WSEAS TRANSACTIONS on CIRCUITS n SYSTEMS Kzui Kgihi Kenihi Ie Miueru Nur Kzuo Torihi Yuhiro Ohiy Hioi Muri Quri Flueny DA Funion Non-unifor Spling Funion for Inerpoling Sple-vlue KAZUKI KATAGISHI KENICHI IKEDA MITSUTERU NAKAMURA KAZUO TORAICHI YASUHIRO OHMIYA HITOMI MURAKAMI Grue Shool of Sye n Inforion Engineering Univeriy of Tuu Tennoui -- Tuu-hi Iri - JAPAN gii@uujp {ie nur yohiy}@wlriuujp hp:wwwwlriuujpinex-ehl Flueny R&D Lorory Univeriy of Tuu Tennoui -- Tuu-hi Iri - JAPAN orihi@iluujp hp:wwwwlriuujpinex-ehl Fuly of Siene n Tehnology Seiei Univeriy -- Kihijoji-ihi Muhino-hi Toyo - JAPAN hi-uri@eieijp Ar: - Inerpolion for ple-vlue wih non-unifor pling poin i require for vriou e of ignl proeing In uh e pling funion re ueful o inerpole ple-vlue n hen o genere ignl liner oinion of he pling i weighe y equene of he ple-vlue Thi pper propoe pling funion for non-unifor pling poin eh of whih i opoe wih pieewie polynoil of egree We ne he pling funion he flueny DA funion of egree The flueny DA funion genere ooh n unule ignl fro equene of ple-vlue Key-Wor: - Flueny inforion heory Flueny DA funion Inerpolion Non-unifor pling funion Pieewie polynoil Inrouion Muliei uh uio ill ige n vieo whih exi in he rel worl i generlly ree nlog ignl In orer o re he nlog ignl in he opuer worl hey u e onvere ino igil ignl The igil ignl re onvere ino nlog ignl n hen originl uliei i reproue Therefore oh of nlog-o-igilad onverer n igil-o-nlogda one ply iporn role in ignl proeing In he onvenionl ignl nlyi n proeing Inforion Couniion n TehnologieICT uh AD n DA ehnologie hve een eigne in he nlyi funion pe S upe of ypil Hiler pe L where L i he pe pnne y qure inegrle funion Shnnon unifor pling heore whih gurnee ioorphi eween n-liie nlog ignl pe n igil ignl one of equene of ple-vlue i well-nown n i lo oniere in he nlyi funion pe S One of uhor propoe n elihe Flueny Inforion Theory h generlize Shnnon pling heore The Flueny Inforion Theory e ignl nlyi n proeing re oniere in he ul pe for he funion pe pnne y pieewie polynoil Thi pper propoe pling funion for non-unifor pling poin eh of whih i opoe wih pieewie polynoil of egree ISSN: 9- Iue Volue Jnury 9
WSEAS TRANSACTIONS on CIRCUITS n SYSTEMS Kzui Kgihi Kenihi Ie Miueru Nur Kzuo Torihi Yuhiro Ohiy Hioi Muri We ne he pling funion he flueny DA funion of egree The flueny DA funion re eigne e on geoeri rierion of urve The flueny DA funion genere ooh n unule ignl fro equene of ple-vlue Preliinrie Signl Spe D opoe of Pieewie Polynoil of Degree - In he onvenionl ignl nlyi n proeing Inforion Couniion n TehnologieICT uh AD n DA ehnologie hve een eigne in he nlyi funion pe S upe of ypil Hiler pe L where L i he ignl pe pnne y qure inegrle funion Dir el funion hve een ofen ue in ing iuion on ioorphi propery eween nlog ignl n igil one Moreover in n o funion hve een lo ue in DCT-e uliei oing lie JPEG n MPEG However hee funion for ignl nlyi n proeing o no elong o L So in reing hee in of funion i i neery o expn he onvenionl ignl pe L If X i funion pe we n efine i ul pe X o e he e of oninuou liner funion T fro X o