Tarea 1

Assume that k is fixed and no prior knowledge about the position of the UE is available. Then, in LOS, minimizing the sum of the squared residuals (2.4) yields the optimal solution. However, since hm(·) is nonlinear, even a quadratic cost function (as in least-

squares estimation) can result in several local minima and local search algorithms [31], e.g. the Gauss-Newton algorithm [11] can lead to biased estimates. In other words, Equation (2.5) from Section 2.1.2 does not have a unique solution which requires good initialization for the local search algorithms. These algorithms iteratively determine the location of the UE based on a first-order approximation of the measurement model (3.1). However, outliers due to NLOS propagation or a starting point too far away from the true position can result in convergence problems and position accuracy can substantially be reduced.

One way to cope with the NLOS problem is to identify the NLOS FTs and use the remaining ones for positioning based on the above mentioned methods. NLOS detec- tion algorithms are proposed in [10, 36, 83, 101] and [91] provides an overview of NLOS identification techniques. The approaches developed in [10,83] are based on hypothesis testing where a Gaussian pdf for the NLOS error is assumed. The authors in [101] use more general statistical LOS/NLOS identification schemes where outlier detection is performed based on higher order moments or tests for Gaussianity. A statistical non- parametric NLOS detection approach is suggested in [36] where the error pdf of the observations from one FT is estimated using KDE and statistical distance measures are used to compare the pdf estimate with the Gaussian pdf. This approach requires several observations per FT for performing KDE and a FT is accepted to be LOS if the distance of its empirical error pdf to the Gaussian pdf is smaller than a certain threshold. However, the concept of detecting the NLOS FTs and discarding them for positioning has two major limitations. If a mis-detection occurs, higher positioning errors are expected using standard LS techniques because outliers have a deleterious effect on these estimates. In contrast, a false alarm reduces the set of LOS FT which increases the positioning error as well since less observations are left for estimation. In general, problems with this approach occur when a small amount of FTs remain (e.g. a number of FTs smaller than three, leading to ambiguities) or when the remaining FTs are spaced in a disadvantageous geometric constellation, e.g., when all remaining FTs lay approximately on a straight line. Therefore, we concentrate on approaches

that use the observations from all FTs for positioning.

In [18,22,40] NLOS mitigation algorithms based on TOA measurements are proposed. The three approaches consider grouping of range measurements to obtain an LS posi- tion estimate from each subgroup. The position estimates are then combined in differ- ent manners so that deleterious estimates are underweighed with respect to accurate position estimates. All references mentioned above avoid imposing a specific NLOS error distribution, which is convenient since it is unknown in practice. Chen [22] sug- gests an NLOS mitigation algorithm based on residual weighting. The observations are formed into different subgroups and LS estimation is used for each group to determine the position of the UE and its residual error (the sum of the squared residuals from the obtained position estimate). The overall position estimate is a weighted combination of the different estimates where the weights are determined as the inverse of the residual error, i.e., high residual errors result in small weights and low residual errors result in larger weights. Since the final position estimate is the weighted sum of the different position estimates, an error in the residual weights can have a large impact on the final position estimate.

Apart from the grouping, which is done to obtain M₃
subgroups, the authors in [18] suggest a different approach to overcome this problem. For each position estimate from a subgroup the median of the squared residuals is calculated. Then the final solution is the position estimate corresponding to the minimum of the medians from each subgroup. Taking the median (instead of the mean which corresponds to the sum in [22]) ensures that the position estimates corresponding to very large or very small squared residuals are not taken into account. This is because the median is the number that separates the higher half of the sample from the lower half which is achieved by ordering the data.

However, the performance of both algorithms [18, 22] strongly depends on the opti- mization algorithms used for each subgroup to obtain the LS position estimate. While local search algorithms can fail when outliers occur or initialization is erroneous, more sophisticated and thus more complex optimization tools [11] rather find the global minimum instead of falsely selecting a local one. Given that the latter techniques are available, one could also use a different penalty function to decrease the impact of NLOS outliers and search for the global minimum to obtain robust position esti- mates. That means one can use a penalty function that is increasing less severely than a quadratic one, e.g., ρc1(v) in Equation (2.16) is increasing linearly beyond a certain

threshold, such an approach is applied to geolocation in [72].

An alternative is to estimate the penalty function non-parametrically and determine the position of the UE by searching for the global minimum or minimizing a non- parametric estimate of the entropy of the residuals [98], as in [110] (semi-parametric approaches). Another alternative is to use constraint optimization techniques [106]

that take into account that the NLOS error is always positive and thus search for the minimum in a restricted area. However, in many applications computational power is limited and therefore numerically simpler solutions are preferred and considered in this thesis.

To prevent convergence problems, when using local search algorithms, the signal model is linearized to obtain a closed-form solution and robust and semi-parametric regression techniques for linear models are applied.