International Journal of Systems Science volume 34, number 7, 10 June 2003, pages 439–452 Fixed-point smoothing with non-independent uncertainty using covariance information S. NAKAMORIy*, R. CABALLERO-A ´ GUILAz, A. HERMOSO-CARAZO and J. LINARES-PE ´ REZx Recursive filtering and fixed-point smoothing algorithms, using covariance information, are designed in systems with uncertain observations, when the variables describing the uncertainty are not necessarily independent. It is assumed that the observations are perturbed by white plus coloured noises, and the autocovariance functions of the signal and coloured noise are given in a semidegenerate kernel form. The estimators are obtained by the orthogonal projection technique and using an invariant imbedding method. The algorithms can be applied for estimating stationary and non-stationary signals. 1. Introduction Usually, the estimation theory in linear stochastic systems assumes that the signal to be estimated is always present in the observations. However, in many practical situations, the systems are inherently vulnerable to changes in their measurements due, for instance, to component or interconnection failures, environment changes etc.; these kind of systems can be modelled including a multiplicative noise component in the mea- surements in addition to the additive noise. These sys- tems with multiplicative noise are also appropriate to describe many situations of fading or reflection of the transmitted signals from the ionosphere and, also, certain cases involving sampling, gating, or amplitude modula- tion. Nahi (1969) was the first to study the optimum linear recursive filtering problem in discrete-time systems, assuming that the observation matrix is multiplied by a scalar binary-valued white noise. Monzingo (1975) extended it to an optimal linear smoother, and later on, Hermoso and Linares (1994, 1995) generalized these results to the case in which the additive noises of the state and observation are correlated. However, there exist many practical situations, such as transmission of data in multichannels or remote- sensing situations, in which the independence assump- tion on the observation multiplicative noise is not realistic. An alternative model for this multiplicative noise using a two-state Markov chain was considered by Jaffer and Gupta (1971), deriving non-linear estima- tion algorithms. Hadidi and Schwartz (1979) proved that, in this general model, the optimal linear estimators are not recursive; they then found a necessary and sufficient condition on the sequence which describes the uncertainty in the observations for the existence of linear recursive estimators of the signal. Wang (1984) reached the same conclusions by a different approach, and Monzingo (1981) studied the optimal linear smoother for these systems. Also, the estimation problem in models with stochastic parameters, which include multiplicative noise, has been treated by several authors such as De Koning (1984), Yaz (1992), and Costa (1994), to mention just a few. In all the aforementioned works, the estimation algorithms are derived assuming a full knowledge of the state-space model for the signal process. In both linear continuous-time systems (Nakamori 1990, 1991) and linear discrete-time systems (Nakamori 1992), recursive least-squares estimators are obtained for signals perturbed by white plus coloured additive noises using the autocovariance functions of the signal and coloured noise, without requiring the state-space model of the signal. Received 25 January 2002. Revised 24 July 2003. Accepted 8 October 2003. y Department of Technology. Faculty of Education, Kagoshima University, 1-20-6, Kohrimoto, Kagoshima 890-0065, Japan. z Departamento deEstadı´stica e Investigacio´ n Operativa, Universi- dad de Jae´n, Paraje Las Lagunillas, s/n, 23071 Jae´n, Spain. x Departamento deEstadı´stica e Investigacio´ n Operativa, Universi- dad de Granada, Campus Fuentenueva, s/n, 18071 Granada, Spain. * To whom correspondence should be addressed. e-mail: nakamori@edu.kagoshima-u.ac.jp International Journal of Systems Science ISSN 0020–7721 print/ISSN 1464–5319 online ß 2003 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/00207720310001636390