Efficiency, accuracy, and stability issues in discrete time simulations of single reed instruments

A quantitative study of discrete-time simulations for a single reed physical model is presented. It is shown that when the continuous-time model is discretized, a delay-free path is generated in the computation. A general solution is proposed to this problem, that amounts to operating a geometrical transformation on the equations. The transformed equations are discretized using four different numerical methods. Stability properties of each method are assessed through analysis in the frequency domain. By comparing the discrete and continuous frequency responses, it is studied how the physical parameters are mapped by each method into the discrete-time domain. Time-domain simulations are developed by coupling the four digital reeds to an idealized bore model. Quantitative analysis of the simulations shows that the discrete-time systems produced by the four methods have significantly different behaviors, even when high sampling rates are used. As a result of this study, a general scheme for accurate and efficient time-domain simulations of the single reed model is proposed.