目錄

ABSTRACT

INTRODUCTION

LINEAR TOA LOCALIZATION

MULTIDIMENSIONAL SIMILARITY ANALYSIS

SUBSPACE BASED LOCALIZATION

SIMULATION RESULTS

CONCLUSION

REFERENCES

ABSTRACT

A novel noise subspace based method is applied to the minimum localization system using time-of-arrival (TOA) measurements from three base stations (BS). Since the distance measurement between the mobile station (MS) and the BS bears analogy to the multidimensional similarity (MDS) between their coordinates, we express the MS coordinate as the linear combination of the BSs’ coordinates, where the weight vector lies in the noise subspace of the MDS matrix. It is proved that this weight vector is the area coordinate of the MS when the triangle formed by the three BSs serves as the reference frame.
Because the dimension knowledge of the localization problem is utilized to estimate the noise subspace and to mitigate the errors in TOA measurements, the proposed method is superior to the ordinary linear localization method in most of the enhanced quadrants of the area
coordinates system.

基於噪聲子空間的新方法應用於來自三個基站（BS）的到達時間（TOA）測量的最小定位系統。 由於移動臺（MS）和BS之間的距離測量類似於它們座標之間的多維相似性（MDS），我們將MS座標表示為BS座標的線性組合，其中權重向量位於噪聲中 MDS矩陣的子空間。 證明了當由三個BS形成的三角形作為參考框架時，該權重向量是MS的區域座標。

由於利用定位問題的維數知識來估計噪聲子空間並減輕TOA測量中的誤差，所提出的方法在區域座標系的大多數增強象限中優於普通線性定位方法。

INTRODUCTION

Recently, positioning of the mobile station (MS) in cellular systems attracted large interest and the number of applications based on the location information grew rapidly. Large-scale deployment of such applications usually requires methods for positioning that are accurate
and simple enough to be used in mobile phones.
Many linear methods have been proposed to estimate the MS position in closed-form.

The first and simplest of the existing methods is the Cell ID (CID) method where the position estimate is the coordinate of the serving BS.

The second of the existing methods is described in [1], the MS position is calculated as the average (centroid) of the positions of all N BSs whose beacons the MS can decode. It is called UnWeighted Centroid (UWC) method.

The CID method can be regarded as a special case of UWC with N=1. In [2], three approximated linear methods were proposed to estimate the position of the MS. The first, Path-Gain Weighted Centroid (PGWC) method, is based on signal strength, the second, Time Weighted Centroid (TWC) method, is based on time, and the third, PGWC+TA, is a hybrid between time and signal-strength methods.

Finally, the third benchmark method uses circular multilateration of propagation delays. This is sometimes known as Time-of-Arrival (TOA) [3]. In our implementation of TOA localization, the ordinary method calculates the MS positions from the MS-BSi propagation delays using a linear and closed-form solution [4].

In this paper, we propose a simple but quite accurate localization method and compare it with the ordinary method by means of simulation. Our results show that in the noisy measurement environment, the proposed algorithm is superior to ordinary linear localization methods in most of the enhanced quadrants of area coordinates system. Unlike the enhanced quadrant-aware localization method in [5] and [6], the linear weight vector in the proposed method is derived from multidimensional similarity analysis.
Our proposed methods are very simple and do not require complicated calculations as opposed to some iterative methods [7]-[9]. The performance measures are coordinate bias and cumulative distribution function of the root mean square location error, and the evaluation is done for seven quadrants in the area coordinate system.
Because the dimension knowledge of the localization problem is utilized to estimate the noise subspace and to mitigate the errors in range measurements, the proposed method is superior to the ordinary linear localization method in most of the enhanced quadrants.

注：紅色字型部分是論文所做的工作！

LINEAR TOA LOCALIZATION

Consider the problem of MS location using range measurements from three BSs. Assume that the BSs locate at   and the MS locates at    in the system of rectangular coordinates.
The MS position can be expressed as

It is the linear combination of BSs’ position, where the weight vector  is not unique. Introducing the following constraint

Equation (1) becomes

From Cramer’s determinant formula, we have

It can be proved that the absolute value of  is the normalized area (with respect to the area of triangle formed by BS1, BS2 and BS3) of triangle formed by (MS, BS2, BS3), (MS, BS1, BS3) and (MS, BS1, BS2), respectively.

