# A Hybrid AOA and TDOA-Based Localization Method Using Only Two Stations

Authors: Xiaolong Li; Yi Xing; Zhenkai Zhang
Journal: International Journal of Antennas and Propagation
 (2021)

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## Abstract

Target localization plays an important role in the application of radar, sonar, and wireless sensor networks. In order to improve the localization performance using only two stations, a hybrid localization method based on angle of arrival (AOA) and time difference of arrival (TDOA) measurements is proposed in this paper. Firstly, the optimization model for localization based on AOA and TDOA are built, respectively, in the sensor network. Secondly,the majorization-minimization (MM) method is employed to create surrogate functions for solving the multiple objective optimization problem. Next, the hybrid localization problem is solved by the projected gradient decent (PGD) method. Finally, the Cramer–Rao lower bound (CRLB) for the joint AOA and TDOA method is derived for the comparison. Simulations proved that the proposed method has improved localization performance using AOA and TDOA measurements from only two base stations.


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## Body

## 1. Introduction

Source localization in the sensor network is a fundamental problem which has received an upsurge of attention in recent years. A number of separated sensors in the network will be employed to measure the emitted or reflected signal from the target [1]. Many algorithms have been presented in literature to determine the source position, based on the time of arrival (TOA), time difference of arrival (TDOA), angle of arrival (AOA), and received signal strength (RSS) method.Localization problem based on the TOA method in the quasi-synchronous network is addressed in [2], in which a two-step linear algorithm is presented for estimating the passive object position. TOA-based localization problem in an adverse environment is solved in [3], which employs two convex relaxation methods for mitigating non-line-of-sight (NLOS) effect. Using a single transmitter and multiple receivers, the optimal geometries for TOA localization configuration in the two-dimensional environment are analyzed in [4]. With the help of floor plan information, Liu et al. [5] propose an effective geometrical localization method based on TOA in the non-line-of-sight (NLOS) environment without considering multiple-bound scattering signal paths.A coarse position estimation algorithm-based TDOA is presented in [6], which is robust with regard to the initial position without redundant receivers. As a further step, a low-complexity nonlinear expectation maximization localization algorithm is presented in [7]. Target location is estimated by the TDOA data in [8], where empirical mode decomposition is used to decompose seismic signals, and multiple features of intrinsic mode functions are extracted. The work of [9] transforms a TDOA model into a TOA model, by presenting a semidefinite programming method. The TDOA method is also employed in passive radar source localization [10], taking advantage of the characteristics of the same sensor in different positions.For obtaining the best localization performance, the angular separation requirements for AOA sensors are established in [11]. A closed-form solution for 3D localization using AOAs is presented in [12] to handle the effect of sensor position errors in a wireless sensor network. In the AOA localization system, an evaluation function by frame theory is derived and proved in [13], in order to solve the sensor augmentation problem.Localization algorithm based on the RSS method in NLOS environment is presented in [14], by correcting the NLOS measurements. A novel fingerprinting paradigm is introduced for RSS localization in [15], which obtains a significant gain for multiple sources and multipath environments. In [16], RSS localization methods based on convex optimization are presented to solve the noncooperative and cooperative problems in sensor networks.However, localization based on the TOA method needs an accurate time synchronization. TDOA-based approach requires at least four located sensors. RSS algorithms are based on known exact received signal without multipath effect, and the AOA method can perform well only when the target is not far away from the sensors [17]. There are many advantages with the hybrid localization method [17, 18], which has already attracted many attentions. Using the hybrid-bearing AOA and TDOA sensors [19], the constrained optimization method is employed to estimate the maximum likelihood values. The RSS-assisted TOA-based localization method is presented in [20], which improves the efficiency and localization accuracy of the indoor geolocation system. In adverse NLOS environment, a new mitigation method is proposed using the combined RSS and TOA measurements in [21]. Combining TDOA and RSS measurements, a two-step position estimator is presented in [22] for the visible light positioning system. The hybrid AOA- and RSS-based localization method is presented in [23] for 3D wireless sensor network without central processor. Using the same hybrid method, a simple closed-form solution method is proposed by using the spherical coordinate conversion in [24]. Using the hybrid measurements of TDOA and AOA methods, the mean square error matrix is derived under small error condition in [25], and a closed-form solution based on two base stations (CFS-2BSs) is proposed. In order to reduce the sensors for target localization, we propose a hybrid TDOA/AOA localization approach only with two sensors, taking advantage of the complementary property between the TDOA- and AOA-based localization methods.This paper investigates the localization problem based on AOA and TDOA measurements from two stations. The main contributions of this paper are summarized as follows:(1)
A localization model using only two stations is developed, which is converted to be a multiple optimization problem.(2)
The MM method is used to solve this optimization problem, which maximizes the similarity between unknown and known information for AOA and TDOA measurements.(3)
A single objective optimization model is constructed after the derivation based on the MM. The PGD method is employed to solve this problem effectively.(4)
Simulations prove the superior localization performance using only two stations in the sensor network.This paper is organized as follows. Section2 describes the basic TDOA localization model. Section 3 proposes the hybrid localization method based on AOA and TDOA data from two base stations. CRLB is derived for evaluating the localization performance in Section 4. Section 5 shows the simulation results based on the proposed method. Section 6 gives the conclusion of this paper.Notations. Bold uppercase (e.g., H) and lowercase (e.g., b) letters represent the matrices and vectors, respectively. The notations ⋅T and ⋅H stand for transpose and Hermitian of their argument, respectively. ⋅2 denotes the l2 norm of a vector. The gradient of f at x is denoted by ∇fx.


