An Optimal Linear Fusion Estimation Algorithm of Reduced Dimension for T-Proper Systems with Multiple Packet Dropouts

This paper analyses the centralized fusion linear estimation problem in multi-sensor systems with multiple packet dropouts and correlated noises. Packet dropouts are modeled by independent Bernoulli distributed random variables. This problem is addressed in the tessarine domain under conditions of T1 and T2-properness, which entails a reduction in the dimension of the problem and, consequently, computational savings. The methodology proposed enables us to provide an optimal (in the least-mean-squares sense) linear fusion filtering algorithm for estimating the tessarine state with a lower computational cost than the conventional one devised in the real field. Simulation results illustrate the performance and advantages of the solution proposed in different settings.

For multi-sensor systems, the potential of fusion estimation techniques to produce consistent and accurate estimators has been demonstrated. Thus, these techniques have also been applied to multi-sensor systems with multiple packet dropouts, giving rise to centralized as well as distributed fusion estimation algorithms (see, e.g., [4,10,[23][24][25][26][27]). In general, centralized fusion methodology yields optimal estimators, but the computational load involved can be a handicap in practical applications.
Alternatively, 4D hypercomplex-based signal processing has been satisfactorily applied as a dimension reduction approach in multi-sensor fusion estimation problems with uncertainties [28][29][30][31][32][33][34][35]. Effectively, the benefit of using hypercomplex algebras is twofold: first, they may provide a compact representation of multidimensional signals and a better insight into the structure of the problem than that provided by a traditional or real formalism, and second, the characterization of certain properness properties related to the vanishing of some correlation or pseudo correlation functions means the dimension of the processes involved may be reduced. Then, even though the optimal processing in the 4D hypercomplex field is the widely linear (WL) processing, which implies working on simulated values and the other one on realistic phenomena, illustrate the theoretical results obtained. The paper finishes with the concluding remarks in Section 6.

Notations
The following standard notation is used throughout this paper: scalars are denoted by lightface letters, while boldface lowercase and boldface uppercase letters represent the vectors and matrices, respectively. The symbol 0 n×m (respectively, 0 n ) stands for the n × m matrix (respectively, n column vector) whose elements are all zeros, I n is the n × n identity matrix, and 1 n denotes the n column vector of ones.
Z, R, and T denote, respectively, the set of integer, real, and tessarine numbers. R n (respectively, T n ) is the set of all n-dimensional real (respectively, tessarine) vectors, and R n×m (respectively, T n×m ) refers to the set of all n × m-dimensional real (respectively, tessarine) matrices. Moreover, the superscripts "*", "T", and "H" symbolize the tessarine conjugate, transpose, and Hermitian transpose, respectively.
The notation E[·] represents the mathematical expectation, Cov(·) is the covariance operator, and diag(·) denotes the diagonal (or block diagonal) matrix with the input arguments on the main diagonal. Finally, δ t,s represents the Kronecker delta function, and the Hadamard and Kronecker product operators are symbolized by "•" and "⊗", respectively.

Definitions and Preliminaries
This section is devoted to stating the core concepts and results in the tessarine domain that will be used throughout the paper.
Unless otherwise indicated, we shall assume that all random variables have zero mean.
In the tessarine domain, the second-order statistical properties of x(t) ∈ T n are completely described from the augmented tessarine signal vector where x * (t) is the conjugate of x(t) defined as T be the real vector formed by the components x ν (t) ∈ R n , ν = r, η, η , η , of x(t) ∈ T n . The following relationship can be established: Notice that T H T = I 4n .
It should be highlighted that the properness profile of a tessarine random signal plays a key role in the choice of the suitable type of linear processing that leads to a reduction in the dimension of the problem. This properness profile is characterized by the degree of correlation between the imaginary components and the real component. In particular, two interesting types of properness can be defined in the tessarine domain [36,37]. Definition 2. Let x(t) ∈ T n be a tessarine random signal vector. It is said that: Likewise, let x(t) ∈ T n 1 and y(t) ∈ T n 2 be two tessarine random signal vectors. It is said that: • x(t) and y(t) are cross T 1 -proper, if and only if Γ xy ν (t, s) = 0, for ν = * , η, η , and ∀t, s ∈ Z, • x(t) and y(t) are cross T 2 -proper, if and only if Γ xy ν (t, s) = 0, for ν = η, η , and ∀t, s ∈ Z, • x(t) and y(t) are jointly T k -proper, for k = 1, 2, if and only if they are T k -proper and cross T k -proper.

