| Résumé |
Accurate parameter estimation methods have been developped for synthesis by signal models (e.g. additive synthesis) and have been very successful by allowing high quality and very flexible processing. On the contrary, there is a lack of similar estimation methods for most of physical models, particulary those describing instruments with self-sustained sound. Such a method would allow to recover the time-varying input parameter values of the instrument from a given sound, such that the same sound is obtained when these input parameters are fed into the model. Typical input parameters are bow velocity, lip pressure on a reed, blowing pressure or tension of the lips. Input parameters will be simply called parameters in the following. In this paper, we present a method which allows to estimate the time-varying parameters of a physical model. We focus on a physical model of trumpet-like instruments. This model includes a single mass oscillator (representing the lips) nonlinearly coupled with a linear resonator (representing the bore). In a preliminary stage, we have proved that this model is non invertible: the knowledge of the produced sound is not sufficient to reconstruct the original parameters' evolution. As a matter of fact, the set of solutions proved to be infinite, non countable, and even possibly multi-dimensional, since solutions live on a differential manifold. Moreover, there is no obvious reason to choose a particular point of the manifold as a solution. Then, in order to palliate this difficulty, a strongly restrictive but physically meaningful constraint is chosen: the minimal variation of the parameters is looked for. Indeed, we can consider that the internal dynamic of the model evolves much faster than the musician's control parameters. Thus, for a short-enough time window, the looked-up parameters can be considered as steady. Now, the differential manifold on which the solutions live is a function of the measured signal; and since the signal evolves on the considered time interval, the manifold is altered. Therefore, under the precedent hypothesis, the constant looked-up solution is necessarly, at each time, a point of the manifold. In the case of our trumpet model, it appears that only one point lies into the physical range of parameters. Existence and uniqueness are then obtained. Practically, we use a penalty function which measures the adequacy of the looked-up constant parameters to the measured output through the equations of the model. We look for the solution which minimizes this penalty function. Thus, we counterbalance the potential presence of numerical errors, and eventually of non modelled phenomena. Moreover, slowly-varying parameters become acceptable. For example, in the simplified case where the internal mouth pressure is fixed, the looked-up parameters live on a time-varying parabola (the parameters correspond to the tension and to the viscosity of the lips, which appeared to be musically crucial in the course of real-time simulations). It has been demonstrated that seeking after the minimum of the penalty function was equivalent to fitting a straight line (or more generally an hyperplane) to the set of points calculated according to the measured output. Because of numerical analysis problems, some aberrant points appear. So, in order to improve the robustness of the method, we developped an algorithm based on a statistical approach (likelyhood). This last step finally results in an efficient estimation method. We present examples of estimations in various playing conditions such as attacks, vibratos, and slowly varying gestures. In spite of restrictions inherent to the method (local constancy of the parameters), satisfying results are obtained. Moreover, promising investigations have been carried out, in order to allow more flexibility on the hypothesis (rapidly varying parameters) while keeping an effective method. |
