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About this paper
Jake Araujo-Simon
This work develops a compositional theory of nonlinear audio signal processing by reformulating Volterra series as endofunctors and their interrelationships as natural transformations, establishing a monoidal category, `Volt`. The framework successfully extends Volterra series representations to higher-order Polynomial Time-Frequency Distributions.
Key insights
- Category theory provides a robust mathematical framework for unifying and generalizing nonlinear system theory, allowing for a compositional and diagrammatic approach.
- Redefining Volterra series as functors enables a formal understanding of how linear transformations applied to inputs affect the outputs of nonlinear systems.
- The proof of associativity for series composition is a foundational step for developing a rigorous, modular design and analysis language for interconnected nonlinear systems.
- Nonlinearity can be constructively understood through the Volterra series as weighted interactions between input signal samples, moving beyond its negative definition.
Summary by alphaXiv.
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