ( wih fudml priod of 4. g( / / h Fourir sris rprsio c b obid dircly usig h lysis quio (3.33. W my lso us h rlio of g ( o h symmric priodic squr wv x ( discussd o pg 8. Rfrrig o h xmpl, 4 d, g ( /. (3.56 h im-shif propry idics h if h Fourir sris cofficis of x ( r dod by h Fourir sris cofficis of x ( c b xprssd s b /. (3.57 h Fourir cofficis of h dc offs i g (, h is h rm / o h righ-hd sid of Eq. (3.56 r giv by, for c. (3.58, for Applyig h liriy propry, w coclud h h cofficis for g ( c b xprssd s d /,, for, (3.59 for rplcig si( / /, h w hv d si( / /,, for. (3.6 for 8/3 Yo

19 ELG 3 Sigls d Sysms Chpr 3 Exmpl: h rigulr wv sigl x ( wih priod 4 ω / is show i h figur blow., d fudml frqucy h driviv of his fucio is h sigl g ( i h prvious prcdig xmpl. Doig h Fourir sris cofficis of g ( by d, d hos of x ( by, bsd o h diffriio propry, w hv d ( /. (3.6 his quio c b xprssd i rms of xcp wh. From Eq. (3.6, d si( / /. (3.6 ( For, c b simply clculd by clculig h r of h sigl udr o priod d divid by h lgh of h priod, h is /. (3.63 Exmpl: h propris of h Fourir sris rprsio of priodic ri of impuls, x ( δ (. (3.64 W us Eq. (3.33 d slc h igrio irvl o b / /, voidig h plcm of impulss h igrio limis. / ( / ( d δ. (3.65 / All h Fourir sris cofficis of his priodic ri of impuls r idicl, rl d v. 9/3 Yo

20 ELG 3 Sigls d Sysms Chpr 3 h priodic ri of impuls hs srighforwrd rlio o squr-wv sigls such s g ( o pg 8. h driviv of g ( is h sigl q ( show i h figur blow, g( q( which c lso irprd s h diffrc of wo shifd vrsios of h impuls ri x (. h is, q. (3.66 ( Bsd o h im-shifig d liriy propris, w my xprss h Fourir cofficis b of q ( i rms of h Fourir sris coffici of ; h is b ω ω [ ] ω ω, (3.67 Filly w us h diffriio propry o g b ωc, (3.68 whr c is h Fourir sris cofficis of g (. hus /3 Yo

21 ELG 3 Sigls d Sysms Chpr 3 b si( ω si( ω c,, (3.69 ω ω ω c c b solv by ispcio from h figur: c. (3.7 Exmpl: Suppos w r giv h followig fcs bou sigl x (. x ( is rl sigl.. x ( is priodic wih priod 4, d i hs Fourir sris cofficis. 3. for >. / 4. h sigl wih Fourir cofficis b is odd. 5. ( 4 x d 4 Show h h iformio is suffici o drmi h sigl x ( o wihi sig fcor. Accordig o Fc 3, x ( hs mos hr ozro Fourir sris cofficis :, d. Sic h fudml frqucy ω / / 4 /, i follows h / /. (3.7 Sic x ( is rl (Fc, bsd o h symmry propry is rl d *. Cosquly, / ( * { / }. (3.7 / R Bsd o h Fc 4 d cosidrig h im-rvrsl propry, w o h corrspods o. Also h muliplicio propry idics h muliplicio of h Fourir sris / by corrspods o h sigl big shifd by o h righ. W coclud h h cofficis b corrspod o h sigl x ( (, which ccordig o Fc 4 mus b odd. Sic x ( is rl, x ( mus lso b rl. So bsd h propry, h Fourir sris cofficis mus b purly imgiry d odd. hus, b, b b. Sic im rvrsl d im shif co chg h vrg powr pr priod, Fc 5 holds v if x ( is rplcd by x (. h is 4 4 x ( d. (3.73 /3 Yo

