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.

翻译：我们发展了一种基于Volterra级数范畴化的非线性音频信号处理组合理论。我们将Volterra级数的经典定义扩展为关于某个基范畴的函子，该范畴的对象为缓增分布，态射为特定的线性变换。由此导出的公式描述了当输入信号首先经过线性处理时非线性变换结果所受的影响。进而研究非线性音频系统如何变化，并提出Volterra级数态射作为其数学模型。我们展示了态射的参数化方法及其用于生成带索引的Volterra级数族的过程，这类级数族特别适合建模非平稳或时变非线性现象。我们阐述了Volterra级数及其态射如何构成一个函子范畴Volt，其中对象为Volterra级数，态射为自然变换。揭示了Volterra级数的和、积与级数组合运算作为Volt上的幺半群结构，并分别指出其对应的泛性质，特别证明了Volterra级数的级数组合满足结合律。随后将我们的理论框架与音频信号处理的核心主题——时频分析——建立联系：证明某类二阶Volterra级数与双线性时频分布（TFDs）之间的等价性可推广至高阶Volterra级数与所谓多项式TFDs之间的等价关系。最后展望未来工作，包括融合非线性系统辨识技术以及将本理论扩展至组合图和拓扑音频信号处理领域。