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Digital Signal Processing 13 (2003) 552–568

www.elsevier.com/locate/ds

Linear estimation from uncertain observationswith white plus coloured noisesusing covariance information

S. Nakamori,a,∗ R. Caballero-Águila,b A. Hermoso-Carazo,c

and J. Linares-Pérezc

a Department of Technology, Faculty of Education, Kagoshima University, 1-20-6, Kohrimoto,Kagoshima, 890-0065, Japan

b Departamento de Estadística e Investigación Operativa, Universidad de Jaén, Paraje Las Lagunillas23071 Jaén, Spain

c Departamento de Estadística e Investigación Operativa, Universidad de Granada, Campus Fuentenue18071 Granada, Spain

Abstract

This paper considers the least mean-squared error linear estimation problems, using coinformation, in linear discrete-time stochastic systems with uncertain observations for the cwhite plus coloured observation noises. The different kinds of estimation problems treated ione-stage prediction, filtering, and fixed-point smoothing. The recursive algorithms are deriemploying the Orthogonal Projection Lemma and assuming that both, the signal and the conoise autocovariance functions, are given in a semi-degenerate kernel form. 2003 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 fromobservations has been widely treated when the observation sequence contains thebe estimated with probability one.

* Corresponding author.E-mail addresses:[email protected] (S. Nakamori), [email protected]

(R. Caballero-Águila), [email protected] (A. Hermoso-Carazo), [email protected] (J. Linares-Pérez).

1051-2004/03/$ – see front matter 2003 Elsevier Science (USA). All rights reserved.doi:10.1016/S1051-2004(02)00026-X

S. Nakamori et al. / Digital Signal Processing 13 (2003) 552–568 553

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However, in many practical situations, the signal vector enters in the observationtion in a random manner. In these cases, there is a positive probability (false alarmbility) that the observation in each time does not contain the signal to be estimated.situations are described by an observation equation which includes not only an anoise, but also a multiplicative noise component, modelled by a sequence of Bernoudom variables whose values, one or zero, indicate the presence or absence of the sthe observation.

This can occur in many practical situations, for example, in problems where thereintermittent failures in the observation mechanism, fading phenomena in propagationnels, target tracking, accidental loss of some measurements, or inaccessibility of thduring certain times; that is, problems where, due to different reasons, the measuset of the signal can contain observations which are only noise.

The least mean-squared error linear estimation problem for these linear discretsystems, when the uncertainty is modelled by independent Bernoulli variables, hatreated by different authors.

Assuming a full knowledge of the state-space model for the signal process and tfalse alarm probability is known a priori, Nahi [5] and Monzingo [4] were the first wtreated the estimation problem. Later on, Hermoso and Linares [1,2] extended theirto 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 uncobservations when the probability of the presence of the state in the observation is unbut fixed throughout the time interval of interest. By using a bayesian approach, theyan estimator of the false alarm probability which provides an adaptive algorithm fostate estimators. These results are extended in Sawaragi et al. [11] to the case ofwith uncertain observations with stationary Markov interrupted observation mechawhen the transition probabilities of the Markov chain are unknown but fixed. Markovusing a quasi-Bayes procedure, consider the estimation of the unknown probability acurrent state of the system based upon the entire observation sequence without knoof which observations contain or not the state.

In Porat and Friedlander [8], the estimation technique of the power spectral debased on nonlinear optimization of a weighted squared-error criterion for the struof the ARMA model, is proposed from missing observed data in the case of obsernoise free. In the technique, the information of the false alarm probability is not reqIn Rosen and Porat [9], the general formulas for the second-order moments of thecovariances are derived for the ARMA model with missing observed data in the caobservation noise free. In the relation with the studies by Porat and Friedlander [8], fcase of the uncertain observed values including additive observation noises, the estof the signal would provide very important information.

In linear stationary discrete-time systems with uncertain observations, Nakamoproposed a recursive estimation technique using as information the autocovariance fof the signal, and assuming that the probability that the signal exists in the observatavailable. Recursive algorithms for the least mean-squared error linear filter and fixedsmoother are derived, without requiring the complete state-space model of the signnecessary information for the state-space model is obtained by using a factorization mof the autocovariance of the signal.