22 ELG 3 Sigls d Sysms Chpr 3 Usig Prsvl s rlio, b b /. (3.74 Sic b b, w obi b /. Sic b is ow o b purly imgiry, i mus b ihr b / or b /. Filly w rsl h codiios o b d b io h quivl sm o d. Firs, sic b, Fc 4 implis h. Wih, his codiio implis h / b b b. hus, if w b /, /, from Eq. (3.7, cos( /. Alrivly, if w b /, h /, d hrfor, cos( /. 3.6 Fourir Sris Rprsio of Discr-im Priodic Sigls h Fourir sris rprsio of discr-im priodic sigl is fii, s opposd o h ifii sris rprsio rquird for coiuous-im priodic sigls 3.6. Lir Combiio of Hrmoiclly Rld Complx Expoils A discr-im sigl x [] is priodic wih priod if x [ ] x[ ]. (3.75 h fudml priod is h smlls posiiv for which Eq. (3.75 holds, d h fudml frqucy is ω /. h s of ll discr-im complx xpoil sigls h r priodic wih priod is giv by ω ( /,,,,... φ [ ] ± ±, (3.76 All of hs sigls hv fudml frqucis h r mulipls of hrmoiclly rld. / d hus r hr r oly disic sigls i h s giv by Eq. (3.76; his is bcus h discr-im complx xpoils which diffr i frqucy by mulipl of r idicl, h is, φ [ ] φ [ ]. (3.77 r /3 Yo

23 ELG 3 Sigls d Sysms Chpr 3 h rprsio of priodic squcs i rms of lir combiios of h squcs φ [ ] is x[ ] ω [ ] ( / φ. (3.78 Sic h squcs φ [ ] r disic ovr rg of succssiv vlus of, h summio i Eq. (3.78 d iclud rms ovr his rg. W idic his by xprssig h limis of h summio s. h is, x[ ] ω ( / φ [ ]. (3.79 Eq. (3.79 is rfrrd o s h discr-im Fourir sris d h cofficis s h Fourir sris cofficis. 6. Drmiio of h Fourir Sris Rprsio of Priodic Sigl h discr-im Fourir sris pir: x[ ] ω ( / φ [ ], (3.8 ω ( /. (3.8 x[ ] x[ ] Eq. (3.8 is clld syhsis quio d Eq. (3.8 is clld lysis quio. Exmpl: Cosidr h sigl x [ ] siω, (3.8 x [] is priodic oly if / ω is igr, or rio of igr. For h cs h wh / ω is igr, h is, wh ω, (3.83 [] x is priodic wih h fudml priod. Expdig h sigl s sum of wo complx xpoils, w g 3/3 Yo

24 ELG 3 Sigls d Sysms Chpr 3 x[ ] ( / ( /, (3.84 From Eq. (3.84, w hv,, (3.85 d h rmiig cofficis ovr h irvl of summio r zro. As discussd prviously, hs cofficis rp wih priod. h Fourir sris cofficis for his xmpl wih 5 r illusrd i h figur blow. Wh / ω is rio of igr, h is, wh M ω, (3.86 Assumig h M d do o hv y commo fcors, x [] hs fudml priod of. Agi xpdig x [] s sum of wo complx xpoils, w hv x[ ] M ( / M ( /, (3.87 From which w drmi by ispcio h M ( /, M ( /, d h rmiig cofficis ovr o priod of lgh r zro. h Fourir cofficis for his xmpl wih M 3 d 5 r dpicd i h figur blow. 4/3 Yo

25 ELG 3 Sigls d Sysms Chpr 3 5/3 Yo Exmpl: Cosidr h sigl 4 cos 3cos si ] [ x. Expdig his sigl i rms of complx xpoil, w hv x / ( / / ( / / ( / ( 3 ( 3 ( ] [. hus h Fourir sris cofficis for his sigl r, 3 3, 3 3,,. wih for ohr vlus of i h irvl of summio i h syhsis quio. h rl d imgiry prs of hs cofficis for, d h mgiud d phs of h cofficis r dpicd i h figur blow.

