# An Efficient Estimator for Target Localization in a Multistation Redundancy System without Matrix Inversion

Authors: Xiwen Zhang; Jiyan Huang; Yiyuan Wang; Yineng Zhou
Journal: Journal of Sensors
 (2018)

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## Abstract

This paper focuses on target localization problem in a multistation redundancy system which finds broad applications in sonar, radar, and location-based service. Previous solutions can only be applied to the minimum system, such as the TOA method with three sensors and the AOA method with two sensors or need matrix inversion. To solve this problem, we propose a simple closed-form solution for a multistation redundancy localization system by using the estimation variance as the weighting coefficient to compute an average of each group’s localization result. The proposed method, with simple algebraic solution, requires no matrix inversion and can be used for low-cost hardware devices. We derive the method in TOA solution and AOA solution, respectively. The proposed method can also be extended to other locating technologies. Numerical examples are provided to illustrate the performance of the proposed method in root-mean-square error. The positioning accuracy of the proposed method is close to Cramér-Rao low bound both in TOA solution and AOA solution.


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## Body

## 1. Introduction

Target localization has a wide range of applications in sonar, radar, and location-based service [1]. It is often of concern for many scholars to enhance the localization accuracy and make the localization algorithm widely applicable to all kinds of situations [2]. A large variety of range-based localization methods have been proposed with respect to a set of sensors with known positions [3, 4]. Several range-based geolocation techniques have been used to estimate the target position including the time-of-arrival (TOA) [5–7], the time-difference-of-arrival (TDOA) [8, 9], the angle-of-arrival (AOA) [10–12], the received signal strength (RSS) [13–16] based techniques, or hybrid location techniques [17, 18]. Among these location techniques, TOA and AOA have attracted much attention because of their high positioning accuracy [16, 18]. This paper focuses on TOA and AOA location techniques.There are algebraic method and matrix method for the multistation localization system [5]. Algebraic methods mainly include the TOA method [19] which uses the circles produced by range measurements at other sensors to find the position of the transmitter at the intersection of circles in two dimensions or spheres in three dimensions and the AOA method [10] which calculates the target position from the point of intersection of bearing lines. It should be noted that algebraic methods are designed for the minimum location system, such as the TOA method with three anchor sensors, and the AOA method with two anchor sensors. The main advantage of algebraic methods is that they can provide a simple algebraic solution without the computation of matrix inversion which makes those methods can be applied in a low-cost location system especially for wireless sensor networks (WSNs) [5, 20]. Since the algebraic methods are based on local information of a minimum system, the direct extension for those methods to a multistation redundancy system may not obtain optimal performance since those solutions cannot make use of extra measurements to improve position accuracy [8].To improve the localization accuracy, matrix methods [5, 6, 8, 11, 21–25] based on global information of measurements from all sensors were developed for a multistation redundancy system. The closed-form solution which requires solving a set of linear equations including all the measurements was given by Friedlander [21]. In order to obtain optimal estimation, Foy proposed an iterative localization method [26] based on Taylor series expansion. It starts with an initial guess and improves the estimate at each step by determining the local linear least-squares (LS) solution. Moreover, a proportion of solutions concentrates on transforming the localization problem into a manageable form. The LS method [6] based on TOA measurements is provided by Caffery. It uses the least square procedure to solve geometric equations which represent the lines of positions. Chan’s method [8], using a two-step weighted LS approach, provides higher accuracy which is close to CRLB when the number of sensors is greater than 3. Since the above matrix methods are based on global information and consider the relationships among all of measurements, they have better performance than the algebraic method and may achieve the CRLB. Unfortunately, the high computational burden caused by matrix inversion makes it hard for matrix methods to be applied for a low-cost location system such as WSNs with several Mhz oscillators [20, 27].It is necessary to develop a location scheme with high positioning accuracy for a low-cost location system. In this paper, a novel and efficient estimator is designed for a low-cost location system. The proposed method first divides all of anchor sensors into several groups. Each group is a minimum location system and can provide a position estimate using an algebraic method. Subsequently, theoretical variances of position estimates for every group are derived and are used as weights to obtain the final solution. The proposed method utilizes groups to avoid matrix inversion and uses theoretical variances as weights to improve the system performance. In this paper, the proposed method is applied in TOA and AOA location techniques. Moreover, the theoretical variance of the proposed method is also derived for performance analysis. Simulation results verify the proposed method. Compared with current studies, the proposed method not only provides a simple algebraic solution without matrix inversion but also has the similar performance with the matrix method.To illustrate, the paper is organized as follows. Section2 formulates the system model of the localization problem in a multistation redundancy system using TOA measurements and AOA measurements in a 2-D plane, respectively. The previous method is described and analyzed. Section 3 presents the proposed method in TOA solution and AOA solution, respectively. The performance of the proposed method is analyzed by deriving the theoretical estimate variance in Section 4. Simulation results are presented in Section 5 to illustrate the localization accuracy compared with CRLB and other methods. Section 6 concludes this paper.


