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Movable Antenna Enabled Anti-Jamming: From Local Optimality Guaranteed Design to a Unified Antenna Position Optimization Framework

Lebin Chen, Ming-Min Zhao, Lipeng Zhu, Zhenyu Xiao, Min-Jian Zhao, Rui Zhang

arXiv:2610.35438v1eess.SPcs.ITmath.IT

TL;DR

Fixed antenna geometries limit receivers’ ability to adapt their spatial response to jamming. This paper jointly optimizes movable-antenna positions and receive beamforming, using DPGD for a LoS single-jammer case and a ridge-LS-based SAPG framework for general settings. Numerical results report higher SINR for optimized movable-antenna schemes than fixed-array baselines.

  • Problem

    Fixed-position arrays cannot adapt their geometry to the interference environment, limiting the receiver’s ability to reshape its spatial response for jamming suppression.

    Source: §I · p. 1

  • Method

    The paper develops DPGD for the LoS single-jammer case and a ridge least-squares-based SAPG framework for multipath and/or multiple jammers, applicable to MVDR and ZF reception.

    Source: §I · p. 2, A. Ridge LS Reformulation · p. 8

  • Results

    The numerical results show that optimized movable-antenna schemes maintain higher SINR than fixed-array baselines as jammer-to-noise ratio increases.

    Source: §V · p. 11

  • Takeaways & Limitations

    The framework supports finite-aperture antenna-position optimization for MVDR and ZF reception beyond the special closed-form null-steering configurations considered in.

    Source: §I · p. 2

    The analysis does not cover the near-field setting, which the paper identifies as a direction for future work.

    Source: A. MVDR Objective in Terms of Steering… · p. 4

Abstract

from arXiv · show

Anti-jamming capability is of paramount importance in modern wireless communications, yet conventional receivers are severely constrained by their fixed array geometries, which limit the available spatial degrees of freedom and leave performance highly susceptible to strong interference. By contrast, movable antennas (MAs) provide additional spatial degrees of freedom through antenna-position optimization. This paper investigates MA-enabled anti-jamming by jointly optimizing the antenna-position vector (APV) and receive beamformer to maximize the output signal-to-interference-plus-noise ratio (SINR). For the canonical setting with a single jammer and a single line-of-sight (LoS) propagation path on the legitimate link, we develop a damped projected-gradient descent (DPGD) algorithm with low computational complexity, which scales linearly with the number of receive MAs. We establish a generic active strict-saddle landscape for the resulting steering-coherence minimization problem and prove that DPGD converges almost surely to a local minimizer rather than merely to a Karush-Kuhn-Tucker (KKT) point. For the general setting with multiple channel paths on the legitimate link and/or multiple jammers, we derive an equivalent ridge least-squares formulation and develop a low-complexity soft-active projected-gradient (SAPG) algorithm, which also scales linearly with the number of receive MAs. We further show that the proposed antenna position optimization framework applies to both minimum variance distortionless response (MVDR) and zero-forcing (ZF) reception. Numerical results validate the proposed algorithms, demonstrating substantial gains in both anti-jamming performance and computational efficiency.

I. INTRODUCTION

Fixed array geometry limits a receiver’s ability to reshape its spatial response against jamming, so this paper jointly optimizes movable-antenna positions and receive beamforming. It develops DPGD for the LoS single-jammer case and a general finite-aperture framework for multipath and/or multiple jammers with MVDR or ZF reception.

I. INTRODUCTION

  • Fixed array geometry limits spatial response adaptation, while movable antennas add position optimization as a degree of freedom for suppressing jamming.Antenna repositioning can reduce spatial coupling between desired and jammer channels before or jointly with digital beamforming.Source: §I · p. 1
  • For the LoS single-jammer MVDR case, DPGD minimizes steering coherence with linear per-iteration complexity and converges almost surely to a local minimizer under mild conditions.The analysis establishes a generic active strict-saddle landscape, making non-minimizing critical points unstable under the proposed dynamics.Source: §I · p. 2
  • For multipath and/or multiple jammers, SAPG uses an equivalent ridge least-squares formulation to support low-complexity MVDR and ZF antenna-position optimization.The framework targets finite-aperture APV optimization beyond the special closed-form null-steering configurations considered in.Source: §I · p. 2
  • The proposed methods achieve output SINR comparable to IPM with substantially lower runtime, while general-setting SAPG outperforms fixed-array AO and conventional projected-gradient benchmarks.These comparisons are reported for the canonical and general settings, respectively.Source: §I · p. 2

A. System Model

  • The system model assumes narrow-band, quasi-static channels and jammer AoA and received-power estimates that remain approximately unchanged during one repositioning interval.The legitimate channel may have multiple paths, and the receiver has Nr movable antennas.Source: A. System Model · pp. 2–3

B. Problem Formulation

The problem formulation minimizes output interference-plus-noise power while enforcing a distortionless response and feasible antenna geometry. Eliminating the beamformer yields a nonconvex APV-only problem addressed by the paper’s specialized algorithms.

