We develop a compositional theory of nonlinear audio signal processing based
on a categorification of the Volterra series. We augment the classical
definition of 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 the notion of morphism of Volterra series. 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 describe how Volterra series and their
morphisms organize into a functor category, Volt, whose objects are Volterra
series and whose morphisms are natural transformations. 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.Comment: Master's thesi