554 S. Nakamori et al. / Digital Signal Processing 13 (2003) 552–568

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In all the aforementioned works, the estimation algorithms are derived provided th(uncertain) observations are perturbed by white noise. In a previous paper, Nakamtreated the estimation problem of the signal from observations (without uncertaintyturbed by white plus coloured noise. Under the assumption that the covariance funof the signal and the coloured observation noise are expressed in a semi-degeneraform, and by employing an invariant imbedding method, recursive estimation algorwere derived without using any realization technique for the state-space model of thefrom the covariance information.

This paper is concerned with the generalization of the algorithms proposeNakamori [6] to systems with uncertain observations, that is, we treat the estimproblems using covariance information in linear discrete-time systems with uncobservations. By employing the Orthogonal Projection Lemma, we derive the recalgorithms for the one-stage prediction, filtering, and fixed-point smoothing estimathe case of white plus coloured observation noise. It is assumed that the autocovfunctions of the signal and the coloured observation noise are expressed in the forsemi-degenerate kernel.

Since the semi-degenerate kernel is suitable for expressing autocovariance funcnon-stationary or stationary signal processes, the proposed estimators provide estiof 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)+ v0(k), (1)

wherez(k) is then× 1 signal vector andy(k) represents then× 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 functKz(k, s)=E[z(k)zT (s)], is expressed in a semi-degenerate kernel form, that is,

Kz(k, s)={A(k)BT (s), 0 � s � k,

B(k)AT (s), 0 � k � s,(2)

whereA andB are boundedn×M ′ matrix functions.H2. The noise process{v(k); k � 0} is a zero-mean white sequence with autocovaria

functionE[v(k)vT (s)] =R(k)δK(k − s), beingδK the Kroneckerδ function.H3. The process{v0(k); k � 0} is a zero-mean coloured noise sequence and its au

variance function,K0(k, s) = E[v0(k)vT0 (s)], is given by a semi-degenerate kern

form,

K0(k, s)={α(k)βT (s), 0 � s � k,

β(k)αT (s), 0 � k � s,(3)

whereα andβ aren×N ′ matrix functions.H4. The multiplicative noise{u(k); k � 0} is a sequence of independent Bernoulli rand

variables withP [u(k) = 1] = p(k).


signallns.

-l to

g theering

of

in the

H5. {z(k); k � 0}, {u(k); k � 0}, {v(k); k � 0}, and{v0(k); k � 0} are mutually inde-pendent.

If we denote �K0(i, s) = E[(u(i)z(i) + v0(i))(u(s)z(s) + v0(s))T ] and Ky(i, s) =

E[y(i)yT (s)], as a consequence of the above hypotheses, we have

�K0(i, s) =E[u(i)u(s)

]Kz(i, s)+K0(i, s) (4)

and

Ky(i, s)= �K0(i, s)+R(s)δK(i − s). (5)

We are interested in obtaining the least mean-squared error linear estimator of thez(k) based on the observations{y(1), . . . , y(L)}. This estimator,z(k,L), is the orthogonaprojection ofz(k) on the space ofn-dimensional linear transformations of the observatioSo,z(k,L) is given by

z(k,L)=L∑i=1

h(k, i,L)y(i), (6)

whereh(k, i,L), i = 1, . . . ,L, denotes the impulse-response function.The Orthogonal Projection Lemma (OPL) assures thatz(k,L) is the only linear combi

nation of the observations{y(1), . . . , y(L)} such that the estimation error is orthogonathem, that is,

E

{(z(k)−

L∑i=1

h(k, i,L)y(i)

)yT (s)

}= 0, s � L.

