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Page 00000001 Analysis-Synthesis of Flute Sounds Using a Non-Linear Digital Waveguide Model S0lvi Ystad'~ and Thierry Voinier(2) (1) NTNU - Norges Teknisk Naturvitenskapelige Universitet, 7034 Trondheim, Norway and CCRMA - Department of Music, Stanford University, Stanford, CA 94305-8180, USA (2) CNRS-LMA, 31 Chemin J. AIGUIER, 13411 Marseille, France email: email@example.com & firstname.lastname@example.org Abstract Flute sounds are produced by an air jet fluctuating around the labium of the embouchure of the instrument creating standing waves in the body of the instrument. Thus, the flute can be considered as a source-resonator system, where the source (corresponding to the interaction between the air jet and the labium) is considered as a non-linear system and the resonator (corresponding to the medium in which the waves propagate) is considered as a linear system. These two parts mutually interact, leading to a natural model which consists in a looped system containing both linear and non-linear elements. The construction of the synthesis model should be done in two steps - first the parameters of the non-linear source model and the linear resonator model should be identified. Then, in order to couple the two parts by retroaction, the parameters of a lowpass filter should be calculated in order to assure the stability of the non-linear digital waveguide. 1 Introduction By combining both physical and signal models, hybrid models can be constructed to synthesize sounds. Such models have the advantage of responding to both subjective and objective criteria within the same concept. A hybrid model which simulates flute sounds has been constructed with a physical model simulating the resonator of the instrument and a signal model simulating the source of the instrument. The physical model consists in a digital waveguide model, while the signal model partly has been constructed using a waveshaping technique. These two models are then simply connected by injecting the source model into the resonator model, leading to a hybrid flute model. The digital waveguide parameters are extracted from the analysis of real sounds while the parameters of the waveshaping function are found by perceptual criteria. The combination of these two models leads to good resynthesis and allows an easy control of the instrument. Even though this model has given satisfactory results, it doesn't mirror the physics of the real instrument since the interaction between the standing waves in the resonator and the air jet fluctuating around the labium has been ignored so far. The consequence is that some behaviours of the real instrument (such as the octaviation in the flute case) can not be reproduced. Thus, in a second step, we propose to take into account this interaction by connecting the output of the resonator to the input of the source system through a linear filter. This filter selects one or just a few spectral components of the output signal making the re-injected signal close to the one which so far piloted the waveshaping model. Stability criteria of such systems will be presented and applied to our particular case. 2 The Resonator Model In order to model the resonator we have used a digital waveguide model (Smith 1992). Such a model consists in simulating the behavior of the waves propagating in a medium by a looped system with a delay line and a filter taking into account dispersion, dissipation and boundary conditions. The parameters of this system have been found from the analysis of real flute sounds (Kronland-Martinet, Guillemain and Ystad 1997). The main parameters which are extracted for this purpose
Page 00000002 are the damping factors and the frequencies of the modes. Figure 1: The digital waveguide. 3 The Source Model As already mentioned, the flute source behaves non-linearly. To analyse the source behavior, we have supposed that the excitator and the resonator are independent, which means that the source can be identified by deconvolution (Ystad 1998). The transfer function of the physical waveguide model (representing the resonator) being an all-pole filter, this is a legitimate operation. The analysis of the flute source shows that the source signal contains both spectral lines and a broadband noise. For convenience, these two contributions have been splitted in a deterministic and a stochastic contribution by an adaptive filtering method (Widrow 1985). A FY synthesized sound coincides with the one of the real sound. Unfortunately it is not possible to reconstruct any spectral evolution by changing the index, since this representation of signals is not complete. We have therefore estimated the index so that it satisfies perceptual criteria. The tristimulus criterion (Pollard and Jansson 1982) has been found suitable for this purpose. This criterion consists in considering the loudness in three different parts of the spectrum. Thus, by minimizing the difference between the tristimulus of the synthetic sound and the natural sound, we find how to vary the index of distortion to get a spectral evolution of the synthetic signal similar to the spectral evolution of the real signal (Ystad 2000). 