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Movable Antenna-Enabled Multi-Target Sensing: Sidelobe Suppression via Position Optimization
Chengzhi Ye, Ruoyu Zhang, Lipeng Zhu, Qingqing Wu
TL;DR
Unknown multi-target directions and sidelobe interference complicate movable-antenna position design for sensing. The paper combines a CRB lower bound and SVC in a target-direction-independent position optimization, solved through discretization and SCA-based alternating optimization. The proposed array reaches a 100% estimation success rate at -5 dB and remains successful with eight co-existing targets.
Problem
Multi-target MA position optimization is challenging because array geometry couples with unknown target directions, while existing approaches may require direction distributions or preliminary target estimates.
Source: §I · p. 2Method
The paper derives a direction-decoupled CRB lower bound, minimizes maximum sidelobe-region SVC subject to that bound, and solves the transformed design with SCA-based alternating optimization.
Source: §I · p. 2, §III · p. 3, §VII · p. 11Results
100% estimation success at -5 dB and with K=8 targets; the proposed MA also suppresses SVC sidelobes while outperforming conventional fixed-position arrays in estimation accuracy and robustness.
Source: B. Simulations with Random Targets · pp. 10–11, §VII · p. 11Takeaways & Limitations
The proposed MA configuration combines sidelobe suppression with an estimation-accuracy constraint to provide robust multi-target sensing without relying on target-specific information.
Source: §I · p. 2, §VII · p. 11
Abstract
from arXiv · showhide
Movable antenna (MA) has emerged as a promising technology to unlock spatial degrees of freedom for wireless sensing systems. However, optimizing MA positions for multi-target sensing remains challenging due to the complicated coupling between the array geometry and unknown target directions, as well as severe sidelobe interference. In this paper, we propose an MA-enabled multi-target wireless sensing scheme to achieve both high-precision and robust target parameter estimation. To characterize the sensing performance while maintaining tractability, we introduce two complementary metrics, namely a lower bound (LB) of the Cram\'er-Rao bound (CRB) and the steering vector correlation (SVC). Specifically, by investigating the partial order relationship of the Fisher information matrix, we derive a closed-form LB of the multi-target CRB, which decouples the unknown target directions. Meanwhile, the SVC is employed to characterize the spatial correlation and sidelobe-induced interference among sensing signals over different directions. Based on these two metrics, we formulate an MA position optimization problem that minimizes the SVC while constraining the LB below a predefined threshold. To render the resulting non-convex problem tractable, we exploit the spatial symmetry of the continuous SVC function and discretize the sidelobe region, thereby transforming the spatial minimax objective into a finite set of constraints. Meanwhile, the LB constraint is equivalently reformulated as a spatial variance constraint on the MA positions. Building on these transformations, we develop a successive convex approximation (SCA) algorithm within an alternating optimization (AO) framework to iteratively optimize the horizontal and vertical MA coordinates.
I. INTRODUCTION
Movable antennas offer adaptable array geometry, but designing for unknown multi-target directions remains difficult because existing approaches depend on target-direction distributions or preliminary target estimates. The paper develops a target-direction-independent sensing design; its receiver collects target-scattered signals with movable antennas and estimates directions from MUSIC spectrum peaks.
I. INTRODUCTION
- Movable antennas add continuous position variables that let array geometry adapt to the propagation environment and sensing objective, unlike fixed sparse-array geometries.The paper proposes minimizing maximum sidelobe-region SVC while constraining a CRB lower bound, and develops an SCA-based alternating optimization algorithm for the antenna coordinates.Source: §I · pp. 1–3
- Existing multi-target MA designs rely on target-direction distributions or preliminary target estimates, motivating array design that suppresses spatial ambiguity without target-specific information.High spatial correlation can create sidelobes and ambiguous responses that may mask weak targets or produce false peaks, especially at low SNR.Source: §I · p. 2
A. System Model
- The bistatic system uses a single-antenna transmitter to illuminate K targets and a receiving base station with N movable antennas to collect their scattered signals.MUSIC estimates target directions by searching for the K largest peaks in its spatial spectrum.Source: A. System Model · p. 3
III. PERFORMANCE METRICS
The paper pairs a CRB lower bound with steering vector correlation to assess estimation accuracy and sidelobe interference. The closed-form bound decouples unknown target directions, but does not capture mutual target interference.
