A Closed-Form Localization Algorithm and GDOP Analysis for Multiple TDOAs and Single TOA Based Hybrid Positioning

: Cellular communication systems support mobile phones positioning function for Enhanced-911 (E-911) location requirements, but the positioning accuracy is poor. The fifth-generation (5G) cellular communication system can use indoor distribution systems to provide accurate multiple time-difference-of-arrival (TDOA) and single time-of-arrival (TOA) measurements, which could significantly improve the indoor positioning ability. Unlike iterative localization algorithms for TDOA or TOA, the existing closed-form algorithms, such as the Chan-Ho algorithm, do not have convergence problems, but can only estimate position based on one kind of measurement. This paper proposes a closed-form localization algorithm for multiple TDOAs and single TOA measurements. The proposed algorithm estimates the final position result using three-step weighted least squares (WLSs). The first WLS provides an initial position for the last two steps. Then the algorithm uses two WLSs to estimate position based on heteroscedastic TDOA and TOA measurements. In addition, the geometric dilution of precision (GDOP) of the proposed hybrid TDOA and TOA positioning has been derived. The analysis of GDOP shows that the proposed hybrid positioning has lower GDOP than TDOA-only positioning, which means the proposed hybrid positioning has a higher accuracy limitation than TDOA-only positioning. The simulation shows that the proposed localization algorithm could have better performance than closed-form TDOA-only positioning methods, and the positioning accuracy could approximate Cramer-Rao lower bound (CRLB) when the TDOA measurement errors are small.


Introduction
Indoor positioning is important for location-based service, internet of things and health care [1,2]. Many wireless systems, such as Ultra-Wideband, Bluetooth and cellular communication systems, have been used for indoor positioning. As one of the most widely used wireless systems, the cellular communication system supports mobile phone positioning from 2G to 5G [3]. In recent years, with the development of the location service market, the cellular communication system based positioning can be used in many new fields such as vehicle positioning [4,5], location-aware communications [6] and location-based applications [7]. The 3rd Generation Partnership Project (3GPP) also puts higher requirements on the positioning accuracy of the 5G system [8,9]. measurement errors match Gauss-Markov theory. However, the random errors of TDOA and TOA measurements are not homoscedastic. It means those algorithms cannot be used for the hybrid TDOA and TOA positioning directly.
This paper aims to improve the positioning accuracy of the indoor distribution system by using multiple TDOAs and single TOA based hybrid positioning. In this paper, we establish a three-step WLS closed-form localization algorithm for heteroscedastic TDOA and TOA measurements. The first WLS provides an initial position for the last two steps. Then the algorithm uses two-step weighted least square (WLS) to estimate the final position result. We also derived the GDOP of the proposed hybrid positioning. We performed simulations in a typical indoor hall using an indoor distribution system. As shown in the simulation result, the GDOP of the hybrid positioning is lower than the GDOP of TDOA-only positioning. The simulations also show that the proposed method could approximate the CRLB when TDOA measurements errors are small enough and have the potential to achieve a better performance than two widely used closed-form TDOA-only positioning methods.
This paper is organized as follows: Section 2 describes the system model of hybrid TDOA and TOA positioning. In Section 3, a new closed-form localization algorithm is developed for the nonlinear and heteroscedastic hybrid TDOA and TOA positioning equations. Section 4 analyzes the GDOP of the hybrid positioning based on multiple TDOAs and single TOA. Simulation results and analysis are presented in Section 5. Finally, we draw our conclusions in Section 6.

