Optimal Anchor Placement for Wireless Localization in Mixed LOS and NLOS Scenarios

Abstract

We develop a unified Fisher-information framework for localization in environments with both Line-of-Sight (LOS) and Non-Line-of-Sight (NLOS) paths, focusing on diffraction-dominated NLOS propagation characteristic of Outdoor-to-Indoor (O2I) signal propagation. The model couples anchor geometry with a physically grounded path-loss law that is continuous across the LOS/NLOS boundary and serves as an optimization objective for our optimal anchor placement problem. As the first step, we analyze single-target anchor placement and derive the classical A-, D-, and E-optimality criteria. Under a specific path-loss assumption, these criteria collapse to a polygon-closure condition in the complex plane: A-, D-, and E-optimal designs coincide, yielding necessary and sufficient conditions for optimal placement. Next, we extend the notion of optimal anchor placement with respect to a single target to optimality over a feasible region (multi-target setting) using a general formulation that explicitly includes a realistic path loss model. This is achieved by recasting the anchor placement as a combinatorial anchor-selection problem with provable guarantees. Next, we specify E- and D-optimal objectives over multiple targets in a predefined feasible target region and show that E-optimality straddles A-optimality (within a constant factor), while D-optimality provides looser bounds. These insights yield two practical algorithms, both mixed-integer second-order cone programs (MISOCP) with exact E-optimal and exact D-optimal objectives that produce robust, region-wide designs under mixed LOS/NLOS conditions.

Figures (5)

• Figure 1: In the O2I scenario, we have $K$ anchors transmitting orthogonal signals which are received by the $n^{\text{th}}$ target inside the building. The received signal includes several MPCs, from which the ranging measurement corresponding to the diffraction path length $\bm{A_k}\bm{Q}_e\bm{N}_n$ is extracted. A bandwidth of $200\,\text{MHz}$ is assumed, ensuring that all MPCs are resolvable.
• Figure 2: For a fixed target $n=1$ with ranging-information weights $\{\lambda_{k,1}\}_{k\in \{1,2,3,4\}}=\{1.5,\,2,\,2.3,\,2.5\}$, the generalized triangle inequality \ref{['lemma_necessary_sufficient_condition_polygon_construction']} is satisfied, hence the optimal angles according to our polygon closure argument are $2\psi_{1,1}=138.2^\circ$, $2\psi_{2,1}=314.7^\circ$, $2\psi_{3,1}=17.2^\circ$, and $2\psi_{4,1}=186^\circ$, achieving A-, D-, and E-optimality simultaneously. The corresponding anchor positions can be obtained by solving the transcendental equation in \ref{['eq_psi_definition']}, similar to the approach in sadeghi2020optimal.
• Figure 3: Polygon closure for $12$ targets spread across $3$ floors of a building for the optimal solution obtained using E-opt-MISOCP. The optimization algorithm attempts to simultaneously close the polygon as well as reduce their perimeters by selecting the best $4$ anchors so as to optimize the E-opt objective for the worst performing target.
• Figure 4: Localization performance of the E-opt-MISOCP, and D-opt-MISOCP formulations with $K=4$ anchors versus sampling distance $d$. Each boxplot summarizes variability across Monte Carlo trials: the horizontal line marks the median; the box spans the 25%--75% Inter-Quartile-Range (IQR) range; whiskers extend to $1.5\times$ IQR; and dots indicate outliers beyond the whiskers. Hence, shorter boxes and whiskers indicate tighter dispersion (greater robustness), while a lower median indicates better typical performance. For larger $d$, the solver attains optimality within the 70 s time limit; for denser dictionaries ($d<5$ m), solutions are high quality but not provably optimal within the time limit. Among the objectives, D-opt-MISOCP achieves the lowest CER, E-opt-MISOCP attains the smallest MAD, and both also reduce PEB, consistent with the theoretical relationships among A-, D-, and E-opt criteria in Proposition \ref{['proposition_A_D_E_single_node']}. The baseline performance is set by the random search over all three optimality criteria.
• Figure 5: Finer sampling (smaller $d$) for the anchor-target dictionary in remark \ref{['remark_anchor_node_dictionary']} yields more candidate anchors and longer solve times; a $70\,\text{s}$ per-trial time limit is imposed. If the solver does not terminate before this limit, it returns the best incumbent solution found up to that point.

Theorems & Definitions (17)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 1
• Lemma 1
• proof
• Remark 2
• Remark 3
• Theorem 1