## 2. Basic Localization Methods Based on TDOA

LetAi=xi,yiT denotes the position of ith base station and ϕ=x,yT represents the unknown position of the master station. The distance di between the ith base station and the master station can be written as(1)di=x−xi2+y−yi22=ϕ−Ai2,i=1,2,…,M,where M is the number of base stations. Time difference Δti,1 from the ith and 1th base station can be defined as(2)Δti,1=dic−d1c,where c is the propagation speed. Distance difference can be derived as(3)Δdi,1=cΔti,1.Taylor iteration method and Chan’s method are proposed, respectively, as the classical localization methods based on the time difference.An iterative method is employed to solve the hyperbolic equations for estimating the MS position in the Taylor iteration method, where a local linear least square (LS) solution is selected based as the deviationΔx,Δy of each iteration. The position of master station can be written as(4)x=x0+Δx,y=y0+Δy,where x0,y0 is initial coordinate values for the iteration and Δx and Δy can be derived as(5)Δx,ΔyT=GtTC−1Gt−1GtTC−1ht,where C is the covariance matrix for the time difference estimation and Gt and ht are written as, respectively, as follows:(6)Gt=x1−xd1−x2−xd2y1−yd1−y2−yd2x1−xd1−x3−xd3y1−yd1−y3−yd3……x1−xd1−xM−xdMy1−yd1−yM−ydM,ht=Δd2,1Δd3,1…ΔdM,1.Besides, Chan’s method is proposed to obtain a noniterative solution. Letϕc=ϕ,d1 be the solution, which can be derived as(7)ϕc=GcTC−1Gc−1GcTC−1hc,where(8)hc=12Δd2,12−x22+y22x12+y12Δd3,12−x32+y32+x12+y12…ΔdM,12−xM2+yM2+x12+y12,Gc=x2−x1y2−y1d2,1x3−x1y3−y1d3,1…xM−x1yM−y1dM,1.


## 3. Hybrid Localization Method Based on AOA and TDOA

### 3.1. Multiple Objectives’ Design

The unknown location of the target from the noisy measurements of angle of arrival (AOA) can be estimated, as there is a nonlinear relationship between the AOA and target location. Only two base stations with two anglesθ1,θ2 are considered, C1=cosθ1,cosθ2T, which is assumed to be known in Figure 1.Figure 1
Angles of MS and BSs.In Figure1, β1 and β2 are the azimuth angles of the base stations. θ1 can be written as follows:(9)θ1=β1−β2.Based on the AOA measurements of every BS, the first fitness function is designed as follows:(10)minϕF1−C122,where the vector F1 is designed as follows:(11)F1i=cosβ1,sinβ1cosβ2,sinβ2T=AiTAi2ϕ−Aiϕ−Ai2=AiTϕ−AiAi2ϕ−Ai2.Using the TDOA data from the two base stations, the sum of squared range difference errors is used to design the second fitness function, which is defined as(12)minϕF2−C222,where(13)F2i=ϕ−Ai2−ϕ−A12,C2i=Δdi,1.


