Movable Antenna (MA) Arrays

• Movable Antenna Arrays are reconfigurable systems that adjust element positions and beamforming weights to enhance performance in communication and sensing.
• Their geometric flexibility enables improved beam pattern shaping, reduced channel correlations, and elevated physical-layer security.
• Advanced optimization methods, such as alternating optimization and projected-gradient algorithms, facilitate efficient design and robust operational gains.

Movable antenna (MA) arrays are antenna arrays whose element locations are reconfigurable within prescribed spatial regions, so that array geometry becomes an optimization variable alongside conventional beamforming weights. In the terminology used in recent work, the antenna-position vector (APV) specifies the coordinates of the movable elements, while the antenna-weight vector (AWV) specifies their complex excitation; by jointly optimizing APV and AWV, an MA array can reshape beam patterns, channel correlations, and estimation sensitivity in ways unavailable to fixed-position antennas (FPAs) (Ma et al., 31 Aug 2025). The resulting literature spans one-dimensional, two-dimensional, and three-dimensional movement regions, transmitter-side and receiver-side mobility, communication and sensing objectives, and a growing set of reduced-complexity architectures such as movable subarrays, cross-linked arrays, and two-layer arrays.

1. Geometric and architectural models

The most common MA formulations begin with a finite movement region and a minimum inter-element spacing. In linear arrays, element positions are typically written as $x=[x_1,\dots,x_N]^T$ on a segment such as $[0,D]$ or $[0,A]$, subject to ordering and spacing constraints like $0\le x_1<\cdots<x_N\le D$ and $x_n-x_{n-1}\ge D_0$ or $d$ (Ma et al., 2023). Two-dimensional variants place elements in a square or planar region, with coordinates $\mathbf q_n=[x_n,y_n]^T$ or $\mathbf r_n=[y_n,z_n]^T$, again with pairwise distance constraints to avoid coupling (Ma et al., 2024). Three-dimensional variants appear in UAV-oriented formulations, where each element position is a full Cartesian coordinate constrained by per-slot displacement and a bounding cuboid (Yu et al., 14 Aug 2025).

This geometric flexibility has produced several distinct architectures. In sub-connected hybrid beamforming, the base station is divided into movable sub-arrays, each driven by one motor and translated within a local region $\Omega_i$ (Zhang et al., 2024). In the two-layer movable-antenna (TL-MA) architecture, the absolute position of antenna $a$ in subarray $[0,D]$0 is $[0,D]$1, where $[0,D]$2 controls large-scale subarray movement and $[0,D]$3 provides fine-tuning within the subarray (Yao et al., 19 Nov 2025). In the cross-linked movable-antenna (CL-MA) architecture, an $[0,D]$4 planar grid is actuated by only $[0,D]$5 motors rather than $[0,D]$6, because horizontal tracks move collectively along one axis and vertical tracks move collectively along the other (Zhu et al., 6 May 2025). This suggests that MA-array research has evolved from element-wise motion toward hardware-aware aperture reconfiguration.

A further distinction concerns the mobility granularity. Some formulations move each radiating element independently; others move an entire linear array as a rigid body over a continuous interval $[0,D]$7, with fixed relative spacing $[0,D]$8 between elements (Siriwardana et al., 16 May 2026). The latter model is analytically convenient for outage and level-crossing analyses, whereas independently movable arrays are more common in beamforming, secrecy, and sensing optimization.

2. Channel models, objective functions, and array-dependent statistics

The central mathematical feature of an MA array is that element positions enter the channel model explicitly. A generic narrowband field-response formulation writes the channel to a receiver at location $[0,D]$9 as

$[0,A]$0

with a far-field LoS specialization

$[0,A]$1

Because $[0,A]$2 appears inside the phase terms, geometry directly controls channel correlation and spatial separability (Ma et al., 31 Aug 2025).

