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Power Optimization for Network Localization

Yuan Shen, Wenhan Dai, Moe Z. Win

arXiv:1309.4531v1cs.IT

TL;DR

Accurate active and passive localization requires power allocation that balances localization performance with energy use and remains reliable under uncertain network parameters. The paper unifies WNL and RNL power allocation, exploits SPEB properties to obtain SOCPs, and develops robust asymptotically optimal and near-optimal algorithms. Simulations report that the proposed algorithms significantly outperform uniform allocation and provide efficiency and robustness.

  • Problem

    Power allocation must minimize transmit energy while satisfying localization requirements, despite uncertainty in network parameters that can make direct solutions unreliable or infeasible.

  • Method

    The paper unifies active and passive localization power allocation by exploiting SPEB convexity and low-rank structure to formulate robust SOCP-based algorithms.

  • Results

    The proposed power-allocation algorithms significantly outperform uniform allocation in simulations, while robust asymptotically optimal and near-optimal methods address uncertain parameters.

  • Takeaways & Limitations

    The framework provides a common SOCP-based basis for efficient and robust power allocation in active and passive localization networks.

Abstract

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Reliable and accurate localization of mobile objects is essential for many applications in wireless networks. In range-based localization, the position of the object can be inferred using the distance measurements from wireless signals exchanged with active objects or reflected by passive ones. Power allocation for ranging signals is important since it affects not only network lifetime and throughput but also localization accuracy. In this paper, we establish a unifying optimization framework for power allocation in both active and passive localization networks. In particular, we first determine the functional properties of the localization accuracy metric, which enable us to transform the power allocation problems into second-order cone programs (SOCPs). We then propose the robust counterparts of the problems in the presence of parameter uncertainty and develop asymptotically optimal and efficient near-optimal SOCP-based algorithms. Our simulation results validate the efficiency and robustness of the proposed algorithms.

I. INTRODUCTION

The paper studies power allocation for accurate active and passive network localization, balancing localization requirements with energy consumption under uncertain network parameters. It unifies WNL and RNL formulations, converts them to SOCPs, and develops robust algorithms.

  • Motivation: Power allocation must balance localization accuracy with energy consumption because transmit power affects ranging quality, network lifetime, and throughput.The paper formulates minimizing total transmit power subject to localization requirements.
  • Problem: The paper addresses two questions: minimizing transmit power for a required accuracy and guaranteeing that requirement under parameter uncertainty.Using uncertain parameter estimates directly can produce suboptimal, infeasible, or unreliable solutions.
  • Scope: A unifying framework covers wireless network localization with active agents and radar network localization with passive targets.WNL uses anchor-to-agent measurements, while RNL uses signals reflected by the target between transmit and receive antennas.
  • Method: The proposed formulations exploit localization-metric properties to transform power allocation problems into second-order cone programs.The paper specifically identifies convexity and low-rank structure as enabling properties.
  • Robust design: Robust SOCP-based algorithms are developed that are asymptotically optimal or efficient near-optimal under uncertain network parameters.The paper also characterizes the convergence rate of the asymptotic algorithms.

B. Performance Metric

The paper uses the SPEB, derived from the EFIM, as a localization-accuracy metric for power allocation. The WNL and RNL EFIMs share a canonical topology-dependent structure that supports a unified treatment.

  • Performance metric: The SPEB lower-bounds mean squared position error and is asymptotically achievable by maximum-likelihood estimators in high-SNR regimes.The paper therefore adopts SPEB for high-accuracy localization design and analysis.
  • Performance metric: The EFIM reduces the Fisher-information dimension while retaining the information needed to derive the position-error bound.This provides the information basis for the SPEB used in the power-allocation problems.
  • Wireless Network Localization: For WNL, the EFIM characterizes agent position information from anchor measurements, with equivalent ranging coefficients determined by channel, bandwidth, and noise parameters.The WNL EFIM is stated in Proposition 1.
  • Radar Network Localization: For RNL, the EFIM characterizes target position information from transmit-to-receive measurements reflected by the target.The corresponding ranging coefficients depend on channel parameters, signal bandwidth, and noise power.
  • Unified structure: Both EFIMs are weighted sums of rank-one matrices whose weights encode ranging information and whose directions encode network topology.This canonical structure applies the same analytical framework to WNL and RNL.

C. Power Allocation Formulation

The formulation minimizes total transmit power subject to localization requirements and linear power constraints for both WNL and RNL. Shared SPEB structure yields convex programs and supports the paper’s unified SOCP treatment.

