(Torus Adaptive Filters)

Signal Processing and Filtering


Signals are everywhere : sounds, images, videos, radio waves, biological signals, stock prices, etc. They can be acquired from sensors, measuring instruments, or communication devices. 

Signal processing is a multi-disciplinary field that aims to improve the quality of signals, extract and manipulate information from them, make them more understandable for humans or machines, etc.

Filtering is a central part of signal processing : the filters are mainly used to filter out noises and unwanted information, keep and enhance useful signals, and correct the errors.

For example, GNSS signals from satellites are embedded in noise, and measurements from local sensors such as IMUs are also very noisy. Filtering algorithms play a crucial role in achieving accurate, robust, and reliable geopositioning of mobile receivers.

Most filters currently in use are based on the classical Kalman Filter (KF) or its extended versions (EKF). Theoretically, under certain hypotheses, these Kalman filters are optimal. However, in practice, these hypotheses are not satisfied, and Kalman filters do not give the best results in general. 

Torus Adaptive Filters

The patent-pending Torus Adaptive Filters (TAF) developed by our team is a new generation of filters that are much better than the Kalman filters in many ways.  See below for a table of comparisons. 

The first two contracts for our adaptive filters, awarded in 2022-2023, are: the BPI (Banque Publique d’Investissement) contract for geopositioning of agricultural robots (GEOSUR consortium), and the CNES (the French version of NASA) initial contract for testing our algorithms on PPP  (precise point positioning). Our filters can integrate data from many different sensors, such as GNSS receivers (GNSS-RTK), inertial sensors, odometers, LIDARS, etc.

Our Torus Adaptive Filters can also be applied to many other fields, such as autonomous  vehicles, medical devices, geology and mining, financial markets, etc. 

We are currently developing Torus Neural Filtersa new version of our adaptive filters, that replace and/or enhance some of the steps in our current version of Torus Adaptive Filters by neural networks. These next-generation neural filters show promising preliminary results in fault detection and reduction of integrity risks, among other things.

Meet the Team

Our Signal Processing team consists of 10+ highly-qualified mathematicians and computer scientists, some of them are at professor level,  with many years of experience in GNSS, geology, meteorology, epidemiology, medical devices, etc. We’re looking to expand our team and take on new challenges!


About GNSS Geolocation

The principle of Global Navigation Satellite System (GNSS) geolocation is based on estimating the distances between a receiver and satellites with known coordinates. For this purpose, each satellite emits a signal encoding the time of its transmission. The receiver captures this information, theoretically allowing the deduction of the distance between the receiver and the emitting satellite, and consequently its position.

In practice, GNSS geolocation relies on a detailed modeling of the relationship between the pseudo-ranges measured by a GNSS receiver and its coordinates, phase carrierscoordinates of visible satellites, as well as multiple error sources such as satellite clock desynchronization, ephemeris data, atmospheric phenomena, code interference, receiver clock biases relative to GNSS time, etc.


Differences between Kalman Filters (KF) and Torus Adaptive Filters (TAF)

Kalman Filters (KF)

  • At each moment, solve a matrix equation to update the gain.
  • Algorithmic complexity in O(n3), where n is the size of the state vector.
  • Optimization in the functional space, of infinite dimension, of all realizations.
  • Need to solve matrix equations in real time.
  • Optimality does not imply stability.
  • Accurate knowledge of model error statistics and observations is required.
  • Non-linear systems must be linearized.

Torus Adaptive Filters (TAF)

  • At each time, use a powerful stochastic algorithm to update the coefficients of the gain.
  • Algorithmic complexity in O(n2), where n is the size of the state vector.
  • Optimization in the space of realizations = for each of the trajectories realized.
  • No need to solve matrix equations.
  • Optimality implies stability.
  • Accurate knowledge of model error statistics and observations is NOT required.
  • No need to linearize non-linear systems.


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