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Abstract

To enhance the effectiveness and accuracy of mobile station location estimation, author utilizes time of arrival measurements from three base stations and one angle of arrival information at the serving base station to locate mobile station in non-line-of-sight environments. This article makes use of linear lines of position, rather than circular lines of position, to give location estimation of the mobile station. It is much easier to solve two linear line equations rather than nonlinear circular ones. Artificial neural networks are widely used techniques in various areas due to overcoming the problem of exclusive and nonlinear relationships. The proposed algorithms employ the intersections of three linear lines of position and one angle of arrival line, based on Levenburg–Marquardt algorithm, to determine the mobile station location without requiring a priori information about the non-line-of-sight error. The simulation results show that the proposed algorithms can always provide much better location estimation than Taylor series algorithm, hybrid lines of position algorithm as well as the geometrical positioning methods for different levels of biased, unbiased, and distance-dependent non-line-of-sight errors.

Keywords

Time of arrival angle of arrival non-line-of-sight Levenburg–Marquardt algorithm

Introduction

The basic concepts of positioning system are to gather measurement information to form a location estimate. Wireless location techniques include signal strength (SS), angle of arrival (AOA), time of arrival (TOA), and time difference of arrival (TDOA). The SS scheme conveys the path loss attenuation to distances through a pre-determined model. For AOA schemes, using antenna arrays at the serving base station (BS), the arrival angle of the mobile station (MS) to the BS can be measured. TOA location scheme measures the propagation time of the radio wave, which is traveling between the MS and a BS with velocity of light. The TDOA estimates MS location by ascertaining the time difference of the arriving radio signal sent by BSs. Both time-based and angle-based schemes have their advantages and disadvantages. The angle-based schemes require only two BSs to determine the MS location and do not require synchronization among the BSs. The time-based schemes require at least three BSs. The time-based schemes generally can give better accuracy than angle-based schemes. The mobile positioning techniques can provide some location-based services. Applications of wireless location-based services include the wireless emergency services E-911 and the intelligent transportation system (ITS).¹

The major concern that affects positioning accuracy is the non-line-of-sight (NLOS) propagation effect. If line-of-sight (LOS) propagation between the MS and every BS is existent, MS location estimation can be with high accuracy. Due to the reflection and/or diffraction, even when multipath is absent and high-resolution techniques are used, NLOS propagation will be also with bias angle and/or time measurement errors. Thus, for angle-based location systems, the measured angles are not equal to the true angles to the MS. In time-based location systems, the signal may transmit with a longer path from BS to MS, and a positive error is produced. Mogi and Ohtsuki² proposed a new positioning algorithm using TOA and received-signal-strength methods to estimate MS location. The proposed algorithm does not need to distinguish the LOS or NLOS propagation environment. This algorithm does not need to distinguish the LOS or NLOS propagation condition. A novel algorithm, geometry-constrained location estimation (GLE) using two-step least square (LS) method, was proposed in this article to improve the accuracy of the MS location estimation.³ There are two kinds of system models, one is using two TOA and one AOA measurements and the other is using three TOA measurements. A Markov-transitioned fuzzy-tuned hybrid framework is proposed for modeling the moving of a MS when received LOS or NLOS measurements.⁴

Hearability has been used to analyze the environment type and is an issue in rural cellular communication systems.^1,5,6 In rural areas, the hearability of an MS for neighboring BSs is more serious. The coverage in urban areas is much higher than in rural areas. Therefore, in rural areas, it is difficult to detect three or more BSs for MS location estimation. To achieve better accuracy and hearability problem, combining more different schemes for MS location estimation is reasonable. Hybrid technology improves the availability of the location service. Hybrid location methods can be categorized as follows: (1) fusion of global positioning system (GPS) and terrestrial-based schemes and (2) terrestrial-based hybrid algorithms that combine various measurements such as SS, AOA, TOA, and TDOA. The fusion of current assisted global positioning system (A-GPS) and TDOA/TOA methods for location prediction is proposed in Zhao.⁷ By combining TDOA and AOA, a hybrid algorithm that provides more accurate positioning for wideband code division multiple access (CDMA) cellular systems as proposed in Cong and Zhuang.⁸ The scheme combined the TDOA and AOA measurement to estimate the position of MS has been proposed in NLOS transmission environment.⁹ The results show that the accuracy of positioning can be improved even in the serious NLOS propagation environments. The location of MS can be estimated by at least one BS which can measure the TOA, AOA, and Doppler-shifted frequency (DSF).¹⁰

