After having automated the wireless localization system, its core—the localization algorithm—has to be explained. As shown in Figure 1, the localization algorithm takes as input arguments the RTT measurements carried out between the MU and each AP in range, and the position of these APs that are assumed to be previously known. As a result, the output of the localization algorithm is an estimation of the MU position. Figure 2 shows the flowchart that explains the process followed by the novel localization algorithm to estimate the MU position. Firstly, a location estimator is obtained from the RTT measurements. That value will be used in a model that characterizes the distance between the MU and the corresponding AP. As the transmitted signal could only reach the receiver through a path different from the direct one, the PNMC technique is used to correct the positive bias that introduces the NLOS error. Once the effect of NLOS error is corrected, the model that relates the actual distance to the RTT estimator in LOS is used. Finally, when the distance from the MU to more than two APs is estimated and assuming known the positions of these APs, an MUltilateration method that linearizes the problem of finding the MU position is implemented.
3.1. Statistical Estimators and Robust Linear Regression in LOS
According to [21], the range resolution is determined by the bandwidth of the transmitted signal when RTT measurements are used. Furthermore, when using a 44 MHz clock as input of the measuring system to quantify the RTT measurements, the maximum resolution achievable, if only one sample is taken, is about 6.8 m. Moreover, the RTT measurements have a random behavior due to the error introduced by the standard noise from electronic devices, that is always present. To overcome these limitations several RTT measurements have to be performed at each distance and a representative value, called the location estimator, from this group of RTT measurements has to be selected. That selection is based on the model that relates the location estimator to the distance that separates the MU and the AP in LOS.
The location of a random variable distribution can usually be presented by a single number, the location estimator. In [22], several location estimators of a random variable are analyzed. The mean, median, mode and the scale parameter of the Weibull distribution (scale-W) are examples of the location of a random variable. They have been analyzed and compared as location estimators of the RTT measurements in terms of the coefficient of determination, 
. This coefficient measures how much of the original uncertainty in the RTT measurements is explained by the model [23]. In this paper, a simple linear regression function is assumed to be the model that relates the actual distance between the two nodes involved in RTT measurements to the location estimators in LOS. Analytically,
where, 
and 
are the estimated and the actual distance between the MU and the AP in LOS, respectively, 
is the location estimator of RTT measurements, 
and 
are the intercept and slope of the simple linear regression function, respectively, and 
is the error introduced by 
. The error term 
has been modeled as a zero-mean Gaussian random variable, because the estimators used are asymptotically Gaussian and a large amount of measurements have been used, so
In this case, as the model is a simple linear regression function, 
is simply the square of the correlation coefficient, 
.
The parameters 
and 
that characterize the simple linear regression function do not depend on the environment where the wireless localization system is going to be deployed, but on the communication system used, that is, the MU and the AP. These parameters are computed so as to give a best fit of the location estimators to the actual distance. Most commonly, the best fit is evaluated by using the least squares method, but this method is actually not robust in the sense of outlier-resistance. Hence, robust regression has been performed as it is a form of regression analysis designed to circumvent some limitations of least squares estimates for regression models [22].
Assuming LOS between the MU and the AP without any scatter nearby and guaranteeing the first Fresnel zone clearance of the link between both nodes, three campaigns of 300 RTT measurements were conducted for several distances from 0 to 40 m. Figure 3 shows the robust linear regression function which best fits each location estimator to be analyzed. Each location estimator has been computed from each group of 50 RTT measurements at each distance. The different location estimators are analyzed and compared in terms of the coefficient of determination value, 
.
The mode (
) is the value that is most likely to be sampled, thereby it could be a good candidate for the location estimator, but the value that occurs the most frequently in a data set is a discrete value. Therefore, the resolution achieved, 
, is not enough for indoor localization systems. The same resolution is achieved with the median (
) as it is a discrete value separating the higher half of a data set. Figures 3(a) and 3(b) show that the Gaussian distributions that characterize the errors 
of the mode and the median are the widest, with 
m being 
and 
m being 
.
