Probabilistic chirp instantaneous frequency estimation using Gaussian processes

Companion implementation of the paper "Probabilistic Estimation of Instantaneous Frequencies of Chirp Signals".

The preprint is available at http://arxiv.org/abs/2205.06306.

Please cite as follows to use the codes.

@article{ZhaoZheng2023Chirp,
	title = {Probabilistic Estimation of Instantaneous Frequencies of Chirp Signals},
	journal = {IEEE Transactions on Signal Processing},
	author = {Zheng Zhao and Simo S\"{a}rkk\"{a} and Jens Sj\"{o}lund and Thomas B. Sch\"{o}n},
	year = {2023}, 
	note = {In press}
}

Problem formulation

Consider a chirp signal of the form

$$ Y_k = \sin\biggl( \phi_0 + 2 \, \pi \int^{t_k}_0 f(s) \, \mathrm{d}s \biggr) + \xi_k, \quad \xi_k \sim \mathrm{N}(0, \Xi_k). $$

In the equation above, $\phi_0$ stands for the initial phase, $\xi_k$ is a Gaussian noise, $Y_k$ stands for the noisy chirp measurement at time $t_k$, and $f$ is the instantaneous frequency function that we aim to estimate. Importantly, we would like to estimate $f$ in terms of posterior distribution.

The idea is to put priors on both $f$ and the chirp signal jointly by using a class of non-linear stochastic differential equations -- which gives some conditional Gaussian processes. This leads to a non-linear state-space model in which we can estimate the posterior distribution by using stochastic filters and smoothers (e.g., EKFS).

To have a sense how well this method works, please see the following example figure taken from the paper.

Installation

This implementation is based on JaX, as we need auto-differentiation. Please refer to this official guidance to install JaX. Then open a terminal and use the following commands to install our chirpgp package:

1. git clone [email protected]:spdes/chirpgp.git
2. cd chirpgp
3. pip install -r requirements.txt
4. python setup.py install or python setup.py develop

Examples

There are a plenty of demos in folder ./demos. These demos are written self-contained and pedagogical, helping you understand how the method works.

Try read and run, for example, the script ./demos/ghfs_mle.py.

Project structure

The following list explains what the folders in this repository are.

• ./chirpgp: This is the Python package folder.
• ./demos: This folder contains runnable demos for a synthetic toymodel. It is very recommended to check this out.
• ./docs: A placeholder for future documentation.
• ./paper_plots_tables: Scripts in this folder reproduce the plots and tables in our paper.
• ./real_application: Real applications.
• ./test: Unit tests.
• ./tetralith: If you would like to exactly reproduce the numerical results in our paper, you need to run the scripts in this folder. Please refer to the README.md within for how to do so.

Filtering and smoothing module

This package also ships with a submodule for stochastic filtering and smoothing written in ./chirpgp/filters_smoothers.py. It supports:

• Kalman filter and RTS smoother in discrete time.
• Extended Kalman filter and smoother in discrete time and continuous-discrete time.
• Sigma-points filters and smoothers in discrete time and continuous-discrete time. Supports Gauss--Hermite and spherical cubature sigma points, and the 4th-order Runge--Kutta integration.
• Posterior Cramér--Rao lower bound.

Contact

Zheng Zhao, Uppsala University (2022)

https://zz.zabemon.com

[email protected], [email protected]

License

The GNU General Public License v3.0 or later.