Fixed-interval smoothing algorithm based on covariances with correlation in the uncertainty

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1 Digital Signal Processing 15 (2005) Fixed-interval smoothing algorithm based on covariances with correlation in the uncertainty S. Nakamori a,, R. Caballero-Águila b, A. Hermoso-Carazo c, J. Linares-Pérez c a Department of Technology, Faculty of Education, Kagoshima University, Kohrimoto, Kagoshima , Japan b Departamento de Estadística e Investigación Operativa, Universidad de Jaén, Paraje Las Lagunillas, s/n, Jaén, Spain c Departamento de Estadística e Investigación Operativa, Universidad de Granada, Campus Fuentenueva, s/n, Granada, Spain Available online 12 November 2004 Abstract A least-squares linear fixed-interval smoothing algorithm is derived to estimate signals from uncertain observations perturbed by additive white noise. It is assumed that the Bernoulli variables describing the uncertainty are only correlated at consecutive time instants. The marginal distribution of each of these variables, specified by the probability that the signal exists at each observation, as well as their correlation function, are known. The algorithm is obtained without requiring the state-space model generating the signal, but just the covariances of the signal and the additive noise in the observation equation. The covariance function of the signal must be expressed in a semidegenerate kernel form, assumption which covers many general situations, including stationary and non-stationary signals Elsevier Inc. All rights reserved. Keywords: Stochastic systems; Uncertain observations; Least-squares estimation; Fixed-interval smoothing; Covariance information * Corresponding author. addresses: nakamori@edu.kagoshima-u.ac.jp (S. Nakamori), raguila@ujaen.es (R. Caballero-Águila), ahermoso@ugr.es (A. Hermoso-Carazo), jlinares@ugr.es (J. Linares-Pérez) /$ see front matter 2004 Elsevier Inc. All rights reserved. doi: /j.dsp

2 208 S. Nakamori et al. / Digital Signal Processing 15 (2005) Introduction The problem of estimating a discrete-time signal from noisy observations in which the signal can be randomly missing is considered. To describe this situation, the observation equation is formulated multiplying the signal at any sample time by a binary random variable taking the values one and zero; the value one indicates that the measurement at that time contains the signal, whereas the value zero reflects the fact that the signal is missing and, hence, the corresponding observation is only noise. So, the observation equation involves both an additive and a multiplicative noise which models the uncertainty about the signal being present or missing at each observation. It is assumed that, for each particular observation, the probability of containing the signal is known for the observer. In many practical situations, the variables modelling the uncertainty in the observations can be assumed to be independent and, then, the distribution of the multiplicative noise is fully determined by the probability that each particular observation contains the signal. As it was shown by Nahi [6] and Monzingo [5] (who were the first analyzing the least-squares linear estimation problem in this kind of systems), if the state-space model of the signal is available, the knowledge of the aforementioned probabilities allows to derive estimation algorithms with a similar recursive structure to those obtained when the signal is always present in the observations. More general situations in which the above independence assumption is not valid were subsequently investigated by some other authors. The main difficulty arising in theses cases is that, generally, the estimators of the signal cannot be obtained in a recursive way and some conditions on the distribution of the multiplicative noise must be imposed to obtain the estimators in a simple way. For example, in Hadidi and Schwartz [2] a necessary and sufficient condition for the existence of a recursive leastsquares linear filter, which extends the one proposed by Nahi [6], is established. A different situation, in which the variables modelling the uncertainty are only correlated at consecutive instants, had been previously considered by Jackson and Murthy [3] who, using also a state-space approach, derived a least-squares linear filtering algorithm which allows to obtain the estimator at any time from those in the two preceding instants. In the last years, the estimation problem in all the aforementioned situations has been investigated under a more general approach which does not require the state-space model, but only the autocovariance function of the signal. Assuming that this function can be expressed in a semi-degenerate kernel form, algorithms with a simpler structure than the corresponding ones when the state-space model is known have been obtained for different estimation problems (see Nakamori et al. [7] for the linear filter and fixed-point smoother when the uncertainty is modelled by independent random variables and Nakamori et al. [8], where the noise modelling the uncertainty satisfies the condition in [2]). The situation considered by Jackson and Murthy [3] has been also treated in [9] under a covariance approach and filtering and fixed-point smoothing algorithms have been derived for this uncertain observation model. The aim in this paper is to propose a fixed-interval smoothing algorithm based on covariance information for this last model. The fixed-interval smoothing problem appears when all the measurements of the signal inside a time-interval are available before proceeding to the estimation. Fixed-interval smoothing techniques have been applied to stochastic signal processing problems (Ferrari- Trecate and De Nicolao [1], Young and Pedregal [11]) as well as to the estimation of