## 2. Problem Formulation

The target localization problem in a multistation redundancy system which containsN anchor sensors is depicted in Figure 1. For notational convenience and ease of illustration, we model the problem in a 2-D plane. Extension to three dimensions is straightforward. Assume that N anchor sensors are located at known positions xi,yi,i=1,2,…,N and the unknown target position is x,y. The objective of the target localization is to identify the target location from the known observers’ positions and the measurements, such as TOAs and AOAs.Figure 1
Localization in a multistation redundancy system.Assume that TOA and AOA measurements have already been measured as shown in Figure1. The measured distance between the target and ith sensor is ri and the estimation of azimuth for the target is αi. Then, we can use these measurements to establish TOA equations [28]
(1)ri2=xi−x2+yi−y2,i=1,2,…,N,and AOA equations [10]
(2)y−yi=tanαix−xi,i=1,2,…,N.Letki=xi2+yi2, and r=x2+y2. So that (1) can be written as
(3)ri2−ki=−2xix−2yiy+r,i=1,2,…,N,which is a set of linear equations with unknowns x, y, and r. From (2) we can obtain
(4)sinαix−cosαiy=sinαixi−cosαiyi,i=1,2,…,N.Since TOA and AOA measurements have error, which consists of the true measurements corrupted by additive independent zero-mean Gaussian noise. So with TOA and AOA noise, the error vector [11, 28] derived from (3) and (4) is
(5)φ=h−Gzwhere
(6)h=r12−k1r22−k2⋮rN2−kN,G=−2x1−2y11−2x2−2y21⋯⋯⋯−2xN−2yN1,z=xyr,in TOA case,
(7)h=sinα1x1−cosα1y1sinα2x2−cosα2y2⋮sinαNxN−cosαNyN,G=sinα1−cosα1sinα2−cosα2……sinαN−cosαN,z=xy,in AOA case.Most of the previous methods [5, 6, 8, 11, 23, 25] are based on the least square procedure. Thus, the source position is computed by
(8)z=GTG−1GTh,or some improvements of it, such as the two-step weighted LS method (Chan’s method) [8]. These methods require matrix inversion that leads to a high requirement for hardware devices. However, low-cost hardware devices, such as some embedded systems, may not support matrix inversion. Hence, it is difficult for these methods to be applied to a low-cost location system. A more widely adapted method needs to be developed to address this situation.


## 3. The Proposed Method

In order to enable low-cost hardware devices to achieve high-precision positioning, we propose a simple closed-form solution for the multistation localization system. Without matrix inversion, the proposed method provides an algebraic solution which can be effectively applied to low-cost hardware devices.We develop the approach by using theoretical variances as weighting coefficients to compute an average of each group’s localization result. As shown in Figure2, first, the anchor sensors are divided into several groups and each group of sensors are used to estimate the target position, respectively. Then, theoretical variances of the position estimates in each group are calculated and regarded as weighted values. Finally, the weighted average of the estimates is obtained as final solution.Figure 2
The process of the proposed method.TOA and AOA measurements have already been measured, and the specific derivations are as follows.


### 3.1. TOA Solution

Assume that there areN anchor sensors distributed arbitrarily in a 2-D plane. We divide these sensors into M groups. Each group has 3 sensors, which are the minimum number of base stations required for the TOA location algorithm. Note that some sensors may be repeated in different groups. In the ith group, the mth observation station is at known position xim,yim and the distance measured between the target and mth sensor is d^im, for i=1,2,…,M.In each group, the localization scenario consists of a source and three observation stations. Assume that the unknown source position is located at (x, y). Then, the TOA measurement equations [19] in i-th group are given by
(9)dim=xim−x2+yim−y2,m=1,2,3,where dim is the true distance between the target and mth sensor.Transform (9) with respect to the first receiver, we obtain
(10)xi2−xi1x+yi2−yi1y=12xi22+yi22−xi12+yi12+di12−di22,(11)xi3−xi1x+yi3−yi1y=12xi32+yi32−xi12+yi12+di12−di32.Using (10) and (11), we obtain
(12)x=yi2−yi1ci3−yi3−yi1ci2xi3−xi1yi2−yi1−xi2−xi1yi3−yi1,(13)y=xi2−xi1ci3−xi3−xi1ci2yi3−yi1xi2−xi1−yi2−yi1xi3−xi1.Since TOA measurements have errors, which consist of the true TOAs corrupted by additive independent zero-mean Gaussian noise, we actually model distance estimates associated with receiverm as
(14)d^im=d^im+nim,m=1,2,3,where nim~N0,σ2 and m=1,2,3.Letx^i and y^i represent the position estimate coordinates in each group, where subscript i denotes the ith group. Then, the location estimate is
(15)x^i=yi2−yi1c^i3−yi3−yi1c^i2xi3−xi1yi2−yi1−xi2−xi1yi3−yi1,y^i=xi2−xi1c^i3−xi3−xi1c^i2yi3−yi1xi2−xi1−yi2−yi1xi3−xi1,where c^i2=1/2xi22+yi22−xi12+yi12+d^i12−d^i22 and c^i3=1/2xi32+yi32−xi12+yi12+d^i12−d^i32.With 3 sensors, the target position can be solved using (15) in each group. We compute theoretical variance by using the differential method [8]. From (12), the differential of xi is
(16)Δxi=ai3Δci3−ai2Δci2,where
(17)ai3=yi2−y1ixi3−xi1yi2−yi1−xi2−xi1yi3−yi1,ai2=yi3−yi1xi3−xi1yi2−yi1−xi2−xi1yi3−yi1,Δci2=di1Δd1i−di2Δdi2, and Δci3=di1Δdi1−di3Δdi3. The differential of yi is
(18)Δyi=bi3Δci3−bi2Δci2,where
(19)bi3=xi2−xi1yi3−yi1xi2−xi1−yi2−yi1xi3−xi1,bi2=xi3−xi1yi3−yi1xi2−xi1−yi2−yi1xi3−xi1.For simplicity, we assume that the signal and noises are mutually independent, zero-mean stationary Gaussian random processes. If the distance estimation variance isσ2, then the theoretical variance of xi is
(20)varxi=EΔxi2=ai32EΔci32−2ai3ai2EΔci2Δci3+ai22Δci22=ai3−ai22di12σ2+ai32di32σ2+ai22di22σ2.The theoretical variance ofyi is
(21)varyi=EΔyi2=bi32EΔci32−2bi3bi2EΔci2Δci3+bi22Δci22=bi3−bi22di12σ2+bi32di32σ2+bi22di22σ2.The variance weighted average TOA solution is given by(22)x^wa_TOA=∑i=1Mx^ivarxi−1∑i=1Mvarxi−1,(23)y^wa_TOA=∑i=1My^ivaryi−1∑i=1Mvaryi−1.From (20) and (21), it is clear that the variance is related to the measurement error, the distance between the sensor and target and the baseline. When measurement error or the distance between the sensor and target increases, the variance increases. While when the baseline becomes longer, the variance decreases. Hence, for the proposed method, it is better to let the target be in the central range of the array and divide sensors with uniform extraction.


