  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.
