Compositional nonlinear audio signal processing with Volterra series

Abstract

We present a compositional theory of nonlinear audio signal processing based on a categorification of the Volterra series. We begin by augmenting the classical definition of the Volterra series so that it is functorial with respect to a base category whose objects are temperate distributions and whose morphisms are certain linear transformations. This motivates the derivation of formulae describing how the outcomes of nonlinear transformations are affected if their input signals are linearly processed--e.g., translated, modulated, sampled, or periodized. We then consider how nonlinear systems, themselves, change, and introduce as a model thereof the notion of morphism of Volterra series, which we exhibit as both a type of lens map and natural transformation. We show how morphisms can be parameterized and used to generate indexed families of Volterra series, which are well-suited to model nonstationary or time-varying nonlinear phenomena. We then describe how Volterra series and their morphisms organize into a category, which we call Volt. We exhibit the operations of sum, product, and series composition of Volterra series as monoidal products on Volt, and identify, for each in turn, its corresponding universal property. In particular, we show that the series composition of Volterra series is associative. We then bridge between our framework and the subject at the heart of audio signal processing: time-frequency analysis. Specifically, we show that a known equivalence, between a class of second-order Volterra series and the bilinear time-frequency distributions, can be extended to one between certain higher-order Volterra series and the so-called polynomial TFDs. We end by outlining potential avenues for future work, including the incorporation of system identification techniques and the potential extension of our theory to the settings of graph and topological audio signal processing.