Linear estimation from uncertain observations with white plus coloured noises using covariance information

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1 Digital Signal Processing 13 (2003) Linear estimation from uncertain observations with white plus coloured noises using covariance information S. Nakamori, a, R. Caballero-Águila, b A. Hermoso-Carazo, c and 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 Abstract This paper considers the least mean-squared error linear estimation problems, using covariance information, in linear discrete-time stochastic systems with uncertain observations for the case of white plus coloured observation noises. The different kinds of estimation problems treated include one-stage prediction, filtering, and fixed-point smoothing. The recursive algorithms are derived by employing the Orthogonal Projection Lemma and assuming that both, the signal and the coloured noise autocovariance functions, are given in a semi-degenerate kernel form Elsevier Science (USA). All rights reserved. Keywords: Covariance information; Stochastic systems; Uncertain observations 1. Introduction The least mean-squared error linear estimation problem of a stochastic signal from noisy observations has been widely treated when the observation sequence contains the signal to be estimated with probability one. * 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) /03/$ see front matter 2003 Elsevier Science (USA). All rights reserved. doi: /s (02)00026-x

2 S. Nakamori et al. / Digital Signal Processing 13 (2003) However, in many practical situations, the signal vector enters in the observation equation in a random manner. In these cases, there is a positive probability (false alarm probability) that the observation in each time does not contain the signal to be estimated. These situations are described by an observation equation which includes not only an additive noise, but also a multiplicative noise component, modelled by a sequence of Bernoulli random variables whose values, one or zero, indicate the presence or absence of the signal in the observation. This can occur in many practical situations, for example, in problems where there exist intermittent failures in the observation mechanism, fading phenomena in propagation channels, target tracking, accidental loss of some measurements, or inaccessibility of the data during certain times; that is, problems where, due to different reasons, the measurement set of the signal can contain observations which are only noise. The least mean-squared error linear estimation problem for these linear discrete-time systems, when the uncertainty is modelled by independent Bernoulli variables, has been treated by different authors. Assuming a full knowledge of the state-space model for the signal process and that the false alarm probability is known a priori, Nahi [5] and Monzingo [4] were the first who treated the estimation problem. Later on, Hermoso and Linares [1,2] extended their results to the case in which the additive noises of the state and observation are correlated. Sawaragi et al. [10] consider the state estimation problem in systems with uncertain observations when the probability of the presence of the state in the observation is unknown but fixed throughout the time interval of interest. By using a bayesian approach, they obtain an estimator of the false alarm probability which provides an adaptive algorithm for the state estimators. These results are extended in Sawaragi et al. [11] to the case of systems with uncertain observations with stationary Markov interrupted observation mechanism, when the transition probabilities of the Markov chain are unknown but fixed. Markov [3], using a quasi-bayes procedure, consider the estimation of the unknown probability and the current state of the system based upon the entire observation sequence without knowledge of which observations contain or not the state. In Porat and Friedlander [8], the estimation technique of the power spectral density, based on nonlinear optimization of a weighted squared-error criterion for the structure of the ARMA model, is proposed from missing observed data in the case of observation noise free. In the technique, the information of the false alarm probability is not required. In Rosen and Porat [9], the general formulas for the second-order moments of the sample covariances are derived for the ARMA model with missing observed data in the case of observation noise free. In the relation with the studies by Porat and Friedlander [8], for the case of the uncertain observed values including additive observation noises, the estimation of the signal would provide very important information. In linear stationary discrete-time systems with uncertain observations, Nakamori [7] proposed a recursive estimation technique using as information the autocovariance function of the signal, and assuming that the probability that the signal exists in the observations is available. Recursive algorithms for the least mean-squared error linear filter and fixed-point smoother are derived, without requiring the complete state-space model of the signal. The necessary information for the state-space model is obtained by using a factorization method of the autocovariance of the signal.