R or C where R n C re he e of rel n oplex nuer repeively Suh pping heelve for nore liner pe uing he operor nor T up Tx x X x x If X Y hen Y X ine here re fewer oninuou funion on lrger funion pe Therefore highly rerie Shwrz funion pe S whih i he e of rpily ereing funion ie he funion x x ifying he n : following wo oniion for eh n <> li x n h very lrge ul pe The ul pe S for he Shwrz funion pe S i lrger hn L However he ul pe S i oo epere We inroue pproprie ignl pe D for he ignl nlyi whih i opoe of pieewie polynoil of egree - wih only - ie oninuou iffereniiliy in hi pper where { } In e of he ignl pe D i funion pe pnne y ioninuou funion In e of he ignl pe D i funion pe pnne y oninuou funion whih re no ifferenile I h een hown h he ignl pe D ienil wih n-liie funion pe whih i ree in he Shnnon unifor pling heore when he preer en o infiniy Be on hi f i ee poile o el wih pieewie polynoil funion pe n n-liie funion one unifie erie of ignl pe of whih hrerii vry wih he preer of egree of he polynoil Thi erie i fluen in he ene h we n hooe ignl pe ou of he erie whih he wih eh purpoe of ignl nlyi n proeing So i w ne flueny The ignl pe D D n D re ienil wih he e of ire polygonl n n-liie funion repeively The Flueny inforion heory-e ignl nlyi n proeing re oniere in he ul pe D for he ignl pe D The ul pe D onin rirry erivive of erin ioninuou funion Figure how ignl pe n i ul pe <> x C ISSN: 9- Iue Volue Jnury 9
WSEAS TRANSACTIONS on CIRCUITS n SYSTEMS Kzui Kgihi Kenihi Ie Miueru Nur Kzuo Torihi Yuhiro Ohiy Hioi Muri L D S Signl pe S D L L ul pe for eh ignl pe Fig Signl pe n i ul pe The ignl pe D n i ul pe D re pproprie funion pe for ignl nlyi n proeing Le e pling poin hen he pling i in he ignl pe D i efine y he funion { DA ifying } u D u u DA Equion give repreenion forul liner oinion of he pling i in D weighe y equene of ple-vlue { u } We ne eh funion of he pling i DA he Flueny DA funion In e h he inervl eween jen pling poin i onn h i h h > pling funion in D re lle y unifor Flueny DA funion of egree - In e h he inervl i no onn hen hoe re lle y non-unifor Flueny DA funion of egree - in hi pper Coply Suppore Unifor Flueny DA funion of Degree We propoe n evelope n ipule repone h i uile for DVD-Auio wih xiu pling re of 9KHz I h een eigne in he ul pe D for he ignl pe D The ipule repone i opoe of he oply uppore unifor Flueny DA funion of egree Prilly DVD-Auio plyer equippe wih he Digil-o-Anlog onverer eigne y he unifor Flueny DA funion of egree hve een oerilize The wr hve een reeive There hve een ny ICT ppliion 9 eigne in he ignl pe D The quri unifor Flueny DA funion DA pling funion in he ignl pe D w eigne i ifie he following oniion <><><> n <> <> I i repreene y he liner oinion of quri B-pline funion <> I i only one ie oninuouly ifferenile <> I onverge o he lef n righ eon pling poin fro he origin h i h n h <> I e he vlue of he origin I e he vlue of pling poin ± h± h Le ϕ enoe he quri B-pline funion efine follow : inπfh j πf ϕ e f πfh The quri B-pline funion i expree y pieewie polynoil of egree Then he oply uppore unifor flueny DA funion of egree DA i repreene in he for of liner oinion of he funion ye { ϕ l } h l follow: h h φ h hφ φ h DA The DA funion w erive DA ISSN: 9- Iue Volue Jnury 9