### 3.2. Hybrid Optimization Model

In order to solve the hybrid optimization problems based on (10) and (12), the minorization-maximization method is employed to create the surrogate functions here. Let Γ1=F1−C122, where Γ1 can be majorized by g1ϕ|ϕt, which is derived as follows:(14)g1ϕ|ϕt=Γ1ϕt+∂Γ1∂ϕ|ϕtTϕ−ϕt,where ϕt is the solution at the t th iteration, and(15)∂Γ1∂ϕ=2∂F1∂ϕF1−C1.LetBi=AiT/Ai2; then,(16)F1i=Biϕ−Aiϕ−Ai2.The gradient ofF1i can be written as(17)∂F1i∂ϕ=BiTϕ−Ai22−Biϕ−Aiϕ−Aiϕ−Ai23,∂F1∂ϕ=∂F11∂ϕ,∂F12∂ϕ.LetΓ2=F2−C222,whereΓ2 can be majorized by g2ϕ|ϕt, which can be written as follows:(18)g2ϕ|ϕt=Γ2ϕt+∂Γ2∂ϕ|ϕtTϕ−ϕt,where(19)∂Γ2∂ϕ=2∂F2∂ϕF2−C2,∂F2∂ϕ=ϕ−Aiϕ−Ai2−ϕ−A1ϕ−A12.Based on (14) and (18), the final objective function for the localization using two stations is designed as(20)Φϕ=g1ϕ|ϕt+λg2ϕ|ϕt,(21)=η+f1Tϕ−ϕt+ξ+f2Tϕ−ϕt,where λ is the scalarization coefficient, and(22)η=Γ1ϕt,(23)f1=∂Γ1∂ϕ|ϕt,(24)ξ=λΓ2ϕt,(25)f2=λ∂Γ2∂ϕ|ϕt.Then, we use the PGD method to solve the optimization problem.ϕk can be calculated iteratively by(26)ϕk=proϕϕk−1−tk∇Φϕk−1,(27)=argminϕϕk−1−tk∇Φϕk−122,where proϕ is the Euclidean projection. The gradient of Φ can be written as(28)∇Φϕ=f1+f2.In conclusion, the hybrid localization procedure based on the TDOA and AOA data is described in Algorithm1.Algorithm1: The joint localization method based on TDOA and AOA information.
(i)
Input: Information of AOA and TDOA, Number of iterations K1,K2(ii)
Result: MS position(1) Sett=0; start from one random position ϕ0(2)repeat(3) Construct surrogate functions by using formulas from (22) to (25).(4) Design objective function (20)(5) Setk=0 and ϕ0 to some initial vector, e.g., ϕ0=ϕtTT(6)repeat(7) Computeϕk using (27)(8)k=k+1untilk≤K2;(9) ObtainϕK2(10)t=t+1untilt<K1;


## 3.1. Multiple Objectives’ Design

The unknown location of the target from the noisy measurements of angle of arrival (AOA) can be estimated, as there is a nonlinear relationship between the AOA and target location. Only two base stations with two anglesθ1,θ2 are considered, C1=cosθ1,cosθ2T, which is assumed to be known in Figure 1.Figure 1
Angles of MS and BSs.In Figure1, β1 and β2 are the azimuth angles of the base stations. θ1 can be written as follows:(9)θ1=β1−β2.Based on the AOA measurements of every BS, the first fitness function is designed as follows:(10)minϕF1−C122,where the vector F1 is designed as follows:(11)F1i=cosβ1,sinβ1cosβ2,sinβ2T=AiTAi2ϕ−Aiϕ−Ai2=AiTϕ−AiAi2ϕ−Ai2.Using the TDOA data from the two base stations, the sum of squared range difference errors is used to design the second fitness function, which is defined as(12)minϕF2−C222,where(13)F2i=ϕ−Ai2−ϕ−A12,C2i=Δdi,1.


