On Minimum Spanning Trees and Determinants

Wright, Perrin

doi:10.1080/0025570x.2000.11996794

Cited by 11 publications

(6 citation statements)

References 2 publications

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“…Because of the incoherent measurement, only I m − 1 phase differences have to be evaluated. Hence, a matrix A m of rank I m − 1, resulting in a spanning tree [29] between all antennas, is sufficient to maintain all information. The Appendix A provides a more detailed discussion.…”

Section: B Measurement Modelmentioning

confidence: 99%

“…In this way, the unambiguity range (−π, π] implicitly provides the information about the maximally allowed displacement of the predicted beacon position. In the cost function of the maximum a posteriori estimator (29), this is reflected in the non-convex measurement part J meas,k , which has, apart from its global minima, multiple local minima due to the 2π phase ambiguity, while the prediction part J pred,k is a quadratic function. To discuss the ambiguity problem in more detail, a closer look is taken at the measurement residual, caused by a wrongly estimated beacon position.…”

Section: Solution Of the Maximum A Posteriori Cost Functionmentioning

confidence: 99%

“…Fig.4. J meas,k in(29) for the two antenna arrangements in Fig.3with antenna distance of 1 λ and 5 λ as well as their combination for a measurement z = 0, depicted for an evaluation along the red line inFig 3. …”

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confidence: 99%

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Phase Difference Based Precise Indoor Tracking of Common Mobile Devices Using an Iterative Holographic Extended Kalman Filter

Brückner

Sippel

Lipka

et al. 2022

IEEE Open J. Veh. Technol.

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The 3D indoor localization of low-cost standard mobile devices represents an important research topic. Since the implementation of ultra-wideband localization systems requires elaborated hardware, a localization concept based on phase-difference-of-arrival (PDOA) evaluation of narrow band communication signals at spatially distributed antennas is favorable in many applications. Typically, PDOA measurements are used to estimate the angle-of-arrival (AOA) at several receivers, which are then combined via multiangulation. However, AOA estimation requires far field conditions, thereby limiting measurement sensitivity, and distorts measurements in a non linear fashion. To overcome these limitations, this paper proposes the iterative holographic extended Kalman filter (IHEKF), which directly evaluates the phase differences between spatially distributed antenna pairs. The IHEKF requires neither a specific waveform nor emitter-receiver synchronization and, therefore, represents a good candidate for localization within communication systems such as 5G/6G. Since the evaluation of phase differences is affected by phase ambiguity, the IHEKF is designed so that closely spaced antenna pairs are evaluated first and then more distant antennas are included successively to improve accuracy. The IHEKF's capabilities are demonstrated via a 24 GHz narrow band measurement setup with strong multipath propagation, providing outstanding localization accuracy in the millimeter range without consuming any notable RF signal bandwidth.

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Section: B Measurement Modelmentioning

confidence: 99%

Section: Solution Of the Maximum A Posteriori Cost Functionmentioning

confidence: 99%

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Phase Difference Based Precise Indoor Tracking of Common Mobile Devices Using an Iterative Holographic Extended Kalman Filter

Brückner

Sippel

Lipka

et al. 2022

IEEE Open J. Veh. Technol.

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show abstract

“…To assure that all antennas' phase information is incorporated, T pre,nR needs to have full rank. Hence, the evaluated phase differences should create a spanning tree between all antennas of the array [42] to assure a full rank of the first N A,nR − 1 rows of T pre,nR . Since the phase difference unambiguity range of an evaluated antenna pair increases with decreasing antenna distance, it is advisable to choose the evaluated phase differences in T pre,nR so that each receive antenna is connected to the spanning tree via the most closely spaced neighboring antenna.…”

Section: B Phase Preprocessingmentioning

confidence: 99%

Quasi-Coherent Phase-Based Localization and Tracking of Incoherently Transmitting Radio Beacons

et al. 2021

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The direct measurement of distance-dependent information between wireless units represents a challenge for wireless locating systems, because it requires the exact time synchronization of separate wireless units. To avoid these synchronization efforts, many wireless locating systems only evaluate phase difference of arrival (PDOA) measurements. While simple PDOA localization techniques rely on multiangulation, advanced PDOA concepts like the holographic extended Kalman filter (HEKF) directly evaluate the measured phases without non-linear preprocessing. However, these differential phase measurement approaches are less sensitive than systems that can measure absolute phase variations, which allow the tracking of much smaller position changes than the signal's carrier wavelength. This paper proposes to extend the HEKF by the evaluation of absolute phases in an incoherent measurement setup, which consists of a continuous wave (CW) beacon and several receivers. The developed quasi-coherent holographic extended Kalman filter (QCHEKF) uses the overdetermined PDOA measurements to estimate the phase-frequency relation between each beacon-receiver pair. Then, the established phase-frequency relations allow the evaluation of absolute phase measurements and, thus, the accurate localization and tracking of a simple, unsynchronized, narrowband CW beacon, even under severe multipath conditions. This novel concept is experimentally validated via 3D localization results in a challenging indoor scenario using a 24 GHz CW measurement setup. Here, the QCHEKF improves the achieved localization accuracy in comparison to the HEKF by 35 % from 0.78 cm to 0.51 cm, while the maximum deviation from the trajectory reduces by 68 % from 5 cm to 1.6 cm. Furthermore, the QCHEKF enables the exact tracking of fast changes in direction, which is usually a significant challenge for standard wireless target tracking systems.INDEX TERMS Radar, Kalman filters, incoherent measurements, localization, array signal processing.

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“…Therefore, it can be solved very efficiently with a standard network simplex algorithm. Moreover, the resulting optimal spanning tree solution U * has full rank (Wright, 2000). The last step is to randomize this binary matrix to obtain the desired matrix.…”

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confidence: 99%