In far-field beamforming and sensing, this reduces to a steering-vector dependence on position. For a 1D MA array, the steering vector is commonly written as

$[0,A]$3

with $[0,A]$4 or a related directional cosine (Ma et al., 2024). Near-field sensing replaces plane-wave steering with spherical-wave or Fresnel-type phase terms. For example, one near-field 1D model uses

$[0,A]$5

so that the same position variable simultaneously influences angle and distance information (Wang et al., 30 Nov 2025).

The performance criteria are application-specific but structurally similar. Communication work frequently optimizes sum rate, minimum user rate, transmit power under rate constraints, or beamforming gain under null constraints (Zhu et al., 2023). Secure-communication work typically uses the secrecy-rate expression

$[0,A]$6

or multiuser variants based on secure channel coding (Cheng et al., 9 Jan 2026). Sensing work uses the Cramér–Rao bound (CRB), whose dependence on position statistics is unusually transparent. In 1D far-field AoA estimation,

$[0,A]$7

so larger spatial spread lowers the bound directly (Ma et al., 2024). In 2D sensing, $[0,A]$8, $[0,A]$9, and $0\le x_1<\cdots<x_N\le D$0 jointly determine the angle-estimation CRBs, making aperture shape and axis decoupling central design parameters (Ma et al., 2024).

A related theoretical line studies directivity with mutual coupling. When coupling is modeled through an impedance matrix $0\le x_1<\cdots<x_N\le D$1, the directivity toward direction $0\le x_1<\cdots<x_N\le D$2 becomes

$0\le x_1<\cdots<x_N\le D$3

and, in the endfire superdirective limit with vanishing spacing, the directivity can reach $0\le x_1<\cdots<x_N\le D$4, versus $0\le x_1<\cdots<x_N\le D$5 for the uncoupled case (Xu et al., 17 Mar 2026). This indicates that MA geometry can be used not only to change the steering vector but also to exploit the coupling matrix itself.

3. Optimization methodologies

Most MA-array problems are non-convex because positions, beamformers, and sometimes decoding order or power variables are coupled through trigonometric, fractional, or determinant expressions. The predominant solver pattern is block-wise decomposition. Alternating optimization (AO) updates beamformers, powers, and positions in turn, often obtaining closed-form or eigenvector-based updates for the beamforming block and numerical updates for the position block (Yu et al., 8 Jul 2025).

Projected-gradient methods are common when the secrecy rate or sum rate is smooth in the position variables. In the hybrid FPA–MA secrecy design of Yu et al., the MA-position update uses Nesterov-momentum projected gradient ascent (NMPGA),

$0\le x_1<\cdots<x_N\le D$6

followed by projection onto box and spacing constraints. The reported motivation is accelerated $0\le x_1<\cdots<x_N\le D$7 convergence in smooth regions under low-latency Internet-of-Vehicles conditions (Yu et al., 8 Jul 2025).

Successive convex approximation (SCA) and majorization–minimization (MM) are used when direct gradient ascent is insufficient. In far-field 2D sensing, the min–max CRB problem is solved by alternately optimizing horizontal and vertical coordinates through convex surrogates, with each inner loop monotonically increasing the information variable $0\le x_1<\cdots<x_N\le D$8 (Ma et al., 2024). In CoMP reception, maximizing the principal eigenvalue of a Hermitian channel matrix is handled by MM: the dominant eigenvector at the current iterate yields a lower bound, and the non-convex cosine terms are replaced by a quadratic surrogate (Hu et al., 2023).

Sampling, swarm, and heuristic search appear when the objective is highly non-convex or discrete structure is helpful. Near-field joint AoA-and-distance sensing discretizes the movement region and sequentially updates each antenna position, with complexity $0\le x_1<\cdots<x_N\le D$9 in 1D and $x_n-x_{n-1}\ge D_0$0 in 2D (Wang et al., 30 Nov 2025). TL-MA employs an AO-based particle swarm optimization (PSO) procedure, with swarm sizes $x_n-x_{n-1}\ge D_0$1, iterations $x_n-x_{n-1}\ge D_0$2, and penalty weight $x_n-x_{n-1}\ge D_0$3 (Yao et al., 19 Nov 2025). Multi-target sensing uses Monte Carlo approximation of the expected CRB trace and a swarm-based gradient descent per antenna (Mao et al., 24 Nov 2025). Directivity maximization under mutual coupling uses a two-stage Greedy Search and Gradient Descent (GS–GD) algorithm: a discrete greedy placement stage followed by continuous gradient refinement (Xu et al., 17 Mar 2026).