可以證明  的絕對值是由（MS，BS2，BS3），（MS，BS1，BS3）和 （MS，BS1，BS2）形成的三角形的歸一化區域（相對於由BS1，BS2和BS3形成的三角形的區域）。

Therefore, we call    area coordinates of the MS. The signs of    are determined by the enhanced quadrant in which the MS lies [5]. As shown in Fig.1, we have seven quadrants in area coordinate system, whereas four in rectangular coordinate system.

因此，我們稱MS為  區域座標。  的符號由MS所在的增強象限決定[5]。 如圖1所示，我們在面積座標系中有七個象限，而在直角座標系中有四個象限。

From (3), we have the following orthogonal property,

which means that the vector    lies in the orthogonal  subspace  spanned  by vectors  and .

If the vector    is estimated from TOA measurements, the MS position can be calculated from (4).
In [5], the absolute value of area coordinate of MS is estimated by using Heron formula, but the sign is determined by the enhanced quadrant information. When this information is not known, it may be determined from an original estimation of MS position. However, when the MS locates close to the axes of the reference triangle, the ambiguity of the enhanced quadrant will degrade the location performance.
In the following two sections, we introduce the multidimensional similarity (MDS) matrix and establish the relationship between the vector   and the null  space of the MDS matrix. Based on this relationship, we can calculate the vector from the null subspace of the MDS matrix without additional quadrant information.

MULTIDIMENSIONAL SIMILARITY ANALYSIS

Assume that the distance between the i-th BS and the MS is  , between the i-th BS and the j-th BS is  . Note that  and  . Define a symmetric matrix as

Denote the element of matrix D by  , i, j=1,2,3.
From cosine formula, we have

where  is  the  angle  between  the  vectors  and . Therefore, we have

It can be seen that the matrix D is semi-positive with the rank equaling to 2 if vectors  and  is not correlated, and , the elements of Matrix D is the correlation between the vectors 

 and

Because measures the similarity between two vectors, matrix D is also called multidimensional similarity matrix [10].

SUBSPACE BASED LOCALIZATION

Let the eigenvalue decomposition of   D be 

where ,  and 

Because the rank of matrix D equals to 2, we have

and 

From (6), we have

Comparing (4) with (7) yields

where

is sum of the elements of vector  . From (7), a novel subspace based position estimation can be given by

SIMULATION RESULTS

In this section, we compare the proposed location method with the ordinary method, which gives the position estimation as [4]

As shown in Fig.1, three base stations locate at (0, 0), (2500, 4330) and (5000, 0), and the range measurement error is Gaussian distributed with zero mean and standard deviation of 30, all the units are meter. To compare the performance of the above two methods, seven mobile stations are fixed in different quadrant of area coordinate system.
Table 1 gives the comparison results of the root mean square location error. They are obtained from 300 runs at each MS position. It can be seen that the improvement of the proposed method is significant in quadrant 2, 4, 5 and 7. Because the root mean square location error is not enough to reflect the distribution of location error, we also plot in Figs.2-5 the estimation bias of x and y coordinates obtained in each run.

如圖1所示，三個基站位於（0,0），（2500,4330）和（5000,0），距離測量誤差為高斯分佈，零均值和標準差為30，所有單位 是米。 為了比較上述兩種方法的效能，將7個移動臺固定在不同的區域座標系的象限中。

表1給出了均方根位置誤差的比較結果。 它們來自每個MS位置的300次執行。 可以看出，所提方法的改進在象限2,4,5和7中是顯著的。由於均方根位置誤差不足以反映位置誤差的分佈，我們也在圖2-5中繪製。 每次執行中獲得的x和y座標的估計偏差。

Comparison of location error in different quadrant of area coordinate system is illustrated in Figs.2-5. The left two columns of these figures are the location bias of x and y coordinates obtained from the proposed method and the ordinary method, respectively, and the right column is the cumulative distribution function (CDF) of the root mean square location error (solid line: the proposed method, dotted line: the ordinary method). Though the location error distributes differently in each quadrant, it can be seen that the improvement of the proposed method is significant in most of quadrants, except quadrant 3.