3 S. Nakamori et al. / Digital Signal Processing 15 (2005) time-variable parameters (Young et al. [12]); also, they have been used in the resolution of some practical problems as reconstruction of aerosol size distribution (Voutilainen and Kaipio [13]) or estimation in econometrics (Pollock [10]). In this paper we treat the least-squares linear estimation problem and the fixed-interval algorithm is derived under an innovation approach. This approach provides an expression for the smoother as the sum of the filter and another term, uncorrelated with it, which can be obtained from a backward-time algorithm. Assuming that the covariance function of the signal is expressed as a semi-degenerate kernel, both the backward and the filtering algorithms perform by using at each iteration the data obtained in the two previous ones, as in [3]. The filtering and fixed-interval smoothing algorithms are applied to a simulated observation model where the signal cannot be missing in two consecutive observations, situation which can be covered by the correlation form considered in the theoretical study. 2. Observations and estimation problem We consider the least-squares (LS) linear estimation problem of a discrete-time signal, {z(k); k 0}, from noisy uncertain observations described by y(k) = θ(k)z(k) + v(k), (1) where the involved processes satisfy the following hypotheses: (i) The signal process {z(k); k 0} has zero mean and its autocovariance function, K z (k, s) = E[z(k)z T (s)], is expressed in a semi-degenerate kernel form, that is, { A(k)B K z (k, s) = T (s), 0 s k, B(k)A T (s), 0 k s, where A and B are known n M matrix functions. (ii) The noise process {v(k); k 0} is a zero-mean white sequence with known autocovariance function, E[v(k)v T (s)]=r v (k)δ K (k s), being δ K the Kronecker delta function. (iii) The multiplicative noise {θ(k); k 0} is a sequence of Bernoulli random variables with P [θ(k)= 1]=θ(k) and autocovariance function { 0, k s 2, K θ (k, s) = E[θ(k)θ(s)] θ(k)θ(s), k s < 2. (iv) The processes {z(k); k 0}, {v(k); k 0}, and {θ(k); k 0} are mutually independent. The purpose is to obtain a fixed-interval smoothing algorithm; concretely, assuming that the observations up to a certain time L are available, our aim is to find recursive formulas which allow to obtain the estimators of the signal, z(k), at any time k L. For this purpose, we will use an innovation approach. If ŷ(k,k 1) denotes the LS linear estimator of y(k) based on {y(1),...,y(k 1)}, ν(k) = y(k) ŷ(k,k 1) represents the