System Model
The ANTs are deployed at the ceiling with same height, high vertical dilution of precision makes the system cannot provide reliable vertical positioning results [29], which is usually obtained by other sensors [30].
Therefore, for the system model, we assumed that ANTs are deployed at the ceiling and the height of UE is already known by other sensors, and only two-dimensional horizontal positioning is considered. The system model of the indoor distribution system based UE positioning is shown in Figure 1. We consider a UE position estimation problem using an indoor distribution system containing ANTs. For the system shown in Figure 1, is equal to 4. For each positioning process, in order to simplify mathematical representations, ANTs are sorted by the distance to the UE. The nearest ANT is numbered 1, and the farthest ANT is numbered . UE could obtain − 1 TDOA measurements and one TOA measurement from the nearest ANT. In Figure 1, the nearest ANT to UE is the ANT 1 and the farthest is the ANT 4. In the following description, the position of UE is denoted by = ( , ) . The system contains ANTs locating at = ( , ) , = 1, 2, 3, … , .
The distance from ANT to UE is expressed as The distance difference from ANT 1 to UE and ANT to UE is expressed as: Considering the TDOA ranging error , , TDOA measurement from ANT 1 and ANT is represented as: where , is TDOA measurement and is the speed of light, , is the distance difference calculated by TDOA measurement, , is the real distance difference and , is the ranging error caused by TDOA measurement error.
TOA measurement of ANT 1 to UE is expressed as: where is TOA measurement, is the distance calculated by TOA measurement, is the real distance from ANT 1 to UE and is the ranging error caused by TOA measurement error. For brevity, we denote as the TDOA ranging error and for = 2,3, … , as TDOA ranging errors between the signal from ANT and signal from ANT 1, which is denoted as , in the above formula.
In a system with ANTs, an equation system containing TDOA and TOA measurements is obtained as:

Three-step WLS Localization Algorithm for Hybrid Positioning
In this section, a three steps WLS localization algorithm is proposed for positioning based on TDOA and TOA measurements shown in Section 2. The algorithm is an improvement of the two-step WLS TDOA-only localization algorithm, called the Chan-Ho algorithm, in Reference [17]. Those two algorithms use the same linearization method, and the third step WLS of the proposed algorithm is the same as the second step WLS of the Chan-Ho algorithm.
Using the following method, a linear equation about TDOA measurement and x, y coordination could be obtained. Moving the from the right side of Equation (2) to the left side and then squaring both sides, we obtain: Expanding (6) we obtain: where: Different to the Chan-Ho algorithm, the proposed algorithm needs to consider the TOA measurement and uses a new solution as follows. Using Equation (8), write Equation (5) in matrix form as: where: The vector φ is the error between h and G . It can be obtained by: In accordance with Equations (3) and (4), the items in vector φ can be written as the following expressions: where the assumption r ≫ e has been used.To minimize , the hybrid positioning problem is cast as the following quadratic optimization Ranging errors of TDOAs can be assumed as an independent, identical and zero-mean Gaussian distribution [31]. The ranging error of TOA can also be assumed to be a zero-mean Gaussian distribution [32]. Because TDOA and TOA measurements are estimated using different techniques, TOA and TDOA ranging errors can be assumed to be independent and have different variances. The parameter is used to represent the ratio of TOA ranging error's standard deviation to TDOA ranging error's standard deviation. The ranging errors have the following statistical properties: The errors in do not meet the Gauss-Markov assumption, ordinary least squares estimator is not the best linear unbiased estimator (BLUE) for this optimization problem. WLS could make it BLUE when each weight is equal to the reciprocal of the variance of the error [33]. The WLS solution of that minimizes is: where is the weighting matrix which could be given by the covariance matrix of . The covariance matrix of is: where ( ) is the inverse of the covariance matrix of , is obtained by prior probability statistics of the ranging errors after the system deployed.
Scaling of does not affect the result of WLS [17]. Therefore, the weighting matrix can be written as: However, the distances from ANTs and UE is unknown, to approximated the matrix , an initial guess of UE position is need to be obtained in first-step WLS.

First-Step WLS
To calculate the initial position, we assume that the distances from ANTs to UE are equal to the ratio k. Rewriting the weighting matrix as : We could obtain = ( , , ) using Equation (16) as It is an ordinary least squares. When the UE positions are different, the deviation of the weighting matrix and the initial positioning error would change. The second-step WLS could mitigate this error and can be repeated again to give an even better estimate.

Second-Step WLS
After achieving the initial position = ( , ), the distance from ANT to can be calculated. Inserting and into Equation (18), the weighting matrix could be rewritten as : Inserting matrix into Equation (16) in the same way as Equation (20), the solution = ( , , ) can be obtained.

