PaperlayerView exact revision on arXiv

Source-linked AI summary

Movable Antenna Enhanced Wireless Sensing Via Antenna Position Optimization

Wenyan Ma, Lipeng Zhu, Rui Zhang

arXiv:2405.01215v1cs.ITeess.SPmath.IT

TL;DR

Fixed-position arrays cannot adapt their geometry to sensing needs, and the fundamental effect of MA positions on sensing performance had not been characterized. The paper derives AoA CRBs and optimizes 1D/2D antenna positions; its designed arrays reduce CRB and actual MSE relative to conventional ULAs/UPAs.

  • Problem

    Fixed-position arrays cannot adapt to different sensing requirements, and the fundamental relationship between MA positions and wireless-sensing performance had not been characterized.

    Source: §I · pp. 1–2

  • Method

    The paper derives AoA CRBs as functions of 1D/2D antenna positions, then optimizes those positions using closed-form 1D placement and 2D analytical and alternating-optimization approaches.

    Source: §I · p. 2, §VI · p. 10

  • Results

    Designed 1D/2D MA arrays significantly decrease AoA estimation CRB and actual MSE compared with conventional ULAs/UPAs using different uniform inter-antenna spacings.

    Source: §I · p. 2, §VI · p. 10

  • Takeaways & Limitations

    Designed MA-array steering vectors have low angular-domain correlation, which reduces ambiguity in angle estimation.

    Source: §I · p. 2

    The 2D position-optimization problem is difficult because its fractional and minimum-distance constraints are non-convex and the horizontal and vertical coordinates are coupled.

    Source: B. AoA Estimation and CRB Minimization · p. 5

Abstract

from arXiv · show

In this paper, we propose a new wireless sensing system equipped with the movable-antenna (MA) array, which can flexibly adjust the positions of antenna elements for improving the sensing performance over conventional antenna arrays with fixed-position antennas (FPAs). First, we show that the angle estimation performance in wireless sensing is fundamentally determined by the array geometry, where the Cramer-Rao bound (CRB) of the mean square error (MSE) for angle of arrival (AoA) estimation is derived as a function of the antennas' positions for both one-dimensional (1D) and two-dimensional (2D) MA arrays. Then, for the case of 1D MA array, we obtain a globally optimal solution for the MAs' positions in closed form to minimize the CRB of AoA estimation MSE. While in the case of 2D MA array, we aim to achieve the minimum of maximum (min-max) CRBs of estimation MSE for the two AoAs with respect to the horizontal and vertical axes, respectively. In particular, for the special case of circular antenna movement region, an optimal solution for the MAs' positions is derived under certain numbers of MAs and circle radii. Thereby, both the lower- and upper-bounds of the min-max CRB are obtained for the antenna movement region with arbitrary shapes. Moreover, we develop an efficient alternating optimization algorithm to obtain a locally optimal solution for MAs' positions by iteratively optimizing one between their horizontal and vertical coordinates with the other being fixed. Numerical results demonstrate that our proposed 1D/2D MA arrays can significantly decrease the CRB of AoA estimation MSE as well as the actual MSE compared to conventional uniform linear arrays (ULAs)/uniform planar arrays (UPAs) with different values of uniform inter-antenna spacing.

I. INTRODUCTION

The paper addresses the lack of a fundamental account of how movable-antenna positions affect wireless sensing, deriving AoA estimation bounds and optimizing array geometry. It models a 1D sensing array and shows how antenna-position spread relates to the CRB.

I. INTRODUCTION

  • Movable antennas add position control that can enlarge array aperture, reduce steering-vector correlation, and adapt array geometry to changing sensing needs.The paper also notes that reconfigurable positions can support functionality switching between communication and sensing in ISAC systems.Source: §I · p. 1
  • Prior MA sensing studies demonstrated performance benefits but did not characterize the fundamental relationship between antenna positions and sensing performance.Large or sparse fixed-position arrays also cannot adapt their geometry to different sensing requirements or exploit signal variation across the movement region.Source: §I · pp. 1–2
  • For 1D arrays, the AoA CRB is inversely proportional to the variance of antenna positions, and the paper gives a globally optimal closed-form placement solution.Maximizing position variance spreads antennas apart and increases array aperture.Source: §I · pp. 1–2
  • For 2D arrays, the paper targets the minimum of the maximum CRBs for horizontal and vertical AoAs, deriving a circular-region solution for certain antenna counts and radii and bounds for arbitrary regions.It also proposes alternating coordinate optimization to obtain a locally optimal placement.Source: §I · p. 2
  • Designed 1D/2D MA arrays decrease AoA estimation MSE relative to conventional ULAs/UPAs with different uniform inter-antenna spacings.The paper also attributes reduced angle ambiguity to low angular-domain correlation between the designed arrays’ steering vectors.Source: §I · p. 2

A. System Model

  • The 1D system places N antennas at ordered positions within a line segment of length A and estimates the target’s spatial AoA u = cos ϑ under a far-field LoS model.The target-receiver channel is assumed static over T snapshots.Source: A. System Model · pp. 2–3

B. AoA Estimation

  • The receiver uses MUSIC to estimate spatial AoA, and the CRB decreases as the variance of the antenna-position vector increases.This makes antenna-position variance the 1D placement objective for reducing the CRB.Source: B. AoA Estimation · p. 3

C. CRB Minimization

For a 1D MA array, the CRB-minimizing antenna positions are obtained by optimizing the antenna positioning vector (APV) under line-segment and minimum-spacing constraints. The closed-form solution places two groups at opposite ends, and the minimum CRB decreases as the movement segment grows.