  • C. Power Allocation Formulation: The WNL and RNL problems minimize total transmit power subject to SPEB requirements for agents or the target and linear constraints on the power-allocation vector.Individual antenna or anchor power limits are examples of the linear constraints.
  • C. Power Allocation Formulation: The WNL and RNL formulations have similar localization constraints because their EFIMs and SPEBs share the same functional structure.This similarity enables a unified development of optimal power-allocation algorithms.
  • III. SPEB PROPERTIES AND SOCP FORMULATION: The paper states that the SPEB properties convert the WNL localization constraint into second-order cone form and make the formulation an SOCP.The same framework is intended for the corresponding RNL formulation.
  • A. SPEB Properties: For WNL, the SPEB representation uses an equivalent-ranging-coefficient matrix and a topology matrix determined by anchor geometry.The analogous RNL representation uses transmit-antenna coefficients and its corresponding topology matrix.
  • A. SPEB Properties: The topology matrices for WNL and RNL have rank at most three.The low-rank property is identified alongside convexity as useful for efficient algorithm development.
  • A. SPEB Properties: The proposed methods can also apply to other power-allocation formulations beyond the displayed WNL and RNL problems.This scope is stated explicitly in the formulation discussion.
  • A. SPEB Properties: The SPEB is a convex function of the power-allocation vector, making each localization requirement a convex constraint.Consequently, both power-allocation problems are convex programs.

B. Optimal Power Allocation

The paper converts optimal power-allocation constraints for wireless and radar network localization into SOCPs, using the shared structure of their localization requirements. This SOCP formulation also supports robust extensions when network parameters are uncertain.

  • Wireless Network Localization: The optimal power-allocation problem for wireless localization is equivalent to a second-order cone program (SOCP).The conversion uses functional properties of the SPEB and introduces transformed variables and SOC constraints.
  • Wireless Network Localization: The resulting SOCP formulation is more computationally favorable than the prior semidefinite-program formulation because SOCP has more efficient solvers.The paper also states that the SOC form enables better relaxation in the robust setting.
  • Radar Network Localization: Radar-network power allocation has a similar structure, allowing an analogous SOCP formulation without repeating the derivation.The common structure between the two scenarios motivates a unifying optimization framework.
  • Uncertainty Models: Robust power allocation seeks to guarantee localization requirements for every agent position and uncertain network parameter within the modeled uncertainty region.The uncertainty models represent parameters through linear sets associated with position, angle, distance, and reflection-coefficient uncertainty.
  • Uncertainty Models: The robust framework applies to both wireless and radar localization because their parameter uncertainty models can be converted to a common form.The paper explicitly identifies corresponding common forms for angular and reflection-coefficient uncertainty.

B. Robust Formulation

The robust formulation replaces nominal localization constraints with worst-case SPEB constraints over uncertain network parameters. Because angular uncertainty lacks an explicit maximization formula, the paper develops sequential bounds that yield tractable relaxation problems.

  • Worst-Case Formulation: Robust power allocation imposes a worst-case SPEB constraint so localization requirements hold throughout the uncertainty region.The formulation applies to both wireless and radar networks, with the radar problem treated analogously to the wireless case.
  • SOCP Relaxations: Both robust relaxation formulations can be solved by SOCPs, while the formulation based on the upper bound is preferred because it guarantees the localization requirement.The robust constraint is imposed using the corresponding worst-case SPEB bound.
  • Worst-Case Formulation: The SPEB is monotone in the reflection-coefficient parameter, allowing that part of the uncertainty maximization to be handled explicitly.The remaining angular maximization is more difficult because the relevant function has no explicit expression.
  • Bound Construction: Sequential lower and upper bounds parameterized by M replace the intractable angular maximization and produce robust relaxation problems.These bounds are constructed for the worst-case SPEB and can be used in the localization requirement.

C. Asymptotically Optimal Algorithm

The asymptotically optimal algorithm uses SOCP relaxations indexed by M whose bounds and solutions converge to those of the original robust problem. Increasing M improves accuracy but increases computational complexity.

  • SOCP Formulation: The relaxed problems are equivalent to SOCPs, enabling the asymptotically optimal robust power-allocation algorithm.The paper establishes equivalent SOCP representations for the relevant lower- and upper-bound formulations.
  • Convergence Analysis: The gap between the lower and upper worst-case SPEB bounds converges to zero at rate O(M^-2).This establishes asymptotic tightness as the relaxation parameter M increases.
  • Convergence Analysis: The optimal solutions of the SOCP relaxations converge to the original robust problem’s solution at rate O(M^-2).The convergence result applies to the corresponding robust relaxation problems and their optimal solutions.
  • Performance–Complexity Tradeoff: The relaxation parameter M therefore provides a practical way to approximate the optimal robust solution using a small value.The paper presents this approximation as useful for implementation.
  • Performance–Complexity Tradeoff: For M ≥ 16, simulations report less than 2% performance loss, while computational complexity grows as O(M^3/2).The number of SOC constraints increases linearly with M, creating a performance-versus-complexity tradeoff.

