3 D Localization Algorithm Based on Linear Regression and Least Squares in NLOS Environments

Based on the positive bias property of the time of arrival(TOA) measurement error caused by the non-line-ofsight(NLOS) propagation, a simple and effective three dimensional(3D) geometrical localization algorithm was proposed, the algorithm needs no prior knowledge of time delay distribution of TOA, and only linear regression was used to estimate the parameters of the relationship between the NLOS distance error and the true distance, thus, the approximate real distance between mobile terminal (MT) and base station (BS) was reduced, then, the 3D geometric localization of mobile terminal was carried out by the least square method. The experimental results shows the effectiveness of the algorithm, and the positional accuracy is far higher than the required accuracy by E-911 in NLOS environments.

always exist under NLOS propagation conditions.In order to reduce the influence of the NLOS propagation on the positioning accuracy of the TOA method, a large number of research were carried out.These localization algorithms in NLOS environments can basically be reduced to three types, namely, the accurate modeling of non sight distance error ( (CHEN，1999；CUI, et al.，2014；HUA, et al.，2014；YU, DUTKIEWICZ，2012).These algorithms have their own advantages, but their shortcomings are obvious.For example, it is particularly difficult to obtain the prior knowledge of the statistical characteristics of signals in actual environments, the estimating accuracy is not high, and it is difficult to be widely used.Therefore, the suppression of NLOS error has become the key to the practicability of wireless localization algorithm.Most of the work is focused on the research and optimization of the two-dimensional localization algorithm, such as the study of the TOA localization algorithm and the dynamic localization algorithm based on the mobile station position.These algorithms makes full use of the resources of the third generation mobile communication network, and achieved basically the E-911 positioning performance under the typical channel environment.However, a number of two-dimensional localization algorithms suitable for NLOS environment can not be applied directly to 3D scenes, and there are few literature to study the problem of 3D localization in NLOS environments (XIAO, CHEN, WANG, LI,& LI,2015) .
The paper transformed the TOA measured in NLOS environments into the distance between the terminal and the base station.Based on the positive bias of the measurement error of the TOA in the mobile communication environment, the measurement error caused by NLOS was suppressed by the linear regression method which estimated the linear relationship between the error and the real distance, then located the terminal's threedimensional position based on the least square principle,the performance of the algorithm was verified by the experimental results.The proposed algorithm can overcome the influence of the NLOS propagation error to a great extent, and it has high localization accuracy.

TOA Time Measurement Model and Localization Principle
In the mobile communication system, due to the influence of measuring equipment and signal propagation environment, there is error in TOA measurement of base station.Assuming that the measurement results of each base station are independent of each other, then the TOA of mobile terminal MT to the ith base station is (1) where is the line of sight(LOS) propagation time of signal between MT and ， is system measurement error that obeys the distribution, it can be reduced with the improvement of timing technology and signal detection technology.and only accounts for a small part of the TOA error, it is generally a Gauss random variable of zero mean, is the error introduced by NLOS propagation and is the main component of TOA error, it can be expressed by random variables of exponential distribution, uniform distribution or delta distribution (CHANG, & LV，2007), M is the number of base stations.Due to the existence of systematic measurement error and NLOS propagation error, (2) where is the speed of light, taking , is the distance between and MT obtained by measuring , is the real distance between the them, is the distance error caused by measuring, is the distance error caused by the NLOS propagation.There are usually in NLOS propagation environments, therefore, the has a positive bias.Establishing a series of equations by the measured distance ,taking each base station as the center of the ball, the as the radius of the sphere, calculating all the intersection lines of any two spherical surfaces,then,the intersection point of all intersection lines is the position of MT, as shown in figure 1. (5) let , , then the matrix form of the ( 5) is (6) therefore, in order to locate MT, at least four base stations are required for TOA measurement, and the 3D spatial position coordinates of MT are obtained by least square method, .
In practice, each in ( 4) is replaced by of (2) respectively, therefore, in LOS propagation environments with only the system error, the precision of the least square method is higher, however, in NLOS propagation environments, the error of the least square solution is larger due to the positive bias error.

Linear Regression Estimation Between the Measured Rrror and the True Distance
In order to avoid using TOA's prior know-ledge of time delay distribution, the between a position determined terminal and several base stations were measured, calculating out the true distance of the test terminal to these base station ,then, the distance error caused by NLOS propagation and so on is (8) The relation between the real distance and the measured distance error of a test terminal is shown in figure 2, the cross coordinates of each point in the graph is the true distance between the test terminal and different base station, and the ordinate is the difference between the measured distance and the real distance，i.e. the measured distance error.Drawing and analyzing the real distance and the measured distance error of different terminal in the same conditions, the result was basically similar to that in figure 2，and the linear regression significance test showed that there was a highly significant linear relationship between the measured distance error and the real distance of the terminal to the base stations,i.e., The formula (9) shows that in the same environment, the distance error from the same terminal to different base stations is positively related to the true distance, and the degree of correlation is basically consistent, then, substituting ( 9) into (8), obtained therefore, so as the appropriate value of k and a can be solved, the error caused by NLOS propagation can be effectively eliminated, and the approximate real distance between the terminal and the base stations can be calculated out by (10).

3D Localization Algorithm Based on the Least Squares Method
Substituting of (10) for of (5) respectively, obtained Substituting into (6), then, using the least square method of (7) to solve out the 3D coordinate of MT.
Synthesizing the above analysis, the 3D geometric localization algorithm based on linear regression and least square principle was designed: Step1：Calculating out the measured distance error by the (8)； Step2:Calculating out the linear regression coefficient k and a of (9)； Step3:Calculating out between the MT and by (10), then, substituting all into (5)，obtained b of (12)； Step4：Substituting b into (7) to solve out the 3D spatial position coordinates of MT.

Experimental Results
Selecting 10 location base stations, their 3D geometric coordinates are shown in Table 1, the 3D coordinates of the two test terminals are shown in Table 2, the TOA of the two test terminals to base stations are shown in Table 3, and the calculated values of k and a are shown in Table 4. Choosing TOA from the test terminal MT 1 and MT 2 to the base stations of four to nine smaller to locate MT 1 and MT 2 respectively, the relationship between the mean positional error and the number of chosen TOA is shown in figure 3 and Table 5, therefore, locating with five TOA can not only reduce computational complexity, but also reduce measurement data and achieve higher positional accuracy in practice.Let values of k and a are the average values of the two groups of k and a in table 4, respectively, for each of the 100 terminals, choosing the TOA from the terminal to the base stations of five smaller to locate the terminal, the statistical results of the localization are shown in Table 6, the comparison of the located position of part of the terminals and their real position is shown in Table 7.

Conclusion
The proposed 3D geometric locating algorithm based on the linear regression and the least square principle in NLOS environments not only does not need to obtain the prior statistical characteristic knowledge of the TOA, but also requires only five TOA to achieve more accurate positional results in practice, and the located accuracy can reach less than 2 meters.Basically, it can overcome the influence of NLOS propagation error, and the positional results basically meet the accuracy requirements of various localization mentioned in the introduction.
funding of this article, and thank the reviewers for their comments on this article, and thank the editorial staff for their hard work .

Figure 1 .
Figure 1.TOA localization schematic Figure 2. Relation between real distance and measured distance error

Figure 3 .
Figure 3.The relationship between the number of chosen TOA and the mean positional error

Table 5 .
The number of selected TOA and the mean positional error

Table 7 .
The real and located position of part of terminals (unit:m)