3 554 S. Nakamori et al. / Digital Signal Processing 13 (2003) In all the aforementioned works, the estimation algorithms are derived provided that the (uncertain) observations are perturbed by white noise. In a previous paper, Nakamori [6] treated the estimation problem of the signal from observations (without uncertainty) perturbed by white plus coloured noise. Under the assumption that the covariance functions of the signal and the coloured observation noise are expressed in a semi-degenerate kernel form, and by employing an invariant imbedding method, recursive estimation algorithms were derived without using any realization technique for the state-space model of the signal from the covariance information. This paper is concerned with the generalization of the algorithms proposed by Nakamori [6] to systems with uncertain observations, that is, we treat the estimation problems using covariance information in linear discrete-time systems with uncertain observations. By employing the Orthogonal Projection Lemma, we derive the recursive algorithms for the one-stage prediction, filtering, and fixed-point smoothing estimates in the case of white plus coloured observation noise. It is assumed that the autocovariance functions of the signal and the coloured observation noise are expressed in the form of a semi-degenerate kernel. Since the semi-degenerate kernel is suitable for expressing autocovariance functions of non-stationary or stationary signal processes, the proposed estimators provide estimations of general signal processes. 2. System model and problem formulation Let us consider a discrete-time observation equation described by y(k) = u(k)z(k) + v(k) + v 0 (k), (1) where z(k) is the n 1 signal vector and y(k) represents the n 1 observation vector. We assume the following hypotheses on the signal process and the noises: H1. 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, { K z (k, s) = A(k)B T (s), 0 s k, B(k)A T (2) (s), 0 k s, where A and B are bounded n M matrix functions. H2. The noise process {v(k); k 0} is a zero-mean white sequence with autocovariance function E[v(k)v T (s)]=r(k)δ K (k s),beingδ K the Kronecker δ function. H3. The process {v 0 (k); k 0} is a zero-mean coloured noise sequence and its autocovariance function, K 0 (k, s) = E[v 0 (k)v0 T (s)], is given by a semi-degenerate kernel form, { α(k)β K 0 (k, s) = T (s), 0 s k, β(k)α T (3) (s), 0 k s, where α and β are n N matrix functions. H4. The multiplicative noise {u(k); k 0} is a sequence of independent Bernoulli random variables with P [u(k) = 1]=p(k).

4 S. Nakamori et al. / Digital Signal Processing 13 (2003) H5. {z(k); k 0}, {u(k); k 0}, {v(k); k 0}, and{v 0 (k); k 0} are mutually independent. If we denote K 0 (i, s) = E[(u(i)z(i) + v 0 (i))(u(s)z(s) + v 0 (s)) T ] and K y (i, s) = E[y(i)y T (s)], as a consequence of the above hypotheses, we have K 0 (i, s) = E [ u(i)u(s) ] K z (i, s) + K 0 (i, s) (4) and K y (i, s) = K 0 (i, s) + R(s)δ K (i s). (5) We are interested in obtaining the least mean-squared error linear estimator of the signal z(k) based on the observations {y(1),..., y(l)}. This estimator, ẑ(k, L), is the orthogonal projection of z(k) on the space of n-dimensional linear transformations of the observations. So, ẑ(k, L) is given by L ẑ(k, L) = h(k, i, L)y(i), (6) where h(k, i, L), i = 1,...,L, denotes the impulse-response function. The Orthogonal Projection Lemma (OPL) assures that ẑ(k, L) is the only linear combination of the observations {y(1),..., y(l)} such that the estimation error is orthogonal to them, that is, {( ) } L E z(k) h(k, i, L)y(i) y T (s) = 0, s L. This condition is equivalent to the Wiener Hopf equation E [ z(k)y T (s) ] L = h(k, i, L)E [ y(i)y T (s) ], s L, useful for determining the impulse-response function h(k, i, L), i = 1,...,L.Usingthe hypotheses H1 H5 on the model, the Wiener Hopf equation can be rewritten as L h(k, s, L)R(s) = p(s)k z (k, s) h(k, i, L) K 0 (i, s), s L. (7) This last version of the Wiener Hopf equation will be used in Section 3 for obtaining the smoothing algorithm; specifically, for determining the one-stage prediction and filtering estimates of the signal. On the other hand, as a consequence of the OPL, denoting by ŷ(l,l 1) the least mean-squared error linear estimator of y(l) based on {y(1),..., y(l 1)} and by ν(l) = y(l) ŷ(l,l 1) the innovation at time L, it can be established that the estimators of z(k) satisfy the following recursive equation ẑ(k, L) =ẑ(k, L 1) + h(k, L, L)ν(L). This equation provides the basis for the fixed-point smoothing algorithm presented in the next section.

