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Session C6: Terrestrial Signals of Opportunity-Based Navigation Systems

Maximum Likelihood Time-Delay Estimation in Multipath Channels with two- and Three-Paths Models Using OFDM
Lucas Alvarez Navarro, Christian C. J. M. Tiberius, Department of Geoscience and Remote Sensing, Delft University of Technology; Gerard J. M. Janssen, Department of Microelectronics, Delft University of Technology

As an alternative to GNSS, particularly in dense built-up and urban environments with signal blockage and multipath effects, terrestrial radio positioning systems can provide improved positioning performance. However, the accuracy of these systems is limited by the bandwidth. The frequency spectrum is already occupied by numerous services, making it difficult to allocate large bandwidths exclusively for positioning. As a result of using reduced bandwidths, similarly as in GNSS, multipath remains a major source of error. To mitigate the impact of multipath in time-delay estimation, we propose Maximum Likelihood estimators that account for the strongest reflections. Specifically, we introduce two models: one that accounts for the strongest multipath component and another that incorporates the two strongest multipath components.
In a multipath channel, the received signal is composed of a sum of time-shifted and attenuated signal replicas of the transmitted signal. For ranging and positioning, the aim is to estimate the propagation delay of the LOS signal, which is taken as the first arriving signal. When the difference in propagation time between the first arriving signal and multipath signals is sufficiently large, i.e., beyond 1/B (where B represents the signal bandwidth), correlation based techniques, such as the matched filter, provide sufficient accuracy. However, for close-in multipath, the matched filter estimate can become biased.
To obtain unbiased time-delay estimates in the presence of close-in reflections, we extend the maximum likelihood estimator (MLE) to account for two and three propagation paths. As a compromise between time-delay estimation accuracy and computational complexity, we restrict the model to three paths, which captures some of the most significant reflections. We derive the cost function expressions for these estimators and provide guidelines for efficient implementation. To gain insight, we express the 2- and 3-paths MLE cost function in terms of the 1-path MLE cost function, providing an interpretation of each term. The MLE cost function for the 2- and 3-paths estimators are evaluated based on a grid-search approach. To reduce computational load, the delay search space for each path is constrained based on the maximum bias observed in the multipath error envelope under a worst-case scenario, i.e., when a multipath signal arrives with high relative power compared to the LOS signal.
Introducing additional parameters to estimate, such as the delay of each signal propagation path, into the time-delay estimation measurement model, can reduce estimation bias. However, this increase in complexity leads to larger estimator variance. We explore this trade-off by analysing the Cramer-Rao Lower Bound (CRLB) and Root Mean Squared Error (RMSE) for maximum likelihood estimators (MLE) that account for up to three signal propagation paths: the LOS signal and the two strongest reflections. Furthermore, we analyse and quantify the performance loss associated with using an estimator that accounts for less signal paths than actually present.
We focus on a terrestrial positioning system with a bandwidth of less than 160 MHz , which employs OFDM modulation, as used in Long Term Evolution (LTE) mobile communications systems. To assess the ranging accuracy for the various MLE estimators that account for multiple paths, we utilize a synthetically generated channel based on the Saleh-Valenzuela model. Additionally, we benchmark the positioning performance of these estimators using channel impulse responses recorded with the SuperGPS-prototype[1] at The Green Village on the TU Delft campus. The SuperGPS-prototype is a terrestrial positioning system composed by six base stations synchronized through a fiber-optic Gigabit Ethernet. These base stations transmit an OFDM ranging signal using 16 contiguous frequency bands , each with a bandwidth of 10 MHz and comprising 64 OFDM sub-carriers. We will not use the entire 160 MHz bandwidth, but only select some adjacent bands to emphasize the effect of multipath. By processing these signals, we evaluate the performance of the MLE-based estimators in terms of the mean and standard deviation of the 2-D positioning error (North and East coordinates) and ultimately quantify the improvements in positioning performance when additional paths are considered in the estimation.
[1] Koelemeij, J. (2022). Data for "A hybrid optical-wireless network for decimetre-level terrestrial positioning" (V1). DataverseNL. https://doi.org/10.34894/GFDJI1

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