UMCS. Annales UMCS Informatica AI 5 (2006) Digital signals analysis with the LPC method

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1 Pobrane z czasosma Annales AI- Informatca htt://a.annales.umcs.l Annales Informatca AI 5 (2006) Dgtal sgnals analyss wth the LPC method Ireneusz Codello *, Wesława Kunszy-Jóźowa Annales Informatca Lubln-Polona Secto AI htt:// Mara Cure-Słodowsa Unversty, Pl. M.Cure-Słodowse, Lubln, Poland Abstract Ths aer concerns the ssue of sgnal analyss by the lnear redctve method. It s not only about frequency analyss, that s obtanng the LPC sectrum and comarng t wth the Fourer one, but also about analysng redcton coeffcents themselves. We wll also dscuss the rogram made by the authors of ths aer, whch besde foregong functonaltes, enables vocal tract vsualzaton, modelled on the bass of LPC coeffcents.. Introducton The LPC method [-4] s based on the fact that consecutve samles of seech sgnal are not random, but they change smoothly. In connecton wth ths, we can aroxmate a consecutve samle (that s the next value of sound fle) wth the revous samles v( n) = α s( n ), = where: s(n) n-th nut samle, v(n) aroxmaton of n-th samle, α redcton coeffcents, redcton order. As t can be seen, the goal of ths method s to fnd a set of α coeffcents whch gves the best aroxmaton. Therefore a redcton error e( n) = s( n) v( n) should be the smallest. We have to remember about the redcton order (whch s equvalent to the samle count used n aroxmaton), whch s a very mortant element. Usually ts value les n the <0, 25> range. Another thng we have to remember about s that the sgnal has to be statonary otherwse there s no ont n usng ths method. Snce human seech does not meet ths condton, we have to slt a sgnal nto ms eces. In such short tme our seech s almost nvarable quas-statonary. * Corresondng author: e-mal address: re.codello@gmal.com

2 Pobrane z czasosma Annales AI- Informatca htt://a.annales.umcs.l 36 Ireneusz Codello, Wesława Kunszy-Jóźowa 2. Alcaton The frst, most obvous alcaton s seech synthess. As an examle we can menton LPC-0 format, where havng redcton coeffcent, gan arameter and nowledge f current seech ece s voced or unvoced, we are able to reconstruct the sgnal. Another alcaton, drectly resultng from the frst one, s data comresson. What s a ont n eeng the orgnal wave fle, whch conssts of samled wndows (t corresonds to ms wndow by Hz samlng frequency) f we can use ts 2-26 numbered reresentaton (t deends on the LPC coeffcent count on whch we wll decde). LPC arameters can be used n frequency analyss. A sectrum, created wth ths method, s very smooth n comarson to ts Fourer equvalent, whch maes t more useful. In addton, on the bass of LPC coeffcents we can model the shae of the vocal tract. 3. Lnear Predcton Algorthm The goal of the LPC algorthm [-5] s to mnmze redcton error E for each wndow, where 2 2 E = e ( m) = ( s( m) v( m) ) = s( m) αs( m ). m m m = We can acheve ths by comutng artal dervatves for consecutve α coeffcents and comarng them to zero. Of a few avalable algorthms we chose the autocorrelaton method. It assumes that samles outsde the range <0, N-> equals zero. In order to acheve ths, we can aly wndowng wth the rectangular wndow. Better results gve more comlcated wndows le Hammng or Hann wndow. Usng the autocorrelaton functon N R( ) = s( n) s( n+ ) N n= 0 whch we ut n the afore-mentoned set of artal dervatves equatons, we obtan a set of equatons whch can be exressed n a matrx form: R( 0) R( ) R( 2 )... R( ) α R( ) R() R( 0) R()... R( 2) α 2 R( 2) R( 2) R( ) R( 0 )... R( 3) α 3 = R( 3) R( ) R( 2) R( 3 )... R( 0) α R( ) Because the matrx of autocorrelaton values s a symmetrc Toeltz matrx, we can tae advantage of the effectve Levnson-Durbn algorthm:

3 Pobrane z czasosma Annales AI- Informatca htt://a.annales.umcs.l Dgtal sgnals analyss wth the LPC method 37 E 0 ( 0) = R() α R( ) / E, = = R α = α = α α, ( ) = 2 E E, where the uer ndex reresents teratons from to and a result s defned as ( α ) = α,, E and arameters, used n ths algorthm, are mortant as well. E coeffcent s the afore-mentoned redcton error, whereas are called PARCOR coeffcents we wll dscuss them further n ths artcle. To ensure that our results are correct, we too advantage of the followng deendences []: R, α E ( 0) α ( ) E R R = = Fg.. From the to: nut sgnal, redcton error e(n), changes of 20 α coeffcents for the samle mleo est zdrowe. Wndow wtdh 29 ms. The screenshot comes from a rogram created by the authors of ths aer α = a () () () () a ( ) + a a a =, 2

