φ ( t ) = φ ( t ). The notation denotes a norm that is usually

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1 7h Europea Sigal Processig Coferece (EUSIPCO 9) Glasgo, Scolad, Augus -8, 9 DESIG OF DIGITAL IIR ITEGRATOR USIG RADIAL BASIS FUCTIO ITERPOLATIO METOD Chie-Cheg Tseg ad Su-Lig Lee Depar of Compuer ad Commuicaio Egieerig aioal Kaohsiug Firs Uiversiy of Sci ad Tech Kaohsiug, Taia cc@ccmsfusedu Depar of Compuer Sci ad Iformaio Egi Chug-Jug Chrisia Uiversiy Taia, Taia lilee@mailcjcuedu ABSTRACT I his paper, he desig of digial IIR iegraor is ivesigaed Firs, he radial basis fucio (RBF) ierpolaio mehod is described The, fracioally delayed sample esimaio of discree-ime sequece is derived by usig RBF ierpolaio approach ex, he umerical iegraio rule ad fracioally delayed sample esimaio are applied o obai he rasfer fucio of digial iegraor Fially, some umerical comparisos ih coveioal digial iegraors are made o demosrae he effeciveess of his e desig approach ITRODUCTIO Digial iegraors are useful devices i he applicaio areas of corol, radar ad biomedical egieerig []-[] The mehods of digial iegraor desig ca geerally be classified io o caegories Oe is he liear phase FIR filer approach i hich he filer coefficies are deermied by usig maximal flaess cosrais [][], he oher is he IIR filer mehod i hich he filer coefficies are obaied direcly from ell-o umerical iegraio rule [][] The ideal frequecy respose of digial iegraor is give by I D ( ω ) e () here I is a prescribed ieger delay The problem is ho o desig a digial filer such ha is frequecy respose fis D (ω ) as ell as possible I [], go preseed a hirdorder digial iegraor hose rasfer fucio is give by I z z 5 z + z F ( z ) () z I [], Tseg ad Lee have used Richardso exrapolaio ad polyphase decomposiio o desig digial iegraors From Eq() i [], he rasfer fucio of a ypical forhorder iegraor is give by F ( z z () I + z 88 9 z + z 998 z From Eq(), i is clear ha he gai of iegraor a zero frequecy ω is ifiiy, so he above rasfer fucios have oe pole a z O he oher had, radial basis fucio (RBF) has bee idely used i mulivariae ierpolaio, eural eor, ime-series predicio, corol of oliear sysems, mesh-free approximaio, ad arge racig i video daa [5]-[7] The early or o he RBF heory ad implemeaio is surveyed i he boo [8] A radial basis fucio () is a real-valued fucio hose value depeds oly o he disace from he origi, ha is, ( ) ( ) The oaio deoes a orm ha is usually ae as be Euclidea Commo used ypes of radial basis fucio iclude Gaussia ad iverse muliquadric hose defiiios are give by σ Gaussia: ( ) e (a) Iverse muliquadric: ( ) + σ here σ is called shape parameer hich ca be used o adjus he shape of fucio () The purpose of his paper is o use RBF ierpolaio mehod o desig digial IIR iegraors As a resul, he desig error ca be reduced by suiably choosig he shape parameer of radial basis fucio RADIAL BASIS FUCTIO ITERPOLATIO I his secio, he radial basis fucio ierpolaio mehod ill be described Give a se of + differe pois {,,, L, } ad a correspodig se of + real umbers { s, s, s, L, s }, he ierpolaio problem is o fid a fucio s () ha saisfies he ierpolaio codiio s ( ) s,,, L, (5) The radial basis fucio ierpolaio mehod cosiss of choosig a fucio s () ha has he folloig form ) ( ) () Tha is, he fucio s () is represeed as a sum of + radial basis fucios, each associaed ih a differe ceer, ad eighed by a appropriae coefficie Subsiuig he ierpolaio codiio of Eq(5) io Eq(), e ge he folloig simulaeous liear equaio EURASIP, 9 88

