  We develop a compositional theory of nonlinear audio signal processing based
on a categorification of the Volterra series. We begin by considering what it
would mean for the Volterra series to be functorial with respect to a base
category whose objects are temperate distributions and whose morphisms are
certain linear transformations. This leads to formulae describing how the
outcomes of nonlinear transformations are affected if their input signals are
first linearly processed. We then consider how nonlinear audio systems change,
and introduce as a model thereof a notion of morphism of Volterra series, which
we exhibit as a kind of lens map. 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. We show, in particular, that the series
composition of Volterra series is associative. We then bridge between our
framework and a subject at the heart of audio signal processing: time-frequency
analysis. Specifically, we show that an equivalence between a certain class of
second-order Volterra series and the bilinear time-frequency distributions
(TFDs) can be extended to one between certain higher-order Volterra series and
the so-called polynomial TFDs. We end with prospects for future work, including
the incorporation of nonlinear system identification techniques and the
extension of our theory to the settings of compositional graph and topological
audio signal processing.