Back-propagation neural network (BPNN) is one of the most widely used artificial neural network (ANN) models and is the most popular technique for classification and prediction.¹¹ BPNN has been used in many applications in different fields such as image processing and signal processing. However, BPNN usually slowly converges and get easily stuck in local minima. The Levenburg–Marquardt (LM) algorithm performs the best manner of convergence in back-propagation training process, since it reaches convergent state continuously but slowly with a series of steepest-descent methods. However, Gauss–Newton algorithm is radically different in this property. Considering both effectiveness and efficiency, this article proposed a novel positioning algorithm based on LM algorithm to estimate the MS location. The LM neural network finds the relation of feasible intersections and the true location in the training process. After constructing the model, we input the new measured feasible intersections in it and estimate the MS location.

Instead of the circular one, linear lines of position (LOP) equation derived from the two-circle intersections are used in this article. It is not necessary to calculate the intersections of various circles, the linear LOP can be used to reduce the computation load. Due to NLOS errors, the MS location is located at the intersections of three linear LOP and one AOA line when three BSs are available. Due to positive errors between MS and BS, the true MS location must lie in the overlap region of the three circles. The intersections in this region are defined as feasible intersections. During the training period, LM is derived to establish the functional relationships between the MS location and the feasible intersections. After training, LM is employed for MS location prediction. Numerical results demonstrate that the proposed algorithm can offer better positioning accuracy even in NLOS conditions of biased, unbiased, and distance-dependent errors.

The rest of this article is organized as follows. Section “Related localization methods” describes Taylor series algorithm (TSA) and hybrid lines of position (HLOP) algorithm, as well as the geometrical positioning methods. Section “Traditional BPNN algorithm and LM algorithm” briefly describes BPNN and LM algorithms. Author proposes the LM-based algorithm with HLOP to determine the MS location in section “Proposed location algorithm based on LM algorithm.” Next, section “Simulation results” discusses the simulations performed to test the proposed algorithm and the comparisons with other methods. Finally, section “Conclusion” draws some conclusions.

Related localization methods

Because of hearability issues, the number of BSs available is set to three. To give a location estimate of the MS, TSA,^12,13 HLOP,¹⁴ and the geometrical positioning methods¹⁵ are commonly used schemes. For ease of presentation, they are briefed as follows.

TSA

First, TSA gives an estimate of MS location as an initial value and then does recursive iterations again and again. However, the inaccurate estimate value may lead to a divergent result after performing iterative processes.

HLOP

We define the true distances between BSi and MS as

$r_{i} = \sqrt{{(x - X_{i})}^{2} + {(y - Y_{i})}^{2}}$ (1)

where $(x, y)$ and $(X_{i}, Y_{i})$ are the locations of MS and BSi, respectively. In addition, the serving BS, that is, the one that is the closest to the MS, can measure the arrival angle to the MS with respect to a reference direction (for instance the x-axis). If BS1 is the serving BS, and $θ$ denotes the angle between MS and its serving BS, then one has

$θ = \tan^{- 1} (\frac{y - Y_{1}}{x - X_{1}})$ (2)

By squaring and subtracting from equation (1), the linear LOP equations can be found. The linear LOP, rather than the nonlinear circular LOP, can be applied to determine the MS location. As shown in Figure 1, we can rotate and shift the coordinate system without loss of generality and simplicity so that BS1 is located at (0, 0) and BS2 and BS3 are located at $(X_{2}, 0)$ and $(X_{3}, Y_{3})$ , respectively. Please refer to Venkatraman and Caffery¹⁴ and Caffery¹⁶ for the detail of LOP algorithm with TOA measurements and HLOP algorithm with both TOA and AOA measurements, respectively. The MS location is determined by

