Spectrum Estimation by Several Interpolation Methods
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1 IJCSNS Itertol Jourl o Computer Scece d Network Securty VOL.6 No.A Februry Spectrum Estmto by Severl Iterpolto Metods Mbu Isr Oym Ntol College o Tecology Oym-S Tocg JAPAN Summry I ts pper te expermetl results o te spectrum estmto o te Jpese speec wveorms // // /u/ /e/ /o/ obted by tree deret pproces o terpolto tt s Akm s pproxmto te sple ucto pproxmto d Lgrge s ormul re reported. I our expermets te speec wves re t rst lyzed by FFT(Fst Fourer Trsorm) tecque d te te FFT spectr re smooted or tted by vryg te umbers o dt pots te requecy rego d pplyg tree terpolto metods respectvely. Tese metods re very vlble to extrct te locl peks te speec spectrum. Te dt vlues obted by te vestgto d comprso or te terpolted spectr dcte tt Akm s pproxmto metod s st clculto o smootg process d sutble to cosder te spectrum evelopes. However sple pproxmto s useul expect or te cse o ll dt pots beg used d Lgrge s terpolto ormul s ot useul or te spectrl estmto our expermets. Key words: Sple s ucto Akm s pproxmto Lgrge s ormul Dgtl Flter. Itroducto A lrge mout o ormto s cluded um voce. As mes o cosderg te voce ormto te voce spectr ve bee studed. Te orgl spectr re smooted by usg te 6 bt persol computer. Revews o teoretcl d expermetl developmets speec spectr re gve by Mur[]. Moreover severl reserc ppers o speec spectr ve recetly bee reported or exmple suc ppers dscussed o te locl peks speec spectr[] d te speec recogto [3][4] uder te ssumpto tt te peks o speec spectr re mportt or te percepto o voce. Sple s pproxmto Te metod o estmto usg eter sple s Akm s pproxmto or Lgrge s terpolto ormul s bsclly curve ttg tecque or smootg lrge umber o rdom dt pots. I te sple pproxmto polyoml o rbtrry degree s used to process dt o speec spectr. Te polyoml coecets re estmted to t set o dt pots betwee djcet kots. I ts pper cubc polyoml pssg troug ll te gve dt pots s utlzed to obt te optml estmtes. Te cubc sple s polyoml S s to be obted to stsy te ollowg reltos tt t ll te dt pots x S ( x ) s equl to ( x ) d te rst dervtve S ( x ) d te secod oe S x ) re cotuous respectvely. S ( x ) S ( x S ( x ) S S ( x ) S ( ) ( x ) ( x ) Te S s represeted s Follows: S [ ( x x ) 6 u u (6 (6 u u u S ( x ) x x )( x x ) )( x x )] ( x x ) Moreover u c be obted by solvg te ollowg equto u 6 ( ) u u 3 Muscrpt receved Februry Muscrpt revsed Februry
2 06 IJCSNS Itertol Jourl o Computer Scece d Network Securty VOL.6 No.A Februry 006 S ( x ) u S 0 0 ( x) 0 u ( ) ( xx0 )( xx ) ( xxk )( xxk ) ( xx ) L x k x k x xk x xk xk xk x 0 k xk x ( xx ) 0 ( ) k xk x ( )( ) ( )( ) ( ) Akm s pproxmto I te Akm s pproxmto metod cubc polyoml s wdely utlzed or curve ttg t tervl ( x x ) betwee djcet two pots ( x ) d ( x ). Let ( x ) 0 L be set o dt pots. Akm s cubc polyoml P t tervl x ) s descrbed s ( x P ( x) ( ) ( ) ( ) 0 x x x x x x 3 x x ( ) m 0 3 m m m m m m m ( 3m ) ( ) m m m m Akm s polyoml s very useul s well s te sple tecque or te estmto o te optml curve. Lgrge s terpolto metod I te Lgrge s terpolto metod te polyoml pssg troug te gve pots s geerl used. Te Lgrge s terpolto polyoml P s represeted s