4 210 S. Nakamori et al. / Digital Signal Processing 15 (2005) innovation contained in the observation y(k), that is, the new information provided by y(k) after its estimation from the previous observations. It is known (Kailath [4]) that the LS linear estimator of z(k) based on the observations {y(1),..., y(j)}, which is denoted by ẑ(k, j), is equal to the LS linear estimator based on the innovations {ν(1),..., ν(j)}. The advantage of considering the innovation approach to address the LS estimation problem comes from the fact that the innovations constitute a white process; then, by denoting Π(i)= E[ν(i)ν T (i)], the orthogonal projection lemma (OPL) leads to with ẑ(k, j) = j S(k,i)Π 1 (i)ν(i) (2) i=1 S(k,i) = E [ z(k)ν T (i) ]. (3) In view of expression (2), we will start by obtaining an explicit formula for the innovations. Afterwards, we will derive recursive formulas for the fixed-interval smoother, ẑ(k, L), k<l, which will include that of the filter, ẑ(k,k). 3. Innovation process When the variables {θ(k); k 0} modelling the uncertainty are independent, or the condition in [2] is satisfied, all the information prior to time k which is required to estimate y(k) is provided by the one-stage predictor of the signal, ẑ(k, k 1) (see [7] and [8]). However, for the problem at hand, the correlation between θ(k 1) and θ(k), which must be considered to estimate y(k), is not contained in ẑ(k, k 1). Therefore, to obtain the current innovation ν(k), we need to find the new form for the one-stage predictor of y(k) which, from the OPL, is expressed by k 1 ŷ(k,k 1) = E [ y(k)ν T (i) ] Π 1 (i)ν(i), k 2, ŷ(1, 0) = 0. i=1 Taking into account the model hypotheses, and E [ y(k)ν T (i) ] = θ(k)e [ z(k)ν T (i) ], i k 2 (4) E [ y(k)ν T (k 1) ] = E [ θ(k)θ(k 1) ] A(k)B T (k 1) θ(k)e [ z(k)ŷ T (k 1,k 2) ], expressions that, substituted into (4) and after some operations, lead to k 1 ŷ(k,k 1) = θ(k) E [ z(k)ν T (i) ] Π 1 (i)ν(i) i=1

5 S. Nakamori et al. / Digital Signal Processing 15 (2005) θ(k)e [ z(k)y T (k 1) ] Π 1 (k 1)ν(k 1) + E [ θ(k)θ(k 1) ] A(k)B T (k 1)Π 1 (k 1)ν(k 1). Then, since E[z(k)y T (k 1)]=θ(k 1)A(k)B T (k 1), we conclude that ŷ(k,k 1) = θ(k)ẑ(k, k 1) + K θ (k, k 1)A(k)B T (k 1)Π 1 (k 1)ν(k 1). (5) Therefore, the innovation is obtained by a linear combination of the new observation, the predictor of the signal and the previous innovation, namely ν(k) = y(k) θ(k)ẑ(k, k 1) K θ (k, k 1)A(k)B T (k 1)Π 1 (k 1)ν(k 1) (6) for k 2, and ν(1) = y(1). Next, we derive a recursive expression to obtain the one-stage predictor of the signal which, from (2), is given by k 1 ẑ(k, k 1) = S(k,i)Π 1 (i)ν(i), k 2. (7) i=1 To calculate the coefficients S(k,i), we substitute expression (6) for ν(i) into (3), obtaining S(k,i) = E [ z(k)y T (i) ] θ(i)e [ z(k)ẑ T (i, i 1) ] K θ (i, i 1)E [ z(k)ν T (i 1) ] Π 1 (i 1)B(i 1)A T (i) then, using (7) in E[z(k)ẑ T (i, i 1)] and taking into account the hypotheses on the model for E[z(k)y T (i)], wehave i 1 S(k,i) = θ(i)a(k)b T (i) θ(i) S(k,j)Π 1 (j)s T (i, j) j=1 K θ (i, i 1)S(k, i 1)Π 1 (i 1)B(i 1)A T (i), 2 i k and S(k,1) = θ(1)a(k)b T (1). This expression for S(k,i) guarantees that S(k,i) = A(k)J (i), i k, (8) where J is a function satisfying i 1 J(i)= θ(i)b T (i) θ(i) J(j)Π 1 (j)s T (i, j) j=1 K θ (i, i 1)J (i 1)Π 1 (i 1)B(i 1)A T (i), 2 i k (9) and J(1) = θ(1)b T (1). So, if we denote O(k) = k J(i)Π 1 (i)ν(i), k 1 (10) i=1

