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On the numerical inversion of the Laplace transform in the context of physical models with realistic damping






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On the numerical inversion of the Laplace transform in the context of physical models with realistic damping


Trabelsi, Karim (auteur)
Matignon, Denis (auteur)
Helie, Thomas (auteur)





This technical report sums up the research carried out under the title: Numerical optimisation of physical models with realistic damping for real-time sound synthesis which was supported by the CONSONNES (CONtrˆole de SONs instrumentaux Naturels Et Synth´etiques - in english: Control of natural and synthetic instrumental sounds) project. The context of this work is the real-time simulation of wind instrument resonators. Some realistic physical models as, for instance, the wave equation with viscothermal losses in a flared duct (The Webster-Lokshin model, cf. [63]; see also [45], [46] and [37]), possess non standard Green (transfer) functions with poles, branchpoints and cuts in Laplace's left halfplane, which entails, in the time domain, impulse responses that decay slowly due to the non-purely-exponential damping. This accounts for the the long memory label tagged to such a model. A straightforward consequence to this phenomenon is the need of simulation for long times. Moreover, the time responses to such systems are obtained through the inversion of the Laplace transform which adds a second numerical issue, since the exponential factor is highly oscillatory on the Bromwich line. Over the years, methods have been devised to deal with the numerical inversion of the Laplace transform. The efficiency of most of these depends on some parameters that are tuned heuristically. Furthermore, most of them are not adapted to the nonstandard Green functions we are concerned with. To our knowledge, the most efficient algorithms may be divided along four directions. The first one is based on Fourier series expansion. The second one uses collocation methods. A third procedure is founded on Talbot's idea [75] which consists in deforming the Bromwich contour into a curve that allows for a better numerical integration. Last but not least is an approach that comes from automatic control and which consists in the approximation of diffusive integral representations of the system. The last two approaches seem more efficient and suitable for our purposes. Therefore, these were the tracks we investigated with the goal of obtaining optimal deformations of the Bromwich contour so as to work out algorithms that are at least as good as the diffusive approach with the advantage of being automatic, i.e., without parameters that have to be tuned or whatsovever.


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2010-09-07 02:00:00

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