$G φ = h$ (3)

where $φ = [\begin{matrix} x \\ y \end{matrix}]$ denotes the MS location

$G = [\begin{matrix} X_{1} - X_{2} & Y_{1} - Y_{2} \\ X_{1} - X_{3} & Y_{1} - Y_{3} \\ \tan θ & - 1 \end{matrix}]$

and

$h = \frac{1}{2} [\begin{matrix} r_{2}^{2} - r_{1}^{2} + (X_{1}^{2} + Y_{1}^{2}) - (X_{2}^{2} + Y_{2}^{2}) \\ r_{3}^{2} - r_{1}^{2} + (X_{1}^{2} + Y_{1}^{2}) - (X_{3}^{2} + Y_{3}^{2}) \\ 2 (X_{1} \cdot \tan θ - Y_{1}) \end{matrix}]$

Figure 1.

Geometric layout of the three circles and a line.

Again, the LS solution to equation (5) is given by

$φ = (G^{T} G)^{- 1} G^{T} h$ (4)

Some geometrical positioning methods

From a geometric point of view as in Figure 1, a circle centered at BS and a line linked between BS and MS are used for TOA and AOA measurements, respectively. When two BSs are available, we have presented hybrid schemes that utilize the solutions of two TOA circles and two AOA lines to determine MS location in Chen et al.¹⁵ The following three equations describe circles for TOA measurements, and the last equation describes a line for AOA

$Circle 1 : {(x - X_{1})}^{2} + (y - Y_{1})^{2} = r_{1}^{2}$ (5)

$Circle 2 : {(x - X_{2})}^{2} + (y - Y_{2})^{2} = r_{2}^{2}$ (6)

$Circle 3 : {(x - X_{3})}^{2} + (y - Y_{3})^{2} = r_{3}^{2}$ (7)

$AOA Line : \tan θ \cdot x - y = \tan θ \cdot X_{1} - Y_{1}$ (8)

The proposed several methods utilize the above equations to find all intersections to locate the MS. The hybrid methods reduce the NLOS errors using the weighted sum of the intersections between three TOA circles and one AOA line. In this article, we choose three representative methods such as the distance-weighted method, threshold method, and sort-averaging method to estimate MS location in the simulations.

Traditional BPNN algorithm and LM algorithm

BPNN algorithm

BPNN has been widely used for acquiring solutions for both linear and nonlinear functions in neural networks.¹¹ BPNN is a feed-forward structure that uses supervised learning methods to model biological processes. All neurons are connected to each other following layers, while those which in the same layer or in the interrupted layers are not connected. A supervised neural network learns from the training data and develops a model representing the input and output variables. For supervised methods, a set of input values and corresponding desired output values are employed as the training dataset.

There is an input layer, output layer, and several hidden layers in BPNN. Input layer receives an exogenous data from the external sources. It passes this information to the next layer of network for further processing. The hidden layer receives information from input layer and determines the mapping relationships between input and output layers. The output layer of a neural network receives and transforms the process information to form the response. In the hidden layer and the output layer, the activation function is set as linear transfer function in this article.

BPNN learning process minimizes the mean-square value of the difference between the actual and the desired vector by a gradient descent method. The error function $F$ is defined as

$F = \frac{1}{2} \sum_{k} {(T_{k} - O_{k})}^{2}$ (9)

where $T_{k}$ is the output vector of the network while $O_{k}$ is the desired output vector. The gradient descent algorithm is used to calculate the weight of the network and to minimize an error function. The basic weight update algorithm generates a new weight value according to the multiplication of learning rate (LR) ε and the negative derivative of error with respect to the previous weight

$Δ w_{ij} = - ε \frac{\partial F}{\partial w_{ij}}$ (10)

The network leads to an oscillatory and is hard to converge if a LR is too high. The network will take a long time to converge and may fall into local optimization if a LR is too low. This operation repeats up to the time when the output is correct for all training data or a certain number of iterations are reached. In order to get the optimal value of LR, trial and error is necessary until the network could provide the correct output for training data. The conventional BPNN often leads to poor efficiency, slow learning process, and trapping into local minimum value easily.

LM algorithm

The LM algorithm has faster convergence and training time than the traditional BPNN algorithm, so it will lead to better performance and efficiency.¹⁷ The algorithm interpolates between the Gauss–Newton method and gradient descent method. By neglecting the Hessian matrix calculation, its processing time is pretty close to the second-order training speed. Therefore, the LM algorithm is more robust than Gauss–Newton when applied in a wide variety of problems.

