We introduce a general formulation for automatic differentiation through direct form filters, yielding a closed-form backpropagation that includes initial condition gradients. The result is a single expression that can represent both the filter and its gradients computation while supporting parallelism. C++/CUDA implementations in PyTorch achieve at least 1000x speedup over naive Python implementations and consistently run fastest on the GPU. For the low-order filters commonly used in practice, exact time-domain filtering with analytical gradients outperforms the frequency-domain method in terms of speed. The source code is available at https://github.com/yoyolicoris/philtorch.
Cite
@article{arxiv.2511.14390,
title = {Accelerating Automatic Differentiation of Direct Form Digital Filters},
author = {Chin-Yun Yu and György Fazekas},
journal= {arXiv preprint arXiv:2511.14390},
year = {2025}
}
Comments
Accepted at the 1st Workshop on Differentiable Systems and Scientific Machine Learning @ EurIPS 2025