### 3.2. AOA Solution

Assume that there areN anchor sensors distributed arbitrarily in a 2-D plane. We divide these sensors into M groups. Each group has 2 sensors, which is the minimum number of sensors required for the AOA location algorithm. Note that some sensors may be repeated in different groups. In the ith group, the mth sensor is located at known position xim,yim and the measured angles are α^im, m=1,2, for i=1,2,…,M. The unknown source position is located at (x, y).Inith group, the true AOAs are given by [10]
(24)y−yim=tanαimx−xim,m=1,2.Using (24), we obtain
(25)x=yi2−yi1+xi1tanαi1−xi2tanαi2tanαi1−tanαi2,y=yi2tanαi1−yi1tanαi2+xi1−xi2tanαi1tanαi2tanαi1−tanαi2.The AOA measurements are subject to independent zero-mean Gaussian noise(26)α^im=αim+nim,m=1,2,where nim~N0,σ2 and m=1,2. Let x^i and y^i represent the position estimate coordinates in each group, where subscript i denotes the ith group, then the position estimate is
(27)x^i=yi2−yi1+xi1tanα^i1−xi2tanα^i2tanα^i1−tanα^i2,y^i=yi2tanα^i1−yi1tanα^i2+xi1−xi2tanα^i1tanα^i2tanα^i1−tanα^i2.With 2 sensors, the target position can be solved using (27) in each group. We compute theoretical variance by using the differential method [8]. The differential of xi is
(28)Δxi=xi1Δti1−xi2Δti2Pi1−Pi3Δti1−Δti2Pi12,where
(29)Δti1=Δαi1cosαi12,Δti2=Δαi2cosαi22,Pi1=tanαi1−tanαi2, and Pi3=yi2−yi1+xi1tanαi1−xi2tanαi2. The differential of yi is
(30)Δyi=yi2Δti1−yi1Δti2+xi1−xi2Δti1ti2+ti1Δti2Pi1−Pi2Δti1−Δti2Pi12,where ti1=tanαi1, ti2=tanαi2, and Pi2=yi2tanαi1−yi1tanαi2+xi1−xi2tanαi1tanαi2. If the angle estimation variance is σ2, then the theoretical variance of xi is
(31)varxi=EΔxi2=Pi12xi12si1+xi22si2−2xi1si1+2xi2si2Pi1Pi3+Pi32si1+si2Pi14,where
(32)Si1=EΔti12=σ2cosαi14,Si2=EΔti22=σ2cosαi24.The theoretical variance of y is
(33)varyi=EΔyi2=Pi12yi22si1+yi12si2+xi1−xi22ti22si1+ti12si2Pi14+2Pi12yi2ti2si1−yi1ti1si2xi1−xi2+Pi22si1+si2−2Pi1Pi2yi2si1+yi1si2+xi1−xi2ti2si1−ti1si2Pi14.The variance weighted average AOA solution is given by(34)x^wa_AOA=∑i=1Mx^ivarxi−1∑i=1Mvarxi−1,(35)y^wa_AOA=∑i=1My^ivaryi−1∑i=1Mvaryi−1.Through the analysis of (31) and (33), we can obtain that when measurement error or the distance between the sensor and target increases, the variance increases. These conclusions are similar to those in TOA solution. Yet the variance increases if the baseline gets longer. Therefore, we would better group sensors with sequential extraction to get a smaller baseline in AOA solution.