III. PERFORMANCE METRICS
- The CRB lower bound sets a theoretical estimation-accuracy limit, while SVC complements it by measuring sidelobe levels and interference suppression.The lower bound omits mutual interference among targets, which motivates the second metric.Source: §III · p. 3
A. CRB Lower Bound
- The derived closed-form CRB lower bound decouples unknown target angles from the coupled CRB and varies inversely with the spatial variances of the antenna coordinates.Constraining the bound below a threshold ensures minimum spatial spread, but does not directly guarantee actual CRB performance.Source: A. CRB Lower Bound · p. 4
- Because the bound relaxes the FIM by omitting cross-terms, it cannot characterize mutual target influence or interference from other search-area directions.The paper therefore introduces SVC to assess spatial resolution and interference suppression in multi-target scenarios.Source: §III · p. 3, A. CRB Lower Bound · p. 4
B. Steering Vector Correlation
SVC measures correlation between a target steering vector and candidate directions, making it a metric for sidelobe interference and spatial ambiguity. The design minimizes its maximum value outside a protected main-lobe region.
B. Steering Vector Correlation
- Sidelobe-region SVC peaks can cause angular ambiguity and false detections, and sidelobes leaking into other targets’ directions can mask weak targets.These effects are especially relevant under low-SNR conditions.Source: B. Steering Vector Correlation · p. 4
A. Sidelobe Region and Problem Formulation
- The SVC is even in the direction offset, and the design excludes the main-lobe protection region while minimizing values in the remaining sidelobe region.The white-shaded region in Fig. 2 is the optimization region; the red dashed box marks the protected main lobe.Source: A. Sidelobe Region and Problem… · p. 4, Figure 2 · p. 5
- The position design minimizes maximum SVC over the sidelobe region while requiring the CRB lower bound to meet a threshold.The formulation also constrains the movable region and requires at least λ/2 inter-antenna spacing to avoid significant mutual coupling.Source: A. Sidelobe Region and Problem… · pp. 4–5
B. Problem Transformation
The paper transforms continuous sidelobe suppression and the CRB lower-bound condition into finite SVC constraints and a spatial-variance constraint. It then uses alternating optimization and successive convex approximation to solve the position design problem.
B. Problem Transformation
- Sampling the sidelobe region turns the continuous minimax SVC objective into finite constraints, with an auxiliary variable upper-bounding the sampled SVC values.The transformed problem remains non-convex because of the SVC function and geometric constraints.Source: B. Problem Transformation · p. 5
- The CRB lower-bound threshold is equivalently reformulated as a constraint on the sum of spatial variances of the antenna coordinates.The lower bound is proportional to the reciprocal of var(y) + var(x).Source: B. Problem Transformation · p. 5
V. PROPOSED SCA-BASED AO ALGORITHM
- An SCA-based alternating optimization algorithm iteratively fixes one coordinate axis while optimizing the other, producing convex subproblems with surrogate bounds for non-convex terms.A derived Lipschitz constant bounds the Hessian magnitude and supports the convex surrogate construction.Source: §V · p. 5
A. Optimizing 𝒙with Fixed 𝒚
With the vertical coordinates fixed, the method optimizes the horizontal coordinates by replacing non-convex terms with convex bounds. A position-independent Lipschitz constant bounds the SVC curvature and supports the SCA surrogate.
A. Optimizing 𝒙with Fixed 𝒚
- The update optimizes 𝒙 while holding 𝒚 fixed, separating the horizontal-coordinate step from the full position optimization.The method constructs the surrogate at the current iterate using the SVC gradient and curvature bound.Source: A. Optimizing 𝒙with Fixed 𝒚 · pp. 5–6
- The fixed-𝒚 subproblem replaces non-convex SVC constraints with convex upper bounds and linearized lower bounds for the remaining constraints.The resulting problem (P6) is convex and can be solved using the CVX toolbox in MATLAB [38].Source: A. Optimizing 𝒙with Fixed 𝒚 · pp. 5–6
- A position-independent Lipschitz constant bounds the SVC Hessian, enabling a quadratic SCA surrogate that globally upper-bounds the objective.The bound supports monotonic convergence of the SCA algorithm.Source: A. Optimizing 𝒙with Fixed 𝒚 · p. 6
B. Optimizing 𝒚with Fixed 𝒙
With horizontal coordinates fixed, the method optimizes the vertical coordinates using a counterpart to the horizontal-coordinate SCA step. Alternating the two coordinate updates yields a convergent procedure with stated per-iteration complexity.