5 556 S. Nakamori et al. / Digital Signal Processing 13 (2003) Recursive fixed-point smoothing algorithm Theorem 1 presents the recursive formulas for the fixed-point smoothing estimate of the signal, when the observation are perturbed by white noise plus coloured noise. This theorem includes the formulas for the filtering estimate. Theorem 1. Let us consider the observation equation (1) given in Section 2, satisfying the hypotheses H1 H5. Then, the fixed-point smoothing estimate of the signal z(k), ẑ(k, L) for L>k, is given by ẑ(k, L) =ẑ(k, L 1) + h(k, L, L)ν(L), (8) where ν(l), the innovation, is ν(l) = y(l) p(l)a(l)o(l 1) α(l)q(l 1). (9) The M 1 and N 1 vectors O(L) and Q(L), respectively, are recursively calculated by O(L) = O(L 1) + J(L, L)ν(L), O(0) = 0, (10) Q(L) = Q(L 1) + I (L, L)ν(L), Q(0) = 0 (11) being J(L, L) = [ p(l) ( B T (L) r(l 1)A T (L) ) c(l 1)α T (L) ] Π 1 (L), (12) I(L,L)= [ β T (L) p(l)c T (L 1)A T (L) d(l 1)α T (L) ] Π 1 (L), (13) where Π(L), the covariance matrix of the innovation, is given by Π(L)= R(L) + p(l) [ B(L) p(l)a(l)r(l 1) α(l)c T (L 1) ] A T (L) + [ β(l) p(l)a(l)c(l 1) α(l)d(l 1) ] α T (L). (14) The functions r, c, and d which appear in (12), (13), and (14),areM M, M N, and N N matrices, respectively, verifying r(l) = r(l 1) + J(L, L) [ p(l) ( B(L) A(L)r(L 1) ) α(l)c T (L 1) ], r(0) = 0, (15) c(l) = c(l 1) + J(L, L) [ β(l) p(l)a(l)c(l 1) α(l)d(l 1) ], c(0) = 0, (16) d(l) = d(l 1) + I(L,L) [ β(l) p(l)a(l)c(l 1) α(l)d(l 1) ], d(0) = 0. (17) The smoothing gain, h(k, L, L), is given by h(k, L, L) = [ p(l) ( B(k)A T (L) E(k,L 1)A T (L) ) F(k,L 1)α T (L) ] Π 1 (L), (18) where E(k,L) and F(k,L) are n M and n N matrices satisfying

6 S. Nakamori et al. / Digital Signal Processing 13 (2003) E(k,L) = E(k,L 1) + h(k, L, L) [ p(l) ( B(L) A(L)r(L 1) ) α(l)c T (L 1) ], E(k,k) = A(k)r(k), (19) F(k,L)= F(k,L 1) + h(k, L, L) [ β(l) p(l)a(l)c(l 1) α(l)d(l 1) ], F(k,k) = A(k)c(k). (20) The filtering estimate, ẑ(k,k), which provides the initial condition for the fixed-point smoothing algorithm, is given by ẑ(k,k) = A(k)O(k). Remark. Let us note that the non-singularity of Π(L) is guaranteed if R(L) is positive definite, but there are some phenomena in practice in which this assumption is violated (for instance, if the observations are not affected by additive noise, then R(L) is equal to zero and, consequently, Π(L) can be a singular matrix). If Π(L) were singular, the Moore Penrose pseudo-inverse could be used. Proof. As we have indicated in Section 2, Eq. (8) is a consequence of the OPL. Then, the problem is to find the innovation ν(l), and the smoothing gain h(k, L, L). (I) The innovation process. Since ν(l) = y(l) ŷ(l,l 1), in order to determine it, it is enough to obtain ŷ(l,l 1). From the independence hypotheses on the model and the OPL we have ŷ(l,l 1) = p(l)ẑ(l,l 1) +ˆv 0 (L, L 1), (21) where ẑ(l, ) and ˆv 0 (L, ) are the one-stage linear prediction estimates of the signal z(l) and the coloured noise v 0 (L), respectively. Both predictors will be obtained by a similar procedure, from the Wiener Hopf equation, using the invariant imbedding method. Firstly, from (6), the signal predictor is given by ẑ(l, L 1) = h(l, i, L 1)y(i) (22) and, from (2), the Wiener Hopf equation (7) for this estimator becomes h(l, s, L 1)R(s) = p(s)a(l)b T (s) h(l, i, L 1) K 0 (i, s), s L 1. (23) If we now introduce a function J(s, L 1), such that J(s, L 1)R(s) = p(s)b T (s) J(i, L 1) K 0 (i, s), s L 1, (24)

