Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 8475693, 23 pages
https://doi.org/10.1155/2018/8475693
Research Article

A Constrained-Total-Least-Squares Method for Joint Estimation of Source and Sensor Locations: A General Framework

1National Digital Switching System Engineering and Technology Research Center, Zhengzhou 450002, China
2Zhengzhou Information Science and Technology Institute, Zhengzhou, Henan 450002, China

Correspondence should be addressed to Ruirui Liu; moc.361@729rr_uil

Received 14 July 2017; Accepted 8 February 2018; Published 5 April 2018

Academic Editor: Filippo Ubertini

Copyright © 2018 Ding Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

It is well known that sensor location uncertainties can significantly deteriorate the source positioning accuracy. Therefore, improving the sensor locations is necessary in order to achieve better localization performance. In this paper, a constrained-total-least-squares (CTLS) method for simultaneously locating multiple disjoint sources and refining the erroneous sensor positions is presented. Unlike conventional localization techniques, an important feature of the proposed method is that it establishes a general framework that is suitable for many different location measurements. First, a modified CTLS optimization problem is formulated after some algebraic manipulations and then the corresponding Newton iterative algorithm is derived to give the numerical solution. Subsequently, by using the first-order perturbation analysis, the explicit expression for the covariance matrix of the proposed CTLS estimator is deduced under the Gaussian assumption. Moreover, the estimation accuracy of the CTLS method is shown to achieve the Cramér-Rao bound (CRB) before the thresholding effect occurs by a rigorous proof. Finally, two kinds of numerical examples are given to corroborate the theoretical development in this paper. One uses the TDOAs/GROAs measurements and the other is based on the TOAs/FOAs parameters.

1. Introduction

Passive source localization has attracted significant attention in the signal processing research due to its importance to many applications including radar, sonar, microphone arrays, navigation, sensor networks, and wireless communications. Common wireless location systems are based on a two-step procedure for target position determination. In the first phase, the intermediate parameters that depend on the locations of the sources are estimated from the received signals. In general, these parameters include direction of arrival (DOA) [13], time of arrival (TOA) [47], time difference of arrival (TDOA) [817], frequency of arrival (FOA) [18], frequency difference of arrival (FDOA) [1928], received signal strength (RSS) [29], and gain ratios of arrival (GROA) [30, 31]. In the second phase, the previously estimated parameters are used to locate the sources. During the past few decades, numerous methods have been proposed for the two active research areas. In this paper, we focus on the latter, that is, emitter location estimation.

It is easy to see that the position determination is equivalent to solving a set of nonlinear equations relating the intermediate parameters to the coordinates of the sources. A number of localization methods are available in the literature. Some of them are iterative algorithms (such as Taylor series (TS) algorithm [2, 12, 20, 23, 32] and constrained-total-least-squares (CTLS) algorithm [3, 6, 15, 17, 2527]) that require proper initial solution guesses, and the others are closed-form solutions (such as total least squares (TLS) solution [1, 11, 22], quadratic constraint least square (QCLS) solution [4, 9, 10, 13, 33], and two-step weighted least square (TWLS) solution [5, 7, 8, 14, 16, 19, 21, 24, 2831]) that are more computationally efficient. Most of these algorithms can reach the corresponding Cramér-Rao bound (CRB) accuracy under moderate level of signal-to-noise ratio (SNR). Moreover, it is worth noting that all the localization algorithms need to transform the nonlinear measurement equations into pseudo-linear equations, except for the TS algorithm.

However, it must be emphasized that the accuracy of source location estimate may be seriously degraded by the sensor location errors, regardless of the specific localization algorithm used. In [21, 34], the source location mean square error (MSE) is derived when the sensor locations are assumed correct but in fact have errors. Generally, there exist two classes of methods that can mitigate the effects of the uncertainties in sensor location. The first class of methods is to incorporate the statistical characteristic of sensor location errors into the position estimation procedure [5, 6, 17, 21, 24, 26, 27], and the second one performs joint estimation of the unknown source locations and the inaccurate sensor positions together [2, 7, 12, 14, 16, 20, 23, 28, 31]. In this work, we concentrate on the latter because it can increase the accuracy of the sensor position estimates and tolerate higher noise level before the thresholding effect caused by nonlinear estimation starts to occur.

