S. Tassart, Ph. Depalle
ICMC 97, Thessalonique
Much work has been done in the field of physical modeling in order to digitally simulate the propagation of acoustical plane waves in a waveguide. In most of the work, acoustical assumptions taken into account result in the linear behaviour of the waveguide. It follows that the acoustical wave speed is constant and independent from the acoustical pressure; it is however a function of the total pressure.
For some kinds of musical wind instruments, such as the trombone, the pressure intensity involved inside the waveguide is so high that the linear assumption is no longer valid. In that case the model must consider nonlinear propagation, that is to say, that the wave speed is a function of the input pressure of the waveguide.
Considering delayed transmission instead of propagation speed implies that the value of the delay in the wave propagation depends on the amplitude of the input pressure signal. Thus time dependant digital filters approximating fractional delay are required to simulate wave propagation in waveguides where these nonlinear phenomena occur.
This physical problem cannot however be directly transposed in terms of signal processing. From an acoustical point of view, it appears that the input signal propagates inside the waveguide and is observed at the output of the waveguide d_s(t) samples later (d_s(t) being a fractional value of samples). Whereas from a signal processing point of view, the signal output is computed as a delayed version from the input signal d_e(t) samples sooner. The relation between d_s(t) and d_e(t) is actually nonlinear, and d_s(t) equals d_e(t) only if both are constant. Thus we must find a way to convert the delay d_s(t) into d_e(t) in order to control the digital fractional delay filter for simulating this nonlinear propagation.
The goal of this paper is to present a new formal description of the nonlinear relationship between d_s(t) and d_e(t). From this description we derive an explicit implementation for computing the sampled values of d_e(t) from the past sampled values of d_s(t). This implementation consists in the use of Lagrange Interpolator Filters whose delay is controlled by a feedback loop. Results will be compared to more intuitive and classical techniques. Finally we will propose a completly digital model for the nonlinear acoustical wave propagation which takes place in the slide of the trombone.
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Tue Oct 14 16:04:38 1997
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