## 3.1. TOA Solution

Assume that there areN anchor sensors distributed arbitrarily in a 2-D plane. We divide these sensors into M groups. Each group has 3 sensors, which are the minimum number of base stations required for the TOA location algorithm. Note that some sensors may be repeated in different groups. In the ith group, the mth observation station is at known position xim,yim and the distance measured between the target and mth sensor is d^im, for i=1,2,…,M.In each group, the localization scenario consists of a source and three observation stations. Assume that the unknown source position is located at (x, y). Then, the TOA measurement equations [19] in i-th group are given by
(9)dim=xim−x2+yim−y2,m=1,2,3,where dim is the true distance between the target and mth sensor.Transform (9) with respect to the first receiver, we obtain
(10)xi2−xi1x+yi2−yi1y=12xi22+yi22−xi12+yi12+di12−di22,(11)xi3−xi1x+yi3−yi1y=12xi32+yi32−xi12+yi12+di12−di32.Using (10) and (11), we obtain
(12)x=yi2−yi1ci3−yi3−yi1ci2xi3−xi1yi2−yi1−xi2−xi1yi3−yi1,(13)y=xi2−xi1ci3−xi3−xi1ci2yi3−yi1xi2−xi1−yi2−yi1xi3−xi1.Since TOA measurements have errors, which consist of the true TOAs corrupted by additive independent zero-mean Gaussian noise, we actually model distance estimates associated with receiverm as
(14)d^im=d^im+nim,m=1,2,3,where nim~N0,σ2 and m=1,2,3.Letx^i and y^i represent the position estimate coordinates in each group, where subscript i denotes the ith group. Then, the location estimate is
(15)x^i=yi2−yi1c^i3−yi3−yi1c^i2xi3−xi1yi2−yi1−xi2−xi1yi3−yi1,y^i=xi2−xi1c^i3−xi3−xi1c^i2yi3−yi1xi2−xi1−yi2−yi1xi3−xi1,where c^i2=1/2xi22+yi22−xi12+yi12+d^i12−d^i22 and c^i3=1/2xi32+yi32−xi12+yi12+d^i12−d^i32.With 3 sensors, the target position can be solved using (15) in each group. We compute theoretical variance by using the differential method [8]. From (12), the differential of xi is
(16)Δxi=ai3Δci3−ai2Δci2,where
(17)ai3=yi2−y1ixi3−xi1yi2−yi1−xi2−xi1yi3−yi1,ai2=yi3−yi1xi3−xi1yi2−yi1−xi2−xi1yi3−yi1,Δci2=di1Δd1i−di2Δdi2, and Δci3=di1Δdi1−di3Δdi3. The differential of yi is
(18)Δyi=bi3Δci3−bi2Δci2,where
(19)bi3=xi2−xi1yi3−yi1xi2−xi1−yi2−yi1xi3−xi1,bi2=xi3−xi1yi3−yi1xi2−xi1−yi2−yi1xi3−xi1.For simplicity, we assume that the signal and noises are mutually independent, zero-mean stationary Gaussian random processes. If the distance estimation variance isσ2, then the theoretical variance of xi is
(20)varxi=EΔxi2=ai32EΔci32−2ai3ai2EΔci2Δci3+ai22Δci22=ai3−ai22di12σ2+ai32di32σ2+ai22di22σ2.The theoretical variance ofyi is
(21)varyi=EΔyi2=bi32EΔci32−2bi3bi2EΔci2Δci3+bi22Δci22=bi3−bi22di12σ2+bi32di32σ2+bi22di22σ2.The variance weighted average TOA solution is given by(22)x^wa_TOA=∑i=1Mx^ivarxi−1∑i=1Mvarxi−1,(23)y^wa_TOA=∑i=1My^ivaryi−1∑i=1Mvaryi−1.From (20) and (21), it is clear that the variance is related to the measurement error, the distance between the sensor and target and the baseline. When measurement error or the distance between the sensor and target increases, the variance increases. While when the baseline becomes longer, the variance decreases. Hence, for the proposed method, it is better to let the target be in the central range of the array and divide sensors with uniform extraction.


## 3.2. AOA Solution

Assume that there areN anchor sensors distributed arbitrarily in a 2-D plane. We divide these sensors into M groups. Each group has 2 sensors, which is the minimum number of sensors required for the AOA location algorithm. Note that some sensors may be repeated in different groups. In the ith group, the mth sensor is located at known position xim,yim and the measured angles are α^im, m=1,2, for i=1,2,…,M. The unknown source position is located at (x, y).Inith group, the true AOAs are given by [10]
(24)y−yim=tanαimx−xim,m=1,2.Using (24), we obtain
(25)x=yi2−yi1+xi1tanαi1−xi2tanαi2tanαi1−tanαi2,y=yi2tanαi1−yi1tanαi2+xi1−xi2tanαi1tanαi2tanαi1−tanαi2.The AOA measurements are subject to independent zero-mean Gaussian noise(26)α^im=αim+nim,m=1,2,where nim~N0,σ2 and m=1,2. Let x^i and y^i represent the position estimate coordinates in each group, where subscript i denotes the ith group, then the position estimate is
(27)x^i=yi2−yi1+xi1tanα^i1−xi2tanα^i2tanα^i1−tanα^i2,y^i=yi2tanα^i1−yi1tanα^i2+xi1−xi2tanα^i1tanα^i2tanα^i1−tanα^i2.With 2 sensors, the target position can be solved using (27) in each group. We compute theoretical variance by using the differential method [8]. The differential of xi is
(28)Δxi=xi1Δti1−xi2Δti2Pi1−Pi3Δti1−Δti2Pi12,where
(29)Δti1=Δαi1cosαi12,Δti2=Δαi2cosαi22,Pi1=tanαi1−tanαi2, and Pi3=yi2−yi1+xi1tanαi1−xi2tanαi2. The differential of yi is
(30)Δyi=yi2Δti1−yi1Δti2+xi1−xi2Δti1ti2+ti1Δti2Pi1−Pi2Δti1−Δti2Pi12,where ti1=tanαi1, ti2=tanαi2, and Pi2=yi2tanαi1−yi1tanαi2+xi1−xi2tanαi1tanαi2. If the angle estimation variance is σ2, then the theoretical variance of xi is
(31)varxi=EΔxi2=Pi12xi12si1+xi22si2−2xi1si1+2xi2si2Pi1Pi3+Pi32si1+si2Pi14,where
(32)Si1=EΔti12=σ2cosαi14,Si2=EΔti22=σ2cosαi24.The theoretical variance of y is
(33)varyi=EΔyi2=Pi12yi22si1+yi12si2+xi1−xi22ti22si1+ti12si2Pi14+2Pi12yi2ti2si1−yi1ti1si2xi1−xi2+Pi22si1+si2−2Pi1Pi2yi2si1+yi1si2+xi1−xi2ti2si1−ti1si2Pi14.The variance weighted average AOA solution is given by(34)x^wa_AOA=∑i=1Mx^ivarxi−1∑i=1Mvarxi−1,(35)y^wa_AOA=∑i=1My^ivaryi−1∑i=1Mvaryi−1.Through the analysis of (31) and (33), we can obtain that when measurement error or the distance between the sensor and target increases, the variance increases. These conclusions are similar to those in TOA solution. Yet the variance increases if the baseline gets longer. Therefore, we would better group sensors with sequential extraction to get a smaller baseline in AOA solution.


## 4. Performance Analysis

In location estimation, there are two main different values that we can use to evaluate the performance of localization algorithms. One is theoretical variance, and the other one is CRLB [29, 30]. Theoretical variance is used to evaluate the practical performance of localization methods, while CRLB establishes a lower bound for any unbiased estimator. Since CRLB has already been derived in [29, 30], here, we derive the theoretical estimate variance of the proposed method in this section to illustrate its performance.


