New design of estimators using covariance information with uncertain observations in linear discrete-time systems Seiichi Nakamori a, * , Raquel Caballero-  Aguila b , Aurora Hermoso-Carazo c , Josefa Linares-P erez c a Faculty of Education, Department of Technology, Kagoshima University, 1-20-6 Kohrimoto, Kagoshima 890-0065, Japan b Departamento de Estad  ıstica e Investigaci on Operativa, Universidad de Ja en, Campus Las Lagunillas, s/n, 23071 Ja en, Spain c Departamento de Estad  ıstica e Investigaci on Operativa, Universidad de Granada, Campus Fuentenueva, s/n, 18071 Granada, Spain Abstract This paper proposes recursive least-squares (RLS) filtering and fixed-point smoothing algorithms with uncertain observations in linear discrete-time stochastic systems. The estimators require the information of the auto-covariance function in the semi-degenerate kernel form, the variance of white observation noise, the observed value and the probability that the signal exists in the observed value. The auto- covariance function of the signal is factorized in terms of the observation vector, the system matrix and the cross-variance function of the state variable, that generates the signal, with the signal. These quantities are obtained from the auto-covariance data of the signal. It is shown that the semi-degenerate kernel is expressed in terms of these quantities. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Wiener–Hopf equation; Linear discrete-time systems; Recursive estimation; Covariance information; Stochastic process * Corresponding author. E-mail address: nakamori@edu.kagoshima-u.ac.jp (S. Nakamori). 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII:S0096-3003(02)00060-7 Applied Mathematics and Computation 135 (2003) 429–441 www.elsevier.com/locate/amc