3.2 Modeling the stochastic contribution The stochastic part of the source signal is supposed to be stationary and ergodic, which means that it can be described by its power spectral density and its probability density function. From a perceptual point of view, the coloring of the noise is mainly related to the Power Spectral Density. Thus, by linear filtering one can generate a noise corresponding to the instrument we want to simulate. 4 The Hybrid Model By combining the source model with the resonator model, a very general sound model can be constructed. If we suppose that the source and the resonator can be considered independently, a complete model for flute sounds can be constructed as seen in figure 3. Figure 2: The source model. 3.1 Modeling the deterministic contribution The evolution of the spectrum of the deterministic contribution as a function of the dynamic level is non-linear, since the spectral components evolve independently. A waveshaping synthesis model (Arfib 1979) (Lebrun 1979) has been chosen to simulate such a non-linear behavior. This method consists in distorting a sinusoidal function with an amplitude I(t) called the index of distortion by a nonlinear function y. The generated signal s(t) can then be written: S(t) = y(I(t) cos(o0t)) = akTk(I(t)cos(wot)) (1) k where Tk corresponds to the Chebyshev's polynomials of the first kind. The challenge is now to find the index of distortion I(t) for which the dynamic evolution of the Figure 3: The hybrid model which combines the physical and the signal model. The advantage of this model is that it takes into account both perceptual and physical phenomena. The combination of the two classes of synthesis models makes it possible to simulate a great number of sound sources. Thanks to the physical part of the model a physical interpretation of the sound transformations can be given, and thanks to the signal part, sounds that normally would be too complicated to
Page 00000003 simulate by physical models can be modeled. 5 From a Hybrid Model to a Nonlinear Model As mentioned in the introduction, when a real flute sound is produced, there is an interaction between the standing waves in the resonator and the air jet fluctuating around the embouchure meaning that the source and the resonator are interacting and cannot be separated. The way these two parts are interacting is not clearly understood apart from the fact that they interact in a non-linear way (Verge 1995). In this article we suppose that only the lower frequencies of the standing waves interact with the air jet. In the previous section we have seen that the non-linear fuction y correctly simulates the non-linear behavior of the instrument. The idea is therefore to conserve this function when constructing the non-linear waveguide model. To take into account the interaction, a retroaction should further be made by reinjecting the output of the resonator model into the input of the non-linear source model. Thus the sinusoidal signal which was injected into the non-linear function (figure 3), will be replaced by the low-pass filtered output signal of the digital waveguide model. /+ - Waveguide white noise LPF\ Figure 4. The non-linear digital waveguide model. The construction of such models causes a great number of mathematical problems, in particular stability problems, since the retroaction implies a looped system containing a non-linear element. In figure 4 we have shown how the interaction between the source and the resonator can be made by reinjecting the resonator's output signal into the non-linear source model via a linear lowpass filter (which we call the retroaction filter). This should give a more realistic model from a physical point of view. In the next section we will discuss such problems and how the parameters of such a model can be calculated to satisfy specific stability criteria. 