C. CRB Minimization

  • The non-convex CRB-minimization problem optimizes x subject to line-segment bounds and minimum separation D, requiring A ≥ (N −1)D for feasibility.Source: C. CRB Minimization · p. 3
  • The corresponding minimum CRB decreases with A in the order of O(A^-2).Source: C. CRB Minimization · p. 4
  • The optimal APV places half the MAs at each end of the line segment, with minimum spacing D within each group.The closed-form positions are (n −1)D for the first group and A −(N −n)D for the second.Source: C. CRB Minimization · p. 4

III. CRB CHARACTERIZATION FOR 2D MA ARRAY

The 2D array models antenna positions in a continuous plane and estimates two spatial AoAs from the target-receiver LoS path. The section characterizes how array geometry affects both AoA CRBs and formulates a min-max position optimization whose constraints and coordinate coupling make it difficult to solve optimally.

A. System Model

  • The system places N MAs at coordinates in a continuous 2D region to estimate the target-receiver LoS path’s AoAs.The physical angles are elevation θ and azimuth φ; the spatial AoAs are u and v.Source: A. System Model · p. 4

B. AoA Estimation and CRB Minimization

C. Optimal Solution for Problem (P2) for Circular Region Ccir

For a circular movement region, the paper derives bounds on the min-max CRB and identifies conditions under which the bound is tight. The analysis also links antenna placement and region size to sensing performance.

C. Optimal Solution for Problem (P2) for Circular Region Ccir

  • The circular-region analysis provides an upper bound on the objective and a corresponding lower bound on the min-max CRB.The supplied theorem passages do not give the bound expressions in readable form.Source: C. Optimal Solution for Problem (P2)… · p. 5
  • When D ≤ 2A sin(π/N) and N = 4K, the circular-region min-max CRB bound is tight for the stated optimal antenna positions.The paper states these conditions for K ∈ Z+ and gives the corresponding positions in (55), (57), and (58).Source: C. Optimal Solution for Problem (P2)… · p. 5
  • Placing antennas on the region border is desirable for maximizing aperture, but more antennas may need to move inside to satisfy minimum separation.For the circular region, the stated border-spacing condition is D ≤ 2A sin(π/N).Source: C. Optimal Solution for Problem (P2)… · p. 5
  • The lower bound on min-max CRB is inversely proportional to the circular movement-region area πA².The passage concludes that increasing the movement-region size can reduce the CRB.Source: C. Optimal Solution for Problem (P2)… · p. 5

D. Performance Analysis for General 2D Region C

For arbitrary continuous 2D movement regions, the paper uses inscribed and circumscribed circles to bound the min-max CRB. It also gives bounds for a square region and a condition under which its lower bound holds.

D. Performance Analysis for General 2D Region C

IV. APV OPTIMIZATION FOR 2D MA ARRAY

For the 2D array, the paper seeks locally optimal antenna positions by alternating between horizontal and vertical coordinates. This section develops the horizontal-coordinate update using successive convex approximation (SCA).

IV. APV OPTIMIZATION FOR 2D MA ARRAY

  • The overall 2D optimization alternates between the horizontal and vertical antenna-position vectors, holding one fixed while optimizing the other.The method seeks a locally optimal solution to problem (P2).Source: §IV · p. 6
  • For ease of optimization, the algorithm assumes a convex 2D movement region so constraint (20d) is convex.The preceding analysis is stated to hold for continuous regions of arbitrary shapes.Source: §IV · p. 6

A. Optimization of x with Given y

  • With y fixed, the x-update applies SCA to relax non-convex constraints, using a first-order Taylor expansion to lower-bound a convex function.The resulting relaxed problem is convex and can be solved with optimization toolboxes such as CVX.Source: A. Optimization of x with Given y · pp. 6–7
  • The x-update also uses convex surrogate constraints and a linear surrogate for antenna-separation constraints.The supplied passages describe lower-bounding G1 and constructing a surrogate for the pairwise-distance constraint.Source: A. Optimization of x with Given y · p. 6

B. Optimization of y with Given x

The y-coordinate update mirrors the x-update using SCA with x fixed. The overall algorithm alternates these updates, guarantees convergence, and has complexity determined by the subproblem iteration counts.

