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

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

pip install torchlpc

or from source

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 preprints¹². 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:

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

$$ \frac{\partial \mathcal{L}}{\partial \bf A} = -\begin{vmatrix} \frac{\partial \mathcal{L}}{\partial x_1} & 0 & \dots & 0 \\ 0 & \frac{\partial \mathcal{L}}{\partial x_2} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \frac{\partial \mathcal{L}}{\partial x_t} \end{vmatrix} \begin{vmatrix} y_0 & y_{-1} & \dots & y_{-N + 1} \\ y_1 & y_0 & \dots & y_{-N + 2} \\ \vdots & \vdots & \ddots & \vdots \\ y_{T-1} & y_{T - 2} & \dots & y_{T - N} \end{vmatrix}. $$

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 $x_t|_{t &gt; 0}$.

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

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

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.

TODO

• Use PyTorch C++ extension for faster computation.
• Use native CUDA kernels for GPU computation.
• Add examples.

Citation

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

@misc{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},
    year={2024},
    eprint={2404.07970},
    archivePrefix={arXiv},
    primaryClass={eess.AS}
}

@misc{ycy2024golf,
    title = {Differentiable Time-Varying Linear Prediction in the Context of End-to-End Analysis-by-Synthesis},
    author = {Yu, Chin-Yun and Fazekas, György},
    year={2024},
    eprint={2406.05128},
    archivePrefix={arXiv},
    primaryClass={eess.AS}
}

Footnotes

1. Differentiable All-pole Filters for Time-varying Audio Systems. ↩

2. Differentiable Time-Varying Linear Prediction in the Context of End-to-End Analysis-by-Synthesis. ↩

3. Singing Voice Synthesis Using Differentiable LPC and Glottal-Flow-Inspired Wavetables. ↩