Carsten Fritsche

TOA-Based Robust Wireless Geolocation and Cramér-Rao Lower Bound Analysis in Harsh LOS/NLOS Environments

We consider time-of-arrival based robust geolocation in harsh line-of-sight/non-line-of-sight environments. Herein, we assume the probability density function (PDF) of the measurement error to be completely unknown and develop an iterative algorithm for robust position estimation. The iterative algorithm alternates between a PDF estimation step, which approximates the exact measurement error PDF (albeit unknown) under the current parameter estimate via adaptive kernel density estimation, and a parameter estimation step, which resolves a position estimate from the approximate log-likelihood function via a quasi-Newton method. Unless the convergence condition is satisfied, the resolved position estimate is then used to refine the PDF estimation in the next iteration. We also present the best achievable geolocation accuracy in terms of the Cramér-Rao lower bound. Various simulations have been conducted in both real-world and simulated scenarios. When the number of received range measurements is large, the new proposed position estimator attains the performance of the maximum likelihood estimator (MLE). When the number of range measurements is small, it deviates from the MLE, but still outperforms several salient robust estimators in terms of geolocation accuracy, which comes at the cost of higher computational complexity.

Cramér-Rao Lower Bounds for hybrid localization of mobile terminals

While in outdoor scenarios the global positioning system (GPS) provides accurate mobile station (MS) location estimates in the majority of cases, in dense urban and indoor scenarios GPS often cannot provide reliable MS location estimates, due to the attenuation or complete shadowing of the satellite signals. The existing cellular radio network (CRN)-based localization methods, however, provide MS location estimates in almost every scenario, but they do not reach the accuracy of the MS location estimates provided by GPS. Hybrid localization methods combine MS location information available from measured values of the CRN with MS location information provided by the measured values of GPS. In this paper, a hybrid localization method is proposed that combines received signal level and timing advance measured values from the global system for mobile communication (GSM) and time of arrival measured values from GPS. The best achievable localization accuracy of the proposed hybrid localization method is evaluated in terms of the Cramer-Rao Lower Bound. It is shown that the hybrid localization method significantly improves the localization accuracy compared to existing CRN-based localization methods.

Robust mobile terminal tracking in NLOS environments using interacting multiple model algorithm

An extended Kalman filter-based interacting multiple model algorithm (IMM-EKF) is proposed for mobile terminal tracking in cellular networks based on time of arrival estimates. The proposed IMM-EKF is able to cope with line-of-sight (LOS) and non-line-of-sight (NLOS) conditions modeled by a Markov chain, where the LOS and NLOS errors are described by different noise models. Road-constraints are included into the IMM-EKF to improve performance. Simulation results show that the IMM-EKF outperforms conventional methods. A comparison to the posterior Cramér-Rao lower bound is given to demonstrate the effectiveness of the IMM-EKF.

Recursive Maximum Likelihood Identification of Jump Markov Nonlinear Systems

We present an online method for joint state and parameter estimation in jump Markov non-linear systems (JMNLS). State inference is enabled via the use of particle filters which makes the method applicable to a wide range of non-linear models. To exploit the inherent structure of JMNLS, we design a Rao-Blackwellized particle filter (RBPF) where the discrete mode is marginalized out analytically. This results in an efficient implementation of the algorithm and reduces the estimation error variance. The proposed RBPF is then used to compute, recursively in time, smoothed estimates of complete data sufficient statistics. Together with the online expectation maximization algorithm, this enables recursive identification of unknown model parameters including the transition probability matrix. The method is also applicable to online identification of jump Markov linear systems(JMLS). The performance of the method is illustrated in simulations and on a localization problem in wireless networks using real data.

EM- and JMAP-ML Based Joint Estimation Algorithms for Robust Wireless Geolocation in Mixed LOS/NLOS Environments

We consider robust geolocation in mixed line-of-sight (LOS)/non-LOS (NLOS) environments in cellular radio networks. Instead of assuming known propagation channel states (LOS or NLOS), we model the measurement error with a general two-mode mixture distribution although it deviates from the underlying error statistics. To avoid offline calibration, we propose to jointly estimate the geographical coordinates and the mixture model parameters. Two iterative algorithms are developed based on the well-known expectation-maximization (EM) criterion and joint maximum a posteriori-maximum likelihood (JMAP-ML) criterion to approximate the ideal maximum-likelihood estimator (MLE) of the unknown parameters with low computational complexity. Along with concrete examples, we elaborate the convergence analysis and the complexity analysis of the proposed algorithms. Moreover, we numerically compute the Cramer-Rao lower bound (CRLB) for our joint estimation problem and present the best achievable localization accuracy in terms of the CRLB. Various simulations have been conducted based on a real-world experimental setup, and the results have shown that the ideal MLE can be well approximated by the JMAP-ML algorithm. The EM estimator is inferior to the JMAP-ML estimator but outperforms other competitors by far.

