Design Principles for Lumped Model Discretisation Using Möbius Transforms

Computational modelling of audio systems commonly involves discretising lumped models. The properties of common discretisation schemes are typically derived through analysis of how the imaginary axis on the Laplace-transform s-plane maps onto the Z-transform z-plane and the implied stability regions. This analysis ignores some important considerations regarding the mapping of individual poles, in particular the case of highly-damped poles. In this paper, we analyse the properties of an extended class of discretisations based on Möbius transforms, both as mappings and discretisation schemes. We analyse and extend the concept of frequency warping, well-known in the context of the bilinear transform, and we characterise the relationship between the damping and frequencies of poles in the s- and z-planes. We present and analyse several design criteria (damping monotonicity, stability) corresponding to desirable properties of the discretised system. Satisfying these criteria involves selecting appropriate transforms based on the pole structure of the system on the s-plane. These theoretical developments are finally illustrated on a diode clipper nonlinear model.