If a function $F (w)$ is to be minimized with respect to the parameter vector $w$ , then Newton’s method would be

$Δ w = - [\nabla^{2} F (w)]^{- 1} \nabla F (w)$ (11)

where $\nabla^{2} F (w)$ is the Hessian matrix and $\nabla F (w)$ is the gradient. If $F (w)$ reads

$F (w) = \sum_{h = 1}^{K} e_{h}^{2} (w)$ (12)

then it can be shown that

$\nabla F (w) = 2 J^{T} (w) e (w)$ (13)

$\nabla^{2} F (w) = 2 J^{T} (w) J (w) + 2 S (w)$ (14)

where $e (w)$ is the error vector, and $J (w)$ is the Jacobian matrix given by

$J (w) = [\begin{matrix} \partial e_{1} (w) / \partial w_{1} & \partial e_{1} (w) / \partial w_{2} & \dots & \partial e_{1} (w) / \partial w_{n} \\ \partial e_{2} (w) / \partial w_{1} & \partial e_{2} (w) / \partial w_{2} & \dots & \partial e_{2} (w) / \partial w_{n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \partial e_{N} (w) / \partial w_{1} & \partial e_{N} (w) / \partial w_{2} & \dots & \partial e_{N} (w) / \partial w_{n} \end{matrix}]$

and

$S (w) = \sum_{h = 1}^{K} e_{h} (w) \nabla^{2} e_{h} (w)$ (15)

For the Gauss–Newton method it is assume that $S (w) \approx 0$ , the Hessian matrix is given by

$\nabla^{2} F (w) = 2 J^{T} (w) J (w)$ (16)

The Gauss–Newton update is given by

$Δ w = - [J^{T} (w) J (w)]^{- 1} J^{T} (w) e (w)$ (17)

The LM algorithm modification to the Gauss–Newton method is

$Δ w = - [J^{T} (w) J (w) + μ I]^{- 1} J^{T} (w) e (w)$ (18)

where $μ$ is a variable small scalar which controls the learning process. The parameter $μ$ is multiplied by decay rate $β (0 < β < 1)$ whenever $F (w)$ is decreasing, where it is divided by $β$ whenever $F (w)$ increases in a new step. When the scalar $μ$ is very small or zero, the LM algorithm becomes Gauss–Newton algorithm, thus it can provide faster convergence. When $μ$ is very large, the LM algorithm approximates the steepest-descent method. Hence, the LM algorithm provides a good compromise between the convergent speed of Gauss–Newton and the guaranteed convergence of steepest descent.

Proposed location algorithm based on LM algorithm

We have proposed the algorithm using the feasible intersections of two TOA circles and two AOA lines, based on resilient back-propagation (Rprop) neural network technique, to estimate the MS location.¹⁸ Author has also proposed a novel positioning algorithm based on neural network to determine MS location when the MS is heard by only three BSs. The proposed algorithm uses the feasible intersections of three TOA circles (and one AOA line) to eliminate NLOS errors.

In order to reduce the computational load, a linear LOP is applied to solve the positioning problem with lower complexity. The linear LOP formed by the intersection points of two circles is used to replace the circular LOP. Simulation results show that the computational loads of using linear LOP is much less than using circular LOP.¹⁹ The proposed algorithm is based on a new approach to TOA measurements. The distance measured from each BS can form a circle, centered at the BS, and each circular intersects with another circular at two points. These two points can determine a linear line equation which passes through those two points. It is simpler to find the solution of two linear LOPs rather than nonlinear circle LOPs. The number of the available BSs is three, so we can generate three different linear LOP. Another issue is that how to obtain these linear line equations determined by three circle equations. By squaring and subtracting the distances obtained by equation (1) for $i = 1, 2, 3$ , the linear line equations can be found as follows

$L_{12} : (X_{2} - X_{1}) x + Y_{2} y = \frac{1}{2} (r_{1}^{2} - r_{2}^{2} + X_{2}^{2} + Y_{2}^{2} - X_{1}^{2})$ (19)