This condition is equivalent to the Wiener–Hopf equation

E[z(k)yT (s)

]=L∑i=1

h(k, i,L)E[y(i)yT (s)

], s �L,

useful for determining the impulse-response functionh(k, i,L), i = 1, . . . ,L. Using thehypotheses H1–H5 on the model, the Wiener–Hopf equation can be rewritten as

h(k, s,L)R(s) = p(s)Kz(k, s)−L∑i=1

h(k, i,L)�K0(i, s), s � L. (7)

This last version of the Wiener–Hopf equation will be used in Section 3 for obtaininsmoothing algorithm; specifically, for determining the one-stage prediction and filtestimates of the signal.

On the other hand, as a consequence of the OPL, denoting byy(L,L − 1) the leastmean-squared error linear estimator ofy(L) based on{y(1), . . . , y(L− 1)} and byν(L) =y(L) − y(L,L − 1) the innovation at timeL, it can be established that the estimatorsz(k) satisfy the following recursive equation

z(k,L)= z(k,L− 1)+ h(k,L,L)ν(L).

This equation provides the basis for the fixed-point smoothing algorithm presentednext section.


ate ofe. This

d

3. Recursive fixed-point smoothing algorithm

Theorem 1 presents the recursive formulas for the fixed-point smoothing estimthe signal, when the observation are perturbed by white noise plus coloured noistheorem includes the formulas for the filtering estimate.

Theorem 1. Let us consider the observation equation(1) given in Section2, satisfying thehypothesesH1–H5. Then, the fixed-point smoothing estimate of the signalz(k), z(k,L) forL> k, is given by

z(k,L)= z(k,L− 1)+ h(k,L,L)ν(L), (8)

whereν(L), the innovation, is

ν(L) = y(L)− p(L)A(L)O(L− 1)− α(L)Q(L− 1). (9)

TheM ′ × 1 andN ′ × 1 vectorsO(L) andQ(L), respectively, are recursively calculateby

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)

(BT (L)− r(L− 1)AT (L)

)− c(L− 1)αT (L)]Π−1(L), (12)

I (L,L) = [βT (L)− p(L)cT (L− 1)AT (L)− d(L− 1)αT (L)

]Π−1(L), (13)

whereΠ(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)cT (L− 1)

]AT (L)

+ [β(L)− p(L)A(L)c(L− 1)− α(L)d(L− 1)

]αT (L). (14)

The functionsr, c, andd which appear in(12), (13), and(14), areM ′ ×M ′, M ′ ×N ′, andN ′ ×N ′ matrices, respectively, verifying

r(L) = r(L− 1)+ J (L,L)[p(L)

(B(L)−A(L)r(L− 1)

)− α(L)cT (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)AT (L)−E(k,L− 1)AT (L)

)− F(k,L− 1)αT (L)

]Π−1(L), (18)

whereE(k,L) andF(k,L) aren×M ′ andn×N ′ matrices satisfying


nt

iolated

n, the

td the

sig-a

thod.

E(k,L)= E(k,L− 1)+ h(k,L,L)

× [p(L)

(B(L)−A(L)r(L− 1)

)− α(L)cT (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,z(k, k), which provides the initial condition for the fixed-poismoothing algorithm, is given byz(k, k)=A(k)O(k).

Remark. Let us note that the non-singularity ofΠ(L) is guaranteed ifR(L) is positivedefinite, but there are some phenomena in practice in which this assumption is v(for instance, if the observations are not affected by additive noise, thenR(L) is equalto zero and, consequently,Π(L) can be a singular matrix). IfΠ(L) were singular, theMoore–Penrose pseudo-inverse could be used.

Proof. As we have indicated in Section 2, Eq. (8) is a consequence of the OPL. Theproblem is to find the innovationν(L), and the smoothing gainh(k,L,L).

