Interactive Processes Underlying the Production of Voice

Transcription

1 APSIPA ASC 11 Xi an Interactive Procee Underlying the Production of Voice Tokihiko Kaburagi Kyuhu Univerity, Fukuoka, Japan Tel/Fax: /45 Abtract Thi paper preent recent progre in reearch examining the mechanim of voice generation a method for phyiologically-baed peech ynthei. The overarching goal of thi reearch i the precie modeling of interaction among phyical ytem involved in the procee underlying voice generation. The interaction between the voice generation ytem (glottal flow the vocal fold) the vocal-tract filter ha been found to involve nonlinear factor in peech, uch a the kewing of glottal flow, unteadine of vocal fold ocillation, tranition of the voice regiter. The reult of a quantitative analyi alo revealed frequency-dependent effect of ource-filter interaction. I. INTRODUCTION In the production of voiced peech, glottal ound ource are generated by ocillation of the vocal fold, which travel through the acoutic filter of the vocal tract. The vocal tract act a a reonator, which form the peak of the peech pectrum called formant. In many model or analyi method of peech, thee two ub-ytem, i.e., the ource generating ytem the vocal tract filter, are thought to be independent becaue of the high acoutic impedance of the glotti. Baed on thi aumption, linear ource-filter theory [1] may explain the nature of the peech production mechanim in human. However, when glottal ound ource travel through the vocal tract, they form a ound field in the vicinity of the glotti. Thi acoutic preure can influence the flow though the glotti the movement of the vocal fold, a phenomenon known a ource-filter interaction (SFI). Thi interaction ha variou nonlinear effect, uch a flow kewing [], enhancement or uppreion of vocal-fold ocillation [3], [4], lowering ocillation threhold preure [5], dicontinuity of change in the fundamental frequency caued by continuou change in vocal fold parameter [6], [7]. Time-domain computer imulation ha been an important tool for invetigating thee effect, reult from thee tudie have been analyzed in the frequency domain a an interrelation between the fundamental frequency of the voice the reonant antireonant frequencie of the vocal tract [6]. It ha been hown that the effect become more intene when the fundamental frequency approache the reonant frequency. In addition, vocal fold ocillation become untable when the fundamental frequency i between the reonant antireonant frequencie. Titze [4] interpreted thi effect in term of the reactance of impedance ariing from the acoutic load of the ub- upra-glottal tract. The reult revealed that the harmonic component of the glottal volume flow waveform become weak when the reactance i negative at that frequency (level one interaction). In addition, the author propoed that the reactance of vocal-tract impedance hould be inertive, while that of the ub-glottal tract hould be compliant for the ocillation of the vocal fold (level two interaction). In the current paper, the procee underlying the generation of voice are invetigated, a mathematical model i preented by focuing on the interactive behavior of three phyical ub-ytem, i.e., the glottal flow, the vocal fold, the vocal tract. With repect to ource-filter interaction, the acoutic tube of the ub- upra-glottal tract are connected to the intermediate part of the glotti, the frequency repone of the acoutic preure near the glotti i expreed relative to the acoutic component of the glottal flow. The reult of a quantitative analyi are preented, including an examination of frequency repone related to SFI a time-domain imulation of voice production. II. FLOW-STRUCTURE INTERACTION Myoelatic-aerodynamic theory i widely conidered the baic principle underlying the production of voice. Thi theory wa propoed by Helmholtz Müller in the 19th century, confirmed experimentally by van den Berg in the 195. Baed on the interaction between air flow paing through the glotti vico-elatic movement of the vocal fold, the theory propoe that the vocal fold are driven by aerodynamic preure of glottal flow. Their movement in turn open cloe the glotti control the volume flow that guhe out of the glotti. Acoutic ound ource are then generated a a reult of the elf-utained ocillation of the vocal fold. By auming that the flow through the glotti i quai-teady, incompreible invicid, it behavior can be decribed by the conervation law of energy, known a the Bernoulli relation, a p(x) + 1 ρv(x) = cont., (1) where p(x) i the preure, v(x) i the velocity, ρ i the air denity. The x axi i taken in many voice production model a the axi of ymmetry between both vocal fold, a hown in Fig. 1. The flow velocity can be written a v(x) = u g S(x) = u g, () h(x)l g where u g i the volume flow rate, S(x) = h(x)l g i the ectional area of the glottal channel, l g i the length of the

