Identification of a continuous linear time-varying system using Haar wavelet with unit energy

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1 WSEAS ANSACINS o CICUIS AND SYSES S.J..J. i d S.W. N Ideifiio of oiuous lie ie-vyig syse usig wvele wi ui eegy S.J..J. i d S.W. N Depe of Eleois d Copue Egieeig yg Uivesiy Seoul -79 oe sw@yg.. p:sp.yg.. As: - I is ppe ideifiio of oiuous-ie lie ie-vyig V syse is poposed wee wvele wi ui eegy is eployed. Fo pupose lgei equio is deived y expdig e ipu-oupu d d e ie-vyig ipulse espose usig olized wveles. Uow wvele oeffiies fo e V syse s ipulse espose e effeively esied y solvig e lgei equio. Filly e ie-vyig ipulse espose of V syse e syesized fo e esied wvele oeffiies. ey-wods: ie ie-vyig syse ipulse espose ideifiio wvele Ioduio e wvele sfo s ee effeively pplied o y fields su s eul ewos ouiio d ige poessig -. oe speipilly syse ideifiio uilizig e wvele sfos s eeived eio i ool egieeig d sigl poessig fields. I piul wvele-sed ppoes fo ideifiio of lie ie-vyig V syses ve ee ddessed i oiuous-ie doi 4. Fo exple Dueies wvele ws pplied s oogol sis. oweve sie o lyi expessio exiss fo Dueies wvele ig opuiol ude is equied fo e syse ideifiio. e oe d syse ideifiio of oiuous-ie V se-spe odel y wvele ws epoed equiig less opuiol ude y Dueies wvele 5. I piul soe popeies of wveles wee eslised 6 d uilized fo se lysis d pee esiio of ilie syses 7. I is ppe ew ppo fo effeive esiio of e ipulse espose of oiuous-ie V syse is poposed equiig elively low opuiol ude. oe speifilly i lgei equio is fisly deived y expdig e ipu-oupu d d e ie-vyig ipulse espose usig olized wveles e ii uow wvele oeffiies fo e V syse s ipulse espose e esied y solvig e lgei equio d filly iii e ie-vyig ipulse espose of V syse e syesized fo e esied quies. e poposed ppo is diffee fo oveiol oes 5-7 i opuiolly effiie expessio fo ulipliio of wveles is uilized y eployig wvele wi ui eegy i.e. olized wi leds o effiie eusive ideifiio of lie ie-vyig syse. is ppe is ogized s follows: Fisly si popeies of olized wvele e osideed i Seio. I Seio e poposed ppo fo ideifiio of usl V syse is desied. Seio 4 povides soe siulio esuls d filly e olusio is dw i Seio 5. Also wo ppedies e iluded fo e poof of soe equios uilized i e poposed ppo. Bsi popeies of wvele. Geel popeies of wvele ogol sis fuios iludig wvele s ee uilized fo e syse ideifiio 5. I piul e pliude of e wvele is ± i soe fiie ievls d zeos elsewee i.e. see - ledig o effeive eduio of e lulio 7. If e slig fuio d e pooype wvele e deoed y d espeively ll oe wvele ses i.e. e geeed fo dilios d slios of d e se is olized wi ui eegy 8: ISSN: Issue 5 Volue 7 y 8

2 < < < q q q < e s defie s goup of e wveles: i i Also digil epeseio of is defied y 4 We ses e e e lges splig ie wiou lisig is 5. I geel sigl usully s soe fiie suppo d us wiou loss of geeliy e sigl duio e olized s e ie ievl s i 5. Aodigly y sque-iegle fuio y i e ievl < e expessed y usig e oogol ses } { 5: i.e. d y y 5 I pie e ppoxiio of y usig oly wveles is s follows: y i i 6 wee. ulipliio of wvele A eusive foul fo e podu ix e expessed s i 7. 7 dig dig dig 8 We 7 is uliplied y veo e followig ix C e oied fo e followig eusive foul: C 9 - dig dig dig C C C I 7- e eusive foul is deeied y o-olized wvele 7. e ix C eled o e ulipliio of wo wveles s ivese e of i 8 d. oweve due o e olized wvele e ivese e i C of 8 d e expessed y See e Appedix A. dig dig dig WSEAS ANSACINS o CICUIS AND SYSES S.J..J. i d S.W. N ISSN: Issue 5 Volue 7 y 8