### 4.1. Theoretical Variance of TOA Solution

Assume that the variance of range measurements isσ2. From (22), we can obtain the differential of xwa_TOA:
(36)Δxwa_TOA=∑i=1MΔxivi+xiΔvi∑i=1Mvi−∑i=1Mxivi∑i=1MΔvi∑i=1Mvi2,where vi=varxi−1. Using (16) and (20), Δxi and Δvi can be written as
(37)Δxi=ai3−ai2di1Δdi1+ai2di2Δdi2−ai3di3Δdi3,(38)Δvi=ti1Δdi1+ti2Δdi2+ti3Δdi3,where ti1=−2varxi−2ai3−ai22di1σ2, ti2=−2varxi−2ai22di2σ2, and ti3=−2varxi−2ai32di3σ2. Then, we have
(39)Δxivi+xiΔvi=ki1Δdi1+ki2Δdi2+ki3Δdi3,where ki1=viai3−ai2di1+xiti1, ki2=viai2di2+xiti2, and ki3=−viai3di3+xiti3.Substituting (37), (38), and (39) into (36), the theoretical estimate variance of xwa_TOA is
(40)varxwa_TOA=EΔxwa_TOA2=σ2∑i=1Mvi2∑i=1Mki12+ki22+ki32+σ2∑i=1Mxivi2∑i=1Mti12+ti22+ti32∑i=1Mvi4−2σ2∑i=1Mvi∑i=1Mxivi∑i=1Mki1ti1+ki2ti2+ki3ti3∑i=1Mvi4.With the similar derivation, the theoretical estimate variance ofywa_TOA is
(41)varywa_TOA=EΔywa_TOA2=σ2∑i=1Mwi2∑i=1Mji12+ji22+ji32+σ2∑i=1Myiwi2∑i=1Msi12+si22+si32∑i=1Mwi4−2σ2∑i=1Mwi∑i=1Myiwi∑i=1Mji1si1+ji2si2+ji3si3∑i=1Mwi4,where wi=varyi−1, si1=−2varyi−2bi3−bi22di1σ2, si2=−2varyi−2bi22di2σ2, si3=−2varyi−2bi32di3σ2, ji1=wibi3−bi2di1+yisi1, ji2=wibi2di2+yisi2, and ji3=−wibi3di3+yisi3.The theoretical estimate variance of the proposed method is then obtained from (40) and (41) as
(42)varwa_TOA=varxwa_TOA+varywa_TOA.Figure3 depicts the theoretical variance of the proposed method in the TOA case as the target position varies, where the sensor positions are x1=0,y1=1000, x2=1000sin40°,y2=1000cos40°, x3=1000sin80°,y3=1000cos80°, x4=1000sin60°,y4=−1000cos60°, x5=1000sin20°,y5=−1000cos20°, x6=−1000sin20°,y6=−1000cos20°, x7=−1000sin60°,y7=−1000cos60°, x8=−1000sin80°,y8=1000cos80°, and x9=−1000sin40°,y9=1000cos40° and σTOA is fixed at 50. It shows that the estimation is more accurate when the target gets closer to the central of the array range.Figure 3
The theoretical variance of the proposed method in TOA case as the target position varies.


