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Page 32 ï~~Estimation Of Parameters Corresponding To A Propagative Synthesis Model Through The Analysis Of Real Sounds S01vi Ystad, Philippe Guillemain, Richard Kronland-Martinet CNRS - Laboratoire de M6canique et d'Acoustique, 31 chemin J.Aiguier, 13402 Marseille Cedex 20, France Abstract Simulation of sounds produced by musical instruments can be achieved by the use of synthesis models that take into account the wave propagation in the medium. Here we address the problem of the estimation of parameters for the simple waveguide model. The filter and delay inside the loop mainly take into account phenomena linked to the propagation (dispersion, losses), while the filter outside the loop takes into account the excitation and radiation aspects. In order to estimate these filters, we propose an approximation of the global transfer function of the system in terms of a sum of independent partials with exponential decay, the amplitude, frequency and bandwidth of which can be directly related to the characteristics of these filters. In the case of plucked string instruments, or flutes excitated with a transient, the bandwidth of each component is related to its amplitude modulation law in the time domain. By the use of timefrequency techniques, we estimate the parameters corresponding to the waveguide model, and compare the datas with the theoretical values coming from the solutions of the movement equations. I) Introduction II.1 - Movement equation in a tube Waveguide models make the simulation of propagation of acoustical or mechanical waves in bounded medias possible. From a theoretical point of view, it is possible to deduce the parameters of the model from the solutions of the movement equations. Nevertheless, these equations are often resulting from approximations, yielding a biased estimation of the parameters. In this paper, we address the problem of the estimation of the model parameters from the analysis of real sounds, and the matching of the measured values with the theoretical ones. In the case of the flute and the guitar, relevant parameters such as dispersion and dissipation can be estimated both by studying the solutions of the movement equations and by the analysis of real sounds. The dispersion phenomenon introduces an inharmonicity between the partials, and the dissipation yields a different decay time of the components. These two effects can be measured on real sounds. In order to use the waveguide synthesis model in practice, we shall propose a way of building the filters of the model that will reproduce the dispersion and dissipation phenomena. II) Movement equations We are working on the propagation of longitudinal waves in fluids (wind instruments) and transversal waves in solid medias (string instruments). When visco-thermal losses are taken into account, the propagation equation of the pressure can be written as  1 a lhv t)EAp- A = c2 at2 - c ADa O where lhv represents characteristic lengths in the air (lhv is of order 10-8 in free field). Its equivalent value in a bounded media like a tube depends on the friction near the body of the instrument, and cannot be easily determined. The solution of the wave equation for a tube which is open at both ends (flute) and submitted to a ponctual dirac source can be calculated using classical techniques . It can be given by a superposition of components, each of them being given by sin(nic-41)xsin (nn) Pn(X,t) = An sin(wnt)e- ant, where L is the length of the tube and x0 is the source position. The frequencies are given by con cn22 2c22n4rt4 2L2-lhvC 4L4 where the inharmonicity depends on the value of lhv Ystad et al. 32 ICMC Proceedings 1996
