- Open Access
Improved positioning algorithm based on linear constraints on scatterers
© Zhou et al.; licensee Springer. 2013
Received: 5 December 2012
Accepted: 7 February 2013
Published: 21 March 2013
Non-line-of-sight (NLOS) error is a bottleneck problem influencing positioning accuracy. However, a large number of scatterers distribute randomly in the surrounding of the mobile station (MS) in the dense multipath environment, such as urban. In most cases, there is no obstacle between scatterer and MS. So, the geographic information of scatterers around MS can be used to restraint NLOS errors and improve the positioning accuracy. If scatterer can be regarded as the virtual base station (BS), the precondition of the positioning algorithm is easier to satisfy than the traditional positioning algorithm, such as circle positioning algorithm. The algorithm proposed in this article selects suitable scatterers with linear constraints by analyzing the Doppler frequency shift of received signals which reflected by scatterers. Thereby, the selected scatterers and only two real BSs form a complete positioning system. In addition, because MS is motionless in most scenarios, BS must be moving to acquire the Doppler frequency shift. The algorithm proposed in this article is adjusted for the scenarios. And the scatterers with linear constraint can also be utilize fully. Simulation results show the algorithm proposed in this article in two different scenarios, not only simplifies the traditional algorithm, but also achieves the higher positioning accuracy.
Non-line-of-sight (NLOS) error badly influences the accuracy of wireless positioning technology in the dense multipath environment, such as urban. In recent years, some algorithms using the information of scatterer have been developed to restraint NLOS errors. Their advantages of performance have attracted the attention of many researchers. These algorithms are mainly divided into two categories. The first category of the positioning algorithm utilizes different scatterer models to obtain some statistic information, e.g., probability density function of time-of-arrival (TOA) or angle-of-arrival (AOA). These statistic information are used to compute parameters which related to positioning. Scatterer models commonly assume that scatterers distribute in the surrounding of mobile station (MS) or base station (BS) [1, 2], giving an Elliptical scattering model which suited for the Microcellular communications system. The model assumed that scatterers uniformly distributed with the ellipse which the focus of ellipse are BS and MS  gives a disk model which suited for the macrocell communications system. In the model, scatterers are assumed to distribute with the disk around MS, and BS is out of the disk [4, 5], giving a type of geographical scattering model with non-uniform distribution of scatterers, for example, Gaussian distribution scattering model, Conical scattering model, Eccentro scattering model, and so on. According to these models, the statistic function, such as AOA, power azimuth spectrum, TOA, and power delay spectrum, can be acquired. Finally, some positioning algorithms based on scatterer model can apply TOA, AOA, or other parameters to compute the position of target [6–8].
The second category of positioning algorithm is independent of scatterer model, and utilizes the position of scatterer to compute the position of MS. The key of these algorithms is that NLOS error can be transformed into fixed factors. And positioning accuracy mainly depends on the accuracy of measurement parameters. Thereby, it is possible to achieve high positioning accuracy of scatterers and MS [9, 10], giving some scatterer positioning algorithms based on nonlinear least square theory. These algorithms utilize synthetically spatial–time-frequency information of received signal to compute the position of scatterer and MS. Because the received multipath signals reflected by scatterers include lots of information about scatterers, the position of scatterer can be computed in theory , giving a network grid search algorithm based on single reflection geometry model, which utilizes the information of multiple BS to restraint NLOS error. In [12–14], several possible results of positioning and partial information of scatterer can be acquired from the priori information of BSs, and the selection of the final correct result can be fulfilled by scatterer information. If BS can measure the AOA and the rate of AOA variation of the measurement signal from scatterer, the position of scatterer can be computed. So, the constraint relation among BS, scatterer, and MS can be designed and can also be used to restraint positioning error. In most cases, the scatterer is assumed to be static. If the position of scatterer is known, scatterer can be regarded as virtual BS. Thereby, a circle fitting algorithm can be proposed based on virtual BS. In fact, the relation among scatterers, MS and BS can be obtained by analyzing AOA and angle-of-departure of signal. Thereby, system of linear equations can be designed from TOA. Positioning algorithms based on the linear equations have much advantage. For example, scatters and MS can be positioned simultaneously. And positioning accuracy only depends on measurement parameter. But the constraint condition of the category algorithm is hard to satisfy. In addition, if we can successfully identify NLOS propagation or LOS propagation from related parameters, positioning accuracy would be improved very much. If the number of LOS is enough, positioning algorithms is independent of NLOS error [15, 16]. However, in a real environment, the number of LOS is too small or zero. So, the category of algorithms is hard to apply in real application. The algorithm proposed in this article is based on the second category of positioning algorithm. Referring to the previous study, multiple scatterers and MS in the same line are assumed to have the same absolute Doppler frequency shifts. Many methods that utilize linear constraint between scatterers and MS are introduced in . Three cases are commonly considered in positioning algorithm: (i) one scatterer and MS are in the same line; (ii) two scatterers and MS are in the same line; (iii) three or more scatterers and MS are in the same line. In the first case, if only one scatterer is in the same line with MS, it is impossible to acquire enough scatterer by multipath signal pairing. So, the positioning algorithm proposed in this article is not effective. The third case is studied by so many researchers. However, the second case is more general than the third case. Thereby, the second case is more valuable in implementing the proposed positioning algorithm. Due to high performance of line constraint positioning and inappeasable precondition, this article proposed an improved positioning algorithm using linear constraint on two BSs [17–20].
