In this chapter, we introduced a new complex-domain algorithmic framework for wire-less localization in hybrid networks. Unlike the isometric embedding-based algorithms, the new framework is based on an entirely different approach that resembles that of maximum-ratio combining and eliminates the need for expensive decompositions altogether.
The new framework was exploited extensively with the proposal of various new algorithms, which were shown to outperform the original SMDS in terms of computa-tional cost, localization accuracy or both simultaneously. Furthermore, the proposed algorithms are specifically designed to suit different scenarios of structured information unavailability and strike a different trade-off between performance and complexity.
In this chapter, we considered two typical cases for structured information erasure, namely singleton and distance-based cooperative case. As for the singleton case, which resembles many practical systems such as Wifi and cellular, no target-to-target communication is available. Based on the partitioning of CD-SMDS complex kernel and recasting the edge-embedding as a simple projection problem, whose solution resembles the MRC operation employed at the reception of wireless signals, two
singleton algorithms dubbed as singleton SMDS and iterative singleton MRC-SMDS are proposed, which are shown to both outperform the original MRC-SMDS under the same distance and angle measurement errors. Moreover, due to the elimination of the expensive decompositions, the algorithms perform excessively faster than the original SMDS.
Furthermore, following a similar approach, three more cooperative localization algo-rithms with different trade-offs between performance and complexity are introduced, in which cooperation amongst target nodes is assumed only in terms of the collection of distance information. The best of these, dubbed Turbo MRC-SMDS, is found to be both faster and more accurate than the original SMDS method.
Apart from the algorithms introduced in this chapter, the new complex-domain MRC framework enables the design of many other possible variations with different accuracy/complexity tradeoffs which can be tailored to better suit specific cases of information availability. Moreover, the far less complex formulation of MRC-SMDS gives it a noticeable performance advantage over the original SMDS in terms of computational efficiency. Therefore, as for the future work, many more new MRC-based algorithms can be designed and implemented.
Furthermore, although the algorithms explicitly described are limited to the two-dimensional case, their generalization to three-dimensions can be done as future work by employing multi-complex numbers. In order to address the three-dimensional scenario, the corresponding complex product of two multi-complex number (edges) has to be well defined and formulated such that it can be constructed from the corresponding distance and angle information.
Appendices: Fisher Information Derivations
A.1 Von Mises Variables
Consider a Von-Mises distribution (Tikhonov distribution) [137], which is defined as p(rtn; g(θt|θn)) = 1
2πI0(ω)eω cos(rtn−µ), (A.20) where −π ≤ rtn ≤ π, ω and µ are the shape and centrality parameters and I0(x) denotes the modified Bessel function of the first kind and 0-th order.
The Fisher information of this distribution is given by
F = Ertn
"
∂ ln p(rtn; g(θt|θn))
∂g(θt|θn)
2#
= Ertn
h(−ωsin (rtn− µ))2i
= ω2Ertn
1− cos(2 (rtn− µ)) 2
=ω2 2 −
Z π
−π
cos (2(rtn− µ))
2πI0(ω) eωcos(rtn−µ)drtn
= ω2 2 .
Here we used the geometric identity sin2(x) = 1−cos(2x)2 and the fact that cos(2x) and exp (ωcos(x)) are two orthogonal functions for which the integral over [0, 2π] vanishes.
A.2 Nakagami Variables
The PDF corresponding to a Nakagami variable [138] is defined as p(rtn; g(θt|θn)) = 2mm the second moment of the distribution and controls the spread and Γ(.) is the Gamma function .
Following equation (2.55), the FIM corresponding to Nakagami-m distribution would be
In the above proof, to derive the first term of the expression, we used the general integral property [139, p361]
and for the second term we employed the formula for the k-th moment of Nakagami distribution E[rktn] = ΥΓ(m+k/2)mΓ(m) [138].
A.3 Gamma Variables
Although there exists multiple parameterisation of the Gamma distribution, we choose to use the one where the corresponding PDF is
p(rtn; g(θt|θn)) = 1
where κ > 0 and υ > 0 are the shape and scale parameter accordingly. Similar to the above sections and following equation (2.55), the FIM corresponding to Gamma
distribution would be
F = Ertn
"
∂ ln p(rtn; g(θt|θn))
∂g(θt|θn)
2#
= Ertn
"
κ− 1 rtn − 1
υ
2#
= (κ− 1)2Ertnrtn−2 −2(κ− 1)
υ Ertnr−1tn + 1 υ2
= (κ− 1)2 1
υ2(κ− 1)(κ − 2)−2(κ− 1)
υ · 1
υ(κ− 1) + 1
υ2 = 1 υ2(κ− 2). In the above proof, to derive the first and second term of the expression, we used the general integral property from equation (A.22).
Own Publications
Journal papers:
• A. Ghods and G. Abreu, “Complex-domain Super MDS: A new framework for wireless localization with hybrid information,” in IEEE Transaction on Wireless Communication, 2018. (Under Review)
Conference papers:
• G. Abreu and A. Ghods, “Complex-domain Super MDS: Hybrid wireless local-ization at low complexity,” IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), 2018. (Under Review)
• G. Abreu and A. Ghods, “Turbo MRC-SMDS: Low-complexity cooperative localization from hybrid information,” IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), 2018. (Under Review)
• A. Ghods and G. Abreu, “Complex domain Super MDS: Computationally efficient localization via ranging and angle information,” in IEEE Wireless Communica-tions and Networking Conference (WCNC18), 2018.
• A. Ghods, S. Severi, and G. Abreu, “MRC implementation of Super MDS for efficient 2D localization,” Proc. IEEE Workshop on Positioning Navigation and Communication, (WPNC 2017), 2017.
• A. Ghods, G. T. F. de Abreu, and S. Severi, “Cholesky MDS: A fast and efficient heterogeneous localizaiton algorithm,” in Proc. IEEE 5th ICC Workshop on Advances in Network Localization and Navigation (ANLN), 2017.
• A. Ghods, S. Severi, and G. Abreu, “Localization in V2X communication net-works,” in 2016 IEEE Intelligent Vehicles Symposium (IV), 2016.
• A. Ghods, S. Severi, G. Abreu, S. V. de Velde, and H. Steendam, “On the structural nature of cooperation in distributed network localization,” in 48th Asilomar Conference on Signals, Systems and Computers, 2014.
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