Third-Step WLS
The former two steps assume no relationship between the UE location ( , ) and , but in fact they are related. Chan-Ho algorithm has the same problem in its first stage minimization and use a WLS to refine the result [17]. In the third-step WLS of this proposed algorithm, the same method as the Chan-Ho algorithm is used to refine the result of former two steps.
The former two steps' results location ( , ) and distance have a relationship shown in Equation (22). Therefore, the third-step WLS is to minimize the sum of the squared errors in: Equation (22) could be rewritten in matrix form as: where: Because elements in vector have different variances, we need to use WLS to estimate the solution ∆ as: where weighting matrix could be obtained by calculating the covariance matrix of . Assuming that the error of the solution in second-step WLS are = ( , , ) , vector could be obtained as: where the second-order error terms are ignored.
Therefore, result ∆ could be calculated by The ∆ is the square of the difference between final positioning result and ANT 1 on the x-axis and the y-axis. As with the Chan-Ho algorithm, is given by the square root of ∆ and ANT 1's location ( , ) .

GDOP of the Proposed Hybrid Positioning
GDOP is the ratio of the accuracy limitation of a position fix to the accuracy of measurements [21]. A positioning system has a small GDOP means that the positioning is accurate. In order to compare with GDOP of TDOA-only positioning, the GDOP of the proposed hybrid positioning is defined as the ratio of the positioning accuracy to the TDOA ranging accuracy.
In this section, the GDOP of the hybrid positioning based on multiple TDOAs and single TOA is derived.
To achieve the GDOP of the hybrid TDOA and TOA positioning of the system described in Section 2, the difference between TDOA measurements and real distances difference when UE is at the location ( , ) can be expressed as: Difference between TOA measurements and real distances when UE is at the location ( , ) is expressed as Using a first order Taylor series to approximate Equations (6) and Equation (7), the approximation of ( , ) at location ( , ) can be obtained as When there are no ranging errors, (9) can be obtained from (8).
To simplify the mathematical derivations, we use to represent the TOA ranging error. Moreover, , = 2, … , , is used to represent TDOA ranging errors between the signal from ANT and signal from ANT 0, which is represented as , in Section 2. As the measurements have ranging errors , Equation (33) can be obtained from (31): where ( ′, ′) is the position estimation result. Let position estimation error vector ∆u = ( , ) , Equation (33) from Equation (32), the equation of and ( , ) can be obtained as: For TOA ranging error , it is easy to obtain partial differential equations: For TDOA ranging error , partial differential equations is: Therefore, Equation (34) can be formulated as the following matrix form: where: Different from the GDOP calculation process in [21], TOA and TDOA measurements ranging errors have different variances. Therefore, in order to make an unbiased and efficient estimator, the vector ∆ cannot be calculated based on the ordinary least square (OLS) but weighted least square (WLS) as the following expression: where the weighting matrix W could be given by covariance matrix of error vector . Therefore, we could get a covariance matrix of error vector E as: Therefore, the weighting matrix W in Equation (39) could be given as Expressing matrix as The mean of could be calculated by: The variance of could be calculated by: It is easy to obtain the mean and variance of using the above equations: where: GDOP was been definite as the ratio of the accuracy of a position fix to the accuracy of measurements [34]. In order to directly compare the GDOP of the proposed hybrid positioning and the GDOP of TDOA-only positioning, we redefine the GDOP of the hybrid positioning using multiple TDOAs and single TOA as the ratio of the accuracy of a position fix to the accuracy of TDOA measurements. The GDOP could be calculated as:

Simulation Results and Analysis
To analyze the GDOP of multiple TDOAs and single TOA based hybrid positioning and the positioning accuracy of the proposed closed-form localization algorithm, we have built a simulation scenario according to the real scene of the zone 1, underground parking lot, Beijing University of Posts and Telecommunications. The real scene is shown in Figure 2 and the top view of the developed simulation scenario is shown in Figure 3.  In Figure 3, a Cartesian coordinate system is constructed in meters. The ANTs of the indoor distribution system are referred to the real antennas' location in the underground parking lot.
In the following parts of this section, we compared the GDOP of the proposed hybrid positioning and the GDOP of TDOA-only positioning. Then, we analyzed the positioning accuracy of the proposed localization algorithm.