Cooperative Localization in WSNs Using Gaussian Mixture Modeling: Distributed ECM Algorithms

We study cooperative sensor network localization in a realistic scenario where the underlying measurement errors more probably follow a non-Gaussian distribution; the measurement error distribution is unknown without conducting massive offline calibrations; and non-line-of-sight identification is not performed due to the complexity constraint and/or storage limitation. The underlying measurement error distribution is approximated parametrically by a Gaussian mixture with finite number of components, and the expectation-conditional maximization (ECM) criterion is adopted to approximate the maximum-likelihood estimator of the unknown sensor positions and an extra set of Gaussian mixture model parameters. The resulting centralized ECM algorithms lead to easier inference tasks and meanwhile retain several convergence properties with a proof of the “space filling” condition. To meet the scalability requirement, we further develop two distributed ECM algorithms where an average consensus algorithm plays an important role for updating the Gaussian mixture model parameters locally. The proposed algorithms are analyzed systematically in terms of computational complexity and communication overhead. Various computer based tests are also conducted with both simulation and experimental data. The results pin down that the proposed distributed algorithms can provide overall good performance for the assumed scenario even under model mismatch, while the existing competing algorithms either cannot work without the prior knowledge of the measurement error statistics or merely provide degraded localization performance when the measurement error is clearly non-Gaussian.

Hybrid GPS/GSM localization of mobile terminals using the extended Kalman filter

In dense urban and indoor scenarios, the Global Positioning System (GPS) often cannot provide reliable mobile terminal (MT) location estimates, due to the attenuation or complete shadowing of the satellite signals. Cellular radio network-based localization methods, however, provide MT location estimates in almost every scenario, but they do not reach the accuracy of MT location estimates provided by GPS. Thus, a promising approach is to combine measured values from the cellular radio network and GPS, which is known as hybrid localization. In this paper, an extended Kalman filter (EKF)-based hybrid localization method is proposed that combines round trip time and received signal strength measured values available from the Global System for Mobile Communication (GSM) and pseudorange (PR) measured values from GPS, in order to track the MTs movement. In contrast to existing hybrid approaches, an EKF-based MT tracking algorithm is proposed that additionally takes into account sectorized BS antennas which are typically employed in existing GSM networks, and it takes into account PR instead of geometric range (or time of arrival) measured values from GPS as GSM is normally not time-synchronized to GPS. Simulation and experimental results show that, compared to an EKF that is based only on GSM measured values, the proposed EKF that additionally incorporates PR measured values yields improved MT location estimates.

On the Performance of Hybrid GPS/GSM Mobile Terminal Tracking

The Global Positioning System (GPS) has become one of the state-of-the-art location systems that offers reliable mobile terminal (MT) location estimates. However, there exist situations where GPS is not available, e.g., when the MT is used indoors or when the MT is located close to high buildings. In these scenarios, a promising approach is to combine the GPS measured values with measured values from the Global System for Mobile Communication (GSM), which is known as hybrid localization method. In this paper, a hybrid MT tracking algorithm based on a Rao-Blackwellized unscented Kalman filter (RBUKF) is proposed that combines pseudoranges from GPS with timing advance and received signal strengths from GSM. Simulation results show that the proposed hybrid method outperforms the GSM method. Furthermore, the performance of the RBUKF is compared to the extended Kalman filter and the corresponding posterior Cramer-Rao lower bounds.

The marginalized auxiliary particle filter

In this paper we are concerned with nonlinear systems subject to a conditionally linear, Gaussian sub-structure. This structure is often exploited in high-dimensional state estimation problems using the marginalized (aka Rao-Blackwellized) particle filter. The main contribution in the present work is to show how an efficient filter can be derived by exploiting this structure within the auxiliary particle filter. Based on a multi-sensor aircraft tracking example, the superior performance of the proposed filter over conventional particle filtering approaches is demonstrated.

On parametric lower bounds for discrete-time filtering

Parametric Cramér-Rao lower bounds (CRLBs) are given for discrete-time systems with non-zero process noise. Recursive expressions for the conditional bias and mean-square-error (MSE) (given a specific state sequence) are obtained for Kalman filter estimating the states of a linear Gaussian system. It is discussed that Kalman filter is conditionally biased with a non-zero process noise realization in the given state sequence. Recursive parametric CRLBs are obtained for biased estimators for linear state estimators of linear Gaussian systems. Simulation studies are conducted where it is shown that Kalman filter is not an efficient estimator in a conditional sense.