### 4.2. Theoretical Variance of AOA Solution

Assume that the variance of angle measurements isσ2. From (28), Δxi can be written as
(43)Δxi=Ri1Δαi1+Ri2Δαi2,where
(44)Ri1=xi1/cosαi12tanαi1−tanαi2−yi2−yi1+xi1tanαi1−xi2tanαi21/cosαi12tanαi1−tanαi22,Ri2=−xi2/cosαi22tanαi1−tanαi2+yi2−yi1+xi1tanαi1−xi2tanαi21/cosαi22tanαi1−tanαi22.From (34), we can obtain the differential of xwa_AOA
(45)Δxwa_AOA=∑i=1MΔxivi+xiΔvi∑i=1Mvi−∑i=1Mxivi∑i=1MΔvi∑i=1Mvi2,where vi=varxi−1. Using (31), Δvi can be written as
(46)Δvi=Ci1Δαi1+Ci2Δαi2,where
(47)Ci1=−varxi−2⋅1Pi18⋅Qi1cosαi12+xi1Qi2cosαi12−4σ2Qi3cosαi15,Ci2=−varxi−2⋅1Pi18⋅−Qi1cosαi22−xi2Qi2cosαi22−4σ2Qi4cosαi25,Qi1=Pi14xi12si1+xi22si2−2xi1si1+2xi2si2Pi3−4LiPi13,Qi2=Pi14−2xi1si1+2xi2si2Pi1+2Pi3si1+si2,Qi3=Pi14Pi1xi12−2xi1Pi1Pi3+Pi32,Qi4=Pi14Pi1xi22−2xi2Pi1Pi3+Pi32,Li=Pi1xi12si1+xi22si2−2xi1si1+2xi2si2Pi1Pi3+Pi32si1+si2.Then, we have
(48)Δxivi+xiΔvi=Bi1Δαi1+Bi2Δαi2,where Bi1=viRi1+xiCi1 and Bi2=viRi2+xiCi2.Substituting (43), (46), and (48) into (45), the theoretical estimate variance of xwa_AOA is
(49)varxwa_AOA=EΔxwa_AOA2=σ2∑i=1Mvi2∑i=1MBi12+Bi22+σ2∑i=1Mxivi2∑i=1MCi12+Ci22−2σ2∑i=1Mvi∑i=1Mxivi∑i=1MBi1Ci1+Bi2Ci2∑i=1Mvi4.With the similar derivation, the theoretical estimate variance ofywa_AOA is
(50)varywa_AOA=EΔywa_AOA2=σ2∑i=1Mwi2∑i=1MGi12+Gi22+σ2∑i=1Myiwi2∑i=1MEi12+Ei22∑i=1Mwi4−2σ2∑i=1Mwi∑i=1Myiwi∑i=1MGi1Ei1+Gi2Ei2∑i=1Mwi4,where
(51)wi=varyi−1,Gi1=wiJi1+yiEi1,Gi2=wiJi2+yiEi2,Ei1=−varyi−2⋅1Pi18⋅fi1cosαi12+fi2yi2cosαi12+fi2xi1−xi2tanαi2cosαi12+4σ2fi3cosαi15sinαi1+fi5cosαi12,Ei2=−varyi−2⋅1Pi18⋅fi1cosαi22−fi2yi2cosαi22+fi2xi1−xi2tanαi1cosαi22+4σ2fi4cosαi25sinαi1+fi6cosαi22,fi1=Pi142Pi1qi1−2Pi2qi2−4qi3Pi13,fi2=Pi142Pi2si1+si2−2Pi1qi2,fi3=Pi14Pi12yi22+xi1−xi22ti22+2xi1−xi2yi2ti2+Pi22−2Pi1Pi2yi2+xi1−xi2ti2,fi4=Pi14Pi12yi12+xi1−xi22ti12−2xi1−xi2yi1ti1+Pi22−2Pi1Pi2yi1−xi1−xi2ti1,fi5=Pi14Pi122xi1−xi22ti1Si2−2xi1−xi2yi1Si2−2Pi1Pi2−xi1−xi2Si2,fi6=Pi14Pi122xi1−xi22ti2Si1+2xi1−xi2yi2Si1−2Pi1Pi2xi1−xi2Si1,qi1=yi22si1+yi12si2+xi1−xi22ti22si1+ti12si2+2yi2ti2si1−yi1ti1si2xi1−xi2,qi2=yi2si1+yi1si2+xi1−xi2ti2si1−ti1si2,qi3=Pi12yi22si1+yi12si2+xi1−xi22ti22si1+ti12si2+2Pi12yi2ti2si1−yi1ti1si2xi1−xi2+Pi22si1+si2−2Pi1Pi2yi2si1+yi1si2+xi1−xi2ti2si1−ti1si2,Ji1=yi2/cosαi12+xi1−xi2tanαi2/cosαi12Pi1−Pi2/cosαi12Pi12,Ji2=−yi1/cosαi22+xi1−xi2tanαi1/cosαi22Pi1+Pi2/cosαi22Pi12.The theoretical estimate variance of the proposed method is then obtained from (49) and (50) as
(52)varwa_AOA=varxwa_AOA+varywa_AOA.The theoretical variance of AOA solution is illustrated in Figure4 as the target position varies, where the sensor positions are x1=−1000,y1=1000, x2=1000,y2=1000, x3=1000,y3=−1000, and x4=−1000,y4=−1000 and σAOA is fixed at 3°. The result is similar to TOA solution that the proposed method can give a more accurate estimation if the target gets closer to the central of the array range.Figure 4
The theoretical variance of the proposed method in AOA case as the target position varies.


## 4.1. Theoretical Variance of TOA Solution

Assume that the variance of range measurements isσ2. From (22), we can obtain the differential of xwa_TOA:
(36)Δxwa_TOA=∑i=1MΔxivi+xiΔvi∑i=1Mvi−∑i=1Mxivi∑i=1MΔvi∑i=1Mvi2,where vi=varxi−1. Using (16) and (20), Δxi and Δvi can be written as
(37)Δxi=ai3−ai2di1Δdi1+ai2di2Δdi2−ai3di3Δdi3,(38)Δvi=ti1Δdi1+ti2Δdi2+ti3Δdi3,where ti1=−2varxi−2ai3−ai22di1σ2, ti2=−2varxi−2ai22di2σ2, and ti3=−2varxi−2ai32di3σ2. Then, we have
(39)Δxivi+xiΔvi=ki1Δdi1+ki2Δdi2+ki3Δdi3,where ki1=viai3−ai2di1+xiti1, ki2=viai2di2+xiti2, and ki3=−viai3di3+xiti3.Substituting (37), (38), and (39) into (36), the theoretical estimate variance of xwa_TOA is
(40)varxwa_TOA=EΔxwa_TOA2=σ2∑i=1Mvi2∑i=1Mki12+ki22+ki32+σ2∑i=1Mxivi2∑i=1Mti12+ti22+ti32∑i=1Mvi4−2σ2∑i=1Mvi∑i=1Mxivi∑i=1Mki1ti1+ki2ti2+ki3ti3∑i=1Mvi4.With the similar derivation, the theoretical estimate variance ofywa_TOA is
(41)varywa_TOA=EΔywa_TOA2=σ2∑i=1Mwi2∑i=1Mji12+ji22+ji32+σ2∑i=1Myiwi2∑i=1Msi12+si22+si32∑i=1Mwi4−2σ2∑i=1Mwi∑i=1Myiwi∑i=1Mji1si1+ji2si2+ji3si3∑i=1Mwi4,where wi=varyi−1, si1=−2varyi−2bi3−bi22di1σ2, si2=−2varyi−2bi22di2σ2, si3=−2varyi−2bi32di3σ2, ji1=wibi3−bi2di1+yisi1, ji2=wibi2di2+yisi2, and ji3=−wibi3di3+yisi3.The theoretical estimate variance of the proposed method is then obtained from (40) and (41) as
(42)varwa_TOA=varxwa_TOA+varywa_TOA.Figure3 depicts the theoretical variance of the proposed method in the TOA case as the target position varies, where the sensor positions are x1=0,y1=1000, x2=1000sin40°,y2=1000cos40°, x3=1000sin80°,y3=1000cos80°, x4=1000sin60°,y4=−1000cos60°, x5=1000sin20°,y5=−1000cos20°, x6=−1000sin20°,y6=−1000cos20°, x7=−1000sin60°,y7=−1000cos60°, x8=−1000sin80°,y8=1000cos80°, and x9=−1000sin40°,y9=1000cos40° and σTOA is fixed at 50. It shows that the estimation is more accurate when the target gets closer to the central of the array range.Figure 3
The theoretical variance of the proposed method in TOA case as the target position varies.