2 Trachea Vocal fold Vocal tract The effective flow velocity i alo determined a v(x) = u g {h(x) δ(x)}l g. (5) h(x) y y 1 y m 1 m x With thee effective value, the glottal volume flow i etimated from (3) a u g = S pf ρ (6) k 1 y the preure within the glotti a k 1 k r 1 r u g p(x) = p F 1 ρ S(x), (7) x Glottal inlet (Flow tagnation point) x e Glottal outlet Fig. 1. Repreentation of the one-dimenional glottal channel the mapring model of the vocal fold. The x axi i taken a the ymmetrical axi of both fold. h(x) i the channel height. vocal fold which i perpendicular to the x y plane hown in the figure. In the tracheal region below the glotti, the channel i widely opened the velocity kinetic energy are very mall compared with thoe in the glotti. At the outlet of the glotti, on the other h, the flow preure i almot equal to the atmopheric preure, hence can be et to zero. From the Bernoulli relation, the following relation i then obtained for the ub-, intra-, upra-glottal region: p F }{{} trachea = p(x) + 1 ρ u g S(x) }{{} glotti = 1 ρ u g S }{{ }, (3) vocal tract where p F i the ub-glottal preure S i the ectional area of the glottal jet formed by the eparation of flow from the wall of the channel. If the point of flow eparation, x, i given, S can be written a S = h(x )l g by auming that the ectional area of the jet i contant. The flow eparation point wa fixed to the glottal exit in the two-ma model of the vocal fold[6]. However in reality, thi trongly depend on the hape of the glotti. It i alo known that the glotti take both the divergent convergent hape during a period of the vocal fold ocillation. Recently, the behavior of vicou flow near the urface of the vocal fold wa analyzed uing the boundary-layer theory [8], [9]. In addition to the flow eparation point, thi analyi method can etimate characteritic quantitie of the boundary layer, uch a the diplacement thickne, momentum thickne, wall hear tre. The effective flow channel i then determined by excluding the diplacement thickne, δ(x), from the geometric hape of the channel a S(x) = {h(x) δ(x)}l g. (4) allowing the driving force of the vocal fold to be determined. The effective channel area at the eparation point i now given a S = {h(x ) δ(x )}l g (8) by conidering the influence of the boundary layer. Movement of the vocal fold i modeled uing the modified two-ma model [1] in thi paper. Each vocal fold i contructed by two cylinder, located at the lower upper part of the fold, three plate, which connect the inlet outlet of the glotti the cylinder. In Fig. 1, mae m 1 m are aigned to each cylinder, are connected to the fixed wall by damper of reitance r 1 r, linear pring with Hooke contant k 1 k. The two mae are joined by another linear pring of contant k 1. The equation of motion i then given a d y 1 m 1 dt + r dy 1 1 dt + k 1y 1 + k 1 (y 1 y ) = f 1 (9) d y m dt + r dy dt + k y + k 1 (y y 1 ) = f, (1) where y 1 y are the diplacement of the mae perpendicular to the x axi. The abolute ma poition i y 1 + y y + y, giving the channel height h(x) = (y 1 + y ) (11) h(x) = (y + y ) (1) at the ma poition, where y i the common ret poition. A uch, vocal fold movement determine the height of the glottal channel give the boundary condition to the flow analyi problem. The actual expreion of the driving force, f 1 f, i explained below, but i baically given by integrating the preure p(x) in (7) along the x axi by multiplying the vocal fold length l g.

3 Z p Input impedance A C Subglottal tract B D Area function u A p A Input volume velocity Z Z 1 A g B g C g Dg Glotti u g Tube length ectional area u A1 p A1 u A1 Input impedance A v B v C v Dv Area function Supraglottal tract Fig.. Acoutic tube model connecting the ub- upra-glottal tract via the glotti. III. SOURCE-FILTER INTERACTION (SFI) When the glottal ound ource i generated a a reult of the flow-tructure interaction, it travel through the vocal tract form a ound field in the vicinity of the glotti. Thi ound field caue a preure gradient between the ub- upraglottal region induce additional volume flow. In addition, the intantaneou ound preure in the glotti can puh or pull the vocal fold. A uch, the acoutic field can influence the glottal flow vocal fold movement, a phenomenon known a ource-filter interaction (SFI). Thi ection decribe how thi interaction can be formulated in a model of voice generation [11]. A. Specific acoutic preure related to SFI SFI repreent an