## 3.2. Hybrid Optimization Model

In order to solve the hybrid optimization problems based on (10) and (12), the minorization-maximization method is employed to create the surrogate functions here. Let Γ1=F1−C122, where Γ1 can be majorized by g1ϕ|ϕt, which is derived as follows:(14)g1ϕ|ϕt=Γ1ϕt+∂Γ1∂ϕ|ϕtTϕ−ϕt,where ϕt is the solution at the t th iteration, and(15)∂Γ1∂ϕ=2∂F1∂ϕF1−C1.LetBi=AiT/Ai2; then,(16)F1i=Biϕ−Aiϕ−Ai2.The gradient ofF1i can be written as(17)∂F1i∂ϕ=BiTϕ−Ai22−Biϕ−Aiϕ−Aiϕ−Ai23,∂F1∂ϕ=∂F11∂ϕ,∂F12∂ϕ.LetΓ2=F2−C222,whereΓ2 can be majorized by g2ϕ|ϕt, which can be written as follows:(18)g2ϕ|ϕt=Γ2ϕt+∂Γ2∂ϕ|ϕtTϕ−ϕt,where(19)∂Γ2∂ϕ=2∂F2∂ϕF2−C2,∂F2∂ϕ=ϕ−Aiϕ−Ai2−ϕ−A1ϕ−A12.Based on (14) and (18), the final objective function for the localization using two stations is designed as(20)Φϕ=g1ϕ|ϕt+λg2ϕ|ϕt,(21)=η+f1Tϕ−ϕt+ξ+f2Tϕ−ϕt,where λ is the scalarization coefficient, and(22)η=Γ1ϕt,(23)f1=∂Γ1∂ϕ|ϕt,(24)ξ=λΓ2ϕt,(25)f2=λ∂Γ2∂ϕ|ϕt.Then, we use the PGD method to solve the optimization problem.ϕk can be calculated iteratively by(26)ϕk=proϕϕk−1−tk∇Φϕk−1,(27)=argminϕϕk−1−tk∇Φϕk−122,where proϕ is the Euclidean projection. The gradient of Φ can be written as(28)∇Φϕ=f1+f2.In conclusion, the hybrid localization procedure based on the TDOA and AOA data is described in Algorithm1.Algorithm1: The joint localization method based on TDOA and AOA information.
(i)
Input: Information of AOA and TDOA, Number of iterations K1,K2(ii)
Result: MS position(1) Sett=0; start from one random position ϕ0(2)repeat(3) Construct surrogate functions by using formulas from (22) to (25).(4) Design objective function (20)(5) Setk=0 and ϕ0 to some initial vector, e.g., ϕ0=ϕtTT(6)repeat(7) Computeϕk using (27)(8)k=k+1untilk≤K2;(9) ObtainϕK2(10)t=t+1untilt<K1;