D. Efficient Algorithms

The paper develops efficient SOCP-based near-optimal algorithms by bounding angular uncertainty, reducing robust constraints while preserving tractability. The same approach extends to radar localization and reduces to the nonrobust formulation when uncertainty vanishes.

  • Wireless Localization: For wireless localization, an upper bound on the worst-case SPEB converts each robust requirement into four SOC constraints.Replacing the original robust constraints with these SOC constraints yields an efficient SOCP relaxation.
  • Robust-to-Nonrobust Reduction: The robust relaxation retains the SOC form and reduces to the nonrobust formulation when parameter uncertainty vanishes.This property is stated for the wireless formulation and also holds for the radar counterpart.
  • Radar Localization: For radar localization, separately bounding transmit- and receive-antenna angular uncertainty yields a tighter worst-case SPEB bound.The tighter construction exploits the fact that pairwise angles are generated from transmit- and receive-antenna angles.

V. DISCUSSIONS

The paper extends its power-allocation framework to prior network knowledge and related localization formulations, preserving SOCP tractability while reducing required transmit power when prior information is available.

  • With prior knowledge, both the nominal and robust power-allocation problems can be transformed into SOCPs.
  • When prior knowledge vanishes, the resulting formulations reduce to the corresponding problems without prior knowledge.
  • Prior position knowledge adds information to the EFIM, so less transmit power is required for the same localization requirement.
  • The same SPEB properties also yield SOCP formulations for minimizing maximum or total localization error under a power constraint.
  • For a single agent without individual power constraints, optimal localization can be achieved by activating only three anchors, or three transmit antennas in radar localization.

C. Achievability of SPEB

The simulations assess SPEB-based power allocation in wireless and radar localization, showing strong gains over uniform allocation and robust behavior under parameter uncertainty.

  • The SPEB is asymptotically achievable by maximum-likelihood estimators in high-SNR regimes, while tighter bounds cover wider SNR ranges but are less tractable.
  • SOCP- and SDP-based algorithms achieve the global optimum and reduce required wireless-localization power by more than 40% versus uniform allocation.
  • As uncertainty increases, required power rises, asymptotically optimal solutions approach the optimum, and nonrobust allocation can violate the localization requirement.
  • 15% and 33% gaps from the optimum are reported for the proposed SOCP-based and SDP-based robust algorithms, respectively, at NUSS = 0.15.
  • For radar localization, SOCP allocation reduces power by over 70% versus uniform allocation with eight transmit antennas, while no improvement occurs with one.
  • 12% and 35% gaps from the optimum are reported for the SOCP-based and SDP-based radar algorithms, respectively, at NUSS = 0.15.

APPENDIX A PROOF OF LEMMA 1

The appendix proves the SPEB properties used to establish convex, SOCP-compatible power-allocation formulations.

  • The SPEB is shown to be a convex function of the power-allocation vector through composition with a nonincreasing convex matrix function.
  • The proof derives the topology-matrix representation underlying the SPEB expression and its subsequent bounds.
  • The finite projection bound discretizes angular maximization and establishes an approximation error that decreases at rate O(M^-2).

APPENDIX C PROOF OF PROPOSITION 6

The appendix establishes convergence of finite-angle robust formulations by bounding their feasible sets and optimal power relative to the continuous problem.

  • The approximation error decreases with M, enabling finite-angle robust formulations to approach the corresponding optimal solution.
  • The proof compares feasible sets and optimal objectives using SPEB power scaling and the finite-projection bound.

APPENDIX E PROOF OF PROPOSITION 8

The proof derives an upper bound for the worst-case squared position error bound under uncertain parameters and shows that the resulting constraint admits a second-order cone representation.

  • The derivation bounds the worst-case SPEB by separately bounding the relevant quadratic terms over the uncertainty set and combining those bounds.The proof uses the uncertainty relation and repeated applications of the triangular inequality to obtain the stated upper bound.
  • The SPEB is expressed through eigenvalues and the angles of corresponding eigenvectors before the upper-bound construction is applied.
  • The resulting constraint P(pk; x) ≤ k can be converted into second-order cone form, with the robust case following analogously.
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