Time-continuous power-balanced simulation of nonlinear audio circuits: realtime processing framework and aliasing rejection
This work addresses the real-time simulation of nonlinear audio circuits. In this thesis, we use the port-Hamiltonian (pH) formalism to guarantee power balance and passivity. Moreover, we adopt a continuous-time functional framework to represent "virtual analog" signals and propose to approximate solutions by projection over time frames. As a main result, we establish a sufficient condition on projectors to obtain time-continuous power-balanced trajectories. Our goal is twofold: first, to manage frequency-bandwidth expansion due to nonlinearities, we consider numerical engines processing signals that are not bandlimited but, instead, have a "finite rate of innovation"; second, to get back to the bandlimited domain, we design "virtual analog-to-digital converters". Several numerical methods are built to be power-balanced, high-order accurate, with a controllable regularity order. Their properties are studied: existence and uniqueness, accuracy order and dispersion, but also, frequency resolution beyond the Nyquist frequency, aliasing rejection, reproducing and Peano kernels. This approach reveals bridges between numerical analysis, signal processing and generalised sampling theory, by relating accuracy, polynomial reproduction, bandwidth, Legendre filterbanks, etc. A systematic framework to transform schematics into equations and simulations is detailed. It is applied to representative audio circuits (for the UVI company), featuring both ordinary and differential-algebraic equations. Special work is devoted to pH modelling of operational amplifiers. Finally, we revisit pH modelling within the framework of Geometric Algebra, opening perspectives for structure encoding.