Page 33 ï~~The damping factors are given by: n2rt2 an= lhvc L2 This model can lead to both physical and additive synthesis models . 11.2 - Movement equation in a string The transverse displacement y(x,t) of a stiff string is described by the equation : a2y(xt xtE 4y(xt) aYxt) St2p xy(xt)+ +R t)-= If we assume that the boundary conditions are: y(O,t)=y(L,t)=y"(O,t)=y"(L,t) = 0, it can be shown that each mode of vibration can be written as: yn(x,t)= 1 flltx -R sin( L-)exP(-)sin(ont) Yxt)(Onsi(L p2 n 2Tnt2 EI n4it4 R2 with O n = +IP L2 P L4 2p2 III - Waveguide model In order to simulate the stationnary waves corresponding to the propagation in a bounded medium, we consider the following model  S(w) = IF1(w))12 1 - 2F2(co)coswd + I F2(co) j2 The maxima of S(o) are obtained when 2nrt cos(cod)=1, that is for o-d" In this case the spectrum is harmonic, but inharmonicity can easily be taken into account by adapting the phase of F2(co). In order to approximate the behaviour of these resonances, we suppose that the peaks are narrow-band enough to be considered as separated in the frequency domain. This assumption is generally reasonable for musical instruments. Then, close to the resonance peaks we can note o - on = << 1 and cos(od) = cos(Ed). By using a limited expansion of cos(cod) around o=0, one can approximate S(o) around each resonance: F1(con)2 S(On) - e32d2 2 1-2F2(cOn)(1- 2-)+F2(On) (1) On the other hand, if one consider an exponentially damped sinusoid s(t) = Cne-Â~ntei(nt, its power spectral density is given by C2 S((o)= 2 2 an + ((0 - (on) Similarily, by considering the local behaviour of S(o) around the resonance frequency (co=con+E), we get: Fig. 1 Waveguide model To match this model with the theoretical equations, or through the analysis of real sounds, we seek a description of the impulse response of the system as a sum of sinusoidal functions exponentially damped. Thus, the amplitudes, frequencies and damping factors of this impulse response will be directly related to Fl, F2 and the delay d-L. c The transfert function of the system is given by Fl (o)e-i'd T(w) = 1 - F2(w)e-id then if one assumes that F2(co) is real-valued, the power spectral density of T(co), S(o) is: Cn 2 S(Â~0n) = an2+E2 (2) By comparing (1) and (2), we find an expression of F2(con) depending on an and d 2+d2an2 - -1 (an2d2+2)2 - 4 F2(wn) = as as The filter F1 in the model takes into account the energy variations between the components, and is directly related to the constant Cn of the expression (2): F1( con) = Cnd F2(on). ICMC Proceedings 1996 33 Ystad et al.
Page 34 ï~~IV - Parameters estimation through the analysis of real sounds. In this section we describe experiments, and the analysis method. The results obtained are compared with the theoretical values. IV. 1 - Experiments The impulse response of the flute has been measured in an anechoic room by rapidly closing a finger hole, without exciting the flute with a jet stream at the embouchure. We have also considered guitar sounds, obtained under normal playing conditions. Actually, if we assume that the linear model is valid, the excitation performed on the string is not important for the parameters inside the loop, since it has an influence only on the filter Fl. IV.2 - Analysis Method In order to compute the filters allowing the resynthesis of a real sound by the use of the waveguide model, we need to estimate the amplitudes, frequencies and damping factors of each partial. For that purpose, we use a spectral lines estimation technique based on timefrequency representations . This method, based on the construction of an automatically matched bank of wavelet packets let us extract accurately the frequencies and amplitude modulation laws, even in the case of highly damped components, suh as the ones we get in the flute case. IV.3 - Results IV.3.1 - Flute case As seen in section III, the filter F2(co) depends on the delay d. In the flute case, the figure 2 shows the theoretical filter F2(wo) for L, L/2, L/4 and L/8, where L=0.58m. fJit4.f F a)f'r df/e.ornt l ngthS of a tuAb -~L I.8 0.70L 1 The relation between the filter and the length (when neglecting the inharmonicity in the tube) is found to be L -> - implies F(co) -> F(-) n. As discussed in section II, the value of lhv in the movement equation of a tube, depends on the friction near the body of the instrument, and is determined by comparing the real and the theoretical damping factors as seen in Fig. 3. 2000 1800 -S1600 1400 1200 -1000 -800 600 400 200 1 2 3 4 5 6 7 8 9 10 1m", num Fig. 3 real(+) and theoretical(*) an From the experience, the estimated ]hv is of order 6*10-5. This value is much higher than the theoretical value in free field, which shows that the friction of the air near the body of the instrument can be seen as responsible of all the losses. Figure 4 displays respectively the theoretical and estimated inharmonicity of the modes. The estimated inharmonicity is more important than the theoretical one. This may come from the way we produce the "impulse response" or from the previous estimation of lhv. realvoa ro mode number Fig. 4 Frequency/(n*fundamental) of a tube of length L=0.58m in the theoretical and real cases. Fig. 2 F2(co) for different tube lengths, L. Ystad et al. 34 ICMC Proceedings 1996