2 L s L s M M M O M M M L s here ) Le vecors S ad W be m (7) ( m T [ s s L s (8a) T [ L (8b) S ] W ] ad Φ deoes a ( + ) -by- ( + ) marix ih eleme m, he Eq(7) ca be rerie as Φ W S (9) If,,, are disic pois, he marix Φ is osigular Thus, he uo vecor W is give by W Φ S () Oce W has bee obaied, he fucio s () i Eq() is o So, s () is compuable for he give Fially, a example is use o illusrae he RBF ierpolaio mehod The daa s is obaied by uiformly samplig he coiuous siusoidal fucio co π ) ih, ha is, π s co ) The radial basis fucio is chose as Gaussia i Eq(a) ih σ The umber of pois are + Fig shos he ierpolaed fucio s () i Eq() ad siusoidal fucio co π ) I is clear ha boh fucios loo almos he same To observe here he errors occur, Fig(c) shos he absolue errors ) co π) Clearly, he errors are very small excep a he edge pois ad FRACTIOALLY DELAYED SAMPLE ESTIMATIO I his secio, e ill use RBF ierpolaio mehod o solve fracioally delayed sample esimaio problem because he proposed IIR iegraor desig mehod is based o his esimaio mehod The problem o be sudied is ho o esimae fracioally delayed sample I d) from he give ieger delayed samples s (), s ( ), s ( ),, ), here I ad are iegers ad d is a fracioal umber i he ierval [,] Ad, I is usually chose i he rage [, ] I his paper, e use eighed average approach o achieve he purpose, ha is, fracioally delayed sample is esimaed by I d ) m h ( m, d ) m ) () o, he remaiig problem is ho o use he RBF ierpolaio mehod i he precedig secio o deermie he eighs h ( d ) To solve his proble le us choose ad s ) The, he RBF ierpolaio formula i Eq() becomes s ( ) ( ) ( ( ) ) () Because ad m m are chose, e have m ( m ) ( m ) () Usig he above expressio ad s ), he simulaeous equaio i Eq(7) reduces o () () () L ( ) ) () () () L ( ) ) () M M M O M M M ( ) ( ) ( ) L () ) This equaio ca be shore as he form Φ W S, as described i Eq(9) Clearly, Φ is a symmeric ad Toepliz marix Le he iverse of marix Φ be deoed by L Φ L (5) M M M O M L he he soluio of simulaeous equaio i Eq() is give by m s ( m ) s ( ) m () s ( ) Φ m s ( m ) M M m M s ( ) m s ( m ) m This resul implies ha m s ( m ),,, L, (7) m Subsiuig Eq(7) io Eq(), e ge s ( ) m ( ( ) ) m s ( m ) ( ( ) ) (8) m ( ( ) ) s ( m ) m Taig I d, he above equaio reduces o I d) m( I d ) (9) m Compared Eq(9) ih Eq(), he eighs h ( d ) are give by ( I d ) h( d ) () Fially, give he radical basis fucio () ih shape parameer σ, ieger, ad delay I + d, he procedure o esimae fracioally delayed sample I d) from m 89

3 he give ieger delayed samples s (), s ( ), s ( ),, ) is summarized belo: Sep : Compue he marix Φ hose elemes are give by m ( m ) Sep: Calculae he iverse marix Φ ih eleme m Sep : Use Eq() o compue he eighs h ( d ) Sep : The fracioally delayed sample is esimaed by I d) m h( d) DESIG OF DIGITAL IIR ITEGRATOR I his secio, he umerical iegraio rule ad RBFbased fracioally delay esimaio mehod are used o desig digial IIR iegraor Whe a sigal s () passes hrough he ideal iegraor ih ieger delay I, is oupu y () is give by y( ) () Seig ad, e have y( ) I Usig he folloig equaliy: I s ) dτ + e ge I I ( τ () I I y( ) + ) dτ τ () Thus, he desig problem reduces o ho o evaluae he defiie iegral of he secod erm i Eq() This problem ca be solved by usig various umerical iegraio rules i exboo [9] If he rapezoidal