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https://github.com/yoyololicon/torchlpc


https://github.com/yoyololicon/torchlpc

ddsp linear-predictive-coding speech-synthesis time-varying-filter time-varying-systems

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README 

        
# TorchLPC

[![PyPI version](https://badge.fury.io/py/torchlpc.svg)](https://badge.fury.io/py/torchlpc)



`torchlpc` provides a PyTorch implementation of the Linear Predictive Coding (LPC) filter, also known as all-pole filter.

It's fast, differentiable, and supports batched inputs with time-varying filter coefficients.



Given an input signal $`\mathbf{x} \in \mathbb{R}^T`$ and time-varying LPC coefficients $`\mathbf{A} \in \mathbb{R}^{T \times N}`$ with an order of $`N`$, the LPC filter is defined as:



$$

y_t = x_t - \sum_{i=1}^N A_{t,i} y_{t-i}.

$$



## Usage



```python



import torch

from torchlpc import sample_wise_lpc



# Create a batch of 10 signals, each with 100 time steps

x = torch.randn(10, 100)



# Create a batch of 10 sets of LPC coefficients, each with 100 time steps and an order of 3

A = torch.randn(10, 100, 3)



# Apply LPC filtering

y = sample_wise_lpc(x, A)



# Optionally, you can provide initial values for the output signal (default is 0)

zi = torch.randn(10, 3)

y = sample_wise_lpc(x, A, zi=zi)

```



## Installation



```bash

pip install torchlpc

```



or from source



```bash

pip install git+https://github.com/yoyololicon/torchlpc.git

```



## Derivation of the gradients of the LPC filter



The details of the derivation can be found in our preprints[^1][^2].

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}

= \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 \bf A}

= -\begin{vmatrix}

\frac{\partial \mathcal{L}}{\partial x_1} & 0 & \dots & 0 \\

0 & \frac{\partial \mathcal{L}}{\partial x_2} & \dots & 0 \\

\vdots & \vdots & \ddots & \vdots \\

0 & 0 & \dots & \frac{\partial \mathcal{L}}{\partial x_t}

\end{vmatrix}

\begin{vmatrix}

y_0 & y_{-1} & \dots & y_{-N + 1} \\

y_1 & y_0 & \dots & y_{-N + 2} \\

\vdots & \vdots & \ddots & \vdots \\

y_{T-1} & y_{T - 2} & \dots & y_{T - N}

\end{vmatrix}.

$$



### Gradients for the initial condition $`y_t|_{t \leq 0}`$



The initial conditions provide an entry point at $t=1$ for filtering, as we cannot evaluate $t=-\infty$.

Let us assume $`A_{t, :}|_{t \leq 0} = 0`$ so $`y_t|_{t \leq 0} = x_t|_{t \leq 0}`$, which also means $`\frac{\partial \mathcal{L}}{\partial y_t}|_{t \leq 0} = \frac{\partial \mathcal{L}}{\partial x_t}|_{t \leq 0}`$.

Thus, the initial condition gradients are



$$

\frac{\partial \mathcal{L}}{\partial y_t} 

= \frac{\partial \mathcal{L}}{\partial x_t}

= -\sum_{i=1-t}^{N} A_{t+i,i} \frac{\partial \mathcal{L}}{\partial x_{t+i}} \quad \text{for } -N < t \leq 0.

$$



In practice, we pad $N$ and $N \times N$ zeros to the beginning of $\frac{\partial \mathcal{L}}{\partial \bf y}$ and $\mathbf{A}$ before evaluating $\frac{\partial \mathcal{L}}{\partial \bf x}$.

The first $N$ outputs are the gradients to $`y_t|_{t \leq 0}`$ and the rest are to $`x_t|_{t > 0}`$.



### Time-invariant filtering



In the time-invariant setting, $`A_{t, i} = A_{1, i} \forall t \in [1, T]`$ and the filter is simplified to



```math

y_t = x_t - \sum_{i=1}^N a_i y_{t-i}, \mathbf{a} = A_{1,:}.

```



The gradients $`\frac{\partial \mathcal{L}}{\partial \mathbf{x}}`$ are filtering $`\frac{\partial \mathcal{L}}{\partial \mathbf{y}}`$ with $\mathbf{a}$ backwards in time, same as in the time-varying case.

$\frac{\partial \mathcal{L}}{\partial \mathbf{a}}$ is simply doing a vector-matrix multiplication:



$$

\frac{\partial \mathcal{L}}{\partial \mathbf{a}^T} =

-\frac{\partial \mathcal{L}}{\partial \mathbf{x}^T}

\begin{vmatrix}

y_0 & y_{-1} & \dots & y_{-N + 1} \\

y_1 & y_0 & \dots & y_{-N + 2} \\

\vdots & \vdots & \ddots & \vdots \\

y_{T-1} & y_{T - 2} & \dots & y_{T - N}

\end{vmatrix}.

$$



This algorithm is more efficient than [^3] because it only needs one pass of filtering to get the two gradients while the latter needs two.



[^1]: [Differentiable All-pole Filters for Time-varying Audio Systems](https://arxiv.org/abs/2404.07970).

[^2]: [Differentiable Time-Varying Linear Prediction in the Context of End-to-End Analysis-by-Synthesis](https://arxiv.org/abs/2406.05128).

[^3]: [Singing Voice Synthesis Using Differentiable LPC and Glottal-Flow-Inspired Wavetables](https://arxiv.org/abs/2306.17252).



## TODO



- [ ] Use PyTorch C++ extension for faster computation.

- [ ] Use native CUDA kernels for GPU computation.

- [ ] Add examples.



## Related Projects



- [torchcomp](https://github.com/yoyololicon/torchcomp): differentiable compressors that use `torchlpc` for differentiable backpropagation.

- [jaxpole](https://github.com/rodrigodzf/jaxpole): equivalent implementation in JAX by @rodrigodzf.



## Citation



If you find this repository useful in your research, please cite our work with the following BibTex entries:



```bibtex

@inproceedings{ycy2024diffapf,

    title={Differentiable All-pole Filters for Time-varying Audio Systems},

    author={Chin-Yun Yu and Christopher Mitcheltree and Alistair Carson and Stefan Bilbao and Joshua D. Reiss and György Fazekas},

    booktitle={International Conference on Digital Audio Effects (DAFx)},

    year={2024},

    pages={345--352},

}



@inproceedings{ycy2024golf,

    title     = {Differentiable Time-Varying Linear Prediction in the Context of End-to-End Analysis-by-Synthesis},

    author    = {Chin-Yun Yu and György Fazekas},

    year      = {2024},

    booktitle = {Proc. Interspeech},

    pages     = {1820--1824},

    doi       = {10.21437/Interspeech.2024-1187},

}

```