acoutic influence of preure near the glotti on the voice generation ytem of the vocal fold glottal flow. Two pecific preure value are ued here to repreent thee effect. One i the difference between the preure at the inlet of the glotti (p A ) at the outlet (p A1 ) p A = p A p A1. (13) The other i the mean acoutic preure inide the glotti Z r p A = p A + p A1. (14) To dicriminate between thee temporal waveform their Fourier tranform, time-domain variable are denoted by lower-cae letter while frequency-domain variable are denoted by capital letter in the following equation. The value of thee pecific acoutic preure are determined on the bai of the model hown in Fig.. From thi model, relationhip P A = Z U A, (15) P A1 = Z 1 U A1, (16) U A1 = U A1 + U g (17) can be obtained, where Z Z 1 repreent the input impedance of the ub- upra-glottal tract, repectively. The input impedance of the glotti een from the vocal tract, Z g, can be expreed a Z g = P A1 U A1 = (B g A g Z ) (D g C g Z ), (18) where A g, B g, C g, D g are element of the tranmiion matrix of the glotti approximated a a tube with a uniform cro ection. From thee relationhip, p A p A are each obtained a the frequency repone of a filter having the acoutic component of the glottal volume flow a it input: Z = P A U g where Z M = P A U g = P A P A1 U g = {B g (A g 1)Z }Z 1 Z D (19) = P A + P A1 U g = {B g (A g + 1)Z }Z 1 Z D, () Z D = (D g C g Z )Z 1 (B g A g Z ). (1) The input impedance of the tract can be obtained a Z = (A Z p + B ) (C Z p + D ) () Z 1 = (D vz r B v ) (A v C v Z r ), (3) where A, B, C, D are the element of the tranmiion matrix for the ub-glottal tract A v, B v, C v, D v thoe for the upra-glottal tract. The value of thee element are calculated by applying an acoutic tube model of the loy vocal tract [1], taking into account data regarding vocal-tract area function [13]. Z p Z r are the terminal impedance for the lung lip, repectively. B. Total voice production model including SFI Fig. 3 how the framework of the voice production model [11] to examine the influence of SFI on the generation of voice. The frequency repone related to SFI, Z Z M, can be determined from the area function data a explained in the preceding ubection. Next, the invere Fourier tranform of the frequency repone, z z M, are convolved with the waveform of glottal volume flow, u g, the value of the pecific acoutic preure, p A p A, are calculated. The volume flow i etimated from the pecific acoutic preure, p A, the tatic lung preure, p F. The total preure difference between the ub- upra-glottal region i determined by umming both preure a p = p F + p A = p F + z u g. (4)

4 Area function data Convolution Frequency repone Invere Fourier tranform Magnitude (cg acoutic ohm) (A) Feedback acoutic preure related to SFI Vocal fold model Glottal volume flow Static lung preure Flow-tructure interaction Glottal flow analyi Magnitude (cg acoutic ohm) Frequency (khz) (B) Difference Mean Frequency (khz) Fig. 3. Framework of the voice production model incorporating the effect of ource-filter coupling. Fig. 4. Computed reult of (A) vocal-tract input impedance (B) frequency repone of the acoutic preure difference Z (olid) mean acoutic preure Z M (broken) for the vowel /i/. Here, the convolution i computed only for the latet time intant. Conidering the Bernoulli relation given in (3), a new preure-flow relation can be obtained: p = 1 ρ u g S. (5) By eliminating p, thee relationhip can be rewritten in a dicrete-time form a K 1 p F + z (k)u g (n k) = 1 ρu g(n), (6) k= where K i the length of z (k), n i the index correponding to the latet time intant. The volume flow i then etimated a where S u g (n) = z ()S + S (z ()S ) + ρp ρ (7) K 1 P = p F + z (k)u g (n k). (8) k=1 Thee equation indicate that the time hitory of the glottal flow waveform affect the flow etimation recurively under the influence of SFI. Next, we conider the influence of SFI on vocal fold movement. In the current tudy, the driving force wa exerted only on the lower, firt ma, f wa et to zero [1]. The entire driving preure i given by umming the aerodynamic acoutic preure a p total (x) = p(x) + p A = p F 1 ρ S(x) + z M u g. (9) u g The convolution i computed here for the latet time intant. The driving force for the lower ma i then given a { x } f 1 = λl g p(x)dx + (x e x ) p A. (3) x x i the inlet of the glotti, x e i the outlet of the glotti, x i the flow eparation point, λ i a parameter pecifying the area upon which the preure i exerted. If the glotti i cloed, f 1 i et to f 1 = λl g (x e x )p F. By ubtituting the value of f 1 into the equation of motion of the vocal fold model, the diplacement