ollows: P ( x) Σ L ( x k ) k 0 x L k ( x)( k 0 L ) Lgrge s terpolto coecets re gve by 3 I ts pper te expermetl results o te spectrum estmto o te Jpese speec wveorms // // /u/ /e/ /o/ obted by tree deret pproces o terpolto tt s Akm s pproxmto te sple ucto pproxmto d Lgrge s ormul re reported. I our expermets[6] te speec wves re t rst lyzed by FFT(Fst Fourer Trsorm) tecque d te te FFT spectr re smooted or tted by vryg te umbers o dt pots te requecy rego d pplyg tree terpolto metods respectvely. Tese metods re very vlble to extrct te locl peks te speec spectrum. Te dt vlues obted by te vestgto d comprso or te terpolted spectr dcte tt Akm s pproxmto metod s st clculto o smootg process d sutble to cosder te spectrum evelopes. However sple pproxmto s useul expect or te cse o ll dt pots beg used d Lgrge s terpolto ormul s ot useul or te spectrl estmto our expermets. Moreover te typcl exmple o te dgtl lter s CIC(Cscded Itegrtor d Comb) lter[7]. It s usg te computto o te movg verge or ts tecque. Ts tecque c be used or te dgtl lter.. Expermetl Results Te legcy computer s moter-bord (CPU:8086 (0MHz) memory 640kByte) s utlzed or speec sgl processg d te system progrm s wrtte bot BASIC d mce lguges. But te progrm or clcultg te polyoml coecets s wrtte BASIC lguge. Te bt A/D coverter s used to dgtze te log speec sgls t te smplg requecy o 0 khz[5].. Sple ucto pproxmto Cubc sple polyoml s bee used to terpolte te dt pots o voce spectr o Jpese ve vowels obted by FFT tecque. Fgure sows spectrl curve obted by tkg dt pots every oter spectrl pot. It s see Fgure tt lmost te sme voce spectr s tose o te orgl sgls ve bee obted.
3 IJCSNS Itertol Jourl o Computer Scece d Network Securty VOL.6 No.A Februry Tme requred or clcultg ts curve ttg s bee bout sec. Fg.3 Relto betwee speec spectrum o vowel o // d sple ucto. (I cse o every seve) Fg. Relto betwee speec spectrum o vowel o // d sple ucto. (I cse o every oter) Fgure sows spectrl curve obted by tkg dt pots every tree spectrl pots. As s see Fgure te e structure o te spectr s bee verged d te smooted evelope s bee obted. Te clcultg tme ts cse s bee bout 5 sec.. Akm s pproxmto Akm s cubc polyoml s bee used to terpolte te dt pots o voce spectr o Jpese ve vowels obted by FFT tecque. Fgure 4 sows spectrl curve obted by dt pots. Te voce spectr Fgure 4 re te sme s tose o te orgl sgls. Tme requred or clcultg ts curve ttg s bee bout sec. Fg. Relto betwee speec spectrum o vowel o // d sple ucto. (I cse o every tree) I Fgure 3 te spectrl curve obted by tkg dt pots every seve spectrl pots s sow. Fe structure o te orgl spectr s verged out but te loctos or te mxml d mml vlues re sucetly dstgused. Te clcultg tme ts cse s bout 3 sec. Fg.4 Relto betwee speec spectrum o vowel o // d Akm ucto. (I cse o sme dte pot) Fgure 5 sows spectrl curve obted by tkg dt pots every oter spectrl pots. It s see Fgure 5 tt lmost te sme voce spectr s tose o te orgl sgls s bee obted. Tme requred or clcultg ts curve ttg s bee bout 6 sec. Fgure 6 sows spectrl curve obted by tkg dt pots every tree spectrl pots. As s see Fgure 6 te e structure o te spectr s bee verged d te smooted evelope s bee obted. Te clcultg tme ts cse s bout 3 sec.