6 212 S. Nakamori et al. / Digital Signal Processing 15 (2005) it is clear, from (7) and (8), that the one-stage predictor of the signal is given by ẑ(k, k 1) = A(k)O(k 1), k 1, (11) where, from (10), the vector O(k 1) is obtained from the recursive relation O(k) = O(k 1) + J(k)Π 1 (k)ν(k), k 1, with O(0) = 0. Next, we proceed to establish a simplified expression for J(k) for k 1. By putting i = k in (9) and taking into account (8), we obtain k 1 J(k)= θ(k)b T (k) θ(k) J(i)Π 1 (i)j T (i)a T (k) i=1 K θ (k, k 1)J (k 1)Π 1 (k 1)B(k 1)A T (k). Then, by denoting r(k) = E [ O(k)O T (k) ] k = J(i)Π 1 (i)j T (i) (12) i=1 we have J(k)= θ(k) [ B T (k) r(k 1)A T (k) ] K θ (k, k 1)J (k 1)Π 1 (k 1)B(k 1)A T (k), where, from (12), the matrix functions r are recursively obtained, by starting from r(0) = 0, by r(k) = r(k 1) + J(k)Π 1 (k)j T (k), k 1. Finally, we derive the expression of the covariance matrix of the innovation ν(k), Π(k)= E [ y(k)y T (k) ] E [ ŷ(k,k 1)ŷ T (k, k 1) ]. From expressions (5) and (11) for the predictors ŷ(k,k 1) and ẑ(k, k 1), respectively, the hypotheses on the model together with (12) lead to Π(k)= θ(k)a(k) [ B T (k) θ(k)r(k 1)A T (k) ] Kθ 2 (k, k 1)A(k)BT (k 1)Π 1 (k 1)B(k 1)A T (k) θ(k)k θ (k, k 1)A(k) [ E [ O(k 1)ν T (k 1) ] Π 1 (k 1)B(k 1) + B T (k 1)Π 1 (k 1)E [ ν(k 1)O T (k 1) ]] A T (k) + R v (k). Using now the recursive relation for O(k 1) and, since the vector O(k 2) is orthogonal to ν(k 1),wehavethatE[O(k 1)ν T (k 1)]=J(k 1); consequently, Π(k)= θ(k)a(k) [ B T (k) θ(k)r(k 1)A T (k) ] Kθ 2 (k, k 1)A(k)BT (k 1)Π 1 (k 1)B(k 1)A T (k) θ(k)k θ (k, k 1)A(k) [ J(k 1)Π 1 (k 1)B(k 1) + B T (k 1)Π 1 (k 1)J T (k 1) ] A T (k) + R v (k).

7 S. Nakamori et al. / Digital Signal Processing 15 (2005) All these results are summarized in the following theorem. Theorem 1. Under hypotheses (i) (iv), the innovation process associated with the observations given in (1) satisfies ν(k) = y(k) θ(k)a(k)o(k 1) K θ (k, k 1)A(k)B T (k 1)Π 1 (k 1)ν(k 1), k 2, ν(1) = y(1), (13) where the vectors O(k) are recursively calculated from being O(k) = O(k 1) + J(k)Π 1 (k)ν(k), k 1, O(0) = 0 (14) J(k)= θ(k) [ B T (k) r(k 1)A T (k) ] K θ (k, k 1)J (k 1)Π 1 (k 1)B(k 1)A T (k), k 2, J(1) = θ(1)b T (1) (15) and Π(k) the covariance matrix of the innovation, which is given by Π(k)= θ(k)a(k) [ B T (k) θ(k)r(k 1)A T (k) ] Kθ 2 (k, k 1)A(k)BT (k 1)Π 1 (k 1)B(k 1)A T (k) θ(k)k θ (k, k 1)A(k) [ J(k 1)Π 1 (k 1)B(k 1) + B T (k 1)Π 1 (k 1)J T (k 1) ] A T (k) + R v (k). (16) The covariance matrix r(k) of the vector O(k) verifies r(k) = r(k 1) + J(k)Π 1 (k)j T (k), k 1, r(0) = 0. (17) 4. Fixed-interval smoothing algorithm Once the innovation process has been determined, we proceed to derive expressions which allow us to obtain the estimators ẑ(k, L) for all k L in a recursive way. For this purpose, noting that, from the general expression (2), ẑ(k, L) =ẑ(k,k) + L S(k,i)Π 1 (i)ν(i), k < L (18) i=k+1 it is clear that the first step in the fixed-interval algorithm is to obtain the filter, ẑ(k,k), for all k L.