$L_{13} : (X_{3} - X_{1}) x + Y_{3} y = \frac{1}{2} (r_{1}^{2} - r_{3}^{2} + X_{3}^{2} + Y_{3}^{2} - X_{1}^{2})$ (20)

$L_{23} : (X_{3} - X_{2}) x + Y_{3} y = \frac{1}{2} (r_{2}^{2} - r_{3}^{2} + X_{3}^{2} + Y_{3}^{2} - X_{2}^{2})$ (21)

We only use two of three LOP, because the third one depends on the first two equations, in other words, these three lines will intersect at only one point, as shown in Figure 2. Under the environment without NLOS error, three LOP and one AOA lines intersect with each other at points near the true MS location. Signal of TOA travels for a longer time and results in radius extension for TOA circles, due to the NLOS effect. Thus, MS is located in the overlapping region of three circles. As mentioned earlier, if the intersecting points are enclosed by this area, they are defined as feasible intersections, as shown in Figure 1. Feasible intersections must satisfy the following equations

${(x - X_{1})}^{2} + (y - Y_{1})^{2} \leq r_{1}^{2}$ (22)

${(x - X_{2})}^{2} + (y - Y_{2})^{2} \leq r_{2}^{2}$ (23)

${(x - X_{3})}^{2} + (y - Y_{3})^{2} \leq r_{3}^{2}$ (24)

Figure 2.

The case having four intersections inside the feasible region.

From the geometric analysis, these four linear equations will form 1–4 feasible intersections inside the enclosed region of the three circles. There is an example in Figure 2, which has four feasible intersections in this case. The number of input variables has to decide first. Here, as shown in Figure 3, we proposed two types for input data collection.

Figure 3.

Structure of the prediction models for (a) Type 1 (divided type) and (b) Type 2 (composite type).

Type 1: divided type

Figure 3(a) shows the subset establishments based on the number of feasible intersections. Measurements of V intersections are assigned to the Vth subset. Therefore, total four subsets are constructed for training and each of them has distinct intersection numbers. Here describes the training steps:

Record the coordinates of V feasible intersections formed by three linear LOP and an AOA line;

If the number of feasible intersection is V, assign the V point(s) to the Vth subset;

Train these four subsets of different values of V.

Type 2: composite type

Figure 3(b) shows the second type of subsets for collecting V feasible intersections. There is only one kind of subset collecting points without considering the value of V. The summation of measurement numbers of four subsets in Type 1 is the same as the number in the only subset in Type 2. The detailed steps are as follows:

Record the coordinates of V feasible intersections in a subset and extend it;

Repeat the subset itself (4/V) times for V = 1, 2 or add an average value of the previous three elements as the fourth point for V = 3;

Place the extended subsets as the input training data for neural network.

After we decide the input data, we start to estimate MS location based on the LM algorithm. Before we start to train a network, the weights and biases should be initialized. Once the network weights and biases have been initialized, the network is ready for training period. The feasible intersections are fed to the input layer, and MS location is the only one variable in the output layer. The network has the following input–output mapping:

Input: V feasible intersections

Output: desired MS location

Instead of being a disadvantage to add a training phase, we propose several input–output patterns as a way to speed up and customize the algorithm to fit our goal in real applications. The feasible intersections and the desired MS location are collected for training purpose. In practical scenario, feasible intersections are analyzed by the trained network model to predict the real MS locations. Weights of the connections between layers are adjusted by the iterations to minimize the measurement difference between the estimated (actual) MS location and the real (desired) MS. After the training, the feasible intersections can pass through the trained LM more quickly, but predict the better appropriate MS location. In addition, we can find from the simulation results that when there are only 1000 pieces of input–output patterns to be used as training data, our proposed algorithm still works better and more precisely in the prediction of the position than the other methods.

Simulation results

We performed the computer simulations to examine the performance of the proposed location methods. The coordinates of the BSs are set to BS1: (0, 0), BS2: (1732 m, 0), and BS3: (866 m, 1500 m).²⁰ Each of the coordinates in the region bounded by BS1, I, J, and K has the same possibility to be selected as the MS location, as shown in Figure 4. We uses trial-and-error method to set up the parameters, such as the numbers of hidden layers and the training iterations before starting the iterating calculations, to reach the optimal settings of neural network. Regarding the NLOS effects in the simulations, three propagation models are adopted in this article, namely, the uniformly distributed noise model,²⁰ the biased uniform random error model,¹⁴ and the distance-dependent NLOS error model.²⁰

Figure 4.