## 4. Derivation of Cramer–Rao Lower Bound

Only Gaussian range sampling error is assumed in the localization here, so Fisher’s information matrix [26] is then given by(29)Jϕ=J11J12J21J22=−Eϕ∂2Γ1∂x2−Eϕ∂2Γ1∂x∂y−Eϕ∂2Γ1∂x∂y−Eϕ∂2Γ1∂y2,where Iϕ is defined as the inverse matrix of Jϕ:(30)Iϕ=I11I12I21I22=J−1ϕ=1J11J22−J12J21J22−J12−J21J11,where Ji,j and Ii,j are, respectively, the element of J and I,i = 1, 2 and j = 1, 2. The localization CRLB based on the AOA method can be computed as follows:(31)σCRLB−AOA2=Eϕϕ^−ϕ2=I11+I22.Based on the AOA measurements, (31) can be derived as(32)σCRLB−AOA2=b1b2−b3,where(33)b1=σR2∑i=2Ncosθi−cosθ12+sinθi−sinθ12,b2=∑i=2Ncosθi−cosθ12∑i=1Nsinθi−sinθ12,b3=∑i=2Ncosθi−cosθ1sinθi−sinθ12,where σR2 is the Gaussian error variance of C2 and θi denotes the angle between the MS and ith BS with respect to a reference direction.Using the same method, CRLB based on the TDOA method can be calculated as(34)σCRLB−TDOA2=a1a2−a3,where(35)a1=σt2∑i=1Ntixi−x2+1xi−x2,a2=∑i=1Ntixi−x2∑i=1N1xi−x2,a3=∑i=1Ntixi−x22,where ti=yi−y/xi−x, where ti is supposed to contain a Gaussian error with average and variance of 0 and σt2, respectively.The answer to the multiobjective optimization problem is a set of solutions that defines the best trade-off between competing objectives. Here, the log-likelihood function for the multiobjective optimization problem can be written as(36)Γ3=Γ1maxΓ1σt2+Γ2maxΓ1σR2=Γ1c1+Γ2c2,where c1=maxΓ1σt2 and c2=maxΓ1σR2.The CRLB for the localization based AOA and TDOA using two base stations can be derived as(37)σCRLB−TDOA−AOA2=2d1d2−d3,(38)d1=1c1∑i=12ti2+1xi−x2+2c21−cosθ2cosθ1−sinθ2sinθ1,(39)d2=1c1∑i=12ti2+1xi−x22c21−cosθ2cosθ1−sinθ2sinθ1,(40)d3=1c2cosθ2−cosθ1sinθ2−sinθ1−∑i=121c1tixi−x2.


## 5. Simulations

To verify the effectiveness of the proposed approach, computer simulations are conducted for comparisons with other methods. As it is shown in Figure2, the coordinates of the BSs are, respectively, set to BS1:0,0Tcm, BS2:60,0Tcm, BS3:0,60Tcm, and BS4:60,60Tcm; the MS location is chosen randomly according to a uniform distribution defined over the region formed by the BSs.Figure 2
Positions of MSs and BSs.In our proposed localization algorithm,BS1 and BS2 are selected as the two stations. Then, 15 TDOA and AOA measurements from the selected BSs are obtained. The root of mean squared error (RMSE) of MS estimation x^k,y^kT from the kth BS is given by(41)RMSEk=x^k−x02−y^k−y02,where x0,y0T denotes the true position of MS.


### 5.1. Comparison with the Classical TDOA Localization Methods Using More BSs

As we know, it is necessary for the classical TDOA-based methods to employ more than 3 stations realize the localization. To start with the comparison of the localization methods, the results of the classical source localization method are presented in Figure3. Based on Taylor iteration method and Chan’s method, it can be observed that the estimated value is much closer to the sensor position whenever more BSs are employed. This is because Taylor iteration and Chan’s methods apply the iterative and noniterative solutions to achieve their optimum performance, respectively. Thus, they are considerably affected by the accuracy of TDOA measurements. Under the same TDOA measurement condition, it can be observed that the proposed algorithm has much better localization accuracy than the classical methods using three BSs, while it uses only two BSs. In some cases, e.g., Chan’s method, using three BSs, a superiority of more than one order of magnitude is observable. Since increasing the number of BSs is expected to improve accuracy, reformulating the proposed method to a higher number of BSs is expected to result in even higher accuracy. However, the reformulation is beyond the topic of this article and can be addressed in future investigations.Figure 3
Performance of the classical source localization methods.