It is well known that the TLS technique is an improved least squares (LS) method to solve an overdetermined set of linear equations when there are errors not only in the observations but in the coefficient matrix as well. In [1, 11, 22], the TLS method is applied to source localization. Note that the TLS solution can be found simply via the singular value decomposition (SVD) technique [35], and therefore, it is computationally attractive. However, the TLS estimator is generally not asymptotically efficient because it assumes that the noise components in and are independent and identically distributed (i.i.d.), which is rarely realistic in practical scenario. The CTLS method, as a natural extension of the TLS method, is able to fully exploit the noise structure in and [36], and hence, the resulting solution is shown to achieve the Cramér-Rao bound (CRB). Indeed, the CTLS method has been successfully applied to wireless location. In [3], the CTLS algorithm is proposed to solve the bearing-only localization problem. In [15], the CTLS localization algorithm using TDOA measurements is presented. Additionally, an efficient CTLS algorithm for determining the position and velocity of a moving source based on TDOA and FDOA measurements is developed in [25]. However, it is noteworthy that none of these CTLS algorithms consider the sensor position errors, which may seriously deteriorate the positioning accuracy. In order to reduce the effects of the uncertainties in sensor positions, the robust CTLS localization algorithms are presented in [6, 26, 27].

While the above-mentioned CTLS algorithms can achieve satisfactory performance, it is necessary to point out that all of them apply only to the single-source scenario, and moreover none of them can provide the joint estimation of source and sensor locations. Furthermore, all the algorithm derivations and theoretical analysis are performed only for some specific measurements, thus leading to the lack of a united framework for this problem. This paper presents an efficient CTLS method that can locate multiple disjoint sources and refine the erroneous sensor positions simultaneously. Different from the existing approaches, the proposed method is derived in a more general framework that is applicable to many different location measurements. First, a modified CTLS optimization problem is formulated after some algebraic manipulations and then the corresponding Newton iterative algorithm is derived to yield the numerical solution. Subsequently, by exploiting the first-order perturbation analysis, the closed-form expression for the covariance matrix of the new CTLS estimator is deduced under the Gaussian assumption. Moreover, the estimation accuracy of the CTLS solution is rigorously proved to reach the CRB before the thresholding effect occurs. Finally, we give two examples to illustrate how to utilize the proposed CTLS method for source localization. One uses the TDOAs/GROAs measurements and the other is based on the TOAs/FOAs parameters. The experiment results support the theoretical development in this paper.

The remainder of this paper is organized as follows. In Section 2, the measurement model for source localization is described and the problem under investigation is formulated. Section 3 derives the modified CTLS optimization model. In Section 4, the Newton iterative algorithm is derived to provide the joint estimation of source and sensor locations. Section 5 provides the closed-form expression for the covariance matrix of the new CTLS estimator and proves its asymptotical efficiency. In Section 6, two examples are given to illustrate how to utilize the proposed CTLS method for source localization. Numerical simulations are presented in Section 7 to support the theoretical development in this paper. Conclusions are drawn in Section 8. The proofs of the main results are shown in the Appendixes (available here).

2. Measurement Model and Problem Formulation

2.1. Nonlinear Measurement Model

We consider the localization scenario where disjoint sources are to be located. Under ideal condition, the location observation equation associated with the th source can be represented in a generic form aswhere is the true measurement vector, is the position and/or velocity vector of the th source, denotes the system parameter which contains the sensor positions and/or velocities, and is the nonlinear function that depends on the specific measurement type used.

