On the asymptotic behavior of linearly constrained filters for robust multi-channel signal processing

• Published in: Signal processing 2022-07, Vol.196, p.108500, Article 108500
• Main Authors: Chauchat, Paul, Vilà-Valls, Jordi, Chaumette, Eric
• Format: Article
• Language: English
• Subjects: Asymptotic performance; Engineering Sciences; Linearly constrained filtering; Model mismatch; Robust adaptive beamforming; Signal and Image processing
• ISSN: 0165-1684; 1872-7557
• DOI: 10.1016/j.sigpro.2022.108500

The Kalman filter (KF) is known to loose its optimality properties when the model used does not perfectly match the true system. Extending the use of linear constraints to this filter recently proved to be efficient to mitigate a large class of parametric model mismatch, through the linearly constrained Kalman filter and minimum variance filter (LCKF and LCMVF). However, the asymptotic performances of these new filters are still an open question. In this work, we bring a first answer to the latter problem in the case of measurement model mismatch. We show that both LCKF and LCMVF are equivalent to unconstrained filters in which the directions of the constraints are cancelled by projection, allowing a better understanding of their asymptotic properties. In particular, the steady-state mean square error, when it exists, is derived. The consistency of the filters with respect to nonlinear mismatches is also improved via new constraints. An array processing example is provided to assess the derived formulas, the consistency and performance of the filters.