- Open Access
Enhancement of wireless positioning in outdoor suburban NLOS environment using hybrid-network-GPS systems
© Al-Jazzar; licensee Springer. 2012
- Received: 20 November 2011
- Accepted: 10 March 2012
- Published: 10 March 2012
This article will introduce a method for locating mobile stations (MSs) in outdoor suburban non-line-of-sight (NLOS) environment. The measurements used to locate the MS are taken from three base stations and a satellite. Such a setup of measurements is named the hybrid-network-GPS system. The proposed method uses constraint nonlinear optimization to minimize the NLOS error. The problem is simplified to three independent nonlinear equations of three unknowns, then it is solved to find the MS location. Numerical simulations are introduced to assess the performance of the proposed method compared with other positioning algorithms.
- constrained minimization
The well-known non-line-of-sight (NLOS) problem in wireless location has gained a great attention in the last decade. An important aspect of this problem is the huge error it induces in locating wireless devices. Thus, it is of interest to develop wireless positioning algorithms that will minimize the NLOS error.
In the literature, many articles addressed the NLOS problem. These articles differed in their approaches to solve the problem. The articles in [1–6] used nonstatistical methods to locate the mobile station (MS) using network based measurements. In , the authors propose using linear-programming method to locate the MS in NLOS environments. The authors in , linearize the inequality of range models corrupted with NLOS errors for wireless positioning. In , the authors propose a constrained-optimization algorithm to locate the MS using the sequential quadratic programming algorithm. The authors in  propose a geometry-assisted location estimation algorithm utilizing the different geometric layouts between the MS and the base stations (BSs). In , the authors propose a constrained optimization technique to locate the MS. The method in  is a constrained nonlinear optimization approach, with constraints derived from the geometry of the cell layout and range measurements. Satellite assisted techniques were also proposed in [7, 8] for wireless positioning but they did not assume NLOS environment and measurements were taken at more than one satellite.
This article uses the hybrid-network-GPS system to minimize the NLOS error when locating the MS. The proposed method is named the hybrid-network-GPS constrained algorithm (HNGPS-CA) which depends on minimizing a constraint objective function to locate the MS. The proposed objective function used in the HNGPS-CA method has the advantage of guaranteed convexity. This advantage is not guaranteed in the regularly used least square (LS) objective function which might be non-convex in some cases as will be clarified in Section 4. The environment considered in this article is the outdoor suburban NLOS environment. We will consider that there are three time of arrival (TOA) measurements available from three BSs and only one TOA measurement from the satellite.
The rest of the article is organized as follows: Section 2 presents the problem formulation. Section 3 presents the HNGPS-CA algorithm. In Section 4, some insight on the convexity of the objective function will be provided. Section 5 shows simulation results for the proposed method. Conclusions are presented in Section 6.
Since the consideration for this article is the outdoor suburban environment, then it is reasonable to assume that it is highly unlikely that there will be obstacles between the MS and the GPS satellite in such an environment. Thus, the measured distance between the MS and the GPS satellite will be assumed to be the same as the true distance (R s ) added to it the measurement noise (μ s ), i.e., . The measurement noise (μ s ) is assumed to be additive white Gaussian noise (AWGN) of zero mean and variance of .
This assumption is not valid for the connection between the MS and the BSs where it is very likely that obstacles exist in the MS-BSs connections. Thus, we will consider that the distance measurements at the BSs will have NLOS error (unlike the measurement made at the MS-GPS connection).
where η i is the effective NLOS component and μ i is the measurement noise, which can be considered as AWGN of zero mean and variance of . Since the NLOS causes the signal to arrive from a path which is longer than the true distance, then η i ≥ 0. It is generally considered that the NLOS error (η i ) is usually much more severe than the measurement noise (μ i ).
This distance can be practically obtained by assuming the MS will receive the signal from the satellite and then transmits the TOA measurement to BS i or retransmits the same signal and the BS i computes the total distance .
Since the GPS satellite is located far from the earth surface (i.e., R s is a huge distance when compared to cells dimensions), then is much greater than l i .
Next, the HNGPS-CA method is presented to locate the MS accurately in the presence of the NLOS error.
where α i (for i = 1; 2; 3) are positive weighting factors for the objective function (i.e., α i > 0). Each positive weighting factor (α i ) can be chosen separately, depending on the measurement accuracy  or the measurement geometry . Recall that R i is function of xms and yms as shown in (1), and they are the variables over which FHNGPS-CA(xms, yms) in (5) is minimized.
is the fact that using the objective function in (5) is assured to be convex. Whereas, the one in (6) might be non-convex in some cases as will be explained in Section 4.
Thus, (7) provides an equality constraint on finding (xms, yms) when applying the HNGPS-CA.
where λ is the Lagrange multiplier.
Thus, (9), (10), and (11) provide three independent nonlinear equations with three unknowns (xms, yms, λ). These equations can be solved to find the unknowns especially the MS location coordinates (xms, yms). We will use numerical techniques to solve these equations. Here, we use fsolve.m in MATLAB to solve the set of nonlinear equations in (9), (10), and (11).
As shown in Section 3, we use the objective function FHNGPS-CA(xms, yms) in (5) rather than the LS obcetive function (FLS(xms, yms)) shown in (6). To verify this choice, let us look at the objective functions FHNGPS-CA(xms, yms) and FLS(xms, yms).
The results in this section are averaged over 2,000 ensemble runs. The BSs locations are (0, 0), (8.66, 0) and (4.33, 7.5) with all units in km. The MS location is (xms, ym s) where xms = 4.33 · (u + 0.5) in km, yms = 0.5 · (7.5 + u) in km and u is a random variable uniformly distributed in the region [0,1]. Two NLOS environment models are assumed. The first case considered to model the NLOS effect is the disk of scatterers (DOS) environments presented in . In the DOS model, the scatterers are located on a solid circular disk of fixed radius R d with the MS at the center. The distance to a scatterer from the MS, rDOS, is uniformly distributed over [0, R d ], and the angle is uniformly distributed over [0, 2π]. The radius R d is set to 300 m. In the second case we assume that the NLOS error is a ratio of the measured distance. The reasoning behind this assumption is the fact that as the wave travels through a larger distance there will be a higher probability that the NLOS error will be larger. Thus, in this case η i = βl i where β is the NLOS error factor and it is assumed to be 0.3 in the simulations. The range measurement error is assumed to have 50 m2 variance, i.e,. The weight factors (α1, α2, α3) in (9) and (10) are all set to 1.
An algorithm, named HNGPS-CA, is proposed and investigated via simulations for mitigating the effect of NLOS error in outdoor suburban NLOS environment. The HNGPS-CA algorithm locates the MS by minimizing an objective function formed from TOA measurements at three BSs and utilizes the equality constraint provided by the GPS TOA measurement. This proposed objective function of the HNGP-CA algorithm has the guaranteed convexity advantage over the regularly used LS objective function. Simulations results showed that the proposed HNGPS-CA algorithm gave better performance than other positioning algorithms.
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