26 ELG 3 Sigls d Sysms Chpr 3 Exmpl: Cosidr h squr wv show i h figur blow. Bcus x [ ] for, w choos h lgh- irvl of summio o iclud h rg. h cofficis r giv ( /, (3.88 L m, w obsrv h Eq. (3.88 bcoms 6/3 Yo

27 ELG 3 Sigls d Sysms Chpr 3 ( / ( m ( / ( / m, (3.89 [ ( / / ] si si( / ( / ( / ( / d,, ±, ±,... (3.9,, ±, ±,... (3.9 h cofficis for 5 r schd for,, d 4 i h figur blow. h pril sums for h discr-im squr wv for M,, 3, d 4 r dpicd i h figur blow, whr 9, 5. W s for M 4, h pril sum xcly quls o x []. I cors o h coiuous-im cs, hr r o covrgc issus d hr is o Gibbs phomo. 7/3 Yo

28 ELG 3 Sigls d Sysms Chpr Propris of Discr-im Fourir Sris Propry Priodic Sigl Fourir Sris Cofficis x[ ] y[ ] Priodicwih priod d fudm l frqucy ω b Priodic wih priod Liriy Ax [ ] By[ ] A Bb im Shifig ( / x[ ] Frqucy shifig M ( / x[ ] M Cougio *[ ] * x 8/3 Yo

29 ELG 3 Sigls d Sysms Chpr 3 im Rvrsl x[ ] im Sclig x m if is muliplof x [ [ / ], viwd s ( m ], if is muliplof m wih priod (Priodic wih priod m Priodic Covoluio Muliplicio x [ ] y[ ] Diffriio [ ] x[ ] Igrio Coug Symmry for Rl Sigls Rl d Ev Sigls Rl d Odd Sigls Ev-Odd Dcomposiio of Rl Sigls 3.7. Muliplicio x [ r] y[ r] b r [ ] b l l l< > ( / x ( priodic m x [ ] (fii vlud d priodic ( / oly if x [] rl * R{ } R{ } Im{ } Im{ } x [] rl d v rl d v x [] rl d odd purly imgiry d odd x[ ] Ev{ x[ ] } [ x[ ] rl] R { } x Od{ x } [ x rl] [ ] [ ] [ ] Prsvl s Rlio for Priodic Sigls < > x[ ] < > Im { } FS x [ ] y[ ] l< > l b l. (3.9 Eq. (3.9 is logous o h covoluio, xcp h h summio vribl is ow rsricd o i irvl of coscuiv smpls. his yp of oprio is rfrrd o s Priodic Covoluio bw h wo priodic squcs of Fourir cofficis. h usul form of h covoluio sum, whr h summio vribl rgs from is somims rfrrd o s Apriodic Covoluio. o, 9/3 Yo

30 ELG 3 Sigls d Sysms Chpr Firs Diffrc x[ ] ( / ( FS x[ ]. ( Prsvl s Rlio x[ ]. < > < > ( Exmpls Exmpl: Cosidr h sigl show i h figur blow. x[] -5 5 x[] -5 5 x[] /3 Yo

31 ELG 3 Sigls d Sysms Chpr 3 h sigl x [] my b viwd s h sum of h squr wv x [ ] wih Fourir sris cofficis b d x [ ] wih Fourir sris cofficis c. b c, (3.95 h Fourir sris cofficis for x [ ] is b si(3 /5, 5 si( / 5 3, 5 for, ± 5, ±,.... (3.96 for, ± 5, ±,... h squc x [ ] hs oly dc vlu, which is cpurd by is zroh Fourir sris coffici: 4 c x[ ], ( Sic h discr-im Fourir sris cofficis r priodic, i follows h c whvr is igr mulipl of 5. si(3 /5, 5 si( / 5 8, 5 for, ± 5, ±,... for, ± 5, ±,... (3.98 Exmpl: Suppos w r giv h followig fcs bou squc x []:. x [] is priodic wih priod x [ ]. 3. ( x [ ]. 4. x [] hs miimum powr pr priod mog h s of sigls sisfyig h prcdig hr codiios. From Fc, w hv o h ( 5 x[ ]. 6 3 ( / 63, w s from Fc 3 h From Prsvl s rlio, h vrg powr i x [] is 7 3( / 3 x[ ] /3 Yo