(I) The innovation process.Sinceν(L) = y(L)− y(L,L− 1), in order to determine it, iis enough to obtainy(L,L− 1). From the independence hypotheses on the model anOPL we have

y(L,L− 1)= p(L)z(L,L− 1)+ v0(L,L− 1), (21)

wherez(L,L−1) andv0(L,L−1) are the one-stage linear prediction estimates of thenal z(L) and the coloured noisev0(L), respectively. Both predictors will be obtained bysimilar procedure, from the Wiener–Hopf equation, using the invariant imbedding me

Firstly, from (6), the signal predictor is given by

z(L,L− 1)=L−1∑i=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)BT (s)−L−1∑i=1

h(L, i,L− 1)�K0(i, s),

s � L− 1. (23)

If we now introduce a functionJ (s,L− 1), such that

J (s,L− 1)R(s)= p(s)BT (s)−L−1∑

J (i,L− 1)�K0(i, s), s � L− 1, (24)
i=1


st ber of

from relations (23) and (24) we conclude that the impulse-response function muh(L, s,L− 1)=A(L)J (s,L− 1). So, from (22), it is clear that the one-stage predictothe signal isz(L,L− 1)=A(L)O(L− 1) with

O(L− 1)=L−1∑i=1

J (i,L− 1)y(i). (25)

In a similar way, it is proved thatv0(L,L− 1)= α(L)Q(L − 1) where

Q(L− 1)=L−1∑i=1

I (i,L− 1)y(i) (26)

andI (s,L − 1) is a function which satisfies

I (s,L − 1)R(s)= βT (s)−L−1∑i=1

I (i,L− 1)�K0(i, s), s �L− 1. (27)

Hence, from (21), we deduce that

y(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 vectorO(L).Taking into account that, from (2), (3), and (4),

�K0(L, s) = p(L)A(L)BT (s)p(s) + α(L)βT (s), s � L− 1,

and using (27) forβT (s), we have

�K0(L, s)= p(L)A(L)BT (s)p(s)+ α(L)I (s,L − 1)R(s)

+L−1∑i=1

α(L)I (i,L − 1)�K0(i, s), s � L− 1. (29)

On the other hand, if we subtract (24) from the equation obtained by puttingL − 1 → L

in (24), we obtain[J (s,L)− J (s,L− 1)

]R(s) = −J (L,L)�K0(L, s)

−L−1∑i=1

[J (i,L)− J (i,L− 1)

]�K0(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)BT (s)p(s)

−L−1∑i=1

[J (i,L)− J (i,L− 1)+ J (L,L)α(L)I (i,L − 1)

]�K0(i, s). (30)

From (30), taking into account (24), we conclude that, fors � L− 1,


wing

g

t-

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 folloequation obtained from (25)

O(L)−O(L− 1)= J (L,L)y(L)+L−1∑i=1

[J (i,L)− J (i,L− 1)

]y(i).

Relation (11) for the vectorQ(L) is obtained in an analogous way.Now, we will prove thatJ (L,L) andI (L,L) satisfy (12) and (13), respectively. Takin

into account the expressions (10) and (11) forO(L) andQ(L), we have

E[O(L)νT (L)

]=E[O(L− 1)νT (L)

]+ J (L,L)E[ν(L)νT (L)

],

E[Q(L)νT (L)

]=E[Q(L− 1)νT (L)

]+ I (L,L)E[ν(L)νT (L)

].

Sinceν(L) is uncorrelated withO(L− 1) andQ(L− 1), the first expectation of the righhand side term of both expressions is zero. So, denotingΠ(L) =E[ν(L)νT (L)], we obtain

J (L,L) =E[O(L)νT (L)

]Π−1(L), I (L,L) =E

[Q(L)νT (L)

]Π−1(L). (32)

Now, we calculate the expectations which appear in (32). From (9), it is clear that

E[O(L)νT (L)

]=E[O(L)yT (L)

]−E[O(L)OT (L− 1)

]AT (L)p(L)

−E[O(L)QT (L− 1)

]αT (L).

Firstly we obtainE[O(L)yT (L)]; using (25) forO(L) and (5) forKy(i,L), we have

E[O(L)yT (L)

]= J (L,L)R(L)+L∑i=1

J (i,L)�K0(i,L)

and by puttings → L and L − 1 → L in (24), we conclude thatE[O(L)yT (L)] =p(L)BT (L). On the other hand, from (10), taking into account thatν(L) is uncorrelatedwith O(L− 1) andQ(L− 1), we obtain

E[O(L)OT (L− 1)

]=E[O(L− 1)OT (L− 1)

]and

E[O(L)QT (L− 1)

]=E[O(L− 1)QT (L− 1)

].