The mean (
) is equivalent to the center of gravity of the distribution and it does not take discrete values, thereby the resolution is improved. Although the mean is rather sensitive in the presence of outliers, the use of a robust regression function circumvents this limitation. Figure 3(c) shows that the errors committed when using the mean as a location estimator are characterized by a Gaussian with 
m, being 
, lower than the error commit when the median. But Figure 3(d) shows that the best location estimator is the scale-W parameter (
) once Weibull distribution is fitted to the RTT measurements. In this case 
is characterized by 
m and 
. Therefore, the assumption of a linear function as the model that relates RTT measurements to distance is corroborated by a value of 
close to the unit. This value indicates that the regression line nearly fits the 
perfectly.
Figure 4 shows the cumulative distribution function (CDF) of 
. As the mode and the median take discrete values, the CDF has a step-shape with large errors. The mean has a good behavior with an error lower than 2 m on average. But the scale-W parameter with an error lower than 3 m for a cumulative probability of 80% achieves the best behavior.
There is no phenomenological explanation for choosing the scale-W parameter as location estimator of the RTT measurements set, but this parameter is another kind of a location estimator since the maximum likelihood estimator (MLE) of the scale-W parameter is the Hölder mean [24], a generalized form of the Pythagorean means, taking as parameter the shape parameter of Weibull distribution (for more detail see Appendix A).
Once scale-W parameter is found as the statistical estimator of the RTT measurements that best fits the actual distance when using a simple robust linear regression function as the model that relates the estimator to the distance, its performance is compared to an RSS-based solution to evaluate the goodness of the proposed one. The same two wireless nodes have been used in the same LOS environment. As it is well known the distance between two wireless devices causes an attenuation in the RSS values. This attenuation is known as path loss and it is modeled to be inversely proportional to the distance between both devices raised to a certain exponent. According to [8], the distance between two wireless nodes can be estimated from RSS measurements by
where 
is the estimated distance between the MU and the AP, 
is the RSS measured in logarithmic units at the reference distance of 
, 
is the average RSS in logarithmic units at the actual distance, and 
is the path loss exponent. According to [20], for any distance under 20 m in LOS, 
is recommended to be 2 while 
for longer distances. Therefore, having taken this value for the path loss exponent and from the RSS values measured between both devices, the distance between the two wireless nodes can be estimated by using the expression (3). In Figure 4 that it can be appreciated the great accuracy obtained by the method presented (square marks), since it outperforms the RSS range based method, specially for cumulative probabilities larger than 50%.
3.2. NLOS Correction
The two sources of range measurement errors in localization techniques are mainly electronic errors and NLOS errors. Electronic errors are inherent to electronic devices and they are commonly modeled as a zero-mean Gaussian distribution. In the previous section, assuming LOS propagation, the effect caused by the electronic error has been minimized by choosing the best location estimator of the RTT measurements, the scale-W parameter. But the assumption of LOS condition is an oversimplification of reality in an indoor environment. Therefore, a method to correct the bias that introduces the NLOS in range measurements has to be implemented to improve the indoor wireless localization system.
The easiest method for dealing with NLOS conditions is simply to place APs at additional locations and select those from LOS, but one of the objectives of this paper is to deploy a wireless localization system in a common and unmodified wireless network. Therefore, the PNMC technique [16] is implemented to correct the NLOS errors in range measurements. The PNMC technique was created to correct the NLOS errors in open areas. Its performance has been evaluated by simulations in [16], but as it will be shown in this paper, it also works under indoor environment conditions. Based on a statistical process, the PNMC technique corrects the NLOS effect from a record of range measurements taken through a time window in a previous stage to the positioning process. This processing relies on the statistical estimate of the NLOS range measurements ratio present in the record. The ratio is used to identify the NLOS recorded range measurements. Subsequently, the NLOS range measurements are classified in segments according to the NLOS statistical distribution. Finally, the correction is carried out by subtracting the expected NLOS errors for each segment. For a detailed explanation on the PNMC technique, see [16].