7 558 S. Nakamori et al. / Digital Signal Processing 13 (2003) from relations (23) and (24) we conclude that the impulse-response function must be h(l, s, L 1) = A(L)J(s, L 1). So, from (22), it is clear that the one-stage predictor of the signal is ẑ(l, L 1) = A(L)O(L 1) with O(L 1) = J(i, L 1)y(i). (25) In a similar way, it is proved that ˆv 0 (L, L 1) = α(l)q(l 1) where Q(L 1) = I(i,L 1)y(i) (26) and I(s,L 1) is a function which satisfies I(s,L 1)R(s) = β T (s) I(i,L 1) K 0 (i, s), s L 1. (27) Hence, from (21), we deduce that ŷ(l,l 1) = p(l)a(l)o(l 1) + α(l)q(l 1) (28) and expression (9) for the innovation is immediately obtained. Next, we will establish the recursive relation (10) for the vector O(L). Taking into account that, from (2), (3), and (4), K 0 (L, s) = p(l)a(l)b T (s)p(s) + α(l)β T (s), s L 1, and using (27) for β T (s),wehave K 0 (L, s) = p(l)a(l)b T (s)p(s) + α(l)i(s,l 1)R(s) + α(l)i(i,l 1) K 0 (i, s), s L 1. (29) On the other hand, if we subtract (24) from the equation obtained by putting L 1 L in (24), we obtain [ ] J(s, L) J(s, L 1) R(s) = J(L, L) K 0 (L, s) [ ] J(i, L) J(i, L 1) K 0 (i, s), s L 1, and using (29), we have [ ] J(s, L) J(s, L 1) + J(L, L)α(L)I (s, L 1) R(s) = p(l)j(l, L)A(L)B T (s)p(s) [ ] J(i, L) J(i, L 1) + J(L, L)α(L)I (i, L 1) K 0 (i, s). (30) From (30), taking into account (24), we conclude that, for s L 1,

8 S. Nakamori et al. / Digital Signal Processing 13 (2003) J(s, L) J(s, L 1) = p(l)j(l, L)A(L)J(s, L 1) J(L, L)α(L)I (s, L 1). (31) So, the recursive relation (10) is immediately obtained substituting (31) in the following equation obtained from (25) [ ] O(L) O(L 1) = J(L, L)y(L) + J(i, L) J(i, L 1) y(i). Relation (11) for the vector Q(L) is obtained in an analogous way. Now, we will prove that J(L, L) and I(L,L)satisfy (12) and (13), respectively. Taking into account the expressions (10) and (11) for O(L) and Q(L), wehave E [ O(L)ν T (L) ] = E [ O(L 1)ν T (L) ] + J(L, L)E [ ν(l)ν T (L) ], E [ Q(L)ν T (L) ] = E [ Q(L 1)ν T (L) ] + I(L,L)E [ ν(l)ν T (L) ]. Since ν(l) is uncorrelated with O(L 1) and Q(L 1), the first expectation of the righthand side term of both expressions is zero. So, denoting Π(L)= E[ν(L)ν T (L)], we obtain J(L, L) = E [ O(L)ν T (L) ] Π 1 (L), I (L, L) = E [ Q(L)ν T (L) ] Π 1 (L). (32) Now, we calculate the expectations which appear in (32). From (9), it is clear that E [ O(L)ν T (L) ] = E [ O(L)y T (L) ] E [ O(L)O T (L 1) ] A T (L)p(L) E [ O(L)Q T (L 1) ] α T (L). Firstly we obtain E[O(L)y T (L)]; using (25) for O(L) and (5) for K y (i, L),wehave E [ O(L)y T (L) ] = J(L, L)R(L) + L J(i, L) K 0 (i, L) and by putting s L and L 1 L in (24), we conclude that E[O(L)y T (L)] = p(l)b T (L). On the other hand, from (10), taking into account that ν(l) is uncorrelated with O(L 1) and Q(L 1), we obtain E [ O(L)O T (L 1) ] = E [ O(L 1)O T (L 1) ] and E [ O(L)Q T (L 1) ] = E [ O(L 1)Q T (L 1) ]. Hence, denoting r(l)= E[O(L)O T (L)] and c(l) = E[O(L)Q T (L)],wehave E [ O(L)ν T (L) ] = p(l)b T (L) p(l)r(l 1)A T (L) c(l 1)α T (L). (33) A similar reasoning for E[Q(L)ν T (L)], denoting d(l) = E[Q(L)Q T (L)], leads to E [ Q(L)ν T (L) ] = β T (L) p(l)c T (L 1)A T (L) d(l 1)α T (L). (34) Substituting (33) and (34) in (32), we obtain (12) and (13).

