Causal Convolutional Neural Networks as Finite Impulse Response Filters

TL;DR

The paper investigates how Causal Convolutional Neural Networks (CCNNs) with quasi-linear activations behave on time-series with sparse spectral content, showing that, when equipped with long kernels and left-padding for causality, they approximate Finite Impulse Response (FIR) filtering and can be collapsed into a single least-squares FIR (LS-FIR) filter. Through spectrum learning, regression on multi-degree-of-freedom (MDOF) systems, and unsupervised VAE-based identification, CCNNs exhibit spectral sparsity around modal frequencies and recover interpretable spectral features, including clear modal peaks, with high linearity (R^2 ≈ 0.975). The real-data Z24 bridge benchmark demonstrates that CCNNs can detect and isolate modal behavior using relatively short data segments, offering an interpretable alternative to traditional frequency-domain methods like FDD. Overall, the work provides a principled link between neural network representations and classical signal processing for dynamic system identification, enabling physically interpretable DL models in structural dynamics and SHM contexts.

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.

Figures (20)

• Figure 1: A spectrally sparse time-series dataset characterized by a few dominant peaks in the frequency domain is used to train a Causal Convolutional Neural Network (CNN). After training, the convolutional weights across all layers are combined into an equivalent single-layer filter. The resulting impulse response is then compared to that of a Finite Impulse Response (FIR) filter, designed using a least-squares optimization over the same target bandwidth.
• Figure 2: Band-pass FIR filters with a peak at 10 Hz. Shorter filters are more akin to low-pass than band-pass filters. The FIR filters are designed using the windowing method, adopting a Hamming window Parks1972.
• Figure 3: Comparison of the spectrum of different filters (with 51 taps) against the reference filter spectrum.
• Figure 4: A comparison of the weights and spectrum the LS-FIR and CCNN filters for a band-pass peak at 23 Hz. The CCNN here is composed of 7 layers with a kernel width of 75.
• Figure 5: A comparison of the weights and spectrum the LS-FIR and CCNN filters for a band-pass peak at 10 Hz. The CCNN here is composed of 5 layers with a kernel width of 75.