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On the general conditions of existence for linear MMSE filters: Wiener and Kalman

Chaumette, Eric and Vilà-Valls, Jordi and Vincent, François On the general conditions of existence for linear MMSE filters: Wiener and Kalman. (2021) Signal Processing, 184. 108052. ISSN 0165-1684

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Official URL: https://doi.org/10.1016/j.sigpro.2021.108052

Abstract

State estimation is a fundamental task in a plethora of applications. Two important classes of linear minimum mean square error (MMSE) estimators are Wiener filters (WFs) and Kalman filters (KFs), the latter being the recursive form of the former for linear discrete state-space (LDSS) dynamic models. The minimum set of uncorrelation conditions regarding states and measurements which allows to formulate a general KF form has been recently introduced in the literature, but for both WFs and KFs a standard key assumption is that the measurement noise covariance is invertible. In this contribution we provide i) an alternative derivation of the WF in the case of a non-invertible measurement covariance matrix, and ii) the proof that under the minimum set of uncorrelation conditions lately released, the KF can always be formulated irrespective of the measurement covariance matrix rank. Notice that these results also apply for Kalman predictors and smoothers, linearly constrained WFs and minimum variance distortionless response (MVDR) filters, therefore being a remarkable result of broad interest.

Item Type:Article
HAL Id:hal-03274713
Audience (journal):International peer-reviewed journal
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Institution:Université de Toulouse > Institut Supérieur de l'Aéronautique et de l'Espace - ISAE-SUPAERO (FRANCE)
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Deposited On:11 Jun 2021 17:43

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