rule is used, secod erm i Eq() ca be approximaed by I ( I ) + I )) (5) I Subsiuig Eq(5) io Eq(), e have y( ) + ( I ) + I )) () Taig he z rasform a boh sides, e obai I z ( + z ) ( (7) S ( z here S (Z ) ad Y (Z ) are he z rasforms of s () ad y () The above iegraor (Z ) is he ell-o rapezoidal iegraor i he lieraure This iegraor has large approximaio error i he high frequecy regio, so RBF ierpolaio mehod ad oher iegraio rules ill be used o improve he desig accuracy To ypical cases are sudied belo: Case : Simpso / rule Usig he Simpso / rule, secod erm i Eq() ca be approximaed by I ) + I ) dτ + I I I ) ) τ (8) Because s ( ) is a uo fracioally delayed sample, he RBF ierpolaio mehod is used o esimae is value from he o ieger delayed samples s (), s ( ),, ) The esimaio formula has bee described i Eq() ad Eq() Thus, usig Eq(), Eq(8) ca be rie as I I ) + h( ) (9) m I + I ) Subsiuig Eq(9) io Eq(), e have y ( ) y ( ) + I ) + I ) () + h( m, ) m ) m Taig he z rasform a boh sides, e obai I I m z + z + h( ) z m ( () S ( z The above ( Z ) is he desiged Simpso / iegraor usig RBF ierpolaio mehod Case : Simpso /8 rule Usig he Simpso /8 rule, secod erm i Eq() ca be approximaed by + I I ) I ) 8 + I ) + ( ) () I s I Usig fracioally delayed sample esimaio formula i Eq(), he above equaio ca be rie as I ) + I ) I () I 8 + g ( m) m here g ( m ) h( m, ) + h( m, ) () Subsiuig Eq() io Eq(), e have y( ) y( ) + I ) + I ) 8 8 (5) + g ( m) 8 m Taig he z rasform a boh sides, e obai I I m z + z + g( m) z m ( () S( 8 z The above ( Z ) is he desiged Simpso /8 iegraor usig RBF ierpolaio mehod 9

4 5 DESIG EXAMPLES AD COMPARISO I his secio, e ill sudy he desig error of he proposed RBF-based iegraor ad compare i ih coveioal iegraors To evaluae he performace, he iegral squares error of frequecy respose is defied by λπ E ( e ) D ( ω ) dω (7) Obviously, he smaller he error E is, he beer he performace of he desig mehod is Example : I his example, e firs sudy he relaio beee desig error E ad shape parameer σ for Gaussia radial basis fucio () i Eq(a) The parameers are chose as, I5, ad λ 95 Fig shos he error curve E of he proposed RBF iegraor (Z ) for σ [5, 5] From hese resuls, i is clear ha he error E is almos same as he error E Ad, boh errors reach he miimum value he σ is chose Moreover, Fig depics he magiude resposes (solid lie) of he Simpso / iegraor ( Z ) for Gaussia RBF ih σ The dashed lie is he ideal magiude respose Obviously, he specificaio is fied ell ω Example : I his example, e ill sudy he relaio beee error E ad parameer σ for iverse muliquadric radial basis fucio () i Eq The parameers are chose as, I5, ad λ 95 Fig shos he error curve E of he iegraor (Z ) for σ [5, 95] From hese resuls, i is clear ha he error E is slighly smaller ha he error E Ad, boh errors almos reach he miimum value 7 he σ 7 is chose Because his miimum value is slighly larger ha he value i Gaussia RBF case, Gaussia RBF is preferred i iegraor desig Moreover, Fig5 depics he magiude resposes (solid lie) of he Simpso / iegraor ( Z ) for iverse muliquadric RBF ih σ 7 The dashed lie is he ideal magiude respose ω Obviously, he specificaio is fied ell Compared Fig ih Fig5, i is easy o see ha he magiude resposes are almos he same for Gaussia ad iverse muliquadric RBF desig cases Example : I his example, e compare Gaussia RBF iegraor ih he go iegraor i Eq() uder he same implemeaio complexiy The