WSEAS TRANSACTIONS on CIRCUITS n SYSTEMS Kzui Kgihi Kenihi Ie Miueru Nur Kzuo Torihi Yuhiro Ohiy Hioi Muri DA h h h < h h h h < h h h h < h h h < h < h h h h < h h h h < h h h h < h oherwie Figure how he quri unifor Flueny DA funion DA Fig Quri unifor Flueny DA funion I i noe h he quri unifor Flueny DA funion DA i only one ie oninuouly ifferenile ± h ± h ± h ± h whih re onneing poin of eh pieewie polynoil The pling i { } DA in he ignl pe D re erive follow DA -h -h -h h h h Crierion for Deigning Coply Suppore Non-Unifor Flueny DA Funion Copoe of Quri Pieewie Polynoil Forulion of Coply Suppore Non-Unifor Flueny DA Funion of Degree The non-unifor Flueny DA funion of egree i eigne y expning he oply uppore unifor Flueny DA funion of egree A i uneroo fro Eq he unifor Flueny DA funion of egree h i DA n e generlly oniere o e opoe of pieewie polynoil in -h h We forule oply uppore non-unifor Flueny DA funion of egree y in hi pper The funion i eigne i ifie he following oniion < >< >< > n < > < > I i repreene y he liner oinion of quri pieewie polynoil < > I i only one ie oninuouly ifferenile < > I onverge o he lef n righ eon pling poin fro he origin h i n < > I e he vlue of he origin I e he vlue of pling poin Ting oun of he ove oniion < > n < > he funion n e forule follow h @ @ ± DA DA Thu he ny ignl u in D i repreene y DA DA u D u u uh h ISSN: 9- Iue Volue Jnury 9
oherwie Figure how generl wvefor of he funion Fig Quri Non-unifor Flueny DA Funion Ting oun of he ove oniion < > he following relion re oine 9 Ting oun of he ove oniion < > he following relion re oine 9 Moreover fro Eq- he following iulneou equion Eq onerning o unnown preer n re erive 9 9 - - WSEAS TRANSACTIONS on CIRCUITS n SYSTEMS Kzui Kgihi Kenihi Ie Miueru Nur Kzuo Torihi Yuhiro Ohiy Hioi Muri ISSN: 9-9 Iue Volue Jnury 9
The unnown preer n re oine y olving he ove iulneou equion Fro Eq n n re oine follow: 9 Fro Eq n n re oine follow: Fro Eq9 n n re oine follow: Fro Eq n n re oine follow: Fro Eq n n re oine follow: Fro Eq n n re oine follow: Fro Eq n n re oine follow: Fro Eq9 n n re oine follow: A he reul i ee h unnown preer n n e oien However preer n efine y Eq n repeively n e rirrily e I i noe h Eq h no een ue in he ove proee Thi en h he preer n re no inepenen By uiuing WSEAS TRANSACTIONS on CIRCUITS n SYSTEMS Kzui Kgihi Kenihi Ie Miueru Nur Kzuo Torihi Yuhiro Ohiy Hioi Muri ISSN: 9- Iue Volue Jnury 9
WSEAS TRANSACTIONS on CIRCUITS n SYSTEMS Kzui Kgihi Kenihi Ie Miueru Nur Kzuo Torihi Yuhiro Ohiy Hioi Muri n for Eq he following relion i oine We onier rierion for eiing ny hree preer ou of n in he nex eion Crierion for eiing Coply Suppore Non-Unifor Flueny DA Funion of Degree We onier how i he oply uppore non-unifor Flueny DA funion of egree oine in he eion in e h he pling inervl i onn h i h A i uneroo fro Eq he quri unifor Flueny DA funion h he propery of DA So in eiing he funion he propery of i lo ue By