### 5.2. Comparison with Single Objective Optimization-Based Methods

In order to prove the effectiveness of the proposed method using only two stations, source localization results using 3 BSs are presented here based on optimization methods, including weighted least squares (WLS-O-3BSs) [7], and optimization using time difference of arrival (TDOA-O-3BSs) [21]. They are then compared with the proposed methods. Figure 4 depicts localization error for a various number of testing points. It can be observed that the results obtained in this work outperform others, being able to achieve a lower localization error. Specifically, an improvement of the order of one degree of magnitude is observable in terms of peak localization error, i.e., the highest deviation in estimation.Figure 4
Performance of single objective optimization-based methods.Using two stations, the TDOA (TDOA-O-2BSs), AOA (AOA-O-2BSs) based on equations (10) and (12), and CFS-2BSs in [25] are realized, respectively. The results are demonstrated in Figure 5. Due to the lack of measurement information, there are large errors in the TDOA and AOA measurements, and the single objective function for optimization cannot perform very well in such an environment. CFS-2BSs, in [25], proves its good localization performance through the simulations, but its performance depends on the covariance matrix of the measurement noise. Interestingly, the proposed method shows very low localization error in a broad range for testing point numbers. When all methods use 2 base stations, the achieved accuracy improvement is about two orders of magnitude compared to TDOA and less than one order of magnitude compared to the AOA method in terms of peak localization error.Figure 5
Performance of localization methods using 2 BSs.


### 5.3. Comparison with Other Multiobjective Optimization-Based Methods

There are many optimization methods for the multiple objectives, such as the conventional multiobjective particle swarm optimization (MOPSO) [27] and nondominated sorting genetic algorithm (NSGAII) [28, 29], which can be used to solve the localization problems using two stations. Here, the proposed algorithm is compared with the MOPSO and NSGAII algorithms to demonstrate its superiority over a range of state-of-the-art investigations. In this regard, the population is set to 100 for all three optimizers, and the maximum number of iterations is set to 200. Four simulations are conducted using different optimization methods for the same multiple objectives in equations (10) and 12), where their results are demonstrated in Figure 6. Besides, the percentages of crossover and mutation populations and the mutation rate of NSGAII are set to 0.7, 0.4, and 0.02, respectively. The root CRLB in equations (34) and (38) is also computed and depicted in the same figure, labeled as AOA-CRLB and TOA-AOA-CRLB, respectively. Under the same conditions, the localization RMSE of the proposed method is smaller than other methods, being significantly close to CRLB.Figure 6
Localization performance comparison of optimization methods.Based on the theory of projected gradient descent, our proposed converges at rateO1/k, where k is the iteration number. The runtime comparison results using the same computer are depicted in Figures 7 and 8. The experiments confirm the effectiveness of the proposed MM-based method in solving optimization and localization problems under the uncertainties of MS positions. This is because the nondominated sorting GA (NSGAII) must sort out all the individuals according to the Pareto-domination relationship and selects the individuals with better ranks to form the next-generation population, but with a lower convergence speed [30]. For conventional MOPSO, the neighborhood of one particle is formed by some nearest particles according to the objective space of one objective, but the best particle in the neighborhood cannot be determined very well by the fitness of the other objective. The CFS-2BSs method proposed by constructing relationships between the hybrid measurements and the unknown source position is not the iterative implementation method, which takes less computations.Figure 7
Runtime comparison of optimization methods.Figure 8
Runtime comparison of optimization methods.


## 5.1. Comparison with the Classical TDOA Localization Methods Using More BSs

As we know, it is necessary for the classical TDOA-based methods to employ more than 3 stations realize the localization. To start with the comparison of the localization methods, the results of the classical source localization method are presented in Figure3. Based on Taylor iteration method and Chan’s method, it can be observed that the estimated value is much closer to the sensor position whenever more BSs are employed. This is because Taylor iteration and Chan’s methods apply the iterative and noniterative solutions to achieve their optimum performance, respectively. Thus, they are considerably affected by the accuracy of TDOA measurements. Under the same TDOA measurement condition, it can be observed that the proposed algorithm has much better localization accuracy than the classical methods using three BSs, while it uses only two BSs. In some cases, e.g., Chan’s method, using three BSs, a superiority of more than one order of magnitude is observable. Since increasing the number of BSs is expected to improve accuracy, reformulating the proposed method to a higher number of BSs is expected to result in even higher accuracy. However, the reformulation is beyond the topic of this article and can be addressed in future investigations.Figure 3
Performance of the classical source localization methods.