The remainder of this article is organized as follows. Section 2 presents and analyzes traditional positioning algorithm, which uses linear constraint on scatterers. The key points multipath pairing and TOA reconstruction are analyzed. Section 3 introduces an improved positioning algorithm based on linear constraint on only two scatterers. Section 4 gives a special and valuable case for the moving BS situation. Then the proposed positioning algorithm in Section 3 is adjusted to suit for the new case. Section 5 provides the simulation and results. Finally, conclusion and the further work are given.
2. Traditional positioning algorithm using linear constraints on scatterers
3. Improved positioning algorithm based on linear constraints on scatterers in two BSs
Figure 1 shows a brief frame of the improved algorithm, which comprises three steps.
Step 1: Multipath signal pairing: The BS receives the multipath signal and measures the Doppler frequency shift of the signal. The signals that reflected from scatterers to BS are then paired. Two scatterers that lie on the same line with MS are selected.
Step 2: TOA reconstruction: After multipath signal pairing and scatterer selection, the chosen scatterers can be positioned based on the AOA measured by the two BSs. The distances between MS and the two scatterers can be computed. Finally, by applying the linear constraints of two scatterers, we can implement TOA reconstruction.
3.1. Multipath signal pairing
Two datasets are paired when EN is close to minimum. Thus, four datasets from two BSs (l1i, β1i) and (l2i, β2i), (l1i, β1i) and (l2j, β2j), (l1j, β1j) and (l2i, β2i), and (l1j, β1j) and (l2j, β2j), are grouped into four pairs. The EN s of the datasets can be computed using Equation (14). The dataset with the smallest EN must be reflected by the same scatterer and must be measured by two BSs separately; that is, the set must have correct pairing. So, the other sets of data measured by the same two BSs are also correct pairing.
3.2. TOA reconstruction
Thus, the line relation among S i , S j , and MS can be determined by L i , L j , and L s .
If L j > L s and L j > bL i , then S i is located between S j and MS.
If L i > L s and L i > L j , then S j is located between S i and MS.
If L s > L i and L s > L j , then MS is located between S j and S i .
Similarly, the resolution of cases (1) and (2) is still in effect in case (3).
3.3. Positioning parameter of mobile computation
The position of MS can also be computed by using the LS algorithm, as shown in Equation (8).
4. Improved algorithm on scatter linear restriction based on moving BS
The BS receives a multipath signal and measures the Doppler frequency shift of that signal. Signals reflected from scatterers to BS are then paired. Among the pairings, two scatterers that lie in the same line with the MS are chosen.
After the paired multipath signal that is reflected by two scatterers is chosen, the positions of the paired signal can be computed by utilizing the circle fitting algorithm [7, 8] on the basis of the trajectory information of a single BS, TOA, and AOA.
The TOA is reconstructed in each measurement point of any mobile trajectory of the BS.
The measurement points of two scatterers and BS in the mobile trajectory are regarded as virtual BSs, and the reconstructed TOA is the TOA of LOS.
Thus, the position of the MS can be computed by employing the LS algorithm. The improved algorithm has two key points: multipath signals pairing and TOA/AOA reconstruction.
4.1. Scatterer position estimation by circle fitting algorithm
During the entire motion estimation, the positions of the two scatterers are regarded as scatter points and are assumed stationary.
Here, (x n , y n ) is the coordinate of the n th measurement point, l i (n) is the TOA of the measurement point, and α i (n) is the AOA of the measurement point.
4.2. TOA reconstruction of measurement point
If L s > L i and L s > L j , then L n LOS can be obtained by Equations (31) or (32).
4.3. Computation of MS position
5. Simulation and analysis
Coordinate of MS
Velocity along the x-axis of MS
Velocity along the y-axis of MS
Coordinate of MS
Velocity along the x-axis of BS
Velocity along the y-axis of BS
RMSE comparison of different algorithms (m)
Multipath signal number
Improved algorithm based on static BS
Improved algorithm based on moving BS
This article proposes an improved traditional positioning algorithm by using scatter constraint information. The improved algorithm suits more common cases and performs better than the traditional algorithm. Further research is necessary to improve this study.
This article mainly analyzes the positioning algorithm based on linear constraint on scatterers and gives the drawback that the precondition of traditional algorithm is hard to be satisfied. So, an improved algorithm is proposed to reduce the impact of the drawback. This article first analyzes the key technology of the algorithm, such as multipath signal pairing and TOA reconstruction. Then in the algorithm proposed this article, the number of scatterer for positioning is reduced to two. The improvement can enhance the application range of algorithm, and would not deteriorate the positioning performance. Second, when MS is stationary, the proposed algorithm is adjusted to suit a special case which BS is moving. Both improved algorithms have the same advantage and the positioning performance. These algorithms mainly focus on the design of positioning algorithm, and the restriction of NLOS error. There are many other factors which influenced positioning accuracy, such as signal form and signal band. We will continue the research in the further work.
This work was supported by the special fund of Chongqing key laboratory (CSTC).
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