18 février 2021

Catégorie : Doctorant

Sujet de thèse au laboratoire SATIE (Université Paris-Saclay) : Design de systèmes de traitements multi-capteurs pour l’observation de l’environnement

PDFs of the subject are available here (in english) and there (in french).

*Note**: This thesis proposal is submitted to a funding by the French government defense agency. It is intended for students from the European Union, Switzerland or United Kingdom only.*

**Keywords****:** array processing – topology of sensors – robust parameter estimation – inverse problems – constrained optimization – modeling – non-asymptotic lower bounds for the mean-squared error – specifications

In this thesis, we focus on multi-sensor systems which provide observations in order to estimate (unknown) parameters of interest. Currently, such information systems are often designed on an ad-hoc basis, due to lack of sound dedicated methodology. We propose to investigate this problem from an inverse-problem point of view, i.e., we establish a statistical model linking observations to the unknown parameters. This model is clearly parameterized by the sought, unknown parameters, but also by the observation system parameters. These formers have necessarily an influence on the performance and thus on the respect of specifications. Consequently, inherent performance of parameter estimation algorithms heavily depend on the observation system parameters. We propose here to model this impact in a statistical framework in order to optimally design the observation system under given specification constraints. Only few works in the literature deal with this issue, especially as artificial intelligence give the illusion that such an analysis can be avoided.

As an illustration of this issue, in radio astronomy, modern instruments such as the LOFAR [1] is composed of 35'000 antennas, while the SKA radio interferometer (expected in 2025) will consist of 2.5 million antennas. Given the price of receivers and limited computational resources, it is not possible to use all the available sensors at the same time, in particular when data processing is to be performed in real time. The same kind of problems is encountered in modern systems such as massive MIMO, cognitive radars, ocean exploration, seismic tomography as well as Magnetic Resonance Imaging (MRI) systems. The issue is to determine, for the problem at hand, the most suitable sensor lay-out so as to satisfy given performance specifications.

In order to soundly design such systems, it is fundamental to properly assess their expected performance to ensure the respect of targeted specifications. Performance encompass several aspects, such as accuracy, sensitivity, resolution, computational cost, as well as robustness with respect to modeling errors. Yet, performance assessment is not an easy task, and it is often done by resorting to computationally intensive methods. The computational cost is usually prohibitive because simulations take into account all the diversity of the observation system parameters. Among these parameters, and depending on applications, one can cite the number of sensors, their geometry, bandwidth, bit rate, polarization, waveforms, allocated powers, perturbations... Clearly, it is not possible to add, besides these already costly performance computations, another optimization step with respect to each aforementioned parameter. For these reasons, due to lack of optimal design methodology, many systems are designed based on empirical or heuristic considerations.

The main purpose of this thesis is to propose, as an alternative, a new design methodology. Although the proposed methodology turns out to be general, it will be applied in the context of geometry optimization of an array of sensors.

The accuracy of a system is generally assessed in terms of mean-squared error (MSE). For instance, in the case of an array of sensors, one may be interested in the MSE on a target localization. However, MSE generally has to be numerically evaluated using Monte-Carlo simulations, whose computational cost grows rapidly with the dimension of the problem under study.

In order to avoid such an increasing computational cost, we propose to compute instead the lowest achievable MSE, and use it as a design criterion for complex systems. This minimal MSE is usually given by lower bounds of the MSE, the most famous of which is the Cramér-Rao bound (CRB). The CRB has already been used as a design criterion, for example in array processing [2], but only in very specific cases. It can also be used to solve an antenna selection problem, as the CRB allows to answer the question: "which sensor subset leads to the smallest minimal MSE?" [3].

However, the CRB is only valid in asymptotical conditions (large number of observations and sensors, high SNR), which do not correspond to real operating conditions. In this thesis, we suggest to work on other bounds than the CRB [4–7], which have wider validity conditions (e.g., low SNR, low number of observations). These bounds are little used in the array processing community, as they are currently difficult to handle. We will aim at developing workable non-asymptotic bounds, and at demonstrating their relevance in our context of sensor array design.

The proposed approach can be decomposed into the following steps:

- Problem modeling;
- Formulation of a design optimization criterion involving non-asymptotic MSE lower bounds, as well as regularization terms relating to problem constraints (minimal resolution, ambiguity diagram, topographic constraints, etc.);
- Design of the system by minimizing the proposed criterion with respect to the observation system parameters. This minimization will be formulated as a (possibly sparse) constrained optimization problem, involving non-convex and non-differentiable functions. If closed-form expressions for the optimal parameters cannot be obtained, corresponding numerical optimization procedures will be implemented. Either way, the proposed methodology will provide a tool that accounts for the influence of the system design parameters on its performance;
- Development and implementation of estimation algorithms in array processing on the observation system thus obtained, in order to demonstrate the validity of the proposed approach.

The candidate is expected to hold a M.Sc. in signal processing or applied mathematics (or equivalent). Having notions on optimization would be an advantage.

This thesis will take place in SATIE laboratory, under the supervision of Pascal Larzabal, Lucien Bacharach and Mohammed Nabil El Korso. The supervisory team is located at University Paris-Saclay. In order to apply, please contact the advisors:

- pascal.larzabal [at] universite-paris-saclay.fr
- lucien.bacharach [at] universite-paris-saclay.fr
- m.elkorso [at] parisnanterre.fr

Applications before April 23rd, 2021.

[1] M. P. Van Haarlem, ``LOFAR: The LOw-Frequency ARray,'' Astronomy & Astrophysics, vol. 556, no. A2, Aug. 2013.

[2] J.-P. Delmas, M. N. El Korso, H. Gazzah, and M. Castella, ``CRB analysis of planar antenna arrays for optimizing near-field source localization,'' Signal Processing, vol. 127, pp. 117–134, Oct. 2016.

[3] H. Zhang, J. Shi, Q. Zhang, B. Zong, and J. Xie, ``Antenna selection for target tracking in collocated MIMO radar,'' IEEE Transactions on Antennas and Propagation, vol. 57, no. 1, pp. 423–436, Feb. 2021.

[4] H. L. Van Trees and K. L. Bell, Eds., Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Tracking. New-York, NY, USA: Wiley/IEEE Press, Sep. 2007.

[5] E. Chaumette, J. Galy, A. Quinlan, and P. Larzabal, ``A new Barankin bound approximation for the prediction of the threshold region performance of maximum likelihood estimators,'' IEEE Transactions on Signal Processing, vol. 56, no. 11, pp. 5319–5333, Nov. 2008.

[6] D. T. Vu, A. Renaux, R. Boyer, and S. Marcos, ``Some results on the Weiss-Weinstein bound for conditional and unconditional signal models in array processing,'' Signal Processing, vol. 95, no. 2, pp. 126–148, Feb. 2014.

[7] L. Bacharach, A. Renaux, M. N. El Korso, and E. Chaumette, ``Weiss-Weinstein bound on multiple change-points estimation,'' IEEE Transactions on Signal Processing, vol. 65, no. 10, pp. 2686–2700, May 2017.

(c) GdR 720 ISIS - CNRS - 2011-2020.