B. Problem Formulation

  • The joint design minimizes output interference-plus-noise power subject to a unit-gain response toward the desired channel and MA geometry constraints.Source: B. Problem Formulation · p. 3
  • The APV is constrained to the finite aperture with minimum inter-antenna spacing, and antenna ordering can be imposed without loss of optimality.The feasible-set assumptions include Dx > (Nr − 1)d.Source: B. Problem Formulation · p. 3
  • For each fixed APV, the MVDR beamformer has a closed-form solution, which can be substituted to eliminate the beamformer and obtain an APV-only problem.Source: B. Problem Formulation · p. 4
  • The resulting APV-only problem is highly nonconvex because desired and interference-plus-noise responses vary nonlinearly with antenna positions.Prior closed-form constructions apply only under specific antenna-number and interference-direction conditions.Source: B. Problem Formulation · p. 4

III. LOS CHANNEL AND SINGLE JAMMER CASE

In the single-LoS-path, single-jammer case, the MVDR objective reduces to steering-coherence minimization, and DPGD is designed to converge almost surely to a local minimizer. The analysis also identifies when antenna repositioning can suppress interference and how steering coherence shapes output-SINR sensitivity.

III. LOS CHANNEL AND SINGLE JAMMER CASE

  • DPGD with bounded PAVA projection converges almost surely to a local minimizer, with linear per-iteration complexity.The guarantee is stronger than convergence merely to a KKT point.Source: §III · p. 4

A. MVDR Objective in Terms of Steering Coherence

  • Maximizing the MVDR gain is equivalent to minimizing steering coherence, which decorrelates the desired and jammer steering vectors.When both signals arrive from the same spatial direction in the far-field model, repositioning cannot provide spatial interference suppression.Source: A. MVDR Objective in Terms of Steering… · p. 4
  • The desired-link SNR scales the MVDR output SINR, while its APV-dependent loss is determined by steering coherence; lower coherence also reduces sensitivity to increasing JNR.The output SINR is less sensitive to JNR when f_a(x) is smaller.Source: A. MVDR Objective in Terms of Steering… · pp. 4–5
  • The same-angle indistinguishability conclusion is limited to the far-field model; different source ranges can produce distinct near-field array responses.Extending the analysis to the near field is identified as future work.Source: A. MVDR Objective in Terms of Steering… · p. 4

B. Landscape Structure of the Reduced APV Optimization Problem

The reduced steering-coherence problem has a generic active strict-saddle landscape: outside an at-most-countable exceptional set of phase slopes, every critical point is a local minimizer or an active strict saddle. This landscape follows from identifying active faces and showing non-minimizing critical points have negative curvature along those faces.

B. Landscape Structure of the Reduced APV Optimization Problem

  • Closed-form solutions cover several special cases, but not the general regime of larger arrays or apertures.The listed cases include Nr ≤ 3, a restricted phase-slope range, and an even-Nr replicated-subarray construction under an aperture condition.Source: B. Landscape Structure of the Reduced… · p. 5
  • The reduced problem is equivalent to the original steering-coherence minimization up to common shifts, and each reduced local minimizer yields an original local minimizer.The feasible set satisfies LICQ at every feasible point.Source: B. Landscape Structure of the Reduced… · p. 5
  • Outside Bsc, strict complementarity identifies each non-minimizing critical point’s active face as a smooth active manifold with sharpness.The exceptional set is at most countable; under a non-atomic distribution for a, the stated properties hold almost surely.Source: B. Landscape Structure of the Reduced… · p. 6
  • On any feasible active affine manifold, every non-minimizing critical point has a strictly negative-curvature direction tangent to that manifold.The restricted objective has a free-phasor form, which supplies the curvature result used in the landscape characterization.Source: B. Landscape Structure of the Reduced… · p. 6
  • For every phase slope a outside Bsc, each critical point is either a local minimizer or an active strict saddle.Bsc is at most countable, so these properties hold almost surely when a is drawn from a non-atomic distribution.Source: B. Landscape Structure of the Reduced… · p. 6

C. Proposed DPGD algorithm

DPGD minimizes the reduced steering-coherence objective by taking a gradient step, projecting onto the feasible antenna-position set, and damping the update. Its bounded isotonic projection and gradient evaluation give linear per-iteration complexity.

