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.

翻译：我们发展了一种基于沃尔泰拉级数范畴化的非线性音频信号处理合成理论。首先考虑沃尔泰拉级数在基范畴上的函子性——该范畴的对象为缓增分布，态射为特定线性变换——从而推导出当输入信号经过线性预处理时，非线性变换输出结果受影响的数学表达式。继而研究非线性音频系统的变化规律，引入沃尔泰拉级数态射概念作为建模工具，并证明该态射可视为一类透镜映射。我们展示了如何参数化态射并生成沃尔泰拉级数的索引族，此类索引族特别适用于建模非平稳或时变非线性现象。随后阐述沃尔泰拉级数及其态射如何构成范畴（记为Volt），揭示求和、乘积与级数复合等运算作为Volt上的幺半积结构，并依次给出各运算对应的泛性质。特别证明了沃尔泰拉级数的级数复合满足结合律。最后将本框架与音频信号处理核心领域——时频分析——建立关联：证明某类二阶沃尔泰拉级数与双线性时频分布（TFDs）间的等价性可推广至高阶沃尔泰拉级数与所谓多项式TFDs的等价关系。文末展望未来研究方向，包括非线性系统辨识技术的融合，以及本理论在合成图与拓扑音频信号处理场景中的拓展。