Hence, denotingr(L)=E[O(L)OT (L)] andc(L)=E[O(L)QT (L)], we have

E[O(L)νT (L)

]= p(L)BT (L)− p(L)r(L− 1)AT (L)− c(L− 1)αT (L). (33)

A similar reasoning forE[Q(L)νT (L)], denotingd(L)=E[Q(L)QT (L)], leads to

E[Q(L)νT (L)

]= βT (L)− p(L)cT (L− 1)AT (L)− d(L− 1)αT (L). (34)

Substituting (33) and (34) in (32), we obtain (12) and (13).


m the

r

Let us now calculate the covariance matrixΠ(L) of the innovation. First of all weobserve that, from the OPL,y(L,L−1) is orthogonal toν(L) andE[y(L,L−1)yT (L)] =E[y(L,L− 1)yT (L,L− 1)]. Hence, the covariance matrixΠ(L) can be expressed as

Π(L) =E[y(L)yT (L)

]−E[y(L,L− 1)yT (L,L− 1)

].

Then, from (28), we have

Π(L)=E[y(L)yT (L)

]− p2(L)A(L)E[O(L− 1)OT (L− 1)

]AT (L)

− p(L)A(L)E[O(L− 1)QT (L− 1)

]αT (L)

− α(L)E[Q(L− 1)OT (L− 1)

]AT (L)p(L)

− α(L)E[Q(L− 1)QT (L− 1)

]αT (L).

Using now expressions (2)–(5), we conclude that

E[y(L)yT (L)

]= p(L)B(L)AT (L)+ β(L)αT (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, fro

OPL, we have

r(L) =E[O(L− 1)OT (L− 1)

]+ J (L,L)Π(L)J T (L,L),

c(L)=E[O(L− 1)QT (L− 1)

]+ J (L,L)Π(L)IT (L,L),

d(L)=E[Q(L− 1)QT (L− 1)

]+ I (L,L)Π(L)IT (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) −z(k,L), satisfiesz(k,L) = z(k,L− 1)− h(k,L,L)ν(L) and, consequently,

E[z(k,L)yT (L)

]=E[z(k,L− 1)yT (L)

]− h(k,L,L)E[ν(L)yT (L)

].

Using again the OPL,E[z(k,L)yT (L)] = 0 andE[ν(L)yT (L)] = Π(L); hence, it is cleathat

h(k,L,L) =E[z(k,L− 1)yT (L)

]Π−1(L). (35)

Now, we obtainE[z(k,L− 1)yT (L)] =E[z(k)yT (L)] −E[z(k,L− 1)yT (L)]. From theindependence hypotheses and sincek � L,

E[z(k)yT (L)

]= p(L)B(k)AT (L).

On the other hand, by applying the OPL,

E[z(k,L− 1)yT (L)

]=E[z(k,L− 1)yT (L,L− 1)

],

and using (28), we have

E[z(k,L− 1)yT (L)

]=E[z(k,L− 1)OT (L− 1)

]AT (L)p(L)

+E[z(k,L− 1)QT (L− 1)

]αT (L).


ausing

that

by thea-

i-

tor

Hence, denotingE(k,L)=E[z(k,L)OT (L)] andF(k,L)=E[z(k,L)QT (L)], we obtain

E[z(k,L− 1)yT (L)

]= p(L)(B(k)AT (L)−E(k,L− 1)AT (L)

)− F(k,L− 1)αT (L)

and substituting in (35), relation (18) for the gain is deduced.Finally, from (8), (10), and (11), we have

E(k,L)= E[z(k,L− 1)OT (L− 1)

]+ h(k,L,L)Π(L)J T (L,L),

F (k,L) =E[z(k,L− 1)QT (L− 1)

]+ h(k,L,L)Π(L)IT (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,z(k, k), which can be obtained, just by using

similar reasoning to that used to obtain the predictor at the beginning of the proof, bythe invariant imbedding method. In fact, from (6),

z(k, k)=k∑

i=1

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)BT (s)−k∑

i=1

h(k, i, k)�K0(i, s), s � k.

