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yoyolicoris committed Apr 15, 2024
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## Derivation of the gradients of the LPC filter

The details of the derivation can be found in our preprint **Differentiable All-pole Filters for Time-varying Audio Systems**[^1].
We show that, given the instataneous gradient $\frac{\partial \mathcal{L}}{\partial y_t}$ where $\mathcal{L}$ is the loss function, the gradients of the LPC filter with respect to the input signal $\bf x$ and the filter coefficients $\bf A$ can be computed as follows:
We show that, given the instataneous gradient $\frac{\partial \mathcal{L}}{\partial y_t}$ where $\mathcal{L}$ is the loss function, the gradients of the LPC filter with respect to the input signal $\bf x$ and the filter coefficients $\bf A$ can be expresssed also through a time-varying filter:

```math
\frac{\partial \mathcal{L}}{\partial x_t}
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