B. Optimizing 𝒚with Fixed 𝒙
- Rotational symmetry lets the vertical-coordinate problem use the same approach as the horizontal-coordinate problem, with 𝒙 fixed.The method derives convex lower bounds for the SVC and other constraints in the vertical-coordinate update.Source: B. Optimizing 𝒚with Fixed 𝒙 · pp. 6–7
- The complete SCA-based AO procedure monotonically decreases its objective and converges to a stationary point.The constructed surrogate functions provide strict global bounds, and the objective is bounded below.Source: B. Optimizing 𝒚with Fixed 𝒙 · p. 7
- The overall computational complexity is O(I_iter(L + N^2)^1.5 N^2), where I_iter is the number of iterations to convergence.Each convex subproblem has N + 1 variables and approximately L + N^2 linear and quadratic constraints.Source: B. Optimizing 𝒚with Fixed 𝒙 · p. 7
VI. SIMULATION RESULTS
The simulations evaluate two target-direction settings and compare sensing performance across array configurations and operating parameters. The supplied passages describe the evaluation design and baseline conditions, but do not provide numerical performance outcomes.
VI. SIMULATION RESULTS
- Two simulation groups use directions fixed across Monte Carlo realizations or independently regenerated for each trial.The evaluations examine RMSE and the CRB lower bound; the dynamic-direction group also measures estimation success rate.Source: §VI · p. 7
- Unless otherwise specified, simulations use 25 MAs, an 8λ × 8λ movable region, 20 snapshots, 10 dB SNR, and four targets.The setup uses 500 Monte Carlo simulations and requires at least 10° separation between target directions.Source: §VI · p. 8
- The comparisons include the conventional half-wavelength URA, the uniform sparse array, and the proposed optimized-MA scheme.The CMA baseline uses the conventional algorithm in [30] without considering multi-target coupling.Source: §VI · p. 8
A. Simulations with Fixed Targets
With fixed target directions, the proposed PMA produces sharper target peaks and lower estimation error than the compared array configurations across the evaluated settings. Its advantage persists as SNR, antenna count, movable-region size, and target count vary.
A. Simulations with Fixed Targets
- PMA produces four distinct, sharp MUSIC peaks with almost no severe sidelobes, whereas CMA and USA produce misleading peaks and URA peaks are less sharp.The CMA spectrum has seven prominent peaks, while the USA has periodic grating lobes that prevent distinguishing true targets from spurious ones.Source: A. Simulations with Fixed Targets · p. 8, Figure 4 · p. 9
- PMA RMSE approaches a low LB at -5 dB SNR, while CMA and USA cannot estimate DOAs accurately and URA fails at low SNR.PMA combines sidelobe reduction with an LB below a predefined threshold; URA approaches the LB only above 10 dB.Source: A. Simulations with Fixed Targets · pp. 8–9, Figure 5 · p. 9
- PMA approaches the CRB with N=20 antennas, compared with N=36 for CMA and N=25 for URA.For a fixed movable region, adding antennas also reduces spatial holes and subsequent inter-target interference; USA remains unable to estimate DOAs at its stated spacing.Source: A. Simulations with Fixed Targets · p. 9, Figure 6 · p. 9
- As the movable region expands, PMA RMSE consistently approaches the CRB, while CMA estimates DOAs successfully only within a relatively small region.URA performance remains constant with region size because its deterministic positions are independent of that size; CMA's enlarged spatial holes intensify mutual interference.Source: A. Simulations with Fixed Targets · p. 9, Figure 7 · p. 9
- As target count increases, PMA continues to approach the LB, while CMA fails when K>3 and URA degrades severely when K>5.USA fails regardless of target count; CMA's spatial holes amplify interference, while URA's restricted aperture limits resolution.Source: A. Simulations with Fixed Targets · pp. 9–10, Figure 8 · p. 9
B. Simulations with Random Targets
Random-target simulations compare estimation success across SNR, antenna count, movable-region size, and target count. The proposed PMA maintains high success across these conditions, including low SNR and many targets.
