Recursive smoothing algorithms for the estimation of signals from uncertain observations via mixture approximations

Abstract

Considering discrete-time systems with uncertain observations when the signal model is unknown, but only covariance information is available, and the signal and the observation additive noise are correlated and jointly Gaussian, we present recursive algorithms for suboptimal fixed-point and fixed-interval smoothing estimators. To derive the algorithms, we employ a technique consisting in approximating the conditional distributions of the signal given the observations by Gaussian distributions, taking successive approximations of the mixtures of normal distributions. The expectation of these approximations provides us with the suboptimal estimators. In a numerical simulation example, the performance of the proposed estimators is compared with that of linear ones, via the sample mean square values of the corresponding estimation errors.

Keywords:

• suboptimal estimation
• uncertain observations
• covariance information
• signal–noise correlation
• discrete time systems
• recursive estimation

Acknowledgements

This work was partially supported by the Ministerio de Educación y Ciencia and the Junta de Andalucía through the projects MTM2008-05567 and P07-FQM-02701, respectively.

Notes

1. (H ₁|H ₂) denotes a matrix partitioned into two sub-matrices H ₁ and H ₂.

2. f(y ₁/γ₁ = i, Y ⁰) denotes the conditional pdf f(y ₁/γ₁ = i), i = 0, 1.