32 ELG 3 Sigls d Sysms Chpr 3 5 P. Sic ch ozro coffici coribus posiiv mou o P, d sic h vlus of d r spcifid, h vlu of P is miimizd by choosig. I follows h x [ ] 3 ( which is show i h figur blow., 4 5 / x[] / Fourir Sris d LI Sysms W hv s h h rspos of coiuous-im LI sysm wih impuls rspos h ( o complx xpoil sigl s is h sm complx xpoil muliplid by complx gi: y whr s ( H ( s, s h( τ τ dτ H ( s, (3.99 ω I priculr, for s ω, h oupu is y( H ( ω. h complx fucios H (s d H ( ω?r clld h sysm fucio (or rsfr fucio d h frqucy rspos, rspcivly. By suprposiio, h oupu of LI sysm o priodic sigl rprsd by Fourir sris ω ( / is giv by 3/3 Yo

33 ELG 3 Sigls d Sysms Chpr 3 ω ( H( ω. (3.99 y h is, h Fourir sris cofficis b of h priodic oupu y ( r giv by b H ( ω, (3. Similrly, for discr-im sigls d sysms, rspos h [] o complx xpoil sigl ω is h sm complx xpoil muliplid by complx gi: y ω [ ] H( ω, (3. whr H ω ω ( h[ ]. (3. Exmpl: Suppos h h priodic sigl h( u(, d 3 3 wih,, is h ipu sigl o LI sysm wih impuls rspos 3 o clcul h Fourir sris cofficis of h oupu y (, w firs compu h frqucy rspos: τ ωτ τ ωτ H( ω dτ ω ω, (3.3 h oupu is y( 3 3 b, (3.4 whr b H ( ω H (, so h b, b, b, /3 Yo

34 ELG 3 Sigls d Sysms Chpr 3 34/3 Yo 4 4 b, 4 4 b, b, b. Exmpl: Cosidr LI sysm wih impuls rspos ] [ ] [ u h α, < < α, d wih h ipu x cos ] [. (3.5 Wri h sigl ] [ x i Fourir sris form s x / ( / ( ] [. Also h rsfr fucio is ( ω ω ω ω α α α H (. (3.6 h Fourir sris for h oupu ( ( H H y / ( / ( / ( / / ( / ] [ ω ω α α. (3.7

35 ELG 3 Sigls d Sysms Chpr Filrig Filrig o chg h rliv mpliud of h frqucy compos i sigl or limi som frqucy compos irly. Filrig c b covily ccomplishd hrough h us of LI sysms wih pproprily chos frqucy rspos. LI sysms h chg h shp of h spcrum of h ipu sigl r rfrrd o s frqucy-shpig filrs. LI sysms h r dsigd o pss som frqucis ssilly udisord d sigificly u or limi ohrs r rfrrd o s frqucy-slciv filrs. Exmpl: A firs-ordr low-pss filr wih impuls rspos h( u( cus off h high frqucis i priodic ipu sigl, whil low frqucy hrmoics r mosly lf ic. h frqucy rspos of his filr H ( ω τ ωτ dτ. (3.7 ω W c s h s h frqucy ω icrs, h mgiud of h frqucy rspos of h filr H ( ω dcrss. If h priodic ipu sigl is rcgulr wv, h h oupu sigl will hv is Fourir sris cofficis b giv by b b si( ω H ( ω, (3.8 ( ω H (. (3.9 h rducd powr high frqucis producd oupu sigl h is smohr h h ipu sigl. 35/3 Yo