Let 
be the actual distance between the MU and the AP, thus
where 
and 
are the estimated and the actual distances between the MU and the AP, respectively. The term 
denotes the error. This error is the sum of two independent errors, 
, where 
describes the electronic errors, while 
is the error due to the lack of direct sight between the MU and the AP. On one hand, the term 
has been evaluated in the previous section and it was found as a zero-mean Gaussian with 
m. On the other hand, the term 
can follow different statistical distributions, Gaussian, Exponential, Gamma, and so forth [16]. But, regarding the distribution of 
, it can be characterized by its mean and standard deviations. These parameters, as well as the distribution type of 
, depends on the particular environment, but it can be assumed that the NLOS propagation conditions do not change significantly in the time window that contains the record of range measurements, so the mean and standard deviation of 
can be assumed to be constant. Moreover, the parameters that characterize 
can be obtained previously to the localization process by the estimates performed in the environment where the localization system is going to be deployed. For simplicity, the Exponential distribution has been chosen for the term 
. Therefore,
where the 
parameter is fixed previously to the localization process.
In order to show the feasibility of the PNMC technique in an indoor environment, a campaign of measurements in the second floor of the Higher Technical School of Telecommunications Engineering (ETSIT) at the University of Valladolid has been carried out. Specifically, the PNMC technique is applied to the range measurements computed between an AP and an MU 14 m away who is moving 5 m straight perpendicularly to the path that joins the AP and the MU. As 
, the probability density function (PDF) of the term 
is the convolution of the Gaussian PDF caused by the 
errors and the Exponential PDF caused by the 
errors. Figure 5(a) shows the histogram of the distance estimates record and the PDF of the term 
that best fits these estimates, where the value of the parameter 
that best fits the data is 
m
. Once the term 
is statistically characterized, the PNMC technique can be applied. Figure 5(b) shows the result of applying the PNMC technique to the original range measurements computed in a time window equivalent to 5 m walking. In this scenario, the ratio of 
errors from the record of range measurements has been 52%. Subsequently, these NLOS range measurements have been corrected by subtracting the expected NLOS errors for each segment according to the Exponential distributions. The accuracy improvement of applying the PNMC technique to find the MU position in an indoor environment is shown in Section 4.
3.3. Multilateration
In two-dimensions, multilateration is defined as a method for determining the intersections of 
circles with 
. The circles are defined by their centers 
, corresponding to the known positions of the APs, and the radii 
, corresponding to the distance estimates between the MU and each AP. This means that to infer the position of the MU, a system of 
quadratic equations has to be solved. As the distance estimates between the MU and each AP do not usually match the actual distances, the circles will not cut each other in a single point. Hence, the MU position 
of the localizing wireless node can be estimated by finding 
satisfying
Solving (6) problem requires significant complexity and is difficult to analyze. In order to simplify the resolution of the expression (6), an alternative way to find the location of the MU is defined. Instead of using the circles as the equations to determine the MU location, the radical axes will be used. The radical axis of two circles is the locus of points at which tangents drawn to both circles have the same length. The radical axis is always a straight line and it is always perpendicular to the line connecting the centers of the circles, albeit closer to the circle of the larger radius. Let
be the equations of two circles corresponding to different APs (
). Then, the equation of the radical axis will be the result of subtracting the two involved circles' equations. Analytically,
If 
circles cut each other in a single point then, the 
radical axes performed among all the pairs of circles without repetition will cut each other in the same single point. Therefore, once the radical axes among all the pairs of circles are performed, the problem of solving a system of 
quadratic equations is reduced to solve a system of 
linear equations. In the common case, as the 
circles do not cut each other in a single point, the linear system is solved in a least-square sense. Let
be the linear equation system defined by the radical axes with
where 
is a matrix of 
rows and 2 columns described only by the APs coordinates, while 
is a vector of 
components represented by the distance estimates together with the AP coordinates. In the least-squared sense the solution for the expression (9) is done via
where 
is an estimate of the actual MU position.
Figure 6 shows a graphical example of the method used to multilaterate. In that scenario, the MU has four APs in range whose positions are known 
. After the distance between the MU and each AP is estimated through the RTT measurements, 
with 
, the four circles are well defined. Then, the six radical axes are performed from all the combinations of pairs of circles, 
. The MU position, 
, is obtained as the result of solving the expression (11).