8 214 S. Nakamori et al. / Digital Signal Processing 15 (2005) The same reasoning performed to obtain the one-stage predictor of the signal leads to a similar expression for the filter; specifically, substituting now (8) into (2) with L = k and taking into account (10), we obtain ẑ(k,k) = A(k)O(k). Hence, only the sum at the right-hand side term in (18) must be determined; for this purpose, we begin by calculating the coefficients S(k,i),fori k + 1. First, the following expression is obtained as the one for i k established in Section 3, i 1 S(k,i) = θ(i)b(k)a T (i) θ(i) S(k,j)Π 1 (j)s T (i, j) j=1 K θ (i, i 1)S(k, i 1)Π 1 (i 1)B(i 1)A T (i), i k + 1. Now, taking into account (8) and (12), this relation is rewritten as S(k,i) = θ(i) [ B(k) A(k)r(k) ] A T (i) θ(i) i 1 j=k+1 S(k,j)Π 1 (j)s T (i, j) K θ (i, i 1)S(k, i 1)Π 1 (i 1)B(i 1)A T (i), i k + 2, S(k,k + 1) = θ(k + 1) [ B(k) A(k)r(k) ] A T (k + 1) K θ (k + 1, k)a(k)j (k)π 1 (k)b(k)a T (k + 1) which allows to express S(k,i) as S(k,i) = [ B(k) A(k)r(k) ] 1 (k, i) + A(k)J(k) 2 (k, i), i k + 1 (19) being 1 and 2 matrix functions verifying 1 (k, i) = θ(i)a T (i) θ(i) i 1 j=k+1 1 (k, j)π 1 (j)s T (i, j) K θ (i, i 1) 1 (k, i 1)Π 1 (i 1)B(i 1)A T (i), i k + 2, (20) and 1 (k, k + 1) = θ(k + 1)A T (k + 1) 2 (k, i) = θ(i) i 1 j=k+1 2 (k, j)π 1 (j)s T (i, j) K θ (i, i 1) 2 (k, i 1)Π 1 (i 1)B(i 1)A T (i), i k + 2, (21) 2 (k, k + 1) = K θ (k + 1,k)Π 1 (k)b(k)a T (k + 1). If we define now L q 1 (k, L) = 1 (k, i)π 1 (i)ν(i), k < L i=k+1

9 S. Nakamori et al. / Digital Signal Processing 15 (2005) and q 2 (k, L) = L i=k+1 it is clear, from (19), that L S(k,i)Π 1 (i)ν(i) i=k+1 2 (k, i)π 1 (i)ν(i), k < L = [ B(k) A(k)r(k) ] q 1 (k, L) + A(k)J(k)q 2 (k, L), k < L (22) and then, substituting (22) into (18), the estimators are expressed as ẑ(k, L) =ẑ(k,k) + [ B(k) A(k)r(k) ] q 1 (k, L) + A(k)J(k)q 2 (k, L), k < L. Next, to obtain a recursive formula for q 1 (k, L) we subtract (20) in k + 1 from (20) in k and, comparing the expression obtained with the ones obtained from (20) and (21) for 1 (k + 1,i)and 2 (k + 1,i), it is derived that 1 (k, i) = [ I M 1 (k, k + 1)Π 1 (k + 1)J T (k + 1) ] 1 (k + 1,i) + 1 (k, k + 1) 2 (k + 1,i), i k + 2 being I M the M M identity matrix. In a similar way, the following expression for 2 (k, i) is obtained 2 (k, i) = 2 (k, k + 1)Π 1 (k + 1)J T (k + 1) 1 (k + 1,i) + 2 (k, k + 1) 2 (k + 1,i), i k + 2. Taking into account these last relations, the following recursive expressions, by starting from q s (L, L) = 0, s = 1, 2, are derived from the definitions of q 1 (k, L) and q 2 (k, L), for k L 1, q 1 (k, L) = [ I M 1 (k, k + 1)Π 1 (k + 1)J T (k + 1) ] q 1 (k + 1,L) + 1 (k, k + 1)q 2 (k + 1,L)+ 1 (k, k + 1)Π 1 (k + 1)ν(k + 1), q 2 (k, L) = 2 (k, k + 1)Π 1 (k + 1)J T (k + 1)q 1 (k + 1,L) + 2 (k, k + 1)q 2 (k + 1,L)+ 2 (k, k + 1)Π 1 (k + 1)ν(k + 1). In Theorem 2, the fixed-interval smoothing algorithm is summarized. Theorem 2. Assuming hypotheses (i) (iv), the estimators of the signal z(k) from the observations y(1),...,y(l), with k L, are given by ẑ(k, L) =ẑ(k,k) + [ B(k) A(k)r(k) ] q 1 (k, L) + A(k)J(k)q 2 (k, L), k < L, ẑ(k,k) = A(k)O(k), k L, (23) where q 1 (k, L) and q 2 (k, L) can be recursively calculated, from q 1 (L, L) = 0 and q 2 (L, L) = 0,by

