Christiaan C. J. M. Tiberius

Multipath propagation represents a dominant error source limiting the accuracy of radio-based ranging and positioning in urban environments. Conventional ranging estimators typically estimate only the range of the first arriving path while neglecting the multipath components. This introduces a bias and causes the estimator’s variance to diverge from the Cramér-Rao Bound (CRB). While extensive research has established performance bounds for estimators that jointly estimate all paths, the specific impact on the variance of ignoring these components remains largely unexplored. This paper derives bounds on the bias, variance, and mean square error (MSE) for ranging in multipath channels for the purpose of positioning. We consider Orthogonal Frequency Division Multiplexing (OFDM) signals in multipath channels with deterministic path delays and gains. The paper focuses on time-delay estimation (TDE) in the mid-tohigh signal-to-noise ratio (SNR) regime under two scenarios: (i) when the receiver jointly estimates all propagation paths, and (ii) when the receiver underestimates the number of paths. Specifically, for the latter case, we derive a semi closed-form expression for the variance of the time-delay estimator that considers a single-path when in reality there are L paths. This derivation is based on the misspecified Cramér-Rao bound (MCRB). Although unmodelled multipath has been traditionally viewed as detrimental to time-delay estimation, we reveal that in some cases the estimation variance improves. For a two-path channel we show that the variance depends on the relative gain, carrier phase, and separation of the paths. Additionally, the estimation bias is upper-bounded by constructive and destructive path interference. Finally, we empirically process a six-path simulated channel frequency response with fixed path delays and gains, to demonstrate that the derived MSE bound is tight for mid-to-high SNRs. This work characterizes the impact of multipath propagation on the variance of time-delay estimation, which is essential for designing accurate ranging signals and estimators in urban scenarios.

This book, Exercises, contains hundreds of exercises, including answers and worked examples, for studying and practicing the theory. Python scripts illustrate how to perform spectral signal analysis on a computer.

The Detection, Identification, and Adaptation (DIA)-estimator integrates parameter estimation and hypothesis testing for model misspecifications. This contribution presents a positioning safety analysis approach grounded in the DIA-estimator framework, with a particular emphasis on multidimensional model misspecifications, such as simultaneous outliers in the observations. While recent work has focused on the performance of the detection and identification of multidimensional model misspecifications, we turn our attention to how they affect the probability density function (PDF) of the DIA-estimator and, consequently, the probability of positioning failure–an indicator relevant for safety-of-life applications (e.g., automotive, aviation, rail, maritime). This work formulates and quantifies the probability of positioning failure and its conditional components. A representative simulation-based study is presented for a UAV equipped with a GPS receiver configured to achieve performance comparable to Technical Standard Order (TSO)-certified receivers. The analysis is carried out for two scenarios: a fixed GPS satellite geometry at a single time snapshot, and for a varying GPS satellite geometry over a 24-hour period over an authorized UAV airspace region in the Netherlands using real satellite ephemeris data. Together, these scenarios provide insights into the structure of the DIA-estimator’s PDF, such as multimodality and orientation with respect to the chosen positioning safety region, and support comprehensive evaluation of positioning safety. Although the current focus is on GPS-based positioning, the presented approach is general and can be extended to include multisensor configurations, additional GNSS constellations, and applied to other safety-critical applications, which are subjects of future work.

Engineering Signal Analysis is an introductory textbook on the analysis of signals in time and frequency domain. It discusses how to characterize, model, analyze, interpret and operate on signals in time and frequency. In the first part of the book the basic theoretical concepts are introduced in continuous time, covering the Fourier Series and Fourier Transform. The second part introduces discrete-time signals, addressing sampling and finite signal duration, and their implications on spectral analysis. The third part elaborates on spectral estimation, covering the basic periodogram and more advanced methods. The fourth part concludes the book with an introduction to linear system theory, and addresses in particular signal filtering.

Positioning technologies are widely used in automotive, aviation, rail, and maritime safety-critical applications. Therefore, the computation of the probability of positioning failure for vehicles, which is the probability that the position estimator is outside a safety region, is of interest for positioning safety analyses. Since parameter estimation and statistical hypothesis testing for model misspecifications are commonly employed in positioning algorithms, the resulting position estimator is conditioned on the statistical hypothesis testing outcome. Hence, the probability density function (PDF) of the vehicle position estimator that accounts for the dependence between the two inference concepts should be used in the computations. In this contribution, we propose a method to compute the probability of positioning failure using the PDF of the vehicle position estimator, which accounts for the aforementioned dependence and is based on rare event simulation techniques, specifically Importance Sampling and the Cross-Entropy method. We apply the proposed method to a satellite-based positioning scenario, in decimeter precision, of an automated vehicle. The results show that the proposed method enables extensive positioning safety-analyses giving insights that can be used in the development of positioning algorithms and deciding whether safety targets and/or requirements are met. Finally, we discuss some limitations of the method and propose several further improvements.

Teunissen (J Geod 98(83):1–16, 2024) proposed the ambiguity-resolved (AR) detection theory for GNSS mixed-integer model validation. In this contribution, we study the performance of the AR detector through analysis and simulation experiments and compare it with the ambiguity-float (AF) and ambiguity-known (AK) detectors. We describe how the detectors can be implemented and how to evaluate their performance by computing the power as functions of the model misspecifications’ size. We present two simulation experiments with single- and dual-frequency GPS models and demonstrate that the AR detector can provide a larger detection power than the AF detector, even if the success rate is not close to one. Then, we obtain power functions over 25 user locations with five observation models and 72 satellite geometries per location per model. We find that the AR detector increases the detection probability of ionosphere and troposphere delays by 47% and 60% on average when the success rate is larger than 97.5% and the level of significance is 0.01. We also find the AR detection power to be larger than that of the AF detector in case of multi-dimensional misspecifications.

The analysis of positioning safety often employs a probability-based formulation. This approach quantifies the probability of positioning failure, which is the probability of the position estimator being outside a safety-region, and compares it against an application specific requirement. The design of positioning algorithms for safety-critical applications, such as automated/autonomous vehicles, should consider the dependence between parameter or state estimation and statistical hypothesis testing for model misspecifications in the evaluation of positioning safety. If this dependence is not considered, as this article shows, the conclusions drawn from the positioning safety analysis might be overly-optimistic. Therefore, this article focuses on the aforementioned dependence through a vehicle positioning scenario based on an Extended Kalman Filter (EKF) and the Detection, Identification, and Adaptation (DIA) method for misspecifications in the motion and measurement models. Grounded in the distributional theory for the DIA method, our positioning safety analysis utilizes the conditional probability density functions (PDFs) of the combined EKF and DIA position error, which are generally nonnormal. We compute the probability of vehicle positioning failure in two cases 1) when the dependence is considered and 2) when it is not, to quantify the over-optimism introduced by ignoring this dependence. Finally, we present our conclusions and recommendations.