Moving User Equipment

Incorporating a motion model for a moving UE decreases positioning errors with respect to the stationary case because of time averaging effects. In this context optimal solu- tions briefly introduced in Chapter 2 are intractable. Instead suboptimal algorithms such as the EKF achieve high precision in LOS environments. However, positioning errors increase severely with the percentage of NLOS errors.

In Chapter 4 we adapt the semi-parametric estimator developed in Chapter 3 to the problem of a moving UE, denoted as EKF-SP-MR. The motivation in doing so is to avoid the trade-off between efficiency and robustness which is inherent for classical robust estimators. For this purpose, the EKF equations are reformulated as a linear regression problem and solved at each time step using the semi-parametric estimation scheme. Since it is not straightforward to establish the asymptotic covariance of the semi-parametric estimator the posterior covariance of the standard EKF is used instead as an approximation.

A parametric alternative [47], denoted as R-IMM, is also developed in Chapter 4 where the EKF equations again rewritten as linear regression problem so that robust estima-

tion techniques can be applied. In general, robust estimators always trade-off efficiency in the nominal cases versus robustness in the NLOS case so that achieving both with a single estimator is not possible. To overcome this problem an EKF, well suited for LOS channels, is run in parallel with a robust EKF (REKF) based on robust regression, well suited for NLOS environments. The second filter is adjusted so that it clips a large amount of the data and consequently behaves robustly in highly contaminated NLOS environments. The likelihood of each filter matching the underlying situation is calcu- lated and incorporated in the final position estimate which is a weighted combination of the state estimates computed by both filters. In LOS environments, the weights of the EKF tend to one, whereas the weights of the REKF tend to zero, meaning the influence of the EKF on the state estimates is increased whereas the influence of the REKF is reduced. In severe NLOS environments the opposite is true.

The R-IMM and the EKF-SP-MR trackers both significantly outperform conventional and robust techniques in NLOS environments and gain up to 200m in terms of position- ing accuracy. In particular for non-Gaussian error statistics which follow an exponential or Rayleigh distribution, the EKF-SP-MR tracker gains in precision with respect to R-IMM because the higher order moments of the noise pdf are incorporated into the estimation procedure via non-parametric pdf estimation.

In contrast, the R-IMM achieves higher precision when the NLOS errors follow a shifted Gaussian pdf since it discards larger outliers whereas the semi-parametric tracker ap- proximates the noise pdf, which becomes difficult when the pdf is multimodal.

While the R-IMM approach achieves performance similar to the EKF in LOS environ- ments, accuracy of the semi-parametric tracker drops by a few meters in this situation. This can be explained by the fact that the latter possesses more degrees of freedom and consequently more uncertainty in the nominal case, as it relies on non-parametric pdf estimation. Furthermore, unlike the EKF-SP-MR, the R-IMM algorithm takes into account the switching of the LOS/NLOS events that are modeled by a two-state Markov chain for each FT and also contributes to the performance gain.

The main limitation of the R-IMM, in particular the REKF used within the R-IMM, is that its precision significantly decreases for large percentages of NLOS occurrences and large magnitudes of NLOS measurements because it is based on robust regression. According to the theory of robust statistics a robust M-estimator breaks down when the degree of outliers achieves 50%. However, since a motion model is incorporated within a Bayesian framework the proposed R-IMM algorithm exhibits numerically sta- ble performances for up to 60% outliers. Increasing ε further leads to high positioning errors given that the magnitude of the NLOS errors is large compared to the sensor noise.

To summarize, the EKF-SP-MR is preferable in environments where the NLOS inter- ference statistics are completely unknown and when a large number of FTs are available

because it relies on non-parametric KDE. However, high positioning accuracy is ob- tained in the simulations for only five FTs. In contrast, the R-IMM is better suited if the NLOS errors statistics are likely to follow a shifted Gaussian distribution, for small to medium ε and also for smaller sample sizes since it is based on parametric robust M-estimation.

While both trackers are roughly comparable in terms of computational complexity, a numerically simpler tracking scheme is developed in Chapter 5.

The MPDA approach presented in Chapter 5 relies on a joint error detection and track- ing scheme and is published in [48]. We propose to construct different subgroups of TOA range measurements together with the corresponding positions of the FTs. Each subgroup provides a least-squares position estimate of the UE together with its covari- ance matrix that are both used in a hypothesis test for NLOS detection. The position estimates computed by NLOS FTs are discarded whereas the accepted measurements are weighted with different probabilities in a Kalman filter framework. Again, signifi- cant improvements can be achieved with respect to conventional and robust techniques in NLOS environments whereas similar performance to the EKF is obtained in LOS environments.

The limitations of this tracker become apparent when the percentage of NLOS outliers is close to 50% and the magnitude of the NLOS outliers is large, such as when they are modeled by a shifted Gaussian pdf. In such cases, the innovation covariance matrix calculated in the Kalman filter recursions does not match the observed innovation se- quences which lead to higher false alarm rates. Thus, useful information is lost because valid position estimates are discarded. If this happens for consecutive time step, the state estimates for these time steps are only based on prediction which lead most often to divergence and loss of the track.

For small ε up to approximately 30% and iid environments the MPDA tracker proposed in Chapter 5 has slightly less positioning accuracy than R-IMM and EKF-SP-MR. How- ever, the difference becomes more pronounced for larger ε and when the LOS/NLOS occurrences are modeled by a Markov chain. In these environments, it is less stable than R-IMM and EKF-SP-MR and produces large positioning errors. For ε > 60% and exponentially distributed NLOS outliers, the semi-parametric tracker outperforms the R-IMM and MPDA algorithms significantly.