10 216 S. Nakamori et al. / Digital Signal Processing 15 (2005) with q 1 (k, L) = [ I M 1 (k, k + 1)Π 1 (k + 1)J T (k + 1) ] q 1 (k + 1,L) + 1 (k, k + 1)q 2 (k + 1,L) + 1 (k, k + 1)Π 1 (k + 1)ν(k + 1), k < L, (24) q 2 (k, L) = 2 (k, k + 1)Π 1 (k + 1)J T (k + 1)q 1 (k + 1,L) + 2 (k, k + 1)q 2 (k + 1,L) + 2 (k, k + 1)Π 1 (k + 1)ν(k + 1), k < L (25) 1 (k, k + 1) = θ(k + 1)A T (k + 1), k < L, (26) 2 (k, k + 1) = K θ (k + 1,k)Π 1 (k)b(k)a T (k + 1), k < L. (27) The innovations, ν(k), and their covariance matrices, Π(k), as well as the vectors O(k), their covariances, r(k), and the matrices J(k) are calculated from the formulas given in Theorem Error covariance matrices The least-squares method uses the covariance matrices of the estimation errors to measure the goodness of the estimators. In this section, an expression to obtain these matrices from recursive formulas as well as these formulas are derived from the algorithm proposed in Theorem 2. First, since the estimation error z(k) ẑ(k, L) is orthogonal to the estimator ẑ(k, L), it is easy to verify that P(k,L)= E [{ z(k) ẑ(k, L) }{ z(k) ẑ(k, L) } T ] = K z (k, k) E [ ẑ(k, L)ẑ T (k, L) ]. Hence, using expression (23) for ẑ(k, L) and taking into account the uncorrelation property between each q s (k, L), s = 1, 2, and ẑ(k,k) we have P(k,L)= P(k,k) [ B(k) A(k)r(k) ] Q 1 (k, L) [ B(k) A(k)r(k) ] T A(k)J(k)Q 2 (k, L)J T (k)a T (k) [ B(k) A(k)r(k) ] Q 12 (k, L)J T (k)a T (k) A(k)J(k)Q T 12 (k, L)[ B(k) A(k)r(k) ] T, where P(k,k), the filtering error covariance matrix, is given by P(k,k) = A(k) [ B T (k) r(k)a T (k) ], k L k<l, being r(k) the covariance matrix of the vector O(k), as defined in (12). The matrices Q s (k, L), fors = 1, 2, and Q 12 (k, L) are the covariance and crosscovariance matrices of q 1 (k, L) and q 2 (k, L). From (24) and (25), together with the uncorrelation property