desig parameers are chose as, I ad λ 95 Fig shos he error curve E of he iegraor ( Z ) From his resul, i is clear ha he error E reach he miimum value 9 he σ is chose Fig shos he frequecy respose error log F ( e ) ) The dashed lie is he error log( ( e ) ) for Gaussia RBF iegraor ih σ Obviously, ( has smaller error ha go iegraor i he frequecy bad [ 8π, π ] oever, go iegraor is beer ha Gaussia RBF iegraor i he lo frequecy bad [,8π ] Example : I his example, e compare Gaussia RBF iegraor ih he Tseg iegraor i Eq() uder he same implemeaio complexiy The desig parameers are chose as, I ad λ 95 Fig7 shos he error curve E of he iegraor ( Z ) From his resul, i is clear ha he error E reach he miimum value 57 he σ 8 is chose Fig7 shos he frequecy respose error log( F ( e ) ) The dashed lie is he error log( ( e ) ) for Gaussia RBF iegraor ih σ 8 Obviously, ( has smaller error ha Tseg iegraor i he frequecy bad [ π, π ] oever, Tseg iegraor is beer ha Gaussia RBF iegraor i he lo frequecy bad [,5π ] COCLUSIOS I his paper, he desig of digial IIR iegraor usig radial basis fucio ierpolaio mehod has bee preseed The fracioally delayed sample esimaio ad umerical iegraio rule are applied o obai he rasfer fucio of digial iegraor The umerical comparisos ih coveioal digial iegraors are also made oever, oly digial iegraor desig is sudied here Thus, i is ieresig o exeded RBF ierpolaio mehod o desig various digial filers i he fuure REFERECES [] B Kumar, D Roy Choudhury ad A Kumar, "O he desig of liear phase FIR iegraors for midbad frequecies," IEEE Tras o Sigal Processig, vol, pp5-5, Oc 99 [] B Kumar ad A Kumar, "FIR liear-phase approximaios of frequecy respose /() for maximal flaess a a arbirary frequecy ω, ( < ω < π )," IEEE Tras o Sigal Processig, vol7, pp77-775, Jue 999 [] Q go, "A e approach for he desig of idebad digial iegraor ad differeiaor," IEEE Tras o Circuis ad Sysems-II, vol5, pp9-9, Sep [] CC Tseg ad SL Lee, "Digial IIR iegraor desig usig Richardso exrapolaio ad fracioal delay," IEEE Tras o Circuis ad Sysems-I, vol55, pp-9, Sep 8 [5] S ayi, eural eor: A Comprehesive Foudaio, Secod Ediio, Preice-all, 999 [] GE Fasshauer, Meshfree Approximaio Mehods ih MATLAB, World Scieific, 7 [7] J Madapura ad B Li, Muli-arge racig based o KLD mixure paricle filer ih radial basis fucio suppor, ICASSP, pp75-78, Apr 8 [8] MD Buhma, Radial Basis Fucios: Theory ad Implemeaios, Cambridge Uiversiy Press, [9] SC Chapra, Applied umerical Mehods ih MATLAB for Egieers ad Scieiss, McGra-ill, 5 9

5 error ) co π ) (c) Fig The radial basis fucio ierpolaio The ierpolaed fucio s ( ) Siusoidal fucio co π ) (c) The absolue errors ) co π) Error E shape parameer σ Fig The error curve mehod for σ [5, 5] magiude respose 8 E of he proposed Gaussia RBF 8 Fig The magiude respose of Simpso / iegraor ( Z ) desiged by Gaussia RBF The dashed lie is he ideal magiude respose Error E shape parameer σ Fig The error curve E of he proposed iverse muliquadric RBF mehod for σ [5, 95] magiude respose 8 8 Fig5 The magiude respose of Simpso / iegraor ( Z ) desiged by iverse muliquadric RBF The dashed lie is he ideal magiude respose Error E frequecy respose error (db) shape parameer σ E Error log F ( e ) ) Fig Error curve The dashed lie is he error log ω ) ( e ) ) Error E frequecy respose error (db) shape parameer σ E Error log F ( e ) ) Fig7 Error curve The dashed lie is he error log ( e ) ) 9