3 WSEAS ANSACINS o CICUIS AND SYSES S.J..J. i d S.W. N Also we e i is uliplied y veo ix e dived fo e followig eusive foul See e Appedix B: dig dig dig 4 Aodigly C e oied wi lowe opuiol ude. I piul d 4 e uilized i is ppe fo ideifiio of e ipulse espose of V syse. Ideifiio of usl V syse I e pevious wo e ideifiio of lie uooous syse is pefoed y usig se spe odel 5-7. oweve i is ppe we oside e pole of ideifyig usl V syse expessed i iegl ovoluio fo. Coside oiuous-ie V syse wose ipu- oupu eliosip is give y o y τ x τ dτ τ 5 I 5 xτ d τ deoe e ipu d e ipulse espose of e V syse 9. Suppose ipu d oupu d e give d e ipulse espose of V syse is uow. e e ie e fixed iy ie. o y τ x τ dτ 6 We τ is poeed oo ses e ipulse espose of e V syse e expessed s follows : τ τ τ 7 I 7 d τ oespod espeively o wvele oeffiies d wvele ses wi iplies τ is o eessily seple wi espe o d τ. I ddiio e ipu sigl x τ e expded i siil wy y ses. x τ τ 8 Now oside e pole of esiig e uow ipulse espose τ. Fo pupose uow oeffiies o see 7 sould e esied fis. Also e oupu y e expessed fo 6-8 s τ τ y dτ 9 Sie is sl 9 e wie y τ y τ τ τ τ dτ dτ Noe τ τ i is fuio ofτ d e desied y ses 6. is fo -4 ee exiss Θ Θ sisfyi τ τ Θ. e eoes τ y Θ τ dτ Sie e wvele possesses fiie pois of disoiuiy o e ouded ie doi e wvele e iegle ove e ievl. e e luled fo e followig iegio of e wvele. τ dτ By susiuig io we ve y Θ Fo siple oio le s deoe Θ y w. Fo 7 we see uow ISSN: Issue 5 Volue 7 y 8

4 WSEAS ANSACINS o CICUIS AND SYSES S.J..J. i d S.W. N oeffiies i sould e esied o ideify τ u we ve oly oe equio. o solve su pole diffee ipus e pplied o e V syse poduig oupus oseved d ledig o e followig equios o solve uow oeffiies i. y y y τ x τ dτ τ x τ dτ τ x τ dτ Θ Θ Θ w w w 4 Fueoe 4 e desied i e followig ix fo: Noe wile e ipulse espose ges pidly wi τ of Exple e se ipu d s i Exple wee used i is siulio. I Fig. e expoeilly dped siusoids i.e. ue oes d ei espeive ppoxiios. 9 d.4 : ee e peseed veifyig e poposed ppo leds o ig-quliy ipulse espose esie eve i se of pidly ie-vyig lie syses. Y W 5 wee y y y Y W w w w Aodigly τ e ieved fo 5 d 7 if W is of full. We W is o of full we eed o se up ipu d uil olu veos of W e liely idepede. 4 Siulio esuls Fig.. e ie-vyig fuio d is ppoxiio: d.5 o deose e pefoe of e poposed ppo ee V syses wi diffee o o eessily seple wi espe o is gues ipulse esposes e osideed. Exple : Coside V syse wose ipulse espose is give y τ osπ τ τ 6 Fo is siulio pieewise-os fuio ws pplied s e ipu o e V syse 6 d e oupu ws oied fo 5. Fig. illuses e ue ipulse esposes. 7 d. d ei ppoxiios esied y e poposed wvele-sed ppo ee. Exple : Coside oe V syse wose ipulse espose is give y si e τ τ π τ 7 Fig.. A ie-vyig fuio d is ppoxiio:. d.7 ISSN: Issue 5 Volue 7 y 8

5 WSEAS ANSACINS o CICUIS AND SYSES S.J..J. i d S.W. N Exple : Coside e pole of esiig e ipulse espose of V syse y vyig e esoluio. oe speifilly wveles wi diffee ues of ses 8 6 d e uilized fo e syse ideifiio d V syse wi e followig ipulse espose is osideed: e τ τ τ 8 As i Exple d Exple e se ipu d e lso uilized. I Fig. e ue ipulse espose. 7 d is ppoxiio oied y e poposed ppo e sow fo wi i e see uliesoluio lysis y wvele wi lge ue of ses e.g. yields ee ppoxiio o e ue ipulse espose oes y wvele wi slle ue of ses e.g ese iludes fue exesio of e poposed ppo o e ideifiio of olie V syses. Appedix A - e poved y usig e followig eil iduio: i.e. i 9 We is is deived fo s follows: Sie e ig side of 9 d is exly se - is ue we is. i.e. i We suppose - is ue we is. Fig.. Appoxiios d uliesoluio lysis y wvele wi 8 6 d. 5 Colusios I is ppe e pole of ideifyig V syse fo ipu d oupu d is osideed weey wvele is eployed o fo lgei equio fo e syse ideifiio fo wi wvele oeffiies fo e ipulse espose e esied. Also sie e wvele posses fiie vlue i ouded ie-doi d wi ui eegy e poposed ppo yields ee opuiol effiiey y oe wvele o y sque fuios su s Wls s. Fuue i.e. i We is e lef side of is epeseed y followig fo: ⅰ ulipliio of d ISSN: Issue 5 Volue 7 y 8