11 S. Nakamori et al. / Digital Signal Processing 15 (2005) between q s (k + 1,L), s = 1, 2, and ν(k + 1), these matrices can be recursively calculated from Q 1 (k, L) = [ I M 1 (k, k + 1)Π 1 (k + 1)J T (k + 1) ] Q 1 (k + 1,L) [ I M 1 (k, k + 1)Π 1 (k + 1)J T (k + 1) ] T + 1 (k, k + 1) [ Q 2 (k + 1,L)+ Π 1 (k + 1) ] T 1 (k, k + 1) + [ I M 1 (k, k + 1)Π 1 (k + 1)J T (k + 1) ] Q 12 (k + 1,L) T 1 (k, k + 1) + 1(k, k + 1)Q T 12 (k + 1,L) [ I M 1 (k, k + 1)Π 1 (k + 1)J T (k + 1) ] T, Q 2 (k, L) = 2 (k, k + 1)Π 1 (k + 1) [ J T (k + 1)Q 1 (k + 1, L)J (k + 1) + Π(k + 1) ] Π 1 (k + 1) T 2 (k, k + 1) + 2 (k, k + 1)Q 2 (k + 1,L) T 2 (k, k + 1) 2 (k, k + 1)Π 1 (k + 1)J T (k + 1)Q 12 (k + 1,L) T 2 (k, k + 1) 2 (k, k + 1)Q T 12 (k + 1, L)J (k + 1)Π 1 (k + 1) T 2 (k, k + 1), Q 12 (k, L) = [ I M 1 (k, k + 1)Π 1 (k + 1)J T (k + 1) ] Q 1 (k + 1, L)J (k + 1)Π 1 (k + 1) T 2 (k, k + 1) + 1 (k, k + 1)Q 2 (k + 1,L) T 2 (k, k + 1) + [ I M 1 (k, k + 1)Π 1 (k + 1)J T (k + 1) ] Q 12 (k + 1,L) T 2 (k, k + 1) 1 (k, k + 1)Q T 12 (k + 1, L)J (k + 1)Π 1 (k + 1) T 2 (k, k + 1) + 1 (k, k + 1)Π 1 (k + 1) T 2 (k, k + 1) for k<l, with initial conditions Q s (L, L) = 0, for s = 1, 2, and Q 12 (L, L) = Computer simulation results This section shows a numerical simulation example to illustrate the application of the recursive linear fixed-interval smoothing algorithm presented in Theorem 2. To model the uncertainty in the observations according to hypothesis (iii) in Section 2, we consider a sequence of independent Bernoulli random variables, {γ(k); k 0}, taking the value one with probability p and we define θ(k)= 1 γ(k 1) + γ(k 1)γ (k), k 1. So, the variables θ(k) are also Bernoulli random variables and, since θ(k) and θ(s) are independent for k s 2, they are uncorrelated and hypothesis (iii) is satisfied. The common mean of these variables is θ = 1 p + p 2 and its covariance function is given by { 0, k s 2, K θ (k, s) = (1 θ) 2, k s < 2.

12 218 S. Nakamori et al. / Digital Signal Processing 15 (2005) Let z(k) be the signal to be estimated and y(k) the observation of this signal, defined as in (1) by y(k) = θ(k)z(k) + v(k), where v(k) represents the measurement noise. Since θ(k) = 0 corresponds to γ(k 1) = 1 and γ(k) = 0, this fact implies that θ(k + 1) = 1 and, hence, the possibility of the signal being missing in two successive observations is avoided. So, the considered observation model covers those signal transmission models with stand-by sensors, in which any failure in the transmission is immediately detected and the old sensor is then replaced. We have assumed that {z(k); k 0} is a scalar signal generated by the following firstorder autoregressive model z(k + 1) = 0.95z(k) + w(k), where {w(k); k 0} is a zero-mean white Gaussian noise with var[w(k)]=0.1, for all k. The autocovariance function of this signal is given in a semi-degenerate kernel form, specifically, K z (k, s) = k s, 0 s k and, according to hypothesis (i), the functions which constitute this kernel are as follows: A(k) = k, B(k)= 0.95 k. The noise {v(k); k 0} in the observation equation has been assumed to be a sequence of independent random variables with P [ v(k) = 1/3 ] = 15 18, P[ v(k) = 1 ] = 2 18, P[ v(k) = 3 ] = 1 18, k 0 and, hence E [ v(k) ] = 0, R v (k) = Finally, the mutual independence of the signal, {z(k); k 0}, and the noises, {v(k); k 0} and {θ(k); k 0}, imposed by hypothesis (iv) in the theoretical study, is also assumed. To show the effectiveness of the algorithm proposed in this paper, we have performed a program in MATLAB, which simulates the signal values and their noisy observations, and provides the filtering and fixed-interval smoothing estimates, as well as the corresponding error variances. Next, we show and compare the results obtained from 100 observations of the signal, using different values of the parameter p. Previously, let us note that the mean and covariance function of the variables θ(k) are the same for the values p and 1 p; for this reason, only the case p 0.5 is considered here. First, the performance of the filter and fixed-interval smoother, measured by the error variances, has been calculated for p = 0.1, 0.3, and 0.5. The results are displayed in Fig. 1 which shows that the estimators (filter and fixed-point smoother) have a better performance as p is smaller, due to the fact that the mean value, θ, decreases with p. Consequently, the