4 Pobrane z czasosma Annales AI- Informatca htt://a.annales.umcs.l 38 Ireneusz Codello, Wesława Kunszy-Jóźowa where the uer ndex reresents teratons from down to and ntally we set ( ) a = α,. α () a = () ( ) ( a ) = a + a, where the uer ndex reresents teratons from to and a result s defned as ( α = a ),. 4. Sectrum Transfer functon reresentng our system s of the form [] S( z) G H( z) = =, U( z) α z where α redcton coeffcents, redcton error, G gan arameter. If we set redcton order (.e. 5) and we comute redcton coeffcents α, we only have to obtan the gan arameter G from formula [] G= E. Frequency characterstcs can be obtaned from a modulus of the transfer functon, when we use the arguments of the form e ω, whch gves ω 2 π f / H e = H e F, = ( ) ( ) where magnary unt, F samlng frequency, f nterestng frequency. The recson of ths sectrum deends on the redcton order. It can be seen by comarng, n decbel scale, the Fourer sectrum 20log 0 S(e ω ) wth the LPC sectrum 20log 0 H(e ω ). On these grahs we can see a frayed Fourer sectrum and a smooth LPC sectrum, whose shae deends on the redcton order. Due to ths smoothness, t s much easer to fnd local maxma, whch, n turn, are essental n forma detectng.

5 Pobrane z czasosma Annales AI- Informatca htt://a.annales.umcs.l Dgtal sgnals analyss wth the LPC method 39 Fg. 2. Comarson of LPC sectrum, for varous redcton orders, wth the Fourer sectrum. In both cases we used Hammng wndow of dentcal wdth. Screenshots come from a rogram created by the authors of ths aer 5. Vocal tract Human seech conssts of three characterstcs []: tye of glottal exctaton: voced(erod mulses) or unvoced, vocal tract shae: modulates exctaton sgnal, Set of arameters affectng emsson of the sound wave.e. mouth radaton, envronment densty. A vocal tract s smly the secton from larynx to ls. Its shae s qute comlcated t deends on the shae of the tongue, ls, harynx, and teeth dlaton. Addtonally walls of the tract have absorton roertes and t forms an arc wth the dlaton angle near 90. We have used a smlfed model of the vocal tract, whch conssts of a row of varous dameter cylnders connected wth each other [,3,4]. They form a lossless tube wth varyng cross sectons.

6 Pobrane z czasosma Annales AI- Informatca htt://a.annales.umcs.l 320 Ireneusz Codello, Wesława Kunszy-Jóźowa A A2 A3 A4 Fg. 3. Vocal tract model. A sgnal roagates from left to rght. A-A7 are consecutve values of cross sectons of the tube The left sde of the tube n fgure 3 reresents the larynx, whle the rght sde reresents ls. On a uncton of every ar of cylnders a artal reflecton of a sound wave aears. It turns out that reflecton coeffcents on a ont of two cross sectons, reresented by the formula r = ( A+ A) ( A+ + A), are related to PARCOR coeffcents through a smle relaton r = []. From these two equatons, t s smle to show that A+ = A. + Therefore we obtaned the relaton between successve cross sectons (as we can see, the number of these sectons s greater by one from the redcton order). We are left wth the ssue of choosng the value of the frst cross secton A or the last A +, and decdng how long a vocal tract should be. In ths aer we assumed, that A + (ls area) wll be equal to cm 2 and the vocal tract length wll have an average length of a full grown man that s 7.5 cm []. A5 A6 A7 Fg. 4. 3D vocal tract model. The rght sde reresents the ls. From the to: vowel u, =5, vowel e, =20, vowel, =20. Screenshots come from the rogram created by the authors of ths aer

7 Pobrane z czasosma Annales AI- Informatca htt://a.annales.umcs.l Dgtal sgnals analyss wth the LPC method 32 Conclusons Our man goal was to create a tool to analyse wave sound fles by the LPC method and to obtan va that tool the largest set of nformaton. Those results, that s LPC, PARCOR coeffcents and sectrum, wll be obtaned from fluent and dysfunctonal seech and then analysed. We hoe to dstngush, owng to ths research, the characterstc features n varous tyes of dysfunctonaltes. We are also lannng further research, based on these analyses.e. seech and seaers recognton. Acnowledgements Scentfc wor artally fnanced from the grant of Vce-Rector of Mara Cure-Słodowsa Unversty. The authors than Natala Fedan for language correctons. References [] Rabner L.R., Schafer R.W., Dgtal Processng of Seech Sgnals, Prentce-Hall, Inc., New Jarsey, (978). [2] Zelńs T.P. Od teor do cyfrowego rzetwarzana sygnałów, Wydzał EAIE AGH, Kraów, (2002), n Polsh. [3] Wata H., Drect Estmaton of the Vocal Tract Shae by Inverse Flterng of Acoustc Seech Waveforms, IEE Transactons on Audo and Electroacoustcs, AU-2(5) (973). [4] Anantharshnan K.S., Comuter aded ronuncaton system (CAPS), Unversty of South Australa, (2003), htt:// [5] Komaee A., Seehr A., Lnear Predcton and Synthess of Seech Sgnals, Deartment of Electrcal and Comuter Engneerng, Unversty of Maryland, htt:// Powered by TCPDF (