Then, puttingL− 1→ k in (24) and comparing with the above equation, it is clearh(k, s, k) =A(k)J (s, k) and consequently,z(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 measuredsmoothing errorz(k,L) = z(k) − z(k,L) and, more specifically, by the covariance mtrices of these errorsP(k,L) =E[z(k,L)zT (k,L)].

In this section we derive a recursive formula to obtainP(k,L), a measure of the estmation accuracy for the fixed-point smoother proposed in Theorem 1.

From the OPL the errorz(k,L) is orthogonal to the estimatorz(k,L); hence we have

P(k,L) =Kz(k, k)−E[z(k,L)zT (k)

].

Using again the OPL,E[z(k,L)zT (k)] =E[z(k,L)zT (k,L)]. So, we obtain

P(k,L) =Kz(k, k)−E[z(k,L)zT (k,L)

]. (36)

If we denoteS(k,L) = E[z(k,L)zT (k,L)] the covariance of the smoothing estimaand we use Eq. (8) forz(k,L), we have

S(k,L) = S(k,L− 1)+ h(k,L,L)Π(L)hT (k,L,L),

where we have taken into account thatν(L) andz(k,L− 1) are uncorrelated.


e

oth-

h-unded,

sented

ions iswith

these

Using now (18), forΠ(L)hT (k,L,L), we obtain

S(k,L)= S(k,L− 1)+ h(k,L,L)[p(L)

(A(L)BT (k)−A(L)ET (k,L− 1)

)− α(L)FT (k,L− 1)

]. (37)

Moreover, since the filter isz(k, k)=A(k)O(k), it is clear that the initial condition for threcursive relation (37) isS(k, k)=A(k)r(k)AT (k) with r(k) given by (15).

In view of (36) and (37), the following recursive expression for the fixed-point smoing error covariance,P(k,L), is immediately obtained,

P(k,L) = P(k,L− 1)− h(k,L,L)[p(L)

(A(L)BT (k)−A(L)ET (k,L− 1)

)− α(L)FT (k,L− 1)

],

with initial condition

P(k, k) =Kz(k, k)−A(k)r(k)AT (k).

Finally, sinceP(k,L) andS(k,L) are semi-definite positive matrices, it is clear that

0 � S(k,L) �Kz(k, k).

Moreover,Kz(k, k) = A(k)BT (k) whereA andB are bounded matrix functions (hypotesis H1). Hence, since the covariance matrix of the estimator is lower and upper bothe proposed smoothing algorithm has an unique solution.

5. A numerical simulation example

The effectiveness of the proposed recursive fixed-point smoothing algorithm, prein Theorem 1, is shown in a numerical example.

We consider the following scalar observation equation

y(k)= u(k)z(k)+ v(k)+ v0(k),

where{v(k)} is a stationary white Gaussian noise and the uncertainty in the observatmodelled by{u(k)}, stationary sequence of independent Bernoulli random variablesP [u(k)= 1] = p.

Let the autocovariance functions of the signal{z(k)} and the coloured noise{v0(k)} begiven, in a semi-degenerate kernel form, by

Kz(k, s) = 1.026× 0.95k−s, 0 � s � k, (38)

and

K0(k, s)= 0.1× 0.5k−s, 0 � s � k, (39)

respectively. According to hypotheses H1 and H3, the functions which constituteautocovariance functions are as follows:

A(k)= 1.026× 0.95k, B(s) = 0.95−s,

α(k) = 0.1× 0.5k, β(s)= 0.5−s . (40)


mates

e-ation

obser-

f theinof theationscertain

Fig. 1. Process of coloured observation noisev0(k) vs k.

