# Accelerating Automatic Differentiation of Direct Form Digital Filters

arXiv: https://arxiv.org/abs/2511.14390 · HTML: https://mlsanity.com/paper/2511.14390/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Nov 2025.

## Built on by (most cited first, 1 of 1)

- [An Interpretable, Controllable Time-Varying IIR Denoiser for On-Device Assistive Hearing](https://mlsanity.com/paper/2603.02794/) (2026)

## Abstract (arXiv)

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.