Two-timescale optimization addresses CSI and motion overhead. In uplink Rician systems, MA positions are optimized from statistical CSI using projected gradient ascent, while ZF, MMSE, or MMSE-SIC receivers are computed from instantaneous CSI after positions are fixed (Hu et al., 2024). Statistical CL-MA and ISAC designs use a similar logic: optimize the array geometry on a slow timescale from channel statistics, then update receive combining or precoding on a fast timescale (Zhu et al., 6 May 2025).

4. Communication and beamforming applications

One of the earliest MA-array communication problems is multi-beam forming. For a linear MA array, the APV and AWV are jointly optimized to maximize the minimum beamforming gain over desired directions under a maximum interference threshold. In a representative case with $x_n-x_{n-1}\ge D_0$4, array length $x_n-x_{n-1}\ge D_0$5, two desired directions, and two null directions, the proposed design reaches $x_n-x_{n-1}\ge D_0$6 of the full gain ($x_n-x_{n-1}\ge D_0$7), while the fixed-array benchmark reaches $x_n-x_{n-1}\ge D_0$8, an alternating position-search benchmark $x_n-x_{n-1}\ge D_0$9, and a one-shot benchmark $d$0. With four antennas, the MA design yields about $d$1 gain improvement over the FPA benchmark, and with six interference directions it still retains about a $d$2 dB advantage (Ma et al., 2023).

Coordinated multi-point (CoMP) reception yields a related eigenvalue formulation. There, the MA-position problem reduces to maximizing the principal eigenvalue of a Hermitian Gram matrix $d$3, and the optimal transmit beamformer is the principal eigenvector scaled by the transmit-power budget. The upper bound $d$4 gives an SNR bound $d$5; numerical results show that MA arrays can approach this bound when the movement range is large and can outperform fixed arrays by about $d$6–$d$7 dB or bits/s/Hz, depending on $d$8 and $d$9 (Hu et al., 2023).

In multiuser systems, MA positioning is frequently used to reduce channel correlation and thereby lower power or raise sum rate. In uplink multiple access with user-side MAs and a fixed BS array, multi-directional descent (MDD) algorithms with ZF or MMSE combining substantially reduce required transmit power. For rate targets above $\mathbf q_n=[x_n,y_n]^T$0 bps/Hz, the MA-MMSE design outperforms antenna-selection MMSE by more than $\mathbf q_n=[x_n,y_n]^T$1 dB, and as the user count approaches the BS antenna count the MA gains grow to about $\mathbf q_n=[x_n,y_n]^T$2–$\mathbf q_n=[x_n,y_n]^T$3 dB (Zhu et al., 2023). On the BS side, a two-timescale Rician-fading design reports that MA+ZF exceeds FPA+ZF by about $\mathbf q_n=[x_n,y_n]^T$4 bits/s/Hz at $\mathbf q_n=[x_n,y_n]^T$5, with most benefits already captured when the span reaches about $\mathbf q_n=[x_n,y_n]^T$6 (Hu et al., 2024).