9 560 S. Nakamori et al. / Digital Signal Processing 13 (2003) Let us now calculate the covariance matrix Π(L) of the innovation. First of all we observe that, from the OPL, ŷ(l,) is orthogonal to ν(l) and E[ŷ(L,)y T (L)]= E[ŷ(L,L 1)ŷ T (L, L 1)]. Hence, the covariance matrix Π(L) can be expressed as Π(L)= E [ y(l)y T (L) ] E [ ŷ(l,l 1)ŷ T (L, L 1) ]. Then, from (28), we have Π(L)= E [ y(l)y T (L) ] p 2 (L)A(L)E [ O(L 1)O T (L 1) ] A T (L) p(l)a(l)e [ O(L 1)Q T (L 1) ] α T (L) α(l)e [ Q(L 1)O T (L 1) ] A T (L)p(L) α(l)e [ Q(L 1)Q T (L 1) ] α T (L). Using now expressions (2) (5), we conclude that E [ y(l)y T (L) ] = p(l)b(l)a T (L) + β(l)α T (L) + R(L) and substituting in the above relation we obtain (14). In order to establish the recursive relations (15) (17) we use (10) and (11) and, from the OPL, we have r(l) = E [ O(L 1)O T (L 1) ] + J(L, L)Π(L)J T (L, L), c(l) = E [ O(L 1)Q T (L 1) ] + J(L, L)Π(L)I T (L, L), d(l) = E [ Q(L 1)Q T (L 1) ] + I (L, L)Π(L)I T (L, L). Then, taking into account (32) and using (33) and (34), we obtain (15) (17). (II) The smoothing gain. From (8), the fixed-point smoothing error, z(k, L) = z(k) ẑ(k, L), satisfies z(k, L) = z(k, L 1) h(k, L, L)ν(L) and, consequently, E [ z(k, L)y T (L) ] = E [ z(k, L 1)y T (L) ] h(k, L, L)E [ ν(l)y T (L) ]. Using again the OPL, E[ z(k, L)y T (L)]=0andE[ν(L)y T (L)]=Π(L); hence, it is clear that h(k, L, L) = E [ z(k, L 1)y T (L) ] Π 1 (L). (35) Now, we obtain E[ z(k, L 1)y T (L)]=E[z(k)y T (L)] E[ẑ(k, L 1)y T (L)]. From the independence hypotheses and since k L, E [ z(k)y T (L) ] = p(l)b(k)a T (L). On the other hand, by applying the OPL, E [ ẑ(k, L 1)y T (L) ] = E [ ẑ(k, L 1)ŷ T (L, L 1) ], and using (28), we have E [ ẑ(k, L 1)y T (L) ] = E [ ẑ(k, L 1)O T (L 1) ] A T (L)p(L) + E [ ẑ(k, L 1)Q T (L 1) ] α T (L).