Remark 1.
In the tessarine domain, the optimal linear processing, the widely linear (WL) processing, is based on an augmented tessarine vector of dimension 4n of the form given in (1). Nevertheless, when T k -properness conditions are satisfied, the WL estimators coincide with the one obtained from a T k -proper linear processing, which uses only the information provided by the processes involved (case k = 1) or the 2n-dimensional augmented vector formed by the signal and its conjugate (case k = 2). Consequently, T k -properness means there is a significant reduction in the dimension of the processes involved [37].
Finally, a new product between two tessarine signal vectors is defined.
Definition 3. Consider x(t), y(s) ∈ T n . The product is defined by the expression Note that given two random tessarine signal vectors x(t), y(s) ∈ T n , the augmented vector of

Problem Formulation
Let x(t) ∈ T n be an n-dimensional tessarine state vector which is assumed to be observed from R sensors perturbed by different additive noises according to the statespace model: F j (t) ∈ T n×n , j = 1, . . . , 4: deterministic tessarine matrices. • u(t) ∈ T n : tessarine white noises with pseudo variances Q(t).
The packets or measured outputs z (i) (t) are assumed to be affected by random packet dropouts characterized by Bernoulli distributed random variables that can be described by the following model: for i = 1, . . . , R, with y (i) (1) = z (i) (1), and the product given in Definition 3. Moreover, at each sensor i = 1, . . . , R, the tessarine random vector γ (i) (t) = [γ (i) j,ν (t) are independent Bernoulli random variables with known probabilities p (i) j,ν (t), for j = 1, . . . , n and ν = r, η, η , η , that indicate whether the corresponding component of the packet or measured output z (i) (t) of sensor i is received at time t (γ (i) j,ν (t) = 1) or it is lost and the latest received previously component, corresponding to the measured output

Remark 2.
Observe that model (2) always considers the latest measurement output received when the current measurement output is lost during transmission. Hence, this model can be used to describe multiple packet dropouts.

Remark 3. Under the hypothesis established for the Bernoulli random variables γ
for any j 1 , j 2 = 1, . . . , n, ν 1 , ν 2 = r, η, η , η and i 1 , In this setting, and based on the information supplied by the received measurements, our aim is to devise efficient algorithms for computing the WL centralized fusion estimators of the signal x(t), under the conditions of T k -properness, for k = 1, 2.
With the purpose of a WL processing, the 4n-dimensional augmented vectors are considered. Then, the centralized fusion estimation problem is addressed by applying the traditional estimation methods on the following WL stacked state-space system: with y(1) = z(1), and where z( Now, the centralized fusion estimation problem is analyzed in a T k -properness setting. The following proposition establishes conditions on system (3)-(5) that guarantee the T k -properness of the processes involved. (3)-(5), and taking into account the T k -properness concepts given in Definition 2, the following properties can be established: 1.

Proposition 1. Given the WL stacked state-space model
x(t) is T 1 -proper if and only if the initial state x(0) and the state noise u(t) are T 1 -proper, and the matrixΦ(t) is block diagonal as described below , ∀t, j = 1, . . . , n, i = 1, . . . , R, then x(t) and y (i) (t) are jointly T 1 -proper. 2.
x(t) is T 2 -proper if and only if the initial state x(0) and the state noise u(t) are T 2 -proper, and the matrixΦ(t) is block diagonal as described below

Remark 4.
It should be observed that the conditions established in Proposition 1 for ensuring the different type of properness on the processes involved in (3)-(5), are similar to the one stated in [34].
Then, under conditions of T k -properness, for k = 1, 2, the measurement Equation (5) in the above WL stacked state-space model can be expressed in the following equivalent form of reduced dimension: with y k (1) = ∆ k z(1), and In addition, and Π (i)

Remark 5.
It is worth noting that T k -properness also allows us to reduce the dimension of Equations (3) and (4) by replacing the 4n-dimensional augmented processesx(t),ū(t)z (i) (t), v (i) (t), and the matrixΦ(t) by the corresponding kn-dimensional vectors x k (t), u k (t), z (i) , and Φ 2 (t) given in Proposition 1, in a T 2 -proper scenario.
Thus, whereas the optimal linear processing in the tessarine domain suggests computing the LLMS filter of the state x(t) ∈ T n from its projection onto the augmented measurements { y(1), . . . y(t)}, under conditions of T k -properness, for k = 1, 2, this estimator can be obtained from the measurements {y k (1), . . . , y k (t)} defined in (6), which gives rise to the so-called T k -proper estimators. This approach supposes a reduction in the dimension of the problem that leads to computational savings that cannot be attained from a real formalism.
This methodology has been recently applied to design recursive fusion estimation algorithms for multi-sensor systems affected by random delays and missing measurements [35]. In this paper, we are interested in extending this methodology to systems affected by random multiple-packet dropouts.