區域座標系不同象限的位置誤差比較如圖2-5所示。 這些圖的左兩列分別是從所提出的方法和普通方法獲得的x和y座標的位置偏差，右列是均方根位置誤差的累積分佈函式（CDF）（實線） ：提出的方法，虛線：普通方法）。 儘管位置誤差在每個象限中的分佈不同，但可以看出，除了象限3之外，所提方法的改進在大多數象限中都很重要。

CONCLUSION

This paper establishes the relationship between the weight vectors of the enhanced quadrant-aware method and the proposed dimension-aware method, which estimate the rectangular coordinates of the mobile station by linear combination of that of the base stations. Unlike the former method, the linear weight vector is estimated from multidimensional similarity analysis and the additional enhanced quadrant information is not required in the proposed method. Simulation results shown that the proposed method performs better than the ordinary method in most of quadrants in area coordinate system.

本文建立了增強象限感知方法的權向量與所提出的維數感知方法之間的關係，該方法通過基站的線性組合來估計移動臺的直角座標。 與前一種方法不同，線性權重向量是從多維相似性分析估計的，並且在所提出的方法中不需要附加的增強象限資訊。 模擬結果表明，該方法在區域座標系中的大多數象限中均優於普通方法。

注：至於模擬我會試一下，將這片論文拿出來做筆記的原因在於覺得這是一種不錯的方法。我們可以觸類旁通，用到其他方面，不僅僅是在通訊中。例如，電子對抗定位中。

模擬地址：https://blog.csdn.net/Reborn_Lee/article/details/84206489

REFERENCES

[1] N. Bulusu, J. Heidemann, D. Estrin, “GPS-less Low-Cost Outdoor Localization for Very Small Devices”, IEEE Personal Communications Mag., Vol. 7, No. 5, pp. 28-34, Oct 2000.

https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=878533&tag=1
[2] M. Berg, “Performance of mobile station location Methods in a Manhattan Microcellular Environment”, Technical Reports from Signal Processing, Royal Institute of Technology, Sweden, Oct 2002.

https://www.researchgate.net/publication/2362632_Performance_Of_Mobile_Station_Location_Methods_In_A_Manhattan_Microcellular_Environment
[3] J.J. Caffery, G.L. Stüber, “Overview of Radiolocation in CDMA Cellular Systems”, IEEE Communications Mag., Vol. 36, No. 4, pp. 38 45, Apr 1998.

https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=667411&tag=1
[4] J. J. Caffery, Wireless Location in CDMA Cellular Radio System, Kluwer Academic Publishers, Boston Dordrecht London, 2000.

http://gpreview.kingborn.net/268000/7c9a7a245ef74bf4b743fed11886cf73.pdf
[5] Q. Wan, S. J. Liu, F. X. Ge, J. Yuan, Y. N. Peng, W.L. Yang, “A Quadrant-Based Range Location Method”, IEEE Conference on VTC, pp. 210-212, Apr 2003

https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1207532
[6] Q. Wan, Y. N. Peng, “An Improved 3-Dimensional Mobile Location Method Using Volume Measurements of Tetrahedron”, IEICE Transactions on Communications, Vol. E85-B, No.9, pp. 1817-1823, Sept. 2002

https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1021473
[7] Y. T. Chan, K. C. Ho, “A Simple and Efficient Estimator for Hyperbolic Location”, IEEE Transaction on Signal Processing, Vol.42, No.8, pp. 1905-1915, 1994.

https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=301830
[8] J. O. Smith, J. S. Abel, “The Spherical Interpolation Method of Source Localization”, IEEE Journal of Oceanic Engineering, Vol. 12, No. 1, pp. 246-252, 1987.

https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1145217
[9] Q. Wan, Y. N. Peng, “Enhanced Linear Estimator for Mobile Location using Range Difference Measurements”, IEICE Transactions on Communications, 2002, E85-B(5): 946-950.

[10] I. Borg, J. Lingoes, Multidimensional Similarity Structure Analysis, Springer-Verlag, New York, 1987.

https://link.springer.com/book/10.1007%2F978-1-4612-4768-5

原論文地址：https://download.csdn.net/download/reborn_lee/10792598