Note that if the vectors and can be accurately obtained, the localization problem is equivalent to solving a set of nonlinear equations. However, and are not known exactly in practice. First, only the noisy version of , denoted as , is available. It can be written aswhere is the measurement noise vector that follows zero-mean Gaussian distribution with covariance matrix . In addition, the known system parameter, denoted as , is also erroneous. It can be modeled aswhere is the noise vector and it is Gaussian distributed with zero-mean and covariance matrix . Besides, and are statistically independent.

2.2. Pseudo-Linear Measurement Model

For some special measurements (e.g., DOA, TOA, TDOA, and GROA), (1) can be transformed into the following pseudo-linear model:where is the pseudo-linear measurement vector and is the coefficient matrix, . Vector function is given bywhere is a known and constant matrix and comprises all the instrumental variables whose number is defined by .

Since every equation in (4) is related to the system parameter , we must combine these equations to perform joint estimation of all the position vectors and the system parameter . In this treatment, we can obtain cooperation gain compared to the approaches which locate the sources individually.

Putting all the equations in (4) together yieldswhereIt is obvious from (7) that vector contains the location vectors of all the emitters. In addition, it can also been seen from (7) that vector comprises the measurement vectors of all the sources. The noisy version of is denoted by , which can be expressed aswhereIt can be readily seen from (9) that the noise vector follows zero-mean Gaussian distribution. Its covariance matrix is defined by . If and are statistically independent for , then we have .

The positioning problem here can be briefly stated as follows: Given the observation vectors and available system parameter , find an estimate of (or ) and as accurate as possible based on the pseudo-linear equation (6).

3. Optimization Model

In (6), the functional forms of and are known, but vectors and are not available and only their noisy values (i.e., and ) can be obtained. In order to establish the CTLS optimization model, we shall perform a first-order Taylor series expansion of and around as well as . It can be verified that whereInserting (10) into (6) leads towhere

Note that the problem addressed herein is the joint estimation of and . Therefore, it is necessary to define an augmented parameter vector as below:Then, by combining (3) and (12), we can get the following programming model:Although (15) has equality constraint, it can be converted into an unconstrained minimization problem over and . The details can be found in the following proposition.

Proposition 1. If is an invertible matrix, then the constrained optimization problem (15) can be recast as an equivalent unconstrained one, which is expressed aswhere

Proof. Define and , and then (15) is equivalent toThe optimal solution to (19) is given bywhere represents the Moore-Penrose inverse. Since is invertible, it can be checked that has full row rank, which leads toCombining (20) and (21) yieldswhich, combined with (19), proves the proposition.

We would like to emphasize that (16) is the CTLS optimization model to jointly estimate source position and system parameter simultaneously. Moreover, it is a generic model that can be applied to many different location measurements. In the next section, the numerical algorithm to solve (16) is derived.

4. Numerical Algorithm

It is obvious that (16) is a nonlinear minimization problem. Therefore, the analytical solution is in general not available, and a numerical technique is required. One widely applied numerical method is Newton iteration, which has two-order convergence rate if the function to be minimized is twice differentiable. Note that, in each iteration step, the gradient and Hessian matrix of the object function must be computed. Hence, we need to derive the explicit expressions for the gradient and Hessian matrix.

For notational convenience, the cost function in (16) is rewritten aswhereFrom (23), the gradient of can be expressed aswhereApplying (25), the Hessian matrix of is given bywhere

in which ,  ,  , and . It follows from (26) thatIt is worth pointing out that all the quadratic terms of are ignored in (31) and (32). The reason is that these terms hardly affect the convergence rate and asymptotic performance of the CTLS method.

Based on the above discussion, the Newton iteration is given bywhere the subscript denotes the th iteration and is a suitable step size. Some remarks on the Newton iteration follow.

Remark 2. The initial value for the iteration can be obtained by the WLS or TLS methods, both of which can provide an approximate closed-form solution.

Remark 3. can be used as the prescribed convergence criterion.

Remark 4. Note that, in (29)–(32), there exist some matrices whose expressions are not yet specified. They includeThe exact expressions for the eight matrices in (34) are provided in Appendix  A.