36 ELG 3 Sigls d Sysms Chpr 3 3. Exmpls of coiuous-im Filrs Dscribd By Diffril Equios I my pplicios, frqucy-slciv filrig is ccomplishd hrough h us of LI sysms dscribd by lir cos-coffici diffril or diffrc quios. I fc, my physicl sysms h c b irprd s prformig filrig oprios r chrcrizd by diffril or diffrc quio. 3.. A simpl RC Lowpss Filr h firs-ordr RC circui is o of h lcricl circuis usd o prform coiuous-im filrig. h circui c prform ihr Lowpss or highpss filrig dpdig o wh w s h oupu sigl. ( v r ( v s - ( v c If w h volg cross h cpcior s h oupu, h h oupu volg is rld o h ipu hrough h lir cos-coffici diffril quio: dvc ( RC vc ( vs (. (3. d ω vs (, w ω c ( H( ω. Subsiuig hs xprssios io Eq. (3., w Assumig iiil rs, h sysm dscribd by Eq. (3. is LI. If h ipu is mus hv volg oupu v hv RC or d d ω ω ω [ H ω ] H( ω (, (3. RCω H ω ω ω ( ω H ( ω, (3.3 36/3 Yo

37 ELG 3 Sigls d Sysms Chpr 3 h w hv H ( ω. (3.4 RCω mpliud d frqucy rspos H ( ω is show i h figur blow. W c lso g h impuls rspos / RC h( u(, (3.5 RC d h sp rspos is / RC h( ( u(, (3.6 h fudml rd-off c b foud by comprig h figurs: o pss oly vry low frqucis, / RC should b smll, or RC should b lrg. o hv fs sp rspos, w d smllr RC. h yp of rd-off bw bhviors i h frqucy domi d im domi is ypicl of h issus risig i h dsig lysis of LI sysms. 37/3 Yo

38 ELG 3 Sigls d Sysms Chpr A Simpl RC Highpss Filr If w choos h oupu from h rsisor, h w g RC highpss filr. 3. Exmpls of Discr-im Filr Dscribd by Diffrc Equios A discr-im LI sysm dscribd by h firs-ordr diffrc quio y [ ] y[ ] x[ ]. (3.6 Form h igfucio propry of complx xpoil sigls, if y H ω ω ω [ ] (, whr ( H is h frqucy rspos of h sysm. x ω [ ], h H ω (. (3.7 ω h impuls rspos of h sysm is x[ ] u[ ]. (3.8 h sp rspos is s[ ] u[ ]. (3.9 From h bov plos w c s h for. 6 h sysm cs s Lowpss filr d.6, h sysm is highpss filr. I fc, for y posiiv vlu of <, h sysm pproxims highpss filr, d for y giv vlu of >, h sysm pproxims 38/3 Yo

39 ELG 3 Sigls d Sysms Chpr 3 highpss filr, whr corols h siz of bdpss, wih brodr pss bds s i dcrsd. h rd-off bw im domi d frqucy domi chrcrisics, s discussd i coiuous im, lso xiss i h discr-im sysms orcursiv Discr-im Filrs h grl form of FIR orcursiv diffrc quio is M y [ ] b x[ ]. (3. I is wighd vrg of h ( M vlus of x [], wih h wighs giv by h cofficis b. O frquly usd xmpl is movig-vrg filr, whr h oupu of y [] is vrg of vlus of x [] i h viciiy of - h rsul corrspodig smooh oprio or lowpss filrig. y. (3. 3 A xmpl: [ ] ( x[ ] x[ ] x[ ] h impuls rspos is h[ ] ( δ [ ] δ[ ] δ[ ], (3. 3 d h frqucy rspos H 3 ω ω ( ω (. (3.3 39/3 Yo

40 ELG 3 Sigls d Sysms Chpr 3 A grlizd movig vrg filr c b xprssd s M y [ ] b x[ ]. (3.4 M h frqucy rspos is H( M M [( M / ] si[ ω( M / ] si( ω / M ω ω ω. (3.5 h frqucy rsposs wih diffr vrg widow lghs r plod i h figur blow. FIR orcursiv highpss filr A xmpl of FIR orcursiv highpss filr is 4/3 Yo

41 ELG 3 Sigls d Sysms Chpr 3 x[ ] x[ ] y [ ]. (3.6 h frqucy rspos is ω H ( ω ω / ( si( ω /. (3.7 4/3 Yo

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