MA arrays also integrate naturally with hybrid beamforming and NOMA. In sub-connected hybrid beamforming, movable sub-arrays are jointly optimized with analog and digital beamformers; the MA-aided scheme exceeds its fixed-array counterpart, and with sufficiently large movable regions it can even surpass a fully-connected FPA array (Zhang et al., 2024). In downlink NOMA, joint beamforming and MA-position optimization reduces the transmit power needed to reach a target sum rate: to achieve $\mathbf q_n=[x_n,y_n]^T$7 bps/Hz, MA-NOMA requires about $\mathbf q_n=[x_n,y_n]^T$8 dBm, while fixed-antenna NOMA requires about $\mathbf q_n=[x_n,y_n]^T$9 dBm and OMA about $\mathbf r_n=[y_n,z_n]^T$0 dBm; across the reported settings, MA-NOMA exceeds fixed NOMA and OMA by $\mathbf r_n=[y_n,z_n]^T$1–$\mathbf r_n=[y_n,z_n]^T$2 in sum rate (Li et al., 13 Jun 2025).

Recent wideband evaluations add an important qualification. Under OFDM, EVM-based distortion, and both uplink and downlink processing, MA gains are strongest in low-impairment, high-user-load, LoS-dominated settings. With EVM $\mathbf r_n=[y_n,z_n]^T$3, the gain over fixed arrays reaches up to $\mathbf r_n=[y_n,z_n]^T$4–$\mathbf r_n=[y_n,z_n]^T$5, and one reported LoS case gives about $\mathbf r_n=[y_n,z_n]^T$6 bit/s/Hz for MA versus $\mathbf r_n=[y_n,z_n]^T$7 bit/s/Hz for a compact UPA and $\mathbf r_n=[y_n,z_n]^T$8 bit/s/Hz for a staggered URA. However, the same study reports that gains diminish in rich scattering, under hardware-impairment ceilings, and in wideband or FDD settings; at high SNR, all schemes share the ceiling $\mathbf r_n=[y_n,z_n]^T$9 (Irshad et al., 23 Mar 2026). This suggests that MA arrays are not uniformly superior to fixed arrays; their advantage is strongly regime-dependent.

5. Sensing, near-field estimation, and integrated sensing and communication

In sensing, MA arrays have been analyzed most extensively through CRB minimization. For far-field 1D AoA estimation, the optimal geometry maximizes $\Omega_i$0 subject to spacing and aperture constraints, yielding a closed-form edge-cluster solution: $\Omega_i$1 For $\Omega_i$2, $\Omega_i$3, and $\Omega_i$4, this geometry gives a $\Omega_i$5 MSE reduction versus a half-$\Omega_i$6 ULA, while a full-aperture sparse ULA suffers ambiguity (Ma et al., 2024). In 2D, the min–max CRB problem shows that large $\Omega_i$7, large $\Omega_i$8, and small $\Omega_i$9 are jointly desirable; in a circular region, the best achievable information variable satisfies $a$0, and when $a$1 is divisible by four the tight design places grouped antennas at quarter-cycle angular offsets on the circle (Ma et al., 2024).

Near-field sensing adds distance as an estimation target. For 1D MA arrays, the worst-case CRBs for AoA-only and distance-only estimation lead to the same edge-cluster geometry as far-field sensing, with scaling laws $a$2 and $a$3. Joint AoA-and-distance estimation is qualitatively different: the optimal 1D geometry becomes a three-group structure with edges and center rather than the two-cluster layout used for individual estimation. In a representative setting with $a$4, $a$5, and $a$6, the reported CRB reductions are about $a$7 for AoA-only, about $a$8 for distance-only, and about $a$9 for joint estimation, relative to a half-$[0,D]$00 ULA. In 2D, the reported CRB reduction exceeds $[0,D]$01 relative to fixed UPA benchmarks, and an optimized MA array with $[0,D]$02 outperforms a UPA with $[0,D]$03 (Wang et al., 30 Nov 2025).

