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A Comprehensive Tutorial on Localization: Algorithms and Performance Analysis Tools

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Abstract

This tutorial offers a comprehensive view of technological solutions and theoretical fundamentals of localization and tracking (LT) systems for wireless networks. We start with a brief classification of the most common types of LT systems, e.g. active versus passive technologies, centralized versus distributed solutions and so forth. To continue, we categorize the LT techniques based on the elementary types of position-related information, namely, connectivity, angle, distance and power-profile. The attention is then turned to the difference between active and passive LT systems, highlighting the evolution of the localization techniques. Motivated by the interests of industry and academia on distance-based active localization system, a deep review of the most common algorithms used in these systems is provided. Non-Bayesian and Bayesian techniques will be tackled and compared with numerical simulations. To list some of the proposed approaches, we mention the multidimensional scaling (MDS), the semidefinite programming (SDP) and the Kalman filter (KF) methods. To conclude the tutorial, we address the fundamental limits of the accuracy of range-based positioning. Based on the unifying framework proposed by Abel, we derive the closed-form expressions for the Cramér–Rao lower bound (CRLB), the Battacharyya Bound (BB), the Hammersley–Chapmann–Robbins Bound (HCRB) and the Abel Hybrid Bound (AHB) in a source localization scenario. We show a comparison of the aforementioned bounds with respect to a Maximum-Likelihood estimator and explore the difference between random and regular (equi-spaced anchors) network topologies. Finally, extensions to cooperative scenarios are also discussed in connection with the concept of information-coupling existing in multitarget networks.

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Notes

  1. In this work it is assumed that targets are points in the space without physical dimensions.

  2. Also the accuracy of AOA-system degrades as the source moves farther from the measuring unit.

  3. As explained later, states are the variables of interest that are to be computed, e.g., coordinates, velocity and so forth.

  4. These methods, also known as Fisher methods have a lower bound on the error covariance matrix which is given by the inverse of the Fisher information matrix (FIM), namely the CRLB [8].

  5. Often the cost function is non-convex and therefore the solution is only a local minima which in turn implies the non-optimality of the solution.

  6. In the more general case of multiple observations affected by zero mean Gaussian noise with different standard deviations the ML is equivalent to a weighted least squares (WLS) estimator.

  7. Although the algorithms would require the analytical expression of the gradient, numerical approximations are also possible [33].

  8. Quasi-Newton methods avoid to recompute the Hessian at each step.

  9. Notice that the ML solution can be interpreted as the output of a MAP estimator with uniform a priori distribution for x.

  10. For multiple observations the estimates for x can be recursively updated on the basis of the new observations [8].

  11. Usually the dynamic models are restricted to be probabilistic Markov sequences [8].

  12. Approximations are necessary to make the computation involved in the Bayesian framework feasible.

  13. Strictly speaking the two SONAR architectures are usually referred to as active and passive systems. This should not be confused with the more general definition of active and passive system given in Sect. 4 and 5 which is used in this work.

  14. Only the current set of information is used to make the assignment.

  15. These solutions consider as tracks the sequence of measurements originating from the same point in the space [74].

  16. The MHT is intrinsically an MTT filter although it is possible to modify it to work in the STT scenario.

  17. Labeling-algorithms can be used in conjunction with this solution to recover this information [80].

  18. Under such circumstances the dimension of the targets state vector becomes an unknown random number.

  19. Since the PHD does not integrate to one it is not a pdf.

  20. The latter used to further refine the search-space for \(\hat{{\bf D}}\).

  21. The C-MDS uses the double-centred Euclidean kernel constructed from (14a) setting \({\bf a} = \frac{1}{N}\cdot{\bf 1}_N\) and using all the pairwise distances between the N points in X. Then \(\hat{{\bf X}}\) is recovered from K as

    $$ \hat{{\bf X}} ={\bf U}_{\eta} \cdot \varvec{\Upsigma}_{\eta}^{\circ\frac{1}{2}}, $$
    (28)

    followed by the Procrustes transformation [16]. This solution represent the best approximation of K [23] that can be obtained under the constraint of rank(K) = η and it is exact provided that \({\varvec{\Updelta}}\in{\mathbb{EDM}}^N\).

  22. Namely single target scenarios.

  23. In [37], it is shown how to compute α in the case of w i  = 1, ∀ i, namely σ i  = σ,  ∀ i, which is also proved to be the optimal solution to the problem.

  24. Notice that generally the SDP solution is used as an initial estimate for X and it is generally refined by means of other optimization, e.g. standard non-linear least square (NLS) optimizations.

  25. Although not considered in this work, models for continuos-discrete and continuos-time systems are known in the literature and can be found in [8, 97] while recent advances in the area are offered in [98].

  26. Implicitly it also defines the noise statistics.

  27. The following marginals can also be computed:

    • prediction distribution \(p(\user2{x}_{\bar{k}+\bar{n}}| \user2{z}_1,\dots, \user2{z}_{\bar{k}})\), marginal of the future states;

    • smoothing distribution \(p(\user2{x}_{\bar{k}}| \user2{Z}^{\bar{T}})\), marginal of \(\user2{x}_{\bar{k}}\) given the set \(\user2{Z}^{\bar{T}}\).

  28. The distribution \(p(\user2{x}_{\bar{0}})\triangleq p(\user2{x}_{\bar{0}}|\user2{z}_{\bar{0}})\) at \(\bar{k}=0\) is known.

  29. Its computation would require the entire filtering pdf obtained solving the multidimensional integrals in (44) and (46).

  30. Also notice that \({\bf P}_{\user2{x},\bar{k}|\bar{k}}=\left[{\bf I}-\user2{K}_{\bar{k}}\cdot \user2{G}_{\bar{k}}\right]\cdot{\bf P}_{\user2{x},\bar{k}|\bar{k}-1}\).

  31. The derivation of the KF can be obtained using a LS argument [8].

  32. An anchor is a node whose location is known a priori, while a target is a node whose location is yet to be determined.

  33. The information vector \(\varvec{\nu}\) determines the features captured by the corresponding bound. Via appropriate choices of \(\varvec{\nu}\), different bounds better suitable to small, large and both types of estimation errors can be derived.

  34. The former refers to a scenario where anchor nodes are located at the corner of a square, and the latter to the most general assumption where anchors are randomly distributed.

  35. Recall that the CRLB corresponds to lower bound on the variance of an unbiased estimator.

  36. Even in the case of the target node outside the convex-hull of the anchors but within a reasonable distance from it, the results do not differ much from Fig. 11.

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  1. Centre for Wireless Communications, University of Oulu, P.O. Box 4500, 90014, Oulu, Finland

    Davide Macagnano & Giuseppe Destino

  2. School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, 28759, Bremen, Germany

    Giuseppe Abreu

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Macagnano, D., Destino, G. & Abreu, G. A Comprehensive Tutorial on Localization: Algorithms and Performance Analysis Tools. Int J Wireless Inf Networks 19, 290–314 (2012). https://doi.org/10.1007/s10776-012-0190-4

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  • DOI: https://doi.org/10.1007/s10776-012-0190-4

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