C. Proposed DPGD algorithm

  • DPGD takes a gradient step, projects onto the feasible set, then damps between the current and projected points.The fixed points of the iteration coincide with the critical points of the extended objective.Source: C. Proposed DPGD algorithm · p. 7
  • The bounded isotonic projection is implemented with PAVA in O(Nr) complexity.The constraints are transformed to bounded isotonic constraints before projection.Source: C. Proposed DPGD algorithm · p. 7
  • Each DPGD iteration costs O(Nr), so K iterations require O(KNr) operations, including O(Nr) recovery of the MVDR beamformer.The gradient evaluation and bounded PAVA projection each have O(Nr) complexity.Source: C. Proposed DPGD algorithm · p. 7

D. Almost-Sure Convergence to Local Minima

Under the stated stepsize, damping, phase-slope, and initialization conditions, DPGD converges almost surely to a local minimizer. The argument combines saddle avoidance with convergence of the full sequence to a critical point.

D. Almost-Sure Convergence to Local Minima

  • The convergence guarantee is established under an absolutely continuous phase-slope distribution and a suitable stepsize and damping choice.For almost every phase-slope realization, the landscape result rules out non-minimizing critical points other than active strict saddles.Source: D. Almost-Sure Convergence to Local… · p. 8
  • With absolutely continuous a and an independently drawn absolutely continuous feasible initialization, DPGD converges almost surely to a local minimizer under the stated stepsize condition.The result transfers from the reduced variable y to the original antenna-position vector x.Source: D. Almost-Sure Convergence to Local… · p. 8
  • Damping makes the iteration map invertible, enabling the saddle-avoidance argument; trajectories converging to active strict saddles have probability zero.The stepsize condition, compact feasible set, and Kurdyka-Łojasiewicz property ensure convergence of the full sequence to a critical point.Source: D. Almost-Sure Convergence to Local… · pp. 7–8

IV. GENERAL MULTIPATH MULTI-JAMMER CASE

The general multipath and multi-jammer problem is highly nonconvex because antenna positions affect both desired and jammer channels and the interference-plus-noise covariance. An equivalent ridge least-squares formulation supports a low-complexity SAPG framework for MVDR and ZF reception.

IV. GENERAL MULTIPATH MULTI-JAMMER CASE

  • The APV affects the desired channel, all jammer channels, and the interference-plus-noise covariance, making direct coordinate differentiation involve repeated Nr×Nr inversions.The setting allows multiple legitimate-link paths and/or multiple jammers.Source: §IV · p. 8
  • An equivalent ridge least-squares reformulation reduces MVDR gradient evaluation to an I × I positive-definite solve and avoids differentiating covariance inverses.Danskin’s theorem applies because the ridge subproblem has a unique minimizer.Source: §IV · p. 8
  • The resulting SAPG framework extends the antenna-position optimization approach to MVDR and ZF reception.Its bounded slack-dependent metric biases projected steps toward the tangent space of near-active constraints.Source: §IV · p. 8

A. Ridge LS Reformulation

The APV objective is rewritten as an equivalent ridge least-squares value function, allowing evaluation and gradient computation through a reduced positive-definite system. Danskin’s theorem gives the gradient without differentiating the covariance inverse or the inner minimizer.

A. Ridge LS Reformulation

  • The objective can be evaluated through an I × I positive-definite system instead of an Nr × Nr covariance inverse.The reformulation exploits the diagonal-noise-plus-low-rank-jammer structure of the interference-plus-noise covariance.Source: A. Ridge LS Reformulation · p. 8
  • Rewriting the reduced objective as a ridge LS value function lets Danskin’s theorem yield its gradient without differentiating the covariance inverse or the optimizer.The inner problem has a unique minimizer, and the value function is continuously differentiable under the stated conditions.Source: A. Ridge LS Reformulation · pp. 8–9
  • Because each antenna position affects only its corresponding row of V(x) and entry of h0(x), the value-function gradient admits a component-wise expression.The gradient of the original objective follows by scaling the value-function gradient by 1/σ2.Source: A. Ridge LS Reformulation · p. 9

B. Proposed SAPG Algorithm

SAPG uses a slack-dependent metric to shape projected ascent steps toward near-active APV constraint boundaries. The general multipath, multi-jammer setting does not directly inherit the preceding almost-sure saddle-avoidance result.