of the lower upper mae, y 1 y, are calculated. Thee value determine the glottal hape h(x), the entire procedure i repeated for the dynamic imulation of voice production. IV. NUMERICAL RESULTS A. Frequency repone repreenting the effect of SFI The frequency repone of the pecific acoutic preure, Z Z M, were computed from data related to vocal-tract area function taken for the vowel /i/ [13] data related to the ub-glottal tract [14]. The length of the glottal tube (depth of the glotti) wa et to.3 cm, the cro-ectional area wa et to.14 cm. The reult are plotted in Fig. 4. From the top plot, it can be een that the input impedance of the vocal tract, Z 1, exhibited a ignificant peak at approximately 4 khz. Thi peak wa much tronger than thoe correponding to the firt to third formant. An invetigation confirmed that thi high-frequency peak wa due to the reonance of the epiglottal tube, i.e., a local, mall acoutic cavity above the glotti. The figure alo how that each peak in the input impedance form a peak of the ame frequency in the magnitude of Z Z M. Concerning the peak een in the input impedance,

5 18 3 Volume flow (cm /) 3 [.-.4 ] [.-. ] [.3-.3 ] Frequency (Hz) Time () Fig. 5. Spectrogram for the temporal pattern of the glottal volume flow for the vowel /i/. The tenion parameter of the vocal-fold model wa linearly changed in time from two to five in thi imulation. Preure difference (dyn/cm ) 1-1 Mean preure (dyn/cm ) 1-1 Diplacement (cm) Driving force (dyn) 5-5 5m Titze [4] clarified that ource-filter interaction can be enhanced by reducing the cro-ectional area of the epiglottal tube. It hould alo be noted that the magnitude of frequency repone i interpreted a the frequency-dependent trength of ourcefilter interaction, becaue they determine the amplitude of the preure value, p A p A, reulting from the glottal volume flow. B. Dynamic imulation Dynamic imulation wa conducted for the vowel /i/, the reult are plotted in Fig In the imulation, the parameter of the vocal-fold model were et to m 1 =.15 g, m =.5 g, k 1 = 8 dyn/cm, k = 8 dyn/cm, k 1 = 55 dyn/cm, r 1 =. m 1 k 1 = 3.3 dyn /cm, r = 1. m k = 18.6 dyn /cm. For the glotti in a cloed tate, the value of the tiffne parameter were increaed uch that k 1 = 3 k = 3 dyn/cm. The damper were alo increaed uch that r 1 = 57 r = 49.6 dyn /cm. The initial diplacement wa y =.14 cm the vocal-fold length wa l g = 1. cm. λ wa determined o that the effective depth of the lower ma, λ(x e x ), wa approximately.7 cm. The ub-glottal preure wa 8 cmh O the ampling frequency wa khz. To control the fundamental frequency, the mae were divided by a parameter T the pring contant were multiplied by T. The natural frequency of each ma-pring ytem wa then changed uch that k 1 T/(m 1 /T ) = k 1 /m 1 T. The value of T wa changed from two to five linearly with time in thi imulation. The fundamental frequency wa initially below the firt formant of /i/, above the firt formant the firt dip at the end of the imulation. Figure 5 how the pectrogram of the glottal flow, u g, within the frequency range - khz. The horizontal broken line around 35 5 Hz correpond to the firt peak dip, repectively, of Z Z M (ee Fig. 4). The frequency Fig. 6. Reult of a time-domain imulation of voice production for the vowel /i/. The plot how the change over time of the glottal volume flow, acoutic preure difference acro the glotti, mean acoutic preure in the glotti, ma diplacement, driving force on the vocal fold. The diplacement of the lower upper mae are hown by olid broken line, repectively. region encloed by the broken line i therefore untable for phonation [4], [6]. The fundamental frequency reache thi peak frequency at a time of approximately.15. Between.15.8, the pectrogram indicate the appearance of ubharmonic untable behavior. The harmonic pattern then tabilize when, at.8, the fundamental frequency agree with the dip frequency of approximately 5 Hz. A uch, the effect of SFI i apparent in the untable region between 35 5 Hz. In addition, it i noteworthy that the voice regiter change when the fundamental frequency croe thi pecific region [15]: the vocal fold ocillate in the chet regiter when the fundamental frequency i below the untable region. In contrat, when the fundamental frequency i above the region, the vocal fold ocillate in the faletto regiter. Figure 6 alo how waveform of the main variable for the ame imulation. In the firt time ection (..4 ), the fundamental frequency i outide the untable frequency region. In addition, the glottal flow waveform appeared kewed, a table poitive-to-negative change of driving force wa oberved. The acoutic preure difference i poitive when the glotti i cloing, which induce a forward flow a harpening of the waveform of the glottal flow. On the other h, the mean preure driving force to the vocal fold i negative when the glotti i cloing. The glotti i then effectively cloed the vocal fold ocillation i maintained. In the econd ection (.. ), the vocal fold ocillation i untable. In the waveform of the glottal flow other phyical quantitie, we can oberve an additional periodicity occurring for each of two pitch period, reulting in the

