Analytic MMSE Bounds in Linear Dynamic Systems with Gaussian Mixture Noise Statistics

TL;DR

This paper derives analytical lower and upper bounds for the MMSE in linear dynamic systems with Gaussian Mixture noise, aiding in understanding estimation accuracy when noise is non-Gaussian.

Contribution

It introduces tractable analytic bounds for MMSE in systems with GM noise, which were previously difficult to evaluate analytically.

Findings

Bounds are tight for highly multimodal GM noise.

The upper bounds correspond to the MSE of implementable filters.

Simulation results confirm the bounds' validity and convergence behavior.

Abstract

Using state-space representation, mobile object positioning problems can be described as dynamic systems, with the state representing the unknown location and the observations being the information gathered from the location sensors. For linear dynamic systems with Gaussian noise, the Kalman filter provides the Minimum Mean-Square Error (MMSE) state estimation by tracking the posterior. Hence, by approximating non-Gaussian noise distributions with Gaussian Mixtures (GM), a bank of Kalman filters or 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.