Analysis and Comparison of GDOP
To compare the GDOP of the TDOA-only positioning and the GDOP of the hybrid positioning using multiple TDOAs and single TOA in the same indoor scenario. 2889 points is selected every 0.5 m, as shown in Figure 4. We divide the selected points into two groups. One group contains the points within the quadrilateral surrounded by connections between ANTs, which is the green quadrilateral in Figure 4. The other group contains the points outside the quadrilateral. 2292 points are outside the green quadrilateral and 597 points are inside the green quadrilateral. The GDOPs of the hybrid positioning method based on three TDOAs and one TOA with different ratio values are calculated at each points. The is the ratio of TOA ranging error standard deviation to TDOA ranging error standard deviation definite in Section 4. We use four different k values and the result is shown in Figure 5.  Figure 6 shows the GDOP of TDOA-only positioning in the same scenario. The TDOA-only positioning is based on three TDOA measurements which are the same as the proposed hybrid positioning method used above. The GDOP calculation method for TDOA-only positioning is similar to that GDOP calculation method for hybrid positioning described in Section 5 but does not contain the TOA measurement terms. The calculation method is the same as the method described in the definition of GDOP [21].
It can be easily seen from the comparison between Figures 5 and 6 that the GDOP of the TDOAonly positioning is not much different from the hybrid positioning in the center region (the quadrilateral area surrounded by the four ANTs). In other regions, the GDOP of hybrid positioning is significantly smaller than the TDOA-only positioning. However, the area of the center region only accounts for one-fifth of the total area. Assuming that the distribution of UEs is uniform, in most cases, using the hybrid positioning method will increase the positioning accuracy limitation.  From Figure 7, it is obvious that the GDOP values of the proposed hybrid positioning are significantly lower than the TDOA-only positioning. It can be seen in (16) that the greater the ratio k is, the smaller the weight of the TOA ranging error item in the weighting matrix (18) is, and the closer the GDOP calculated is to the GDOP of TDOA-only positioning. The result shows that it is possible to effectively improve the positioning performance by adding a TOA measurement.
To compare the GDOPs of the two kinds of positioning in the two different regions as shown in Figure 4, we separately counted the GDOPs of two kinds of positioning at points in the two groups. The CDFs of the GDOPs are shown in Figure 8. Analysis result shows that adding a TOA measurement to the TDOA measurements to achieve hybrid positioning have potential to greatly improve the positioning ability of the system in the edge area of the network. At the same time, in the core area of network coverage, although hybrid positioning has only a limited improvement in positioning ability, it is also superior to the TDOAonly positioning.

Analysis and Comparison of the Proposed Localization Algorithm
CRLB is the theoretical lower bound on the variance of unbiased estimator [35]. In order to analyze the performance of the proposed three-step WLS closed-form localization algorithm, we compare the algorithm with CRLB of the proposed hybrid positioning, CRLB of TDOA-only positioning and two widely used TDOA-only closed-form localization algorithms, Chan-Ho [17] and BiasRed [18]. CRLB is the trace of the inverse matrix of the Fisher information matrix [19].
where matrix and are defined in Section 3.1, is a matrix as (51)

Simulations at Selected Point
According to the GDOP analysis in Section 5.1, we selected three test points at the right edge, the center, and the left edge of the scenario. Scenario shown in Figure 3 is used, and three test points are selected for the simulation, which is shown in Figure 9. The simulation was performed 10,000 times at each test point. TDOA and TOA ranging errors were the zero-mean Gaussian distribution but with different standard deviations.
5G system could provide a TOA ranging accuracy with standard deviation = 0.3 or = 1.5 [25] and provide a TDOA ranging accuracy with a best = 0.03 [13]. In each simulation, we calculated the Mean Square Error (MSE) of the positioning errors for each test points as    It can be seen from those three figures that the positioning error of Chan-Ho and BiasRed would increases irregularly with the increase of the error after the error becomes larger. This may be caused by the error of the weighting matrix of those algorithms. Those two algorithms use WLS in the calculation process and ignore some items about TDOA ranging errors to build the weighting matrix. When TDOA ranging errors are small, the weighting matrix is closed to be right. However, when the ranging errors are large, the error of the weighting matrix would increase. The positioning error, caused by weighted matrix, is irregular. The proposed algorithm also has this problem, but the item about TOA measurement in weighting matrix is not related with ranging error. This may inhibit the irregularity of the positioning error when TOA ranging error is low. The positioning error of the proposed algorithm would also increases irregularly when TOA ranging error is high.

