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/.

翻译：无限脉冲响应滤波器是许多时变音频系统（如音频效果器和合成器）的核心基础组件。然而，其递归结构阻碍了利用自动微分对这些系统进行端到端训练。尽管先前研究已提出并广泛采用频率采样和基于帧的处理等非递归滤波器近似方法，但这些方法无法准确反映原始系统的梯度信息。我们通过重新设计时变全极点滤波器的梯度反向传播机制来解决这一难题，使得滤波器实现不再受限于自动微分框架的技术约束。该实现可应用于包含极点滤波器的音频系统中，以实现高效的梯度计算。我们通过在移相器、时变减法合成器和压缩器上的实验，展示了该方法在建模真实动态音频系统时具有的训练效率与表达能力。我们在https://diffapf.github.io/web/公开了代码与音频样本，并以VST插件形式提供训练完成的音频效果器与合成器模型。