6 exitence of ubharmonic behavior in the pectrogram. Finally, over.3.3, regular vocal-fold ocillation i recovered, the voice regiter i now changed to the faletto. Note that the fundamental frequency i cloe to the dip frequency of Z Z M. The feedback acoutic preure the effect of SFI are therefore weak. In the faletto regiter, the phae difference of the ocillatory movement of the upper lower mae i not apparent, indicating that a convergent-divergent change in the glottal hape i not attained. The driving force to the vocal fold i mainly given by the aerodynamic preure of the glottal flow, from the figure, it i clear that the force i alway poitive. V. CONCLUSION The production proce of the voice wa examined by focuing on the interaction of three phyical ytem, i.e., the glottal flow, the vocal fold, the vocal tract. The flow-tructure interaction between the glottal flow vocal fold play an eential role in the production of voice. Regarding the influence of the geometry of glottal channel on the behavior of flow, a detailed analyi reult wa reported in [8] [9]. In the preent tudy, the ource-filter interaction between the voice production ytem (glottal flow the vocal fold) the acoutic cavity of the vocal tract wa modeled uing the frequency repone of two type of pecific acoutic preure in the vicinity of the glotti. Quantitative invetigation revealed that thee frequency repone exhibited dominant peak in the high frequency region at approximately 4-5 khz. In addition, SFI wa typically oberved when the fundamental frequency of the vocal fold ocillation approached the reonant frequency of the vocal tract. The trength of the interaction wa alo found to increae near the reonant frequency, imulation reult revealed intability in phonation change of the voice regiter. Thi reearch wa partly upported by the Grant-in-Aid for Scientific Reearch from the Japan Society for the Promotion of Science (Grant No ). [9] T. Kaburagi Y. Tanabe, Low-dimenional model of the glottal flow incorporating vicou-invicid interaction, J. Acout. Soc. Am., vol. 15, pp , 9. [1] X. Peloron, A. Hirchberg, R. R. van Hael, A. P. J. Wijn, Y. Auregan, Theoretical experimental tudy of quaiteady-flow eparation within the glotti during phonation. Application to a modified two-ma model, J. Acout. Soc. Am., vol. 96, pp , [11] T. Kaburagi, Voice production model integrating boundary-layer analyi of glottal flow ource-filter coupling, J. Acout. Soc. Am., vol. 19, pp , 11. [1] M. M. Sondhi J. Schroeter, A hybrid time-frequency domain articulatory peech yntheizer, IEEE Tran. Acout., Speech & Signal Proce., vol. ASSP-35, pp , [13] B. H. Story, I. R. Titze, E. A. Hoffman, Vocal tract area function from magnetic reonance imaging, J. Acout. Soc. Am., vol. 1, pp , [14] E. R. Weibel, Morphometry of the human lung, Berlin: Springer, 1963, pp [15] I. Tokuda, M. Zemke, M. Kob, H. Herzel, Biomechanical modeling of regiter tranition the role of vocal tract reonator, J. Acout. Soc. Am., vol. 17, pp , 1. REFERENCES [1] G. Fant, Acoutic theory of peech production, with calculation baed on X-ray tudie of Ruian articulation (nd printing), The Hague: Mouton, 197. [] M. Rothenberg, Acoutic interaction between the glottal ource the vocal tract, in Vocal Fold Phyiology, Steven Hirano Ed. Tokyo: Univerity of Tokyo Pre, 1981, pp [3] M. Zañartu, L. Mongeau, G. R. Wodicka, Influence of acoutic loading on an effective ingle ma model of the vocal fold, J. Acout. Soc. Am., vol. 11, pp , 7. [4] I. R. Titze, Nonlinear ource-filter coupling in phonation: Theory, J. Acout. Soc. Am., vol. 13, pp , 8. [5] R. W. Chan I. R. Titze, Dependence of phonation threhold preure on vocal tract acoutic vocal fold tiue mechanic, J. Acout. Soc. Am., vol. 119, pp , 6. [6] K. Ihizaka J. L. Flanagan, Synthei of voiced ound from a two-ma model of the vocal cord, Bell Syt. Tech. J., vol. 51, pp , 197. [7] J. G. Švec, H. K. Schutte, D. G. Miller, On pitch jump between chet faletto regiter in voice: Data from living excied human larynge, J. Acout. Soc. Am., vol. 16, pp , [8] T. Kaburagi, On the vicou-invicid interaction of the flow paing through the glotti, Acout. Sci. Tech., vol. 9, pp , 8.