Frame Theory and Optimal Anchor Geometries in Wireless Localization

Abstract

Abstract-We revisit the problem of describing optimal anchor geometries that result in the minimum achievable MSE by employing the Cramer Rao Lower bound. Our main contribution is to show that this problem can be cast onto the whelm of modern Frame Theory, which not only provides new insights, but also allows the straightforward generalization of various classical results for the anchor placement problem. For example, by employing the frame potential for single-target localization, we see that the directions of the anchors, as seen from the target, should optimally be as orthogonal as possible, and that the existence of an optimal geometry for an arbitrary number of anchors is governed by the fundamental inequality in frame theory. Furthermore, the frame-theoretic approach allows for the simple derivation of some properties on optimal anchor placement that prove to be useful in a tractable approach for the more complex, multi-target anchor placement problem.In a more general sense, the paper builds a refreshing bridge between the classical problem of wireless localization and the powerful domain of Frame Theory, with far-reaching potential.

On the use of tight frames for optimal sensor placement in time-difference of arrival localization

“…Notice how in the initial configuration the sensors are approximately concentrated along the bisector of In Figs. 2, 3 and 4 we show the RMSE estimation performance of sensor networks whose sensor placement is defined by different frames. All numerical experiments follows the same setup: we randomly generate via a Gaussian distribution the initial sensor positions in the frame A for which we fix σ 1 = 3 and σ 2 = 1 and when we calculate the new sensor positions in a tight frame B built via (8) and a symmetric tight frame C via (14). The results we show are averaged in the following way: we generate 100 instances of A (and consequently B and/or C) and then for each instant we proceed to estimate 1000 randomly generated sources x from noisy distance measurements.…”

“…4 we compare the random initial positioning with the positioning given by the s-tight frame B (8) closest to the random configuration and the symmetric s-tight frame C (14). The frame C is created starting from the positions of first m/4 sensor from A.…”

“…Previous work in the literature has already considered the use of tight frames for optimal sensor placement [5], [14]. These papers provided an optimal placement strategy given the number of anchor points or proposed heuristics to construct frames whose properties maximize localization accuracy.…”

Sequential sensor placement in two‐dimensional passive source localisation using time difference of arrival measurements

“…angle of arrival/ TOA) has been studied in [24]. An optimisation problem in frame theory that is equivalent to the sensor placement problem is addressed in [25]. Assuming equal signal-to-noise ratios (SNRs) in all receivers is a typical assumption but recently SNR dependent sensor placement has become popular [26].…”