## 4.2. Theoretical Variance of AOA Solution

Assume that the variance of angle measurements isσ2. From (28), Δxi can be written as
(43)Δxi=Ri1Δαi1+Ri2Δαi2,where
(44)Ri1=xi1/cosαi12tanαi1−tanαi2−yi2−yi1+xi1tanαi1−xi2tanαi21/cosαi12tanαi1−tanαi22,Ri2=−xi2/cosαi22tanαi1−tanαi2+yi2−yi1+xi1tanαi1−xi2tanαi21/cosαi22tanαi1−tanαi22.From (34), we can obtain the differential of xwa_AOA
(45)Δxwa_AOA=∑i=1MΔxivi+xiΔvi∑i=1Mvi−∑i=1Mxivi∑i=1MΔvi∑i=1Mvi2,where vi=varxi−1. Using (31), Δvi can be written as
(46)Δvi=Ci1Δαi1+Ci2Δαi2,where
(47)Ci1=−varxi−2⋅1Pi18⋅Qi1cosαi12+xi1Qi2cosαi12−4σ2Qi3cosαi15,Ci2=−varxi−2⋅1Pi18⋅−Qi1cosαi22−xi2Qi2cosαi22−4σ2Qi4cosαi25,Qi1=Pi14xi12si1+xi22si2−2xi1si1+2xi2si2Pi3−4LiPi13,Qi2=Pi14−2xi1si1+2xi2si2Pi1+2Pi3si1+si2,Qi3=Pi14Pi1xi12−2xi1Pi1Pi3+Pi32,Qi4=Pi14Pi1xi22−2xi2Pi1Pi3+Pi32,Li=Pi1xi12si1+xi22si2−2xi1si1+2xi2si2Pi1Pi3+Pi32si1+si2.Then, we have
(48)Δxivi+xiΔvi=Bi1Δαi1+Bi2Δαi2,where Bi1=viRi1+xiCi1 and Bi2=viRi2+xiCi2.Substituting (43), (46), and (48) into (45), the theoretical estimate variance of xwa_AOA is
(49)varxwa_AOA=EΔxwa_AOA2=σ2∑i=1Mvi2∑i=1MBi12+Bi22+σ2∑i=1Mxivi2∑i=1MCi12+Ci22−2σ2∑i=1Mvi∑i=1Mxivi∑i=1MBi1Ci1+Bi2Ci2∑i=1Mvi4.With the similar derivation, the theoretical estimate variance ofywa_AOA is
(50)varywa_AOA=EΔywa_AOA2=σ2∑i=1Mwi2∑i=1MGi12+Gi22+σ2∑i=1Myiwi2∑i=1MEi12+Ei22∑i=1Mwi4−2σ2∑i=1Mwi∑i=1Myiwi∑i=1MGi1Ei1+Gi2Ei2∑i=1Mwi4,where
(51)wi=varyi−1,Gi1=wiJi1+yiEi1,Gi2=wiJi2+yiEi2,Ei1=−varyi−2⋅1Pi18⋅fi1cosαi12+fi2yi2cosαi12+fi2xi1−xi2tanαi2cosαi12+4σ2fi3cosαi15sinαi1+fi5cosαi12,Ei2=−varyi−2⋅1Pi18⋅fi1cosαi22−fi2yi2cosαi22+fi2xi1−xi2tanαi1cosαi22+4σ2fi4cosαi25sinαi1+fi6cosαi22,fi1=Pi142Pi1qi1−2Pi2qi2−4qi3Pi13,fi2=Pi142Pi2si1+si2−2Pi1qi2,fi3=Pi14Pi12yi22+xi1−xi22ti22+2xi1−xi2yi2ti2+Pi22−2Pi1Pi2yi2+xi1−xi2ti2,fi4=Pi14Pi12yi12+xi1−xi22ti12−2xi1−xi2yi1ti1+Pi22−2Pi1Pi2yi1−xi1−xi2ti1,fi5=Pi14Pi122xi1−xi22ti1Si2−2xi1−xi2yi1Si2−2Pi1Pi2−xi1−xi2Si2,fi6=Pi14Pi122xi1−xi22ti2Si1+2xi1−xi2yi2Si1−2Pi1Pi2xi1−xi2Si1,qi1=yi22si1+yi12si2+xi1−xi22ti22si1+ti12si2+2yi2ti2si1−yi1ti1si2xi1−xi2,qi2=yi2si1+yi1si2+xi1−xi2ti2si1−ti1si2,qi3=Pi12yi22si1+yi12si2+xi1−xi22ti22si1+ti12si2+2Pi12yi2ti2si1−yi1ti1si2xi1−xi2+Pi22si1+si2−2Pi1Pi2yi2si1+yi1si2+xi1−xi2ti2si1−ti1si2,Ji1=yi2/cosαi12+xi1−xi2tanαi2/cosαi12Pi1−Pi2/cosαi12Pi12,Ji2=−yi1/cosαi22+xi1−xi2tanαi1/cosαi22Pi1+Pi2/cosαi22Pi12.The theoretical estimate variance of the proposed method is then obtained from (49) and (50) as
(52)varwa_AOA=varxwa_AOA+varywa_AOA.The theoretical variance of AOA solution is illustrated in Figure4 as the target position varies, where the sensor positions are x1=−1000,y1=1000, x2=1000,y2=1000, x3=1000,y3=−1000, and x4=−1000,y4=−1000 and σAOA is fixed at 3°. The result is similar to TOA solution that the proposed method can give a more accurate estimation if the target gets closer to the central of the array range.Figure 4
The theoretical variance of the proposed method in AOA case as the target position varies.


