Non-oversampled physical modeling for virtual analog simulations

Physical modeling of real-world systems involves building computer models from continuous-time differential equations. These equations are derived from the physical principles governing the dynamics of these systems. To do so, it uses a systematic process to build discrete-time differential equations that approximate such dynamics. In audio research, this process is the focus of the field referred to as virtual analog modeling. This thesis presents several contributions to this field in the form of results on the topics of analysis and optimization of discretization methods applied to lumped audio systems, a class of systems which includes most audio electrical circuits as well as many mechanical systems of interest in audio (e.g., loudspeakers). Our contributions are strongly informed by the atypical compromises of virtual analog applications regarding the model update rate, which arise from the need for low-complexity and low-latency methods. In particular, we focus on non-oversampled methods, meaning methods where the update rate is fixed and accuracy must be controlled through alternative methods. The first chapter presents a framework to properly define signal identity in the context of a non-oversampled computer simulation. Indeed, a proper equivalency relationship has to be established between the continuous-time quantities we want to approximate and the discrete-time quantities that we compute in order to unambiguously define an optimal model. Multiple options can be derived for such equivalency, but only one can be used at any one time since they lead to incompatible definitions of optimality if the update rate is kept fixed. Our framework details mathematically how to unambiguously define and apply a chosen equivalency option. A by-product of this framework is a broader definition of the concept of aliasing, attached to the limitation in the information amount encoded by discrete-time sequences, and the resulting implications regarding the analysis of any continuous-time system. The second chapter presents new analytical and empirical developments for the analysis of aliasing error in the numerical models of lumped audio systems. It allows for an advanced and efficient comparison of discretization approaches and antialiasing methods. The framework discusses aliasing distortion in the context of different both static and dynamical nonlinear systems, and for various input periodic waveforms. The proposed empirical analysis method uses an extension of harmonic balance methods in order to efficiently retrieve all the harmonic components of the output waveform of the model and the original system. The third chapter presents a generalization of known one-step discretization methods using the formalism of Möbius transforms. It formalizes particular subsets of these transforms to provide for new discretization approaches for nonlinear lumped systems, such as the alpha-transform and the parametric bilinear transform. It also formalizes several design criteria to choose the most appropriate candidate of that family for a given virtual analog application. In particular, it presents the novel criterion of damping monotonicity. These methods are of particular interest due to two properties. First, they can be readily interpreted as one-to-one mapping functions between the s-plane used to describe continuous-time systems and the z-plane used to describe discrete-time computer models, facilitating the derivation and interpretation of parametrization rules. Second, they preserve the order of the update equations so that the computational costs of models generated with any of these methods are comparable. In particular, this class generalizes common virtual analog methods such as the bilinear transform and the Euler methods. The fourth chapter presents a novel framework to optimize the discretization methods applied to individual elements of a lumped linear audio system in order to systematically obtain a more accurate computer model. Deriving physical models for linear systems can provide access to the equations relative to all the dynamical elements forming the system and its model responses. However, discretization procedures are generally applied globally to the system equations, allowing very limited control on the reproduction error through available free parameters in the discretization. We show how to leverage optimizing such free parameters at the individual element level to generate a much more accurate physical model while preserving the system structure and the model computational costs. In particular, our approach can perform a joint optimization of an arbitrary number of output variables using either a Kirchhoff domain or a wave domain formulation.

Keywords: Music.
Advisor: Smith Julius O. (Julius Orion) 
Abel Jonathan (Jonathan Stuart)
Chafe Chris