pplying he propery n o Eq9 he following relion i oine Moreover for rirry ineger hol goo Fro Eq n he following relion 9 re erive Moreover y uing he propery h he quri unifor Flueny DA funion re yery h i DA DA he relion n hol goo When we pu he following relion i erive A he reul he quri unifor Flueny DA funion i repreene follow: @@@@@@@@@@@@@@@@@@ @ @@@@@@@@@@@@@@@@@ @ @ oherwie Propoiion Uner he oniion of ± ± he quri non-unifor Flueny DA funion i ienil wih in he rierion of { } in DA Proof By uiuing of Eq for he relion { } he following relion {} { } { } { } { } { } { } { } { } 9 i oine A he reul he preer whih 9 iniize Eq i oine When we 9 uiue for of Eq he unifor Flueny DA funion of egree i ienil wih DA erie in ueion QED ISSN: 9- Iue Volue Jnury 9
WSEAS TRANSACTIONS on CIRCUITS n SYSTEMS Kzui Kgihi Kenihi Ie Miueru Nur Kzuo Torihi Yuhiro Ohiy Hioi Muri A i erie in eion he oply uppore unifor Flueny DA funion of egree i ueful for genering nlog ignl oninuou ignl fro igil ignl iree ignl We ue he rierion of { } in o eign he oply uppore non-unifor Flueny DA funion of egree in hi pper Deign of Coply Suppore Non-Unifor Flueny DA Funion of Degree e on Geoeri Crierion of Wvefor In he previou eion he rierion for eigning non-unifor Flueny DA funion of egree i iue We ue he rierion of { } in ou of n o eie ny hree preer Non-Unifor Flueny DA Funion of Degree wih The non-unifor Flueny DA funion of egree i forule y in eion The preer n re no fixe So ing oun of he oniion he following relion re oine fro Eq The non-unifor Flueny DA funion of egree i reue y uiuing n of Eq for Eq9- follow } { { } oherwie A i uneroo fro Eq he preer n re no fixe Non-Unifor Flueny DA Funion of Degree wih n { } in The non-unifor Flueny DA funion of egree i eie y uing he rierion of { } in By uiuing Eq for { } we ge {} { } { } {} Be on he rierion of { } in he preer n re fixe follow: ISSN: 9- Iue Volue Jnury 9
WSEAS TRANSACTIONS on CIRCUITS n SYSTEMS Kzui Kgihi Kenihi Ie Miueru Nur Kzuo Torihi Yuhiro Ohiy Hioi Muri Be on he geoeri rierion of wvefor ll of he four preer n re fixe fro Eq n Therefore he oply uppore non-unifor Flueny DA funion of egree i erive follow: { } { } oherwie Figure eonre he non-unifor Flueny DA Funion of egree for pling poin Sine he non-unifor Flueny DA funion of egree i oine in Eq he inerpolion of ny ignl u wih non-unifor pling poin D i lo forule in Fig An exple of he non-unifor flueny DA funion Propoiion Le DA enoe he non-unifor Flueny DA funion of egree eh non-unifor pling poin ± Then ny ignl u in D i expree DA u D u u where eh Flueny DA funion DA i expree y eigh quri pieewie polynoil follow l l } { { } oherwie ISSN: 9- Iue Volue Jnury 9