4 08 IJCSNS Itertol Jourl o Computer Scece d Network Securty VOL.6 No.A Februry 006 re sucetly recogzed. Te clcultg tme ts cse s bout sec..3 Lgrge s ormul Fg.5 Relto betwee speec spectrum o vowel o // d Akm ucto. (I cse o every oter) Tree kds o terpolto metod ve bee ppled to te voce spectr obted by FFT. Te etures d temporl vrto o te voce spectr ve bee vestgted. As te results t s sow tt te cubc sple polyoml d te Akm s ucto re te optml tecques or curve ttg. It s reveled by te smulto tt te polyoml o Akm s superor to oters te clculto tme. My ttempts or speec recogto usg te locl peks o voce ve recetly bee reported. It s cosdered tt te peks o te voce spectr could probbly be recogzed eve te umbers o dt pots re ew. Fg.6 Relto betwee speec spectrum o vowel o // d Akm ucto. (I cse o every tree) Fg.8 Relto betwee speec spectrum o vowel o // d Lgrge ucto. (I cse o every oter) Fg.7 Relto betwee speec spectrum o vowel o // d Akm ucto. (I cse o every seve) Fg.9 Relto betwee speec spectrum o vowel o // d Lgrge ucto. (I cse o every tree) I Fgure 7 te spectrl curve obted by mkg use o dt pots every seve spectrl oes s sow. Te e structure o te orgl spectr s verged out but te loctos or te mxml d mml vlues o spectr
5 IJCSNS Itertol Jourl o Computer Scece d Network Securty VOL.6 No.A Februry Fg.0 Relto betwee speec spectrum o vowel o // d Lgrge ucto. (I cse o every seve) We clculte t g speed we use te ltest computer. We ted ereter to vestgte detl te relto betwee te umbers o dt pots d te evelope o te voce spectr..4 Clcultg tme Te dt vlues obted by te vestgto d comprso or te terpolted spectr dcte tt Akm s pproxmto metod s st clculto o smootg process d sutble to cosder te spectrum evelopes. We sow te comprso o te computto speed te gure. Te expermet o us c be used s te dgtl lter. Clcultg Tme [Sec] Lgrge's ormul Sple ucto 3 7 Dt Pots Akm's pproxmto Fg. Relto betwee clcultg tme d dt pots 3. Cocludg Remrks Tree kds o terpolto metod ve bee ppled to te voce spectr obted by FFT. Te etures d temporl vrto o te voce spectr ve bee vestgted. As te results t s sow tt te cubc sple polyoml d te Akm s ucto re te optml tecques or curve ttg. It s reveled by te smulto tt te polyoml o Akm s superor to oters te clculto tme. My ttempts or speec recogto usg te locl peks o voce ve recetly bee reported. It s cosdered tt te peks o te voce spectr could probbly be recogzed eve te umbers o dt pots re ew. We clculte t g speed we use te ltest computer. Te expermet o us c be used s te dgtl lter. We ted ereter to vestgte detl te relto betwee te umbers o dt pots d te evelope o te voce spectr. Reereces [] T.Mur etc : Audtory d Speec IECE(983) ( Jpese). [] T.Mtsuok d K.Kdo : Ivestgto o Poemc Iormto o Sttc Propertes o Locl Peks te Speec Spectr J.Accost. So. Vol.3 No. pp.-3 (976) ( Jpese). [3] J.Mw Y.Ntsu S.mko d K.Kdo : Spoke Word Recogto System Usg Gross Fetures o Speec Spectrum d Tese Dymc Propertes J.Accost. Soc. Vol.34 No.3(978) ( Jpese). [4] N.Iked T.Aok H.Kog d K.kdo : Recogto o Vowels Usg te Locl Peks FFT Spectrum J.Accost. Soc. Jp. Vol.4 No. pp (985) ( Jpese). [5] J.Srtk d M.Isr : Speec Alyss System by Use o Persol Computer Reserc Reports o Ikutoku Teccl Uversty prt B No. pp.7-3(987) ( Jpese). [6] M.Isr J.Srtk S.Ier d Y.Skt : Spectrum Estmto o Speec by Severl Iterpolto Metods Proceedgs o Te Frst C-Jp Itertol Symposum o Istrumetto Mesuremet d Automtc Cotrol pp (989). [7] Sogo Nkmur : Dgtl Sgl Processg Tokyo Dek Pub.pp.40-54(004) ( Jpese). Mbu Isr receved te B.E. degrees Electrcl Egeerg rom Ikutoku Teccl Uversty Jp 98 respectvely. He receved te M.E. d P.D. degrees Electrcl Egeerg rom Mese Uversty Jp 984 d 987 respectvely. Now He become Assocte Proessor te Deprtmet o Electrcl d Computer Egeerg Oym Ntol College o Tecology. Hs reserc multmed computer etwork um egeerg. He s member o te IEEE te IEICE te IEE o Jp. d ASJ.
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