10 S. Nakamori et al. / Digital Signal Processing 13 (2003) Hence, denoting E(k,L) = E[ẑ(k, L)O T (L)] and F(k,L)= E[ẑ(k, L)Q T (L)], we obtain E [ z(k, L 1)y T (L) ] = p(l) ( B(k)A T (L) E(k,L 1)A T (L) ) F(k,L 1)α T (L) and substituting in (35), relation (18) for the gain is deduced. Finally, from (8), (10), and (11), we have E(k,L) = E [ ẑ(k, L 1)O T (L 1) ] + h(k, L, L)Π(L)J T (L, L), F(k,L)= E [ ẑ(k, L 1)Q T (L 1) ] + h(k, L, L)Π(L)I T (L, L). So, taking into account (32) and using (33) and (34), we obtain (19) and (20). The initial condition for (8) is the filter, ẑ(k,k), which can be obtained, just by using a similar reasoning to that used to obtain the predictor at the beginning of the proof, by using the invariant imbedding method. In fact, from (6), ẑ(k,k) = k h(k,i,k)y(i) and, from (2), the Wiener Hopf equation (7) for the filter can be written as h(k,s,k)r(s) = p(s)a(k)b T (s) k h(k,i,k) K 0 (i, s), s k. Then, putting L 1 k in (24) and comparing with the above equation, it is clear that h(k,s,k) = A(k)J(s, k) and consequently, ẑ(k,k) = A(k)O(k). The other initial conditions in the algorithm are easily obtained. 4. Fixed-point smoothing error covariance The performance of the fixed-point smoothing estimates can be measured by the smoothing error z(k, L) = z(k) ẑ(k, L) and, more specifically, by the covariance matrices of these errors P(k,L)= E[ z(k, L) z T (k, L)]. In this section we derive a recursive formula to obtain P(k,L), a measure of the estimation accuracy for the fixed-point smoother proposed in Theorem 1. From the OPL the error z(k, L) is orthogonal to the estimator ẑ(k, L); hence we have P(k,L)= K z (k, k) E [ ẑ(k, L)z T (k) ]. Using again the OPL, E[ẑ(k, L)z T (k)]=e[ẑ(k, L)ẑ T (k, L)]. So, we obtain P(k,L)= K z (k, k) E [ ẑ(k, L)ẑ T (k, L) ]. (36) If we denote S(k,L) = E[ẑ(k, L)ẑ T (k, L)] the covariance of the smoothing estimator and we use Eq. (8) for ẑ(k, L),wehave S(k,L) = S(k,L 1) + h(k, L, L)Π(L)h T (k,l,l), where we have taken into account that ν(l) and ẑ(k, L 1) are uncorrelated.

11 562 S. Nakamori et al. / Digital Signal Processing 13 (2003) Using now (18), for Π(L)h T (k,l,l), we obtain S(k,L) = S(k,L 1) + h(k, L, L) [ p(l) ( A(L)B T (k) A(L)E T (k, L 1) ) α(l)f T (k, L 1) ]. (37) Moreover, since the filter is ẑ(k,k) = A(k)O(k), it is clear that the initial condition for the recursive relation (37) is S(k,k) = A(k)r(k)A T (k) with r(k) given by (15). In view of (36) and (37), the following recursive expression for the fixed-point smoothing error covariance, P(k,L), is immediately obtained, P(k,L)= P(k,L 1) h(k, L, L) [ p(l) ( A(L)B T (k) A(L)E T (k, L 1) ) α(l)f T (k, L 1) ], with initial condition P(k,k) = K z (k, k) A(k)r(k)A T (k). Finally, since P(k,L) and S(k,L) are semi-definite positive matrices, it is clear that 0 S(k,L) K z (k, k). Moreover, K z (k, k) = A(k)B T (k) where A and B are bounded matrix functions (hypothesis H1). Hence, since the covariance matrix of the estimator is lower and upper bounded, the proposed smoothing algorithm has an unique solution. 5. A numerical simulation example The effectiveness of the proposed recursive fixed-point smoothing algorithm, presented in Theorem 1, is shown in a numerical example. We consider the following scalar observation equation y(k) = u(k)z(k) + v(k) + v 0 (k), where {v(k)} is a stationary white Gaussian noise and the uncertainty in the observations is modelled by {u(k)}, stationary sequence of independent Bernoulli random variables with P [u(k) = 1]=p. Let the autocovariance functions of the signal {z(k)} and the coloured noise {v 0 (k)} be given, in a semi-degenerate kernel form, by K z (k, s) = k s, 0 s k, (38) and K 0 (k, s) = k s, 0 s k, (39) respectively. According to hypotheses H1 and H3, the functions which constitute these autocovariance functions are as follows: A(k) = k, B(s)= 0.95 s, α(k) = k, β(s)= 0.5 s. (40)