## 5.2. Comparison with Single Objective Optimization-Based Methods

In order to prove the effectiveness of the proposed method using only two stations, source localization results using 3 BSs are presented here based on optimization methods, including weighted least squares (WLS-O-3BSs) [7], and optimization using time difference of arrival (TDOA-O-3BSs) [21]. They are then compared with the proposed methods. Figure 4 depicts localization error for a various number of testing points. It can be observed that the results obtained in this work outperform others, being able to achieve a lower localization error. Specifically, an improvement of the order of one degree of magnitude is observable in terms of peak localization error, i.e., the highest deviation in estimation.Figure 4
Performance of single objective optimization-based methods.Using two stations, the TDOA (TDOA-O-2BSs), AOA (AOA-O-2BSs) based on equations (10) and (12), and CFS-2BSs in [25] are realized, respectively. The results are demonstrated in Figure 5. Due to the lack of measurement information, there are large errors in the TDOA and AOA measurements, and the single objective function for optimization cannot perform very well in such an environment. CFS-2BSs, in [25], proves its good localization performance through the simulations, but its performance depends on the covariance matrix of the measurement noise. Interestingly, the proposed method shows very low localization error in a broad range for testing point numbers. When all methods use 2 base stations, the achieved accuracy improvement is about two orders of magnitude compared to TDOA and less than one order of magnitude compared to the AOA method in terms of peak localization error.Figure 5
Performance of localization methods using 2 BSs.


## 5.3. Comparison with Other Multiobjective Optimization-Based Methods

There are many optimization methods for the multiple objectives, such as the conventional multiobjective particle swarm optimization (MOPSO) [27] and nondominated sorting genetic algorithm (NSGAII) [28, 29], which can be used to solve the localization problems using two stations. Here, the proposed algorithm is compared with the MOPSO and NSGAII algorithms to demonstrate its superiority over a range of state-of-the-art investigations. In this regard, the population is set to 100 for all three optimizers, and the maximum number of iterations is set to 200. Four simulations are conducted using different optimization methods for the same multiple objectives in equations (10) and 12), where their results are demonstrated in Figure 6. Besides, the percentages of crossover and mutation populations and the mutation rate of NSGAII are set to 0.7, 0.4, and 0.02, respectively. The root CRLB in equations (34) and (38) is also computed and depicted in the same figure, labeled as AOA-CRLB and TOA-AOA-CRLB, respectively. Under the same conditions, the localization RMSE of the proposed method is smaller than other methods, being significantly close to CRLB.Figure 6
Localization performance comparison of optimization methods.Based on the theory of projected gradient descent, our proposed converges at rateO1/k, where k is the iteration number. The runtime comparison results using the same computer are depicted in Figures 7 and 8. The experiments confirm the effectiveness of the proposed MM-based method in solving optimization and localization problems under the uncertainties of MS positions. This is because the nondominated sorting GA (NSGAII) must sort out all the individuals according to the Pareto-domination relationship and selects the individuals with better ranks to form the next-generation population, but with a lower convergence speed [30]. For conventional MOPSO, the neighborhood of one particle is formed by some nearest particles according to the objective space of one objective, but the best particle in the neighborhood cannot be determined very well by the fitness of the other objective. The CFS-2BSs method proposed by constructing relationships between the hybrid measurements and the unknown source position is not the iterative implementation method, which takes less computations.Figure 7
Runtime comparison of optimization methods.Figure 8
Runtime comparison of optimization methods.


## 6. Conclusion

In this paper, a significant increase in the positioning accuracy is achieved by using the MM and PGD methods, and the TDOA and AOA information together, while decreasing the number of required base stations. Accordingly, the localization problem is formulated for a two station scenario and is then derived by the MM method and then solved by the PGD method. Then, theoretical CRLB analysis is realized to establish a basis for the accuracy comparison. The extensive performance comparison with several state-of-the-art localization methods and some other optimization methods reveal the superiority of the proposed method. Specifically, they show that the proposed algorithm can achieve a superior localization accuracy using only two base stations compared to other recent methods required three or more base stations.



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