Simulation Results in Different Regions
To compare the positioning accuracy of the three localization algorithms in different regions, the two groups of selected points in Figure 4 were used for simulation.
The positioning simulation is performed 10,000 times at each test point. For each test point group, the MSE of the positioning errors for each group test points was calculated and denoted as   The algorithms performance at each selected points in Figure 4 is also simulated as in Section 5. As shown in Figure 15b, it is obviously that, regardless of = 0.3 or 1.5 m, performance of proposed algorithm is better than Chan-Ho and BiasRed algorithms. Only at 16% points, BiasRed algorithm is more accurate than the proposed algorithm when = 1.5 m. Moreover, the Chan-Ho algorithm is only more accurate at 13% points compared to the proposed algorithm when = 1.5 m. When = 0.3 m, the proposed algorithm has a better performance than the two algorithms at all points outside the green quadrilateral. In Figure 15a, the proposed algorithm still has a better performance than the two algorithms at all points inside the green quadrilateral when = 0.3 m. In the comparison, the two algorithms and the proposed algorithm with = 1.5 have their own advantages. BiasRed is more accurate than the proposed algorithm with = 1.5 at 89% points and Chan-Ho is more accuracy than the proposed algorithm with = 1.5 at 72% points. However, the proposed algorithm with = 1.5 is more reliable than the two algorithm, since the maximum RMSE is only 2.613 and is much lower than the maximum RMSE of the two algorithms. Based on the simulation results at the two groups' points and the three selected test points, the proposed closed-form localization algorithm using multiple TDOAs and single TOA could provide a better positioning accuracy than the two closed-form TDOA-only positioning methods, Chan-Ho and BiasRed, in the indoor distribution system application scenario.

Conclusions
The 5G system could provide accurate multiple TDOA measurements and single TOA measurement, which could improve UE positioning accuracy. However, existing closed-form localization algorithms are designed for measurements having the same variance, and not suitable for heteroscedastic TDOA and TOA measurements.
In this paper, a three-step WLS closed-form localization algorithm is developed for the nonlinear equation set of those hybrid heteroscedastic measurements. The algorithm is an extension of Chan-Ho algorithm [17]. The first WLS provides an initial position for the last two steps. Then the algorithm uses two WLSs to estimate position based on heteroscedastic TDOA and TOA measurements. Moreover, after the GDOP of this hybrid positioning is derived, it was proven that this hybrid positioning has a higher theoretical higher bound of accuracy than TDOA-only positioning.
The simulation scenario is built according to the real situation. By comparing with two TDOAonly closed-formed localization algorithm, Chan-Ho [17] and BiasRed [18], and CRLB [35] of TDOAonly positioning and the proposed hybrid positioning in the same simulation scenario, the results show that the proposed localization algorithm has better performance than the two closed-form TDOA-only positioning methods, and the positioning accuracy approaches CRLB when the TDOA measurement errors are small. In different areas of the scenario, the proposed positioning algorithm has a different improvement in positioning accuracy. In the area surrounded by the ANTs, the proposed algorithm shows only a slight improvement effect. However, in the rest of the area, the accuracy of the proposed algorithm is even higher than the CRLB of TDOA-only positioning. The simulations indicates that using the hybrid TDOA and TOA measurements as well as the localization algorithm proposed in this paper have the potential to effectively improve the positioning accuracy of the indoor distribution system.
The proposed localization algorithm and the two compared algorithms, Chan-Ho and BiasRed, were constructed and simulated according to the ranging errors under the line-of-sight wireless propagation channel. In real scenes, non-line-of-sight channels are ubiquitous, which may cause the positioning accuracy of these algorithms to decrease. In future work, the analysis and improvement of the proposed algorithm in the non-line-of-sight environment can be studied. Meanwhile, the proposed algorithm is only confirmed by simulation for a specific environment. The actual algorithm performance should be investigated in future work.