Analytic Minimum Mean-Square Error Bounds in Linear Dynamic Systems With Gaussian Mixture Noise Statistics

Leila Pishdad,Fabrice Labeau

doi:10.1109/access.2020.2986420

Leila Pishdad, Fabrice Labeau

Open Access

https://doi.org/10.1109/access.2020.2986420

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Abstract

For linear dynamic systems with Gaussian noise, the Kalman filter provides the Minimum Mean-Square Error (MMSE) state estimation by tracking the posterior. Similarly, for systems with Gaussian Mixture (GM) noise distributions, a bank of Kalman filters or the Gaussian Sum Filter (GSF), can provide the MMSE state estimation. However, the MMSE itself is not analytically tractable. Moreover, the general analytic bounds proposed in the literature are not tractable for GM noise statistics. Hence, in this work, we evaluate the MMSE of linear dynamic systems with GM noise statistics and propose its analytic lower and upper bounds. We provide two analytic upper bounds which are the Mean-Square Errors (MSE) of implementable filters, and we show that based on the shape of the GM noise distributions, the tighter upper bound can be selected. We also show that for highly multimodal GM noise distributions, the bounds and the MMSE converge. Simulation results support the validity of the proposed bounds and their behavior in limits.

Highlights

Sequential Bayesian estimation techniques provide probabilistic approximations of the state in dynamic systems
The rest of this paper is organized as follows: In Section II we provide the details of the system model and introduce the notation used in this paper, as well as the details of Gaussian Sum Filter (GSF), which is the Minimum Mean-Square Error (MMSE) filter
In addition to the proposed bounds, we have provided numerically approximated Posterior Cramer-Rao lower bound for the MMSE filter, for comparison purposes

Summary

Introduction

Sequential Bayesian estimation techniques provide probabilistic approximations of the state in dynamic systems. Some examples include event-based state estimation [6], [7], systems with multipath fading or non-line-of-sight conditions (NLOS) [8], systems using multiple measurement sensors [6], or systems with maneuvering targets [2]. In such cases, the Kalman filter cannot provide an optimal MMSE solution [9] as we need higher order statistics to represent the posterior [10]. Approximations should be made to find suboptimal MMSE solutions [4]

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Conclusion