Cell layout showing the relationship between the true ranges and inter-BS distances.

The first NLOS propagation model is called the uniformly distributed noise model,²⁰ in which the TOA measurement error is assumed to be uniformly distributed over $(0, U_{i})$ , for $i = 1, 2, 3$ , where $U_{i}$ is the upper bound of the error. The AOA measurement error is assumed to be $f = ϖ \cdot 5 \circ$ , where $ϖ$ (a random variable) is uniformly distributed in (−1, 1).³ Root-mean-square (RMS) error is used to quantify the positioning accuracy of the proposed algorithm, which is defined as the difference between the actual and the desired MS location. In our simulations, we have only considered single hidden layer, which is the most commonly used. It is well enough to model arbitrarily complex nonlinear functions. Figure 5 shows the relationship between the converged RMS error and the number of training iterations (epochs). The proposed training method reduces the error significantly during the preceding iterations. After the number of epochs increases to more than 1000, the performance almost does not improve.

Figure 5.

Positioning RMS error of convergence versus the number of epochs.

Furthermore, having too few hidden neurons will cause a bigger error. Increasing the number of hidden neurons can alleviate this problem, but it will affect the speed of convergence simultaneously. Some general rules to determine the number of hidden neurons are (1) $0.5 \cdot (p + q)$ , (2) $p$ , (3) $2 p + 1$ , and (4) $3 p + 1$ , where $p$ and $q$ are input neurons and output neurons, respectively.²¹Figure 6 shows the relationship between the converged RMS and number of the hidden-layer neurons. We can see that two hidden-layer neurons are enough to make good prediction performance. Thus, to avoid increasing the computation load, the proposed algorithm with two hidden neurons and 1000 epochs are used in the MS location estimation for the rest of simulations.

Figure 6.

Positioning RMS error versus the number of the hidden-layer neurons.

Figure 7 shows the effect of upper bound of NLOS error on the average location error for various methods. In the simulations, the initial guess of MS location in TSA is the true solution to allow for convergence. Simulations show that at least three iterations are required for TSA to converge. Average location error increases with the upper bound of NLOS error. We can see that HLOP can mitigate the NLOS error to some degree. But under highly NLOS conditions, the performance of the proposed algorithm is much better than TSA, HLOP, and some other geometrical positioning methods.

Figure 7.

Average location error versus the upper bound of NLOS errors for various methods.

To compare the computational burden, computing times of using circular LOP and linear LOP are measured in Table 1. The NLOS errors are assumed to be uniformly distributed. The computational time of employing linear LOP is much less than applying circular LOP. Calculating the intersections of the nonlinear circles has a large amount of complexity. The computational complexity of applying linear LOP is much less than using circular LOP.

Table 1.

Computing times for using circular LOP and linear LOP: Type 1 and Type 2.

	300 m	400 m	500 m	600 m
Type 1
	$U_{i}$
Circular LOP	0.0004373 s	0.0004423 s	0.0004449 s	0.0004356 s
Linear LOP	0.0001878 s	0.0001897 s	0.0001889 s	0.0001872 s
Type 2
Technique	$U_{i}$
Circular LOP	0.0004359 s	0.0004409 s	0.0004435 s	0.0004342 s
Linear LOP	0.0001859 s	0.0001875 s	0.0001867 s	0.0001851 s

$U_{i}$ : The upper bound of error.

BS1 has a better positioning accuracy of TOA as long as it acts as a serving BS for MS. The upper bound of the NLOS error for BS1 is 200 m and those of other BSs are 500 m. Figure 8 shows cumulative distribution functions (CDFs) of the location error for various methods. The proposed algorithm can still yield better performance. In addition, the divided type and the composite type provide nearly identical MS location estimation.

Figure 8.

CDFs of location error when the upper bound is used to model the NLOS errors.

