Causal Convolutional Neural Networks as Finite Impulse Response Filters

TL;DR

This paper reveals that causal CNNs with long kernels behave like FIR filters, capturing spectral features and enabling interpretability in time-series analysis, validated on simulated and real-world data.

Contribution

It demonstrates that trained causal CNNs can be equivalent to FIR filters, providing new insights into their spectral learning capabilities and interpretability.

Findings

Causal CNNs with extended kernels act as FIR filters.

Networks can be reduced to a single FIR-like filter via convolution properties.

Validated on simulated and real-world dynamic system data.

Abstract

This study investigates the behavior of Causal Convolutional Neural Networks (CNNs) with quasi-linear activation functions when applied to time-series data characterized by multimodal frequency content. We demonstrate that, once trained, such networks exhibit properties analogous to Finite Impulse Response (FIR) filters, particularly when the convolutional kernels are of extended length exceeding those typically employed in standard CNN architectures. Causal CNNs are shown to capture spectral features both implicitly and explicitly, offering enhanced interpretability for tasks involving dynamic systems. Leveraging the associative property of convolution, we further show that the entire network can be reduced to an equivalent single-layer filter resembling an FIR filter optimized via least-squares criteria. This equivalence yields new insights into the spectral learning behavior of CNNs trained on signals with sparse frequency content. The approach is validated on both simulated beam dynamics and real-world bridge vibration datasets, underlining its relevance for modeling and identifying physical systems governed by dynamic responses.