Page 35 ï~~IV.3.2 - Guitar case V - Conclusion Fig 5 displays respectively the theoretical and estimated inharmonicity of a steel guitar string. The estimated datas are in good accordance with the model. 1.01 1 5 10 15 20 25 30 35 1.0!1 10 15 20 25 30 35 mode number Fig. 5 Frequency/(n*fundamental) of a steel guitar string in the theoretical and real cases. Figs 6 and 7 displays respectively the Fourier transforms of the two estimated filters Fl and F2 corresponding to a guitar sound. Since the filters have been estimated directly on the sound itself, F l includes the frequency response of the soundboard mechanically coupled with the string, and its radiative characteristics. Though, from a theoretical point of view, the filter F2 should be constant, one can observe greater losses for the high frequencies than for the lower ones. This is consistent with the fact that higher harmonics decrease faster than lower harmonics. This phenomenon can be justified theoretically by a loss coefficient R being a function of the frequency. In this paper, we have presented a technique allowing the estimation of relevant parameters for the resynthesis of real sounds by the use of a waveguide model, and compared the estimated datas with their theoretical values deduced from simple propagative equations. Though we have considered two very different musical instruments, the flute and the guitar, we have seen that they can be processed in the same way, thanks to the similarity of the movement equations of longitudinal waves in fluids and transversal waves in solid beams. In the flute case, we have worked with the response of the air column excitated with a transient, and the sound produced this way has been resynthesized faithfully. However, for a realistic simulation, the source problem (vortex phenomena, turbulence noise, non-linearities) together with the connection between the source and the resonator (the existing field in the tube interacts with the source) still remains. In the guitar case, though the sounds resynthesized this way are of very good quality, some problems still remain, namely the coupling between the string and the soundboard which is hard to modelize theoretically, and the radiation of the soundboard itself. Nevertheless, we believe that the first effect can be taken into account in the movement equations by the use of a loss term being a function of the frequency. In order to get around the second problem, we shall make measurements of the string vibrations by the use of a laser vibrometer. Acknowledgements: This work has been partly supported by The Research Council of Norway. References:  J. Kergomard,"Champ interne et champ externe des instruments a vent",These d'Etat, Universite Paris IV, 2 septembre 1981.  IN. Sneddon, "Fourier Transforms",McGraw-Hill Book Company, 1951.  S. Ystad, R. Kronland-Martinet,"A Synthesis Model for the Flute Transient by Analysis of real sounds Through Time-Frequency Methods" proceedings of ICA, Volume III, p.525-528 Trondheim, Norway July 26-30 1995.  C.Valette, C. Cuesta "M canique de Ia corde vibrante", Hermes, 1993.  J.O.Smith III "Digital Waveguide Models for Sound Synthesis based on Musical Acoustics" proceedings of ICA, Volume III, p.525-528 Trondheim, Norway July 26-30 1995.  Ph. Guillemain, R. Kronland-Martinet, "Characterisation of Acoustics Signals Through Continuous Linear Time-frequency Representations" Proceedings of the IEEE, Special Issue on Wavelets, April 96, Vol. 84 nÂ~4, pp 561-585. 510' Fig. 6 The Fourier transform of the filter Fl corresponding to a guitar sound. 1 0 10 0 30 40 5 00 10 20 30 ' 40 S0 Fig. 7 The Fourier transform of the filter F2 corresponding to a guitar sound. ICMC Proceedings 1996 35 Ystad et al.