We consider the second NLOS propagation model with the biased uniform random error.¹⁴ In words, the measured error of TOA between MS and $BS i$ is $η_{i} = κ_{i} + u \cdot λ_{i}$ , where $κ_{i}$ and $λ_{i}$ are constants, and $u$ is a uniform random variable over [0, 1]. Similarly, the measured error of AOA can be modeled as $| f | = α + ϖ \cdot ρ$ , where $α$ and $ρ$ are constants. The error variables are chosen as follows: $κ_{1} = 50 m$ , $κ_{2} = κ_{3} = 150 m$ , $λ_{1} = λ_{2} = λ_{3} = 200 m$ , $α = 2.5 \circ$ , and $ρ = 5 \circ$ . Figure 9 shows CDFs of the location error. We can see that TSA and HLOP offer the worst performance, and the proposed algorithm with the divided type and composite type really can make more accurate estimations of MS location.

Figure 9.

Comparison of location error CDFs with NLOS errors as biased uniform random variable.

The final NLOS propagation model is based on the distance-dependent NLOS error model. The NLOS range error for the ith range is taken to be $ξ_{i} = χ_{i} \cdot R_{i}$ , where $χ_{i}$ is a proportional constant and $R_{i}$ is the true range between ith BS and MS. The AOA measurement error is assumed to be $f = ϖ \cdot 5 \circ$ , where $ϖ$ (a random variable) is uniformly distributed in (−1, 1).³Figure 10 shows the CDF of the average location error with distance-dependent NLOS error, and the variables are chosen as follows: $χ_{1} = 0.1$ , $χ_{2} = 0.13$ , and $χ_{3} = 0.13$ . Compared with the other methods, the accuracy of MS location can be considerably improved with the proposed algorithms.

Figure 10.

CDFs of the location error with distance-dependent NLOS error.

Considering NLOS effects and measurement errors, simulation results are shown in Figure 11. NLOS errors are using the uniformly distributed noise model,²⁰ $U_{1} = U_{2} = U_{3} = 400 m$ . TOA and AOA measurement errors are assumed in some of the simulations to be independent Gaussian random variables with zero mean. The TOA measurement noise standard deviation is set to 50 m and AOA measurement noise standard deviation is 1°. The proposed LM-based algorithms always predict the better appropriate MS location.

Figure 11.

CDFs of location error when NLOS effects and measurement errors are considered.

The simulation results are obtained for the proposed algorithm and some other methods when four BSs are available. BS1 is located at (0, 0). Figure 12 shows CDF of the average location error of various methods when the errors are using the uniformly distributed noise model,²⁰ $U_{1} = 200 m$ and $U_{2} = U_{3} = U_{4} = 500 m$ . The proposed LM-based algorithms with Type 1 and Type 2 always generate more accurate MS location estimation than the other methods.

Figure 12.

Comparison of error CDFs when four BSs are available.

Conclusion

BPNN has disadvantages of taking long time for learning and possibly being held by a local minimum. This article proposed a novel LM-based algorithm for MS location estimation. This article proposed hybrid algorithms that combine TOA measurements at three BSs and AOA information at the serving BS to estimate the MS location. Using one AOA measurement and computing the linear line equations obtained from the nonlinear circular equations, we can utilize the feasible intersections of three linear LOP and one AOA line to obtain an approximation of MS location. The complexity can be reduced to solve the two linear line equations rather than nonlinear circle ones. After training, the proposed algorithm can reduce NLOS errors. Whenever we need to find the MS location, the feasible intersections can be applied as the trained input, and this model can predict MS locations quickly and precisely as we expected it should be. From the simulation, the proposed algorithms can always yield superior performance for different levels of biased, unbiased, and distance-dependent NLOS errors. Therefore, our proposed algorithm should be able to apply to the wireless communication system in real situations.

Footnotes

Academic Editor: Fei Yu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) received no financial support for the research,authorship,and/or publication of this article.

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Combination linear lines of position and neural network for mobile station location estimation

Abstract

Keywords

Introduction

Related localization methods

TSA

HLOP

Some geometrical positioning methods

Traditional BPNN algorithm and LM algorithm

BPNN algorithm

LM algorithm

Proposed location algorithm based on LM algorithm

Type 1: divided type

Type 2: composite type

Simulation results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References