6 I is e se fo i. eefoe d e e se so is epeseed y ssupio of s followig: ⅱ ulipliio of d e ulipliio of d is epeseed s followig: 4 e ulipliio of d is e se s e ulipliio ewee d e splig of. 5 e ig d side of 5 is ewie y e digil epeseio d : dig dig 6 ⅲ ulipliio of d e ulipliio of d is epeseed s followig 7 I siil wy e ulipliio of d is se s ulipliio of d e splig of. 8 e ig d side of 8 is epeseed y d s followig: dig dig 9 ⅳ ulipliio of d e ulipliio of d is epeseed s followig: 4 Fo siple deivio le e d e fuio is defied s followig: WSEAS ANSACINS o CICUIS AND SYSES S.J..J. i d S.W. N ISSN: Issue 5 Volue 7 y 8

7 q q q 4 eefoe 4 is epeseed s followig: 4 Also 4 e expessed y digol ix. dig 4 By popeies of e wvele sfo e expessed y e lie oiio of 5. F F 44 o fid e ix F d e spled y e se splig e s followig: F F 45 Sie is lwys ivele e ivese ix is oied s followig: dig 46 is digil epeseio of olized wvele d s ivese ix 7. dig dig dig dig eefoe e ix F is oied s followig: F I F I F ix ideiy is 46 eefoe is expessed s 47 Also is epeseed s followig: dig dig 48 Fo -48 is expessed s follows: dig dig dig 49 WSEAS ANSACINS o CICUIS AND SYSES S.J..J. i d S.W. N ISSN: Issue 5 Volue 7 y 8

8 eefoe - e poved. Appedix B oof of -4 e oied y usig e followig eil iduio: i.e. i 5 e ig d side of 5 is equl o of -4 we is. 5 eefoe -4 is ue we is. i.e. i : We suppose -4 is ue we is. 5 i.e. i : We is e lef side of is epeseed y followig fo: 5 ⅰ Alysis of Fo siil eso i ⅰ of Appedix A is epeseed y followig: 54 ⅱ Alysis of Fo siil wy i ⅱ of Appedix A is epeseed y followig: dig dig 55 ⅲ Alysis of : 56 Also 56 e expessed s dig dig 57 WSEAS ANSACINS o CICUIS AND SYSES S.J..J. i d S.W. N ISSN: Issue 5 Volue 7 y 8

9 ⅳ Alysis of : Fo 4 e expessed s follows: dig 58 By is epeseed s dig F 59 I e se wy s e wie s dig F 6 F I F I ix ideiy is dig dig dig eefoe e expessed s followig: dig 6 Fo e epeseed y dig dig dig dig dig dig 6 eefoe -4 is poved y eil iduio. Aowledges is sudy ws suppoed y g of e oe el & D oe iisy of el & Welfe epuli of oe -J-G6-EV8-. efeees:.. osvi A Wvele Bsed Neul Newo fo DGS Coeios ediio WSEAS s. o syses vol. De. 4 pp. 7. Cio. uey Coli C. upy Geelisig Wvele-Bsed Eo Coeio Codig vi olypse Cosis WSEAS s. o syses vol. De. 4 pp eg.. i Ige Copessio I e Wvele Doi Usig A A exue odel Wi Copessed Iiil Codiios WSEAS s. o sigl poessig vol. Fe. 6 pp Ge d F. oeo A wvele-sed ppo fo e ideifiio of lie ie vyig dyil syses J. Soud d Viio vol. 4 o. 4 Jul. pp S.. Ce.C. i d.c. o Ideifiio of lie ie vyig syses y wvele I. J. Syse Siee vol. 7 o.9 Jul. 6 pp C.F. Ce d C.. sio wvele eod fo solvig luped d disiued-pee syse IEE o. Cool eoy Appl. vol. 44 o. J C.. sio d W.J. Wg Se lysis d pee esiio of ilie syse vi wvele IEEE s. o Ciui d Syse I: Fudel eoy d Appliios vol. 47 o. Fe.. WSEAS ANSACINS o CICUIS AND SYSES S.J..J. i d S.W. N ISSN: Issue 5 Volue 7 y 8

10 WSEAS ANSACINS o CICUIS AND SYSES S.J..J. i d S.W. N 8 D.F. ix d.j. leiz Elees of Wveles fo Egiees d Sieiss Jo Wiley & Sos. 9 I.W. Sdeg ie ps d ipulse esposes IEEE s. Ciuis d Syses vol. 5 o. Fe. 998 pp. -6..I. Dooslovči d. F Wvele-sed lie syse odelig d dpive fileig IEEE s. o Sigl poessig vol. 44 o.5 y 996. W. udi iiples of eil Alysis d ed. Gw-ill 976. ISSN: Issue 5 Volue 7 y 8