Let the uncertain probability bep(= p(k)) = 0.95. Substituting the functionsA(k),B(s), α(k), andβ(s) given in (40), together with the probabilityp, into the estimationalgorithm of Theorem 1, we can calculate the filtering and fixed-point smoothing estiof the signalz(k).

Figure 1 illustrates the coloured observation noise process vsk. Figure 2 illustrates theobserved valuey(k) vs k for u(k) simulated from a Bernoulli distribution with paramterp = 0.95 (Binomial distribution with parameters 1 and 0.95), the coloured observnoise of Fig. 1 plus white Gaussian observation noiseN(0,0.32), whereN(0,0.32) repre-sents the Gaussian distribution with mean 0 and variance 0.32.

Figure 3 illustrates the signalz(k) and the filtering estimatez(k, k) vsk for the colouredobservation noise of Fig. 1 plus white Gaussian observation noise. Here, the whitevation noise processes obeys toN(0,0.32) andN(0,1), respectively.

Table 1 summarizes the MSVs of the filtering and fixed-point smoothing errors osignal forN(0,0.32), N(0,0.52), N(0,0.72), andN(0,1) in the cases of the uncertaand certain observations. From Table 1, it is shown that the estimation accuracyfixed-point smoother is superior to the filter for both the uncertain and certain observand that the MSVs for the certain observations noise are less than those for the unobservations except the MSVs of the fixed-point smoothing error forN(0,0.32).


oise

or

pointthe

tion

Fig. 2. Process of observed valuey(k) for the coloured noise process of Fig. 1 plus the white Gaussian nprocess featured byN(0,0.32).

The MSVs are calculated by∑200

i=1(z(i) − z(i, i))2/200, for the filtering errorz(i) −z(i, i), and by

∑200i=1

∑10j=1(z(i)− z(i, i + j))2/2000, for the fixed-point smoothing err

z(i)− z(i, i + j).Figure 4 illustrates the mean-square values (MSVs) of the filtering and fixed-

smoothing errors of the signal vsk when the white Gaussian observation noise obeys

Table 1MSVs of the filtering and the fixed-point smoothing errors for the observation noisesN(0,0.32), N(0,0.52),N(0,0.72), andN(0,1)

White Gaussian MSV of the filtering error MSV of the fixed-pointobservation noise smoothing error

Uncertain Certain Uncertain Certainobservation observation observation observa

N(0,0.32) 0.1804 0.1805 0.1307 0.1511N(0,0.52) 0.2216 0.2196 0.1526 0.1525N(0,0.72) 0.2687 0.2654 0.1836 0.1821N(0,1) 0.3400 0.3353 0.2344 0.2317


ite

sows

d the

simu-so bel

c-

MSVs

-lues inobser-

Fig. 3. Signalz(k) and the filtering estimatez(k, k) for the coloured noise process of Fig. 1 plus the whGaussian noise process featured byN(0,0.32) andN(0,1), respectively.

N(0,0.32) distribution and the algorithm is applied with different values ofp increasingfrom 0 to 1 by 0.01. Estimations ofp can be done as 0.89 or 0.91 from the minimum valueof the MSVs of the filtering or fixed-point smoothing errors, respectively. Table 2 shthe minimum values of the MSVs of the filtering and fixed-point smoothing errors anestimations ofp from them, for white Gaussian observation noisesN(0,0.32), N(0,0.52),N(0,0.72), andN(0,1).

On the other hand, according to the assumptions of Theorem 1, in the abovelation, the signal is uncorrelated with the coloured observation noise. It might alinteresting to see how the proposed algorithms perform in the estimation of a signaz(k)

correlated with the coloured observation noisev0(k). For it, let the crosscovariance funtion of the signalz(k) with v0(k) be represented byKzv0(k, s) = 0.165× 0.95k−s. Underthis assumption, from the estimation algorithms of Theorem 1, Table 3 shows theof the filtering and fixed-point smoothing errors of the signal for white noiseN(0,0.32),N(0,0.52), N(0,0.72), 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 vaTable 1. The MSVs for the certain observations are less than those for the uncertainvations, except the MSVs of the fixed-point smoothing errors forN(0,0.32). The MSVs


ing

r

Fig. 4. MSVs of the filtering and fixed-point smoothing errors of the signal vs the uncertain probabilityp(= p(k))

when the white Gaussian observation noise obeysN(0,0.32).

