ELSEVIER Statistics & Probability Letters 28 (1996) 239-243 On the derivation of a suboptimal filter for signal estimation 1 Juan Carlos Ruiz-Molina a, Mariano J. Valderrama b,, a Department of Statistics and Operations Research, University of JaOn, 23071 JaOn, Spain b Department of Statistics and Operations Research, Campus de Cartuja, University of Granada, 18.071 Granada. Spain Received November 1994; revised May 1995 Abstract A suboptimal filter to estimate a signal corrupted by a white noise, derived from the approximative Karhunen-Lo6ve expansion of the signal, is given. The convergence of the suboptimal filter is showed and a bound on the truncation error is found. Keywords: Causal filter; Approximative Karhunen-Lo~ve expansion; Suboptimal filter 1. Introduction The importance of the Karhunen-Lobve expansion is well-known in the general theory of statistical de- tection of random signal corrupted by a white noise. This expansion allows the building of a set of observ- able coordinates, in such a way, that once their distribution is known, it is possible to solve the problem by means of a likelihood ratio test (e.g. Davenport and Root, 1958; Van Trees, 1968). Likewise, in the associated literature, the Karhunen-Lobve expansion has been applied to solve the linear mean-square esti- mation problem of a corrupted signal by white noise. For example, Van Trees (1968) derives the optimal unrealizable (i.e., not causal) linear filter from such an expansion, Fortmann and Anderson (1973) consider the realizable case and give an approximate realizable linear estimator which approaches the optimal one. Other papers dealing with series representations for random processes in problems of causal least-mean- square estimation and with discussions about the problem are, among others, Cambanis (1973) and Gardner (1973). Let us consider the classic problem that consists in finding the minimum mean-square realizable linear estimate of a random signal corrupted by additive white noise. The model for the unobservable signal x(t) is the following one: y(t) = x(t) + v(t), t E [Ti, 7"/'] * Corresponding author. I This paper was supported in part by Projects No. PS93-0201 and PS94-0136 of DGICYT, Ministerio de Educaci6n y Ciencia, Spain. 0167-7152/96/$12.00 (~) 1996 Elsevier Science B.V. All rights reserved SSD1 0167-7152(95)00130-1