## 5. Simulation Results

Simulations are performed to validate the performance of the proposed method and to compare the relative localization accuracy for different methods.Without loss of generality, we assume that the signal and noises are mutually independent, zero-mean stationary Gaussian random processes. Suppose thatσTOA and σAOA denote TOA standard deviations and AOA standard deviations, respectively. Simulation results are displayed in terms of the root-mean-square error (RMSE) given by Exes−x2+yes−y2 [15] which are obtained from the average of 3000 independent runs.As shown in Figures5(a) and 5(b), the correctness of the theoretical variance we derive is verified in TOA solution and AOA solution, respectively. The source is at an arbitrary position within the range of the array. The Monte Carlo experiment generates a source position randomly in each run. Simulations demonstrated that the theoretical variance fits well with RMSE in both TOA solution and AOA solution.Figure 5
The theoretical variance and RMSE of the proposed method: (a) theoretical variance in TOA solution with 9 sensors asσTOA varies; (b) theoretical variance in AOA solution with 8 sensors as σAOA varies.
(a)
(b)Figure6(a) illustrates the performance of the proposed method as the number of the sensor varies, where the area size is 1000, σTOA is fixed at 30, and the source is at x=−200,y=−100. The proposed method with the simplified formulae instead of matrix inversion still performs well, and its localization accuracy is close to CRLB. In case of AOA solution, the results are given in Figure 6(b), where the area size is 1000, σAOA is fixed at 2°, and the source is at x=100,y=−200. The performance of the proposed method also performs well in AOA solution.Figure 6
Performance of the proposed method as sensor number varies: (a) RMSE of TOA solution (σTOA is fixed at 30 and the source is at x=−200,y=−100); (b) RMSE of AOA solution (σAOA is fixed at 2° and the source is at x=100,y=−200).
(a)
(b)In order to verify dynamic localization performance of the proposed method, simulations of dynamic positioning were carried out. Figures7(a) and 7(b) compare the real target trace and estimated target trace in TOA solution. Two different motion trajectories of W and S are simulated in Figures 7(a) and 7(b), respectively. σTOA is fixed at 40. Both Figures 7(a) and 7(b) show that the proposed method performs well in dynamic localization. The real target trace and the estimated target trace are almost coincident. Similar simulation was conducted in AOA solution as shown in Figures 8(a) and 8(b), where σAOA is fixed at 2°.Figure 7
Dynamic localization performance of the proposed method in TOA solution: (a) the comparison between real target trace and estimated target trace inW trajectory; (b) the comparison between real target trace and estimated target trace in S trajectory.
(a)
(b)Figure 8
Dynamic localization performance of the proposed method in AOA solution: (a) the comparison between real target trace and estimated target trace inW trajectory; (b) the comparison between real target trace and estimated target trace in S trajectory.
(a)
(b)Figures8(a) and 8(b) show that the AOA solution of the proposed method achieves high accuracy in dynamic localization.To give a comparison of the localization accuracy between the proposed method and other methods, Figure9(a) compares the performance of the proposed method, Chan’s method [8], and direct average method in TOA solution with 4 sensors. The sensor positions are x1=−1000,y1=1000, x2=1000,y2=1000, x3=1000,y3=−1000, and x4=−1000,y4=−1000. The source is at an arbitrary position within the range of the array. The Monte Carlo experiment generates a source position randomly in each run. Simulations show the estimation performance of the proposed method as σTOA varies from 10 to 50. Among the three methods, the direct average method performs worst and the proposed method gives a smaller RMSE using variance weighting. The positioning accuracy of the proposed method is close to the Chan’s method and CRLB [29]. Note that no matrix reverse is required for the proposed method; the computation complexity is largely simplified, and the method can be used in low-cost hardware devices.Figure 9
Performance comparison with other methods and CRLB: (a) RMSE of the direct average method, the proposed method, and the Chan’s method in TOA solution asσTOA varies; (b) RMSE of the direct average method, the proposed method, and the WLS method in AOA solution as σAOA varies.
(a)
(b)Similar experiment was conducted in AOA solution as shown in Figure9(b). The sensor positions are x1=0,y1=0, x2=1000,y2=1000, x3=1000,y3=1000, and x4=0,y4=1000. Simulations show the localization accuracy of the proposed method compared with the WLS method [11] and direct average method as σAOA varies from 1° to 3°. The performance of the proposed method remains promising in AOA solution.As seen in all the figures, the proposed method using theoretical variances as weights to improve the system performance provides higher accuracy than the method that uses the direct average of each group’s estimate result. Previous closed-form solutions cannot be used for low-cost hardware devices due to matrix inversion. This method, on the other hand, provides an effective solution which has the similar performance with the matrix method for target localization with low-cost hardware devices.


## 6. Conclusions

We have developed a simple method for target localization in a multistation redundancy system. The proposed method, which utilizes groups to avoid matrix inversion and uses theoretical variances as weights to improve the system performance, gives an algebraic solution which can be applied to low-cost hardware devices. It requires less computation and does not reply on matrix inversion. The performance of the proposed method is verified by the simulation results. A comparison with other localization methods and CRLB was conducted. With theoretical variances used as the weighting coefficients, the proposed method has a significantly reduced RMSE compared with the method that directly averages each group’s localization result. The positioning accuracy of the proposed method is close to CRLB both in TOA solution and AOA solution.This paper introduces the variance weighted average method for target localization in a multistation redundancy system. The proposed method, without matrix inversion, has the similar performance with the matrix method. For many low-cost hardware devices applications, this closed-form solution may itself be sufficiently accurate.



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*Source: 6362939-2018-07-16.xml*