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Update math notation in README.md
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yoyolicoris committed Apr 15, 2024
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Expand Up @@ -10,8 +10,6 @@ $$
y_t = x_t - \sum_{i=1}^N A_{t,i} y_{t-i}.
$$

It's still in early development, so please open an issue if you find any bugs.

## Usage

```python
Expand Down Expand Up @@ -51,11 +49,11 @@ pip install git+https://github.com/yoyololicon/torchlpc.git
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 $x_t$ and the filter coefficients $A_{t, :}$ can be computed as follows:

$$
```math
\frac{\partial \mathcal{L}}{\partial x_t}
= \frac{\partial \mathcal{L}}{\partial y_t}
- \sum_{i=1}^{N} A_{t+i,i} \frac{\partial \mathcal{L}}{\partial x_{t+i}}
$$
```

$$
\frac{\partial \mathcal{L}}{\partial A_{t,i}}
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