12 S. Nakamori et al. / Digital Signal Processing 13 (2003) Fig. 1. Process of coloured observation noise v 0 (k) vs k. Let the uncertain probability be p(= p(k)) = Substituting the functions A(k), B(s), α(k), and β(s) given in (40), together with the probability p, into the estimation algorithm of Theorem 1, we can calculate the filtering and fixed-point smoothing estimates of the signal z(k). Figure 1 illustrates the coloured observation noise process vs k. Figure 2 illustrates the observed value y(k) vs k for u(k) simulated from a Bernoulli distribution with parameter p = 0.95 (Binomial distribution with parameters 1 and 0.95), the coloured observation noise of Fig. 1 plus white Gaussian observation noise N(0, ),wheren(0, ) represents the Gaussian distribution with mean 0 and variance Figure 3 illustrates the signal z(k) and the filtering estimate ẑ(k,k) vs k for the coloured observation noise of Fig. 1 plus white Gaussian observation noise. Here, the white observation noise processes obeys to N(0, ) and N(0, 1), respectively. Table 1 summarizes the MSVs of the filtering and fixed-point smoothing errors of the signal for N(0, ), N(0, ), N(0, ),andn(0, 1) in the cases of the uncertain and certain observations. From Table 1, it is shown that the estimation accuracy of the fixed-point smoother is superior to the filter for both the uncertain and certain observations and that the MSVs for the certain observations noise are less than those for the uncertain observations except the MSVs of the fixed-point smoothing error for N(0, ).

13 564 S. Nakamori et al. / Digital Signal Processing 13 (2003) Fig. 2. Process of observed value y(k) for the coloured noise process of Fig. 1 plus the white Gaussian noise process featured by N(0, ). The MSVs are calculated by 200 (z(i) ẑ(i, i))2 /200, for the filtering error z(i) ẑ(i, i), and by j=1 (z(i) ẑ(i, i + j))2 /2000, for the fixed-point smoothing error z(i) ẑ(i, i + j). Figure 4 illustrates the mean-square values (MSVs) of the filtering and fixed-point smoothing errors of the signal vs k when the white Gaussian observation noise obeys the Table 1 MSVs of the filtering and the fixed-point smoothing errors for the observation noises N(0, ), N(0, ), N(0, ),andn(0, 1) White Gaussian MSV of the filtering error MSV of the fixed-point observation noise smoothing error Uncertain Certain Uncertain Certain observation observation observation observation N(0, ) N(0, ) N(0, ) N(0, 1)

14 S. Nakamori et al. / Digital Signal Processing 13 (2003) Fig. 3. Signal z(k) and the filtering estimate ẑ(k,k) for the coloured noise process of Fig. 1 plus the white Gaussian noise process featured by N(0, ) and N(0, 1), respectively. N(0, ) distribution and the algorithm is applied with different values of p increasing from 0 to 1 by Estimations of p can be done as 0.89 or 0.91 from the minimum values of the MSVs of the filtering or fixed-point smoothing errors, respectively. Table 2 shows the minimum values of the MSVs of the filtering and fixed-point smoothing errors and the estimations of p from them, for white Gaussian observation noises N(0, ), N(0, ), N(0, ),andn(0, 1). On the other hand, according to the assumptions of Theorem 1, in the above simulation, the signal is uncorrelated with the coloured observation noise. It might also be interesting to see how the proposed algorithms perform in the estimation of a signal z(k) correlated with the coloured observation noise v 0 (k). For it, let the crosscovariance function of the signal z(k) with v 0 (k) be represented by K zv0 (k, s) = k s. Under this assumption, from the estimation algorithms of Theorem 1, Table 3 shows the MSVs of the filtering and fixed-point smoothing errors of the signal for white noise N(0, ), N(0, ), N(0, ),andn(0, 1) in the cases of uncertain (p = 0.95) and certain observations. It is noted that the MSVs in Table 3 are less than the corresponding values in Table 1. The MSVs for the certain observations are less than those for the uncertain observations, except the MSVs of the fixed-point smoothing errors for N(0, ).TheMSVs

15 566 S. Nakamori et al. / Digital Signal Processing 13 (2003) Fig. 4. MSVs of the filtering and fixed-point smoothing errors of the signal vs the uncertain probability p(= p(k)) when the white Gaussian observation noise obeys N(0, ). Table 2 Values of uncertain probability p with the minimum values of the MSVs of the filtering and fixed-point smoothing errors for white Gaussian observation noises N(0, ), N(0, ), N(0, ),andn(0, 1) White Gaussian Minimum value of MSVs Minimum value of MSVs observation noise of filtering error of fixed-point smoothing error Value of p for the minimum Value of p for the minimum value of MSVs of filtering error value of MSVs fixed-point smoothing error N(0, ) N(0, ) N(0, ) N(0, 1)