6 Stability of the Nonlinear Model Coupled systems with linear elements are rather easy to implement, and their stability can be assured by well-known stability criteria such as the Nyquist criterion. When a nonlinear element is part of the system as in figure 4, a transfer function of the system can no longer be calculated and new stability criteria have to be found. The stability criterion which has been used in our situation is based upon Liapunov's 2nd method (Balchen 1992) (Slotine and Li 1991) and is called the Circle criterion. This criterion states that a looped system with a non-linear and a linear element is globally asymptotically stable if: - the linear element H(jo) is stable, the non-linear function y(x) is located in a sector characterised by: 0<k_ Y(x)<k2' where x is the input of the non-linear x function, - the curve H(jo) in the complex plane does not enclose nor cut a circle which center is on the negative real axis and which crosses the points -1/k1 and -l/k2. Graphically, this stability criterion can be illustrated as depicted on figure 5. k1 k2 a) -1/k -1 / k2 H(jo) Im [H(jo)] b) Re [H(jo)] Figure 5. a) Section (kl,k2) with the non-linear function and b) the "forbidden circle" and a curve h(jo) in the complex plane. To apply this criterion on our model, it should be divided in two parts - a non linear part (in our case the waveshaping function) and a linear part (in our case the delay line, the loop filter and the retroaction filter) which corresponds to H(jc). Since the retroaction filter is a low-pass filter which output will be close to a sinusoidal function, the linear part of our model can be replaced by a lorentzian function given by: L(o) = P3/(a+i(o-o0))
Page 00000004 where Co represents the selected frequency component, and a and p are related to the amplitude and the width of the function. The characteristics of this function should be adapted to satisfy the stability criterion. Some modifications can also be done to the non-linear function. As explained in section 3.1 the non-linear function is simulated by a waveshaping function (cf eq. (1)). The coefficients Ck are related to the energy of the spectral components of a flute sound. The values of these components can not be changed, but since the sound is periodic, the phase of the sound can be altered without modifying the sound. This means that the signs of the coefficients Ck can be altered to satisfy the stability criterion. Acknowledgments This work was partly supported by the Norwegian Research Council. References Arfib, D. 1979. "Digital synthesis of complex spectra by means of multiplication of non-linear distorted sine waves." Journal of the Audio Engineering Society, 27;757-768. Balchen, J.G. 1992. "Ulineaere systemer og stabilitetsteori", Universitetet i Trondheim, Norges tekniske h0gskole, Institutt for teknisk kybernetikk, 1992. Cook, P. R. "A Meta-Wind-Instrument Physical Model Controller and a Meta-Controller for Real-Time Performance Control." Proceedings of the 1992 International Computer Music Conference. San Jos6, California: International Computer Music Association, pp. 273-276. Kronland-Martinet R., Guillemain, Ph & Ystad S. 1997, "Modeling of natural sounds by time-frequency and wavelet representations", Organised Sound, 2(3): 179-191. Le Brun, M. 1979. "Digital Waveshaping Synthesis." Journal of the Audio Engineering Society, 27; 250-266. Pollard H.F. and Jansson E.V. 1982. "A Tristimulus Method for the Specification of Musical Timbre." Acoustica, Vol. 51;162 -171. Slotine, J-J.E. and W. Li 1991. "Applied non-linear Control." Prentice-Hall International Inc. Smith, J.O. 1992. "Physical modeling using digital waveguides." Computer Music Journal, 16(4); 74-91. Verge M.P. 1995. "Aeroacoustics of Confined Jets with Applications to the Physical Modeling of Recorder-Like Instruments." PhD thesis, Eindhoven University. Widrow B.and Steams S.D. 1985. Adaptive Signal Processing. Englewood Cliffs, Prentice-Hall Inc. Ystad, S. 1998. "Sound Modeling Using a Combination of Physical and Signal Models. " Ph. D. Thesis from the University of AixMarseille II. Ystad, S. 2000. "Sound Modeling Applied to Flute Sounds. " Journal of the Audio Engineering Society, 48(9); 810-825. Ystad, S., Voinier, T. "A Virtually real Flute", Computer Music Journal, Vol 25, N'2,13-24, Summer 2001. S Y H(jo) -w) Figure 6. System to which the stability criteria are applied. 7 Concluding Remarks The authors are currently implementing this model as an external object for the Max-MSP real-time synthesis program. A real flute, specially equipped (Ystad-2001) will be used to control the model in order to check the validity of the model, and the playability of that instrument. The model is still being tested, but sounds promising. In order to take into account non-linear phenomena, a new approach is proposed to derive the parameters of a nonlinear synthesis model. These parameters are obtained from a separate analysis of the source and the resonator of the instrument from real sounds. The non-linear model gives rise to stability problems, which can be analyzed.