13 S. Nakamori et al. / Digital Signal Processing 15 (2005) Fig. 1. Filtering and fixed-interval smoothing error variances for p = 0.1, 0.3, 0.5. Fig. 2. Signal, filtering and fixed-interval smoothing estimates for p = 0.1.

14 220 S. Nakamori et al. / Digital Signal Processing 15 (2005) estimation is more accurate as p is nearer to 0, case in which the signal is always present in the observations. Moreover, this figure shows not only that, for each value of p, the error variances are smaller using the fixed-interval smoother instead of the filter, but also that the improvement with the smoother is highly significant since, even the worst results with the smoother (p = 0.5) are better than the best ones with the filter (p = 0.1). A simulated signal and their filtered and smoothed estimates from 100 observations simulated with p = 0.1 are displayed in Fig. 2. The result, as expected, is that the smoothing estimates are nearer to the signal and, hence, the behavior of the fixed-interval smoother is better than that of the filter. 7. Conclusions In this paper, the least-squares linear fixed-interval smoother is derived from uncertain observations of a signal, when the Bernoulli random variables characterizing the uncertainty in the observations are correlated at consecutive time instants, for the case of white observation additive noise. It is not required the knowledge of the state-space model, but only the first and second-order moments of the processes involved in the observation equation. The fixed-interval smoother is obtained from the filter and the recursive algorithms to obtain these estimators are derived by an innovation approach. The results are applied to a particular model which includes signal transmission models with stand-by sensors for the immediate replacement of a failed unit. Acknowledgments This work has been partially supported by the Ministerio de Ciencia y Tecnología under contract BFM References [1] G. Ferrari-Trecate, G. De Nicolao, Computing the equivalent number of parameters of fixed-interval smoothers, in: Proceedings of the 40th IEEE Conference on Decision and Control, vol. 3, 2001, pp [2] M.T. Hadidi, S.C. Schwartz, Linear recursive state estimators under uncertain observations, IEEE Trans. Automat. Control 24 (6) (1979) [3] R.N. Jackson, D.N.P. Murthy, Optimal linear estimation with uncertain observations, IEEE Trans. Inform. Theory (1976) [4] T. Kailath, Lectures on Linear Least-Squares Estimation, Springer-Verlag, [5] R.A. Monzingo, Discrete optimal linear smoothing for systems with uncertain observations, IEEE Trans. Inform. Theory 21 (3) (1975) [6] N.E. Nahi, Optimal recursive estimation with uncertain observation, IEEE Trans. Inform. Theory 15 (4) (1969) [7] S. Nakamori, R. Caballero, A. Hermoso, J. Linares, Linear estimation from uncertain observations with white plus coloured noises using covariance information, Digital Signal Process. 13 (2003) [8] S. Nakamori, R. Caballero, A. Hermoso, J. Linares, Fixed-point smoothing with non-independent uncertainty using covariance information, Int. J. Syst. Sci. 7 (10) (2003)

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