Multi-target sensing introduces an additional structure beyond simple aperture maximization. The CRB matrix depends on the projected derivative matrix $[0,D]$04 and the orthogonal projector $[0,D]$05, and the design objective becomes the expectation of $[0,D]$06 over random target angles. The reported optimized MA geometry reduces average inter-target sensitivity correlation $[0,D]$07 by about $[0,D]$08 and increases average effective sensitivity power $[0,D]$09 by about $[0,D]$10 relative to dense UPA and single-target MA baselines; the corresponding MUSIC mean-square error improvement exceeds $[0,D]$11. The swarm-based algorithm converges within about $[0,D]$12 AO iterations and improves the objective by about $[0,D]$13 relative to plain gradient descent (Mao et al., 24 Nov 2025).

Integrated sensing and communication (ISAC) formulations combine these CRB ideas with communication-rate objectives. In one statistical-CSI design, the array geometry is chosen to maximize the expected minimum user rate subject to upper bounds on the AoA CRBs. The sensing constraints can be written through quadratic forms involving $[0,D]$14, $[0,D]$15, and $[0,D]$16, while the communication side uses ZF precoding over Monte Carlo samples of user locations. The reported trade-off region is substantially larger than with dense or sparse fixed UPAs: at a moderate reciprocal-CRB threshold $[0,D]$17, the MA-statistical design provides roughly $[0,D]$18–$[0,D]$19 higher minimum rate than a dense UPA and more than $[0,D]$20 gain over a sparse UPA, while the CRB can be up to an order of magnitude smaller (Ma et al., 13 Jan 2025). The stated interpretation is that optimized MA steering vectors exhibit low correlation in the angular domain, aiding both SDMA and target estimation (Ma et al., 13 Jan 2025).

6. Physical-layer security

Security is one of the most active MA-array application areas because antenna movement directly changes the correlation between legitimate and adversarial channels. A common descriptor is the normalized Bob–Eve channel correlation

$[0,D]$21

which measures spatial separability: values near one imply similar spatial signatures and difficult null-steering, while values near zero imply near-orthogonality (Ma et al., 31 Aug 2025). This reframes secrecy enhancement as a geometry-control problem rather than a beamformer-only problem.

Illustrative far-field examples show the effect sharply. In a multi-eavesdropper setup with $[0,D]$22, a 1D movable region of size $[0,D]$23, Bob at $[0,D]$24, and Eves at $[0,D]$25, $[0,D]$26, and $[0,D]$27, the MA array attains perfect nulls at all Eve angles while preserving the full $[0,D]$28 array gain at Bob. Under zero-forcing, the fixed ULA loses about $[0,D]$29 gain in Bob’s direction. In a separate secrecy-rate comparison under $[0,D]$30 dBm, MA arrays outperform dense-FPA and sparse-FPA baselines by up to $[0,D]$31 bps/Hz (Ma et al., 31 Aug 2025).

A more structured secure design is the hybrid FPA–MA architecture, where the fixed array carries the confidential signal and the MA array generates artificial noise. The secrecy-rate maximization jointly optimizes MA positions, FPA beamforming, and MA beamforming under power and spacing constraints, with AO for the beamformers and NMPGA for the positions. In a four-slot Internet-of-Vehicles scenario with two colluding Eves, the proposed FPA–MA co-design yields a $[0,D]$32 secrecy-rate gain over a fixed-antenna system that uses FPA only for artificial noise and a $[0,D]$33 gain over an MA-only design that treats all MAs as information carriers (Yu et al., 8 Jul 2025). The stated mechanism is that MA repositioning sharpens nulls in Eve directions while keeping a deep null at Bob for artificial noise.

A distinct secrecy limitation appears when Eve is aligned with Bob in angle, especially under LoS mmWave/THz propagation. In that regime, MA motion alone is directionally insecure, and frequency-diverse arrays (FDAs) alone cannot resolve same-range angular overlap. The optimized frequency-diverse movable-antenna design therefore combines element positions $[0,D]$34 with small frequency offsets $[0,D]$35, using either closed-form small-perturbation updates or simulated annealing. In the reported results, the general simulated-annealing design almost attains the no-eavesdropper upper bound across the tested antenna counts, while the closed-form perturbative design approaches it as $[0,D]$36 increases; both create a localized beam focus at Bob and nulls at