Remark 6.
Note that although tessarine system is not a Hilbert space, a suitable metric has been defined in [36] to ensure the existence and uniqueness of projections.

T k -Proper Centralized Fusion Filtering Estimation
In this section, based on Kalman filter techniques, an efficient algorithm is provided for the computation of the T k -proper LLMS centralized fusion filterx T k (t|t), for k = 1, 2, of the state x(t) described by the state-space system with packet dropouts given by Equations (3), (4), and (6), as well as its associated error pseudo covariance matrix P T k (t|t). For this purpose, a recursive algorithm is devised under T k -properness conditions for the projection ofx(t) onto the set of measurements {y k (1), . . . , y k (t)}, denoted byx k (t|t), and its error pseudo covariance matrix P k (t|t). Then,x T k (t|t) and P T k (t|t) are determined by the first n components ofx k (t|t) and P k (t|t), respectively. Theorem 1 summarizes the formulas of this T k -proper LLMS centralized fusion filtering algorithm. Theorem 1. The T k -proper LLMS centralized fusion filter,x T k (t|t), for k = 1, 2, is obtained as followsx where for k = 1, 2,x k (t|t) is calculated from the recursive equation andx k (t + 1|t) satisfies the recursive expression with initial valuesx k (1|0) =x k (0|0) = 0 kn . The innovations k (t) are recursively calculated from the formula with initial value k (1) = y k (1), and C k = 1 R ⊗ I kn .
where the matrices Θ k (t) are obtained from the equation with and where Γx(t, t) is given by the recursive expression In addition, where with Γx(t, t) computed in (13), Finally, the T k -proper centralized fusion filtering error pseudo covariance matrix, P T k (t|t), for k = 1, 2, is obtained as follows: where for k = 1, 2, P k (t|t) satisfies the following recursive equation: with initial condition P k (0|0) = P 0 k , and with initial condition P k (1|0) = D k (1) .

Remark 7.
Notice that the computational load of the T k -proper LLMS centralized fusion filtering algorithms, for k = 1, 2, given in Theorem 1 is the same as that of their quaternion domain counterparts, i.e., those derived by using quaternion strictly linear (QSL) and quaternion semiwidely linear (QSWL) processing, respectively. As a consequence, it is noteworthy to see that the proposed T k -proper LLMS centralized fusion filtering algorithm provides estimations of the state that is equivalent to the one obtained from a WL processing or a real vectorial processing, whereas the computational load implied is reduced from O(64R 3 n 3 ) to O(kR 3 n 3 ) for k = 1, 2 [38].

Numerical Example
Our aim in this section is to numerically analyze the performance and benefits of the T k -proper LLMS centralized fusion filtering algorithm proposed in Theorem 1. Two examples are proposed: the first one from simulated values in which a scalar signal is estimated from the observations provided by several sensors; and the second one, a realistic model of a bidimensional tessarine state-space model which described a great amount of experimental phenomena. In both examples, by varying the Bernoulli parameters, different situations are compared in order to illustrate the effectiveness of the proposed algorithm in both T k -proper scenarios, for k = 1, 2.

Example 1
Consider the following multi-sensor tessarine state-space system: for i = 1, . . . , R, with y (i) (1) = z (i) (1), and where F 1 (t) = 0.3 + 0.3η + 0.1η + 0.2η ∈ T. Moreover, u(t) is a tessarine noise such that the covariance matrix of the associated real vector u r (t) is of the form where the parameters a, b, and c take different values depending on the T k -proper scenario considered. Furthermore, to guarantee the correlation hypothesis between the state and observation noises, u(t) and v (i) (t), they are assumed to satisfy the following expression: with α i ∈ R, and where, at each i, w (i) (t) is a tessarine white Gaussian noise independent of u(t), whose real covariance matrix is given by Specifically, the following values of α i and β i , for i = 1, 2, 3, 4, 5, will be considered in our simulations: Additionally, the variance matrix of the real initial state x r (0) is assumed to be of the form whose values d, e, and f will be specified in Sections 5.1.1 and 5.1.2, according to the different T k -proper scenario analyzed.

