# Differentiable All-pole Filters for Time-varying Audio Systems

arXiv: https://arxiv.org/abs/2404.07970 · HTML: https://mlsanity.com/paper/2404.07970/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 29 references that are in the map.
- First posted on arXiv 11 Apr 2024.

## Builds on (most cited first, 14 of 14)

- [Parallel WaveGAN: A fast waveform generation model based on generative adversarial networks with multi-resolution spectrogram](https://mlsanity.com/paper/1910.11480/) (2019)
- [DDSP: Differentiable Digital Signal Processing](https://mlsanity.com/paper/2001.04643/) (2020)
- [LPCNet: Improving Neural Speech Synthesis Through Linear Prediction](https://mlsanity.com/paper/1810.11846/) (2018)
- [dynoNet: a neural network architecture for learning dynamical systems](https://mlsanity.com/paper/2006.02250/) (2020)
- [Style Transfer of Audio Effects with Differentiable Signal Processing](https://mlsanity.com/paper/2207.08759/) (2022)
- [Sinusoidal Frequency Estimation by Gradient Descent](https://mlsanity.com/paper/2210.14476/) (2022)
- [One Billion Audio Sounds from GPU-enabled Modular Synthesis](https://mlsanity.com/paper/2104.12922/) (2021)
- [Lightweight and interpretable neural modeling of an audio distortion effect using hyperconditioned differentiable biquads](https://mlsanity.com/paper/2103.08709/) (2021)
- [Direct design of biquad filter cascades with deep learning by sampling random polynomials](https://mlsanity.com/paper/2110.03691/) (2021)
- [Modulation Extraction for LFO-driven Audio Effects](https://mlsanity.com/paper/2305.13262/) (2023)
- [Differentiable Grey-box Modelling of Phaser Effects using Frame-based Spectral Processing](https://mlsanity.com/paper/2306.01332/) (2023)
- [GELP: GAN-Excited Linear Prediction for Speech Synthesis from Mel-spectrogram](https://mlsanity.com/paper/1904.03976/) (2019)
- [Singing Voice Synthesis Using Differentiable LPC and Glottal-Flow-Inspired Wavetables](https://mlsanity.com/paper/2306.17252/) (2023)
- [DENT-DDSP: Data-efficient noisy speech generator using differentiable digital signal processors for explicit distortion modelling and noise-robust speech recognition](https://mlsanity.com/paper/2208.00987/) (2022)

## Built on by (most cited first, 7 of 7)

- [Designing Neural Synthesizers for Low-Latency Interaction](https://mlsanity.com/paper/2503.11562/) (2025)
- [Modulation Discovery with Differentiable Digital Signal Processing](https://mlsanity.com/paper/2510.06204/) (2025)
- [Inverse Reconstruction of Shock Time Series from Shock Response Spectrum Curves using Machine Learning](https://mlsanity.com/paper/2603.03229/) (2026)
- [Combolutional Neural Networks](https://mlsanity.com/paper/2507.21202/) (2025)
- [Gradient-based Optimisation of Modulation Effects](https://mlsanity.com/paper/2601.04867/) (2026)
- [An Interpretable, Controllable Time-Varying IIR Denoiser for On-Device Assistive Hearing](https://mlsanity.com/paper/2603.02794/) (2026)
- [Physics-Informed Neural Engine Sound Modeling with Differentiable Pulse-Train Synthesis](https://mlsanity.com/paper/2603.09391/) (2026)

## Abstract (arXiv)

Infinite impulse response filters are an essential building block of many time-varying audio systems, such as audio effects and synthesisers. However, their recursive structure impedes end-to-end training of these systems using automatic differentiation. Although non-recursive filter approximations like frequency sampling and frame-based processing have been proposed and widely used in previous works, they cannot accurately reflect the gradient of the original system. We alleviate this difficulty by re-expressing a time-varying all-pole filter to backpropagate the gradients through itself, so the filter implementation is not bound to the technical limitations of automatic differentiation frameworks. This implementation can be employed within audio systems containing filters with poles for efficient gradient evaluation. We demonstrate its training efficiency and expressive capabilities for modelling real-world dynamic audio systems on a phaser, time-varying subtractive synthesiser, and compressor. We make our code and audio samples available and provide the trained audio effect and synth models in a VST plugin at https://diffapf.github.io/web/.