WSEAS TRANSACTIONS on CIRCUITS n SYSTEMS Kzui Kgihi Kenihi Ie Miueru Nur Kzuo Torihi Yuhiro Ohiy Hioi Muri By uing Propoiion ny ignl u wih non-unifor pling poin in D e inerpole Figure eonre n inerpolion y uing Propoiion DA DA Inerpole ignl ple - 9 Fig Inerpolion y uing he non-unifor flueny DA funion Diuion Thi eion iue inerpolion reul y uing he non-unifor Flueny DA funion whih i erive in ueion In orer o evlue heir effeivene inerpolion reul y uing ui pline re lo eonre In he fiel of opuer grphi ui pline hve een ofen ue o inerpole equene of ple-vlue wih non-unifor pling poin Thi i eue ui pline u inerpole he ple-vlue wih he o ooh in he ene of { u } { u } DA { u } in The urve oohne { u } in Eq i n upper oun of qure of urve urvure Figure n eonre inerpolion reul for in of ep funion n for ypil e repeively Inerpole urve y uing he non-unifor Flueny DA funion re rwn wih DA oli line n hen hoe y uing ui pline re rwn wih oe line - propoe eho ui pline ple - - Fig Inerpolion reul for ep funion propoe eho ui pline ple Fig Inerpolion reul for ypil e A re uneroo fro hee reul we rw inerpole urve wih le overhoo or unerhoo Furherore we evlue he inerpole reul fro he view of urve lengh For fig he rio of urve lengh y ui pline o urve lengh y propoe eho i pproxiely 9 Furherore for fig he rio i pproxiely Fro hee reul inerpole urve y non-unifor flueny DA funion p hrough ple-vlue horer hn hoe y ui pline Conluion Thi pper propoe flueny DA funion pling funion for non-unifor pling poin 9 ISSN: 9- Iue Volue Jnury 9
WSEAS TRANSACTIONS on CIRCUITS n SYSTEMS Kzui Kgihi Kenihi Ie Miueru Nur Kzuo Torihi Yuhiro Ohiy Hioi Muri eh of whih i opoe wih pieewie polynoil of egree Eh of he w eigne e on geoeri rierion of urve The flueny DA funion of egree genere ooh n unule ignl fro equene of ple-vlue Anowlegeen Thi reerh w prilly uppore y he reerh fun of R&D uppor hee for funing elee IT propol fro he Nionl Iniue of Inforion n Couniion Tehnology NICT The uhor woul lie o nowlege here he orgnizion Referene: ET Whier On he Funion whih re repreene y he Expnion of he Inerpolion-Theory Pro Royl Soiey of Einurgh 9 pp-9 CE Shnnon Mheil Theory of Couniion Bell Sye Tehnil Journl Vol 9 pp 9- MK KTorihi n RMori Perioi pline orhogonl e J Approx Theory Vol 9 pp - KKgihi KTorihi M O n KW A pril le qure pproxiion e on iorhogonl expnion in he ignl pe of pieewie polynoil Trn IEE of Jpn Vol-C No 99 pp - KTorihi n M K A noe on onneion eween pline ignl pe n n-liie ignl pe Trn IEICE VolJ-A No9 99 pp - KTorihi n K Nur Spling Funion of Degree for DVD-Auio IEEJ Trn EIS Vol No pp 9-9 T Mooy T Kwe K Torihi n K Kgihi New Inegre Deign Approh of RHC wih Apive DA Converer WSEAS Trnion on Sye Iue Vol pp9-9 K Kgihi K Ie M Nur K Torihi Y Ohiy n H Muri Flueny DA Funion Non-unifor Spling Funion for Inerpoling Sple-vlue New Ape of Cirui Proeeing of he h WSEAS Inernionl Conferene on CIRCUITS Herlion Greee July - pp-9 9 M Nur Y Ohiy K K Kgihi Y Moroo K Torihi n H Muri A Seure Coing for Funion-Approxie Ige New Ape of Couniion Proeeing of he h WSEAS Inernionl Conferene on CIRCUITS Herlion Greee July - pp- M Higuhi S Kwi K Kgihi M Nur K Torihi n H Muri A Deign Meho of Nrrow Bn FIR Filer Be on Flueny Spling Funion Copuionl Engineering in Sye Appliion Selee Pper fro he WSEAS Conferene in Herlion Greee July - pp-9 IJShoenerg Conriuion o he prole of pproxiion of equiin y nlyi funion Qur Appl Mh Vol pr A 9 pp-99; pr B9 pp- IJShoenerg Crinl Spline Inerpolion Soiey of Aerin Mhei 9 ISSN: 9- Iue Volue Jnury 9