Table 2Values of uncertain probabilityp with the minimum values of the MSVs of the filtering and fixed-point smootherrors for white Gaussian observation noisesN(0,0.32), N(0,0.52), N(0,0.72), andN(0,1)

White Gaussian Minimum value of MSVs Minimum value of MSVsobservation noise of filtering error of fixed-point smoothing error

Value ofp for the minimum Value ofp for the minimumvalue of MSVs of filtering error value of MSVs fixed-point smoothing erro

N(0,0.32) 0.1199 0.09700.89 0.91

N(0,0.52) 0.1483 0.11180.89 0.91

N(0,0.72) 0.1749 0.14460.89 0.91

N(0,1) 0.2208 0.19140.89 0.91


tion

ors for

the au-

1) and

tions,les, forhms.stemse co-

at themodel

mman noiseprovide

paper

ationcould

Table 3MSVs of the filtering and the fixed-point smoothing errors for the observation noisesN(0,0.32), N(0,0.52),N(0,0.72), andN(0,1), in case of the signal correlated with the coloured noise

White Gaussian MSV of the filtering error MSV of the fixed-pointobservation noise smoothing error

Uncertain Certain Uncertain Certainobservation observation observation observa

N(0,0.32) 0.0705 0.0421 0.0984 0.1141N(0,0.52) 0.0873 0.0691 0.0967 0.0925N(0,0.72) 0.1154 0.1052 0.1209 0.1128N(0,1) 0.1816 0.1669 0.1590 0.1520

of the fixed-point smoothing errors are almost the same as those of the filtering errboth uncertain and certain observations.

For references, a state-space realization for the signal and coloured noise, withtocovariance functions (38) and (39), respectively, can be expressed by

z(k + 1)= 0.95z(k)+ vz(k), E{vz(k)vz(s)

}= 0.1δK(k − s), (41)

and

v0(k + 1)= 0.5v0(k)+ v1(k), E{v1(k)v1(s)

}= 0.075δK(k − s), (42)

respectively. Here,vz(k) andv1(k) are uncorrelated and hence,z(k) andv0(k) becomeuncorrelated.

For the results displayed in Table 3, we have assumed that the input noises in (4(42) are correlated with crosscovariance

√0.0075.

6. Conclusions

The linear state filter and fixed-point smoother for systems with uncertain observawhen the uncertainty in the observations is modelled by independent random variabthe case of white plus coloured observation noises are obtained by recursive algorit

These results extend the algorithms proposed by Nakamori [5] to cope with sywith uncertain observations. The proposed filter and fixed-point smoother use thvariance information of the signal and observation noises and the probability thobservation contains the signal, without requiring the information of the state-spacefor the signal.

The recursive algorithms are derived by employing the Orthogonal Projection Leand assuming that the covariance functions of the signal and the coloured observatioare expressed in the form of a semi-degenerate kernel. So, the proposed estimatorsestimations of non-stationary or stationary signal processes.

A numerical simulation example has shown that the estimators proposed in thisare feasible.

The problem of estimating the false alarm probability using covariance informand the study of the effect of over-estimating or under-estimating this probability


seful

ee forrtially2.

ted dis-

distur-

–847.Trans.

5 (4)

e-time

linear

Trans.

missing

Inform.

serva-

be an interesting problem to be studied in a future. Also, the estimation ofu(k) couldbe an interesting question to be studied in a future, since it would provide very uinformation, for example to discard missing data.

Acknowledgments

The authors would like to express their hearty gratitude to the anonymous referhis invaluable suggestions in improving the original paper. This work has been pasupported by the “Ministerio de Ciencia y Tecnología” under contract BFM2000-060

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