B. Optimization of y with Given x

  • With x fixed, the y-update uses SCA to relax non-convex constraints and solve a convex optimization problem.It modifies the x-update formulation by replacing {x, xp, y} with {y, yq, x}.Source: B. Optimization of y with Given x · p. 7

C. Overall Algorithm

  • The algorithm alternates x- and y-updates until each update's objective increase, and then the overall increase, falls below its threshold.The x- and y-update thresholds are εx and εy; the overall threshold is ε.Source: C. Overall Algorithm · p. 7
  • The objective increases monotonically through the alternating iterations and is upper-bounded, so convergence is guaranteed.The passage states monotonic increase for both subproblems and the outer iterations.Source: C. Overall Algorithm · p. 7
  • The total complexity is O(N^3.5 ln(1/κ)(Ix + Iy)I), based on the subproblem solver and iteration counts.The interior-point complexity for each convex subproblem is O(N^3.5 ln(1/κ)) at accuracy κ.Source: C. Overall Algorithm · p. 7

V. NUMERICAL RESULTS

The 1D experiments compare optimized movable-antenna positions with two fixed ULA layouts using MUSIC estimates and CRBs. The optimized positions reduce MSE and sharpen angular discrimination relative to the benchmarks.

A. 1D MA Array

  • The 1D comparison uses N = 16 antennas on a line segment of length A = 10λ, with true u = 0.71.The benchmarks are ULAH, with half-wavelength spacing, and ULAF, with the largest achievable aperture.Source: A. 1D MA Array · p. 8
  • At 20 dB, the proposed positions reduce MSE by 55.3% over ULAH, and the MUSIC estimate approaches the CRB at high SNR.Fig. 7 compares actual MSE and CRB across SNR.Source: A. 1D MA Array · p. 8, Figure 7 · p. 8
  • The optimized positions yield a narrower steering-correlation main lobe than the benchmarks, corresponding to higher spatial resolution in AoA estimation.The correlation is evaluated over candidate AoAs for a given true u = 0.71.Source: A. 1D MA Array · p. 8
  • For ULAF, the steering-vector correlation equals 1 at the false AoA −0.79 for true u = 0.71, indicating ambiguity that produces high MUSIC MSE.The ambiguity is visible in Fig. 7.Source: A. 1D MA Array · p. 8

B. 2D MA Array

The 2D experiments compare optimized MA positions with fixed UPA layouts for estimating two spatial AoAs. The proposed design lowers MSE, including with few antennas, while balancing estimation performance across both axes.

B. 2D MA Array

  • The optimized positions are symmetric across both axes and lie as far from the square-region center as possible, balancing estimation of u and v while maximizing aperture.The illustrated layouts use N = 36 and N = 100.Source: B. 2D MA Array · p. 9
  • At N = 36, the proposed scheme has lower MSE than UPAH and UPAF, approaches the CRB lower bound, and performs similarly for u and v.Fig. 10 compares the CRBs and actual MSEs versus SNR.Source: B. 2D MA Array · pp. 9–10, Figure 10 · p. 9
  • With N = 8 MAs, the proposed scheme reduces MSE by 97.1% over UPAH and stays close to the CRB lower bound across antenna counts.Fig. 12 plots MSE(u) versus N at SNR = 15 dB; MSE(u) and MSE(v) have similar performance.Source: B. 2D MA Array · p. 10, Figure 12 · p. 10

VI. CONCLUSIONS

  • The conclusion reports that optimized 1D and 2D MA arrays reduce both CRB and actual AoA MSE versus conventional ULAs and UPAs, with lower angular-domain steering-vector correlation reducing ambiguity.For the 2D case, the design targets the min-max CRB of the two AoAs.Source: §VI · p. 10

A. Proof of Theorem 1

The proof establishes the 1D optimum by sequentially moving antennas without violating feasibility and showing that variance never decreases. It also treats a circular 2D region, where grouped antenna placements attain an upper bound under specific conditions.

A. Proof of Theorem 1

  • Sequentially adjusting one antenna at a time preserves the 1D feasibility constraints and never decreases position variance, establishing the optimality of x⋆.The argument applies to any feasible starting position vector.Source: A. Proof of Theorem 1 · pp. 10–11
  • For a circular movement region, the objective is upper-bounded by A^2/2; attaining this bound requires all MAs on the circle’s border and D ≤ 2A sin(π/N).The distance condition ensures the minimum spacing constraint can also hold.Source: A. Proof of Theorem 1 · p. 12
  • For N = 4K, grouping the MAs into K groups of four provides a construction that satisfies the circular-region constraints and attains the objective upper bound.The group positions are parameterized by rotation angles.Source: A. Proof of Theorem 1 · p. 12
  • The circular-region optimal configuration is not unique; for N = 8, two feasible rotation-angle choices are (π/6, π/3) and (0, π/4).Both examples use D = 2A sin(π/12).Source: A. Proof of Theorem 1 · p. 13
Loading 2405.01215v1…