Remark 5. Since the weighting matrix is updated at each iteration step, the proposed CTLS solution is able to yield much smaller estimation bias compared to the TLS and TWLS solutions, as discussed in Section 7.

5. Performance Analysis

In this section, the analytical expression for the covariance matrix of the above CTLS estimator is derived. Furthermore, the CTLS solution is proved theoretically to reach the CRB accuracy before the thresholding effect starts to take place.

5.1. Covariance Matrix of the CTLS Solution

Assuming the convergence results for the Newton iteration are denoted by and , it follows from the iteration termination criteria given in Remark 3 thatThe substitution of (25)-(26) into (35) leads to

Performing a first-order Taylor series expansion of around the true values and produceswhere and are estimation errors. Besides, and , whose expressions are given bywherein which and .

Substituting (37) into (36) and omitting the second- and higher-order error terms yieldswhereIt can be readily deduced from (40) thatThen, the covariance matrix of estimated vector is given by

5.2. Asymptotical Efficiency of the CTLS Estimator

In order to prove that the above CTLS solution is asymptotically efficient, it is necessary to obtain the corresponding CRB. According to [14, 16, 24, 28], we havewhere and . Using the definition of in (9), it is straightforward to show that where and .

By comparing (43) and (44), we get the following proposition.

Proposition 6. One has

Proof. First, combining (41) and the matrix inversion formulas leads toThe proof of (47) is provided in Appendix  B. The substitution of (47) into (43) leads towhereNext, putting into (4) producesDifferentiating both sides of (50) with respect to and , we havewherein whichFrom (11), (13), (53), and (54), it can be verified thatCombining the second equality in (7), the first equality in (38), the first equality in (45), and the first equality in (55) and (51) yieldsPutting the second equality in (7), the second equality in (38), the second equality in (45), and the second equality in (55) and (52) producesThe substitution of (56) and (57) into (49) leads towhich combined with (44) and (48) completes the proof.

Remark 7. It can be concluded from Proposition 6 that the proposed CTLS solution is able to achieve the CRB accuracy at moderate noise level before the thresholding effect due to nonlinear nature of the estimation problem occurs. Moreover, it is worth stressing that different from the existing theoretical analysis in the literature, the proof described above is independent on the specific measurement type used. In other words, it can be considered as a more general analysis framework, which is suitable for many different location measurements. Furthermore, the experiment results in Section 7 show that the proposed method can tolerate higher noise level compared to the other location methods.

Remark 8. It is known that the performance of TLS estimator can be dependent on how the origin of the coordinate system is selected, as shown in [37]. It is obvious that such dependence also exists for the proposed CTLS method. Indeed, it is a valuable research topic and is worthy of further investigation. However, it is not a trivial task because the mathematical analysis is rather elaborate. Hence, we can consider it as an open problem, which will be focused on in our future study.

6. Two Localization Examples

In this section, two localization scenarios are discussed in order to show how to exploit the proposed CTLS method to locate multiple disjoint sources and improve the sensor positions simultaneously.

6.1. Sources Localization Using TDOA and GROA Measurements

Assume that there are static and disjoint sources to be localized using a wireless location system that contains stationary sensors. The unknown source positions are denoted by . The exact sensor position is represented by , and then the system parameter is given by . As previously assumed, the system parameter available for processing, denoted by , has random errors. The TDOA and GROA measurements with respect to the reference sensor, say sensor 1, are exacted from the received signals. Note that the TDOA and GROA measurements are equivalent to the range difference and range ratio measurements, respectively. As a consequence, the corresponding observation equations are given byLet us define the following vectors:Then, the noiseless measurement vector related to the th source is given byThe collection of all the measurements forms the vector as follows:

In order to employ the proposed CTLS method for source localization, we must transform the nonlinear equations in (59) into the pseudo-linear ones by introducing some auxiliary variables. First, it follows from the first equation in (59) that