B. Proposed SAPG Algorithm

  • The canonical case’s almost-sure saddle-avoidance result does not directly extend to the general multipath, multi-jammer problem.On an active face, the restricted objective no longer reduces to the phasor-sum form used in the preceding landscape analysis.Source: B. Proposed SAPG Algorithm · p. 9
  • SAPG weights constraints according to their slacks, biasing search directions toward the tangent space of near-active APV constraint boundaries.Smaller slacks receive larger weights; when neighboring antennas approach minimum spacing, SAPG penalizes their relative displacement.Source: B. Proposed SAPG Algorithm · pp. 9–10
  • The weighted direction is obtained from a positive-definite tridiagonal system, whose solution and bounded PAVA projection require linear operations.SAPG uses backtracking for projected steps and a conventional projected-gradient correction when no trial SAPG point satisfies the acceptance conditions.Source: B. Proposed SAPG Algorithm · pp. 9–10
  • The per-iteration complexity is O(NrI2 + I3 + Nr(I + L)), and the final MVDR beamformer requires only O(Nr) additional operations.The overall complexity over K iterations is O(K[NrI2 + I3 + Nr(I + L)]).Source: B. Proposed SAPG Algorithm · p. 10

C. Extension to ZF Reception

The ZF APV objective has an LS-residual structure that permits the same first-order optimization framework as MVDR. Regularization preserves a unique inner solution and a continuously differentiable value function when jammer steering vectors may lose rank.

C. Extension to ZF Reception

  • The ZF APV objective is an LS residual, so its gradient can be obtained by the same Danskin argument and optimized with SAPG.The ZF receiver for a fixed APV uses projection onto the null space of the jammer steering vectors.Source: C. Extension to ZF Reception · p. 10
  • When jammer steering vectors become linearly dependent, the unregularized LS minimizer may be nonunique and its pseudoinverse may vary nonsmoothly with position.The paper introduces a regularized ZF objective to retain a unique inner solution and continuously differentiable value function.Source: C. Extension to ZF Reception · pp. 10–11
  • The resulting ZF framework has computational complexity of the same order as the preceding SAPG method.The regularized objective preserves the ridge LS structure used for MVDR.Source: C. Extension to ZF Reception · p. 11

V. NUMERICAL RESULTS

The numerical results show that antenna-position optimization maintains strong anti-jamming performance across the tested canonical and general settings, while the proposed methods reduce runtime relative to competing optimization methods. Their advantages vary with jammer conditions, array size, aperture, and desired-path count.

Canonical LoS single-jammer setting

  • In the canonical setting, DPGD overlaps with both the closed-form benchmark and IPM, while optimized MAs maintain higher SINR than fixed arrays as JNR rises and receive-MA count increases.The closed-form comparison uses parameter regimes satisfying its specific conditions; the optimized-MA advantage over fixed arrays narrows as the receive-MA count grows.Source: Figure 3 · p. 11, §V · p. 11
  • At Nr = 32, DPGD is approximately 26 times faster than AO and 13 times faster than IPM, and it has the shortest runtime as array size increases.Its computational advantage over these methods becomes more pronounced for larger arrays.Source: Table I · p. 11, §V · p. 12

General multipath multi-jammer setting

  • In the general setting, SAPG stays comparable to IPM and outperforms fixed-array, AO, and DPGA baselines; at Nr = 32, it is 262 times faster than AO and 31 times faster than IPM.The comparisons cover JNR, jammer count, aperture, and desired-path count; runtime is compared as the number of receive MAs increases.Source: Figure 4 · p. 12, §V · pp. 12–13, Table II · p. 12
  • In the general setting, increasing aperture improves output SINR before gains gradually saturate, and increasing desired-path count also improves it through richer multipath phase structure.The path-count improvement is reported with normalized path gains, rather than an increase in total desired-channel power.Source: Figure 4 · p. 12, §V · pp. 12–13
  • In the general setting, the optimized-MA advantage over fixed arrays becomes more evident with more jammers, while output SINR curves flatten at high JNR.The passage attributes the flattening to strong suppression of the jammer subspace, leaving the desired-channel component outside that subspace to determine most of the remaining SINR.Source: §V · p. 12

VI. CONCLUSION

The paper develops low-complexity methods for jointly optimizing movable-antenna positions and reception for anti-jamming. Numerical results show IPM-level performance at lower computational cost, with the framework also extended to ZF reception.

Conclusion

  • For the LoS single-jammer case, the joint MVDR design reduces exactly to steering-coherence minimization, yielding linear-complexity DPGD with almost-sure convergence to a local minimizer.The convergence analysis relies on the problem's generic active strict-saddle landscape.Source: §VI · p. 13
  • For the general multipath multi-jammer case, a ridge least-squares reformulation enables efficient gradient evaluation for SAPG and extends the optimization framework to ZF reception.Source: §VI · p. 13
  • Numerically, the proposed algorithms attain IPM-level anti-jamming performance at substantially lower computational cost and outperform the considered fixed-array and competing optimization schemes.Extending the approach to two-dimensional arrays and multiuser/multistream reception is identified as future work.Source: §VI · p. 13

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