B. Simulations with Random Targets
- Success is defined by mean Euclidean angle-estimation error across all K targets falling below a tolerance empirically set to 1°; the reported rate aggregates Q Monte Carlo trials.The simulations vary SNR, N, movable-region dimensions, and K with random targets in each trial.Source: B. Simulations with Random Targets · p. 10
- Across movable-region sizes, PMA maintains 100% success, while CMA falls to roughly 15% as the region expands to 10 × 10.The reported comparison is success rate versus region size; CMA's decline is attributed to boundary clustering that expands its central spatial hole.Source: Figure 11 · p. 10, B. Simulations with Random Targets · p. 11
- At K=8 targets, PMA sustains 100% success, compared with approximately 30% for CMA, about 55% for URA, and 0% for USA.The reported comparison is success rate versus target count; the passage attributes CMA's decline to sidelobes and URA's to restricted aperture.Source: Figure 12 · p. 10, B. Simulations with Random Targets · p. 11
- At -5 dB, PMA reaches a 100% success rate; CMA needs 5 dB, URA 20 dB, and USA remains at 0%.The reported comparison is success rate versus SNR.Source: B. Simulations with Random Targets · p. 10
- With N=20 antennas, PMA reaches 100% success; CMA needs at least N=30 for near 100%, while URA is roughly 95% at N=35 and USA stays at 0%.PMA's result is attributed to an optimally distributed movable array; the passage describes success rate versus antenna count.Source: B. Simulations with Random Targets · pp. 10–11
VII. CONCLUSION
The paper combines a CRB lower bound (LB) for estimation accuracy with steering vector correlation (SVC) for sidelobe interference to optimize movable antenna positions. Numerical results report SVC sidelobe suppression alongside superior estimation accuracy and sensing robustness over fixed-position arrays.
VII. CONCLUSION
- A closed-form CRB lower bound decouples unknown target directions from MA positions, while SVC characterizes spatial correlation and sidelobe interference.The position problem minimizes SVC subject to a prescribed LB constraint, transformed into discrete and spatial-variance constraints.Source: §VII · p. 11
- The proposed MA configuration suppresses SVC sidelobes while achieving superior estimation accuracy and sensing robustness over conventional fixed-position arrays.The conclusion describes an SCA-based alternating optimization algorithm for the position problem, with monotonic convergence guaranteed by a position-independent Lipschitz constant.Source: §VII · p. 11
APPENDIX A CRB DERIVATION
Appendix A derives the multi-target CRB from a vectorized observation model and the Fisher information matrix (FIM). It then forms an angle-estimation objective from the CRB matrix.
APPENDIX A CRB DERIVATION
- The derivation vectorizes both sides of the sensing model and treats the resulting observation as Gaussian with a mean and variance.The vectorized signal and noise are denoted by s̃ and ñ in the passage.Source: Appendix A · p. 11
- The unknowns are the K target directions and signal S; the FIM is partitioned into blocks and its entries use derivatives of the mean vector.The passage gives derivatives with respect to the target directions and the real and imaginary parts of the signal.Source: Appendix A · p. 11
- The inverse angle-related CRB block yields the actual CRB matrix, whose trace sums the elevation and azimuth estimation variances into a scalar objective.The derivation invokes the matrix inversion lemma for the inverse CRB block.Source: Appendix A · p. 12
APPENDIX B PROOF OF THEOREM 1
Appendix B proves Theorem 1 by structuring the CRB with a compact matrix and using projection-matrix order to derive a lower bound. The proof expresses that bound through the MA coordinate structure and successive inequalities.
APPENDIX B PROOF OF THEOREM 1
- The proof introduces compact matrix G to simplify the CRB structure, then partitions it into K submatrices with diagonal blocks G_k,k.Source: Appendix B · p. 12
- The x- and y-coordinate matrices D_x and D_y are diagonal matrices of the corresponding MA position coordinates.Source: Appendix B · p. 12
- Because the multi-target null space is a subset of the single-target null space, the associated projection operators satisfy a positive-semidefinite partial order.The proof substitutes this relation into the CRB expression.Source: Appendix B · p. 12
- The proof uses matrix elements and inequalities, including nonnegative coordinate variances, to obtain the CRB lower bound.Source: Appendix B · pp. 12–13