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 any audio system 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 feed-forward compressor. We make our code available and provide the trained audio effect and synth models in a VST plugin at https://christhetree.github.io/all_pole_filters/.

通过重新表达时间变化的全极滤波器以通过自身反向传播梯度，我们消除了在自动微分框架的技术限制下将滤波器实现到这一困难，从而可以在包含极点的任何音频系统中使用此实现以进行高效的梯度评估。我们演示了该实现在相移器、时变减法合成器和前向压缩器上对建模真实动态音频系统的训练效率和表现能力，并在此https URL上提供了我们的代码和经过训练的音频效果和合成模型的VST插件。

可微分时变音频系统的全极滤波器