16 S. Nakamori et al. / Digital Signal Processing 13 (2003) Table 3 MSVs of the filtering and the fixed-point smoothing errors for the observation noises N(0, ), N(0, ), N(0, ),andn(0, 1), in case of the signal correlated with the coloured noise White Gaussian MSV of the filtering error MSV of the fixed-point observation noise smoothing error Uncertain Certain Uncertain Certain observation observation observation observation N(0, ) N(0, ) N(0, ) N(0, 1) of the fixed-point smoothing errors are almost the same as those of the filtering errors for both uncertain and certain observations. For references, a state-space realization for the signal and coloured noise, with the autocovariance functions (38) and (39), respectively, can be expressed by and z(k + 1) = 0.95z(k) + v z (k), E { v z (k)v z (s) } = 0.1δ K (k s), (41) v 0 (k + 1) = 0.5v 0 (k) + v 1 (k), E { v 1 (k)v 1 (s) } = 0.075δ K (k s), (42) respectively. Here, v z (k) and v 1 (k) are uncorrelated and hence, z(k) and v 0 (k) become uncorrelated. For the results displayed in Table 3, we have assumed that the input noises in (41) and (42) are correlated with crosscovariance Conclusions The linear state filter and fixed-point smoother for systems with uncertain observations, when the uncertainty in the observations is modelled by independent random variables, for the case of white plus coloured observation noises are obtained by recursive algorithms. These results extend the algorithms proposed by Nakamori [5] to cope with systems with uncertain observations. The proposed filter and fixed-point smoother use the covariance information of the signal and observation noises and the probability that the observation contains the signal, without requiring the information of the state-space model for the signal. The recursive algorithms are derived by employing the Orthogonal Projection Lemma and assuming that the covariance functions of the signal and the coloured observation noise are expressed in the form of a semi-degenerate kernel. So, the proposed estimators provide estimations of non-stationary or stationary signal processes. A numerical simulation example has shown that the estimators proposed in this paper are feasible. The problem of estimating the false alarm probability using covariance information and the study of the effect of over-estimating or under-estimating this probability could

17 568 S. Nakamori et al. / Digital Signal Processing 13 (2003) be an interesting problem to be studied in a future. Also, the estimation of u(k) could be an interesting question to be studied in a future, since it would provide very useful information, for example to discard missing data. Acknowledgments The authors would like to express their hearty gratitude to the anonymous referee for his invaluable suggestions in improving the original paper. This work has been partially supported by the Ministerio de Ciencia y Tecnología under contract BFM References [1] A. Hermoso, J. Linares, Linear estimation for discrete-time systems in the presence of time-correlated disturbances and uncertain observations, IEEE Trans. Automat. Control 39 (8) (1994) [2] A. Hermoso, J. Linares, Linear smoothing for discrete-time systems in the presence of correlated disturbances and uncertain observations, IEEE Trans. Automat. Control 40 (8) (1995) [3] U.E. Makov, Approximations to unsupervised filters, IEEE Trans. Automat. Control 25 (4) (1980) [4] R.A. Monzingo, Discrete linear recursive smoothing for systems with uncertain observations, IEEE Trans. Automat. Control 26 (3) (1981) [5] N.E. Nahi, Optimal recursive estimation with uncertain observation, IEEE Trans. Inform. Theory 15 (4) (1969) [6] S. Nakamori, Design of recursive fixed-point smoother using covariance information in linear discrete-time systems, Internat. J. Systems Sci. 23 (12) (1992) [7] S. Nakamori, Estimation technique using covariance information with uncertain observations in linear discrete-time systems, Signal Process. 58 (1997) [8] B. Porat, B. Friedlander, ARMA spectral estimation of time series with missing observations, IEEE Trans. Inform. Theory 30 (6) (1984) [9] B. Rosen, B. Porat, The second-order moments of the sample covariances for time series with missing observations, IEEE Trans. Inform. Theory 35 (2) (1989) [10] Y. Sawaragi, T. Katayama, S. Fujishige, Sequential state estimation with interrupted observations, Inform. Control 21 (1972) [11] Y. Sawaragi, T. Katayama, S. Fujishige, Adaptive estimation for a linear system with interrupted observations, IEEE Trans. Automat. Control 18 (1973)