T 1 -Proper Scenario
To guarantee that x(t) and y (i) (t) are joint T 1 -proper, it has been taken a = b = 1, c = −0.5 in (18) and d = e = 4, c = 1.5 in (19). Moreover, it has also been assumed that the components of the multiplicative noise in (17), γ Firstly, the behavior of the estimators proposed is analyzed by considering a different number of sensors. Specifically, Figure 1 shows the T 1 -proper centralized fusion filtering error variances computed from the observations provided by 2, 3, 4, and 5 sensors. As expected, the estimators perform better as the number of sensors increases, which makes sense because the number of observations used to estimate the signal increases. Next, in order to show the computational savings attained with the solution proposed under T 1 -properness conditions, Table 1 presents the computation time required to apply the T 1 -proper centralized fusion filtering algorithms given in Theorem 1, and the conventional one devised from a real-valued linear processing in the cases of 2, 3, 4, and 5 sensors. Then, a reduction in the computation time can be observed when the methodology proposed is used, and this computational saving becomes more significant as the number of sensors increases. Our second objective is to compare tessarine and quaternion signal processing for different probabilities of updated/missing observations under T 1 -properness conditions. For this purpose, the error variances of both T 1 and QSL centralized fusion filters have been calculated for the following cases: Then, the difference between both tessarine and quaternion LLMS centralized fusion filtering error variances, that is, D 1 (t|t) = P QSL (t|t) − P 1 (t|t), have been computed and displayed in Figure 2. In this figure, positive differences can be observed in all the cases, meaning that it can be noted that T 1 -proper fusion estimators perform better than their quaternion counterparts. As expected, the fact that the T 1 -properness conditions are satisfied determines that it is more appropriate to use the T 1 -proper signal processing than the quaternion one, since it yields better estimations. Moreover, these differences become smaller as the probability of updated observations increases. Finally, with the aim of comparing both QSL and T 1 -proper signal processing, they are applied by taking a fixed value for the probabilities of the Bernoulli parameters in all the sensors, but different values of c in (18), that is, c = −0.8, −0.5, −0.2, 0. Note that for c = 0, the state additive noise, u(t), is T 1 besides Q-proper, and as c is further away from 0, the Q-properness conditions are further away. In this setting, the error variances of both T 1 -proper and QSL LLMS centralized fusion filters have been computed, and the mean of the differences between them, MD 1 (t|t) = mean(D 1 (t|t)), have been displayed in Figure 3 for the different values of c. In this figure, tessarine estimators are shown to be more accurate the further the noise u(t) is from the Q-properness conditions. Moreover, as in Figure 2, these differences decrease as the probability of updated observations increases.

Example 2
Let us consider the following general equation of motion [33]: where ϕ is the variable of interest, φ its range of change, and υ the input of the system. Notice that Equation (20) models a great amount of physical phenomena, and it has been used, for example, in bearing-only tracking applications and rotation tracking problems, where υ represents, respectively, the force or acceleration and the torque or angular acceleration.
In discrete-time, by taking x(t) = [ϕ(t), φ(t)] T , it is possible to build a model equivalent to that given in (20), as follows: x(t + 1) = 1 0.04 0 1 x(t) + 0.0008 0.04 (t), t = 1, . . . , 100; where (t) is a tessarine white noise with real covariance matrix: Moreover, the additive noise of single-sensor real observation equation, v(t) = [v 1 (t), v 2 (t)] T , is assumed to be a tessarine white noise with independent components and associated real covariance matrices given by In order to guarantee the T k -properness conditions, the following assumptions and cases about the parameters of the Bernoulli random variables have been considered:

Discussion
The LLMS centralized fusion filtering problem is analyzed in linear systems with multiple sensors and multiple packet dropouts. However, unlike most of the solutions proposed in the literature, a proper hypercomplex-valued signal processing has been employed with the purpose of reducing the dimension of the problem. Specifically, the state-space system is defined in the tessarine domain, and it is assumed that each component of the measurement output at each sensor may present a different packet dropout rate, modeled by using a Bernoulli random variable. Moreover, the state and the measurement noises can be correlated. Under hypotheses of T k -properness, our approach allows us to provide an optimal LLMS fusion filtering algorithm that reduces the computational cost of its counterpart in the real field. The good behavior and benefits of this algorithm have been analyzed in situations of T 1 and T 2 -properness by considering different numbers of sensors. Moreover, a comparative study of the quaternion and tessarine approaches was carried out, showing how the algorithm proposed behaves better than its counterpart in the quaternion domain when T k -properness, k = 1, 2, conditions are satisfied.
As a consequence, our approach based on T k -proper processing presents two main advantages: on the one hand, the tessarine systems offer a suitable framework to model 3D and 4D physical and experimental phenomena, and on the other hand, a considerable reduction of problem dimension is possible when the processes involved are T k -proper, which implies significant computational savings in the implementation of our LLMS fusionfiltering algorithm that cannot be attained from a real formalism of the problem.
In future research, we will approach the estimation problem in other hypercomplex algebras and under different properness conditions by using alternative fusion architectures for the multi-sensor observations with varied uncertainty situations.
Finally, from (A1), it is clear that the pseudo covariance matrixP(t|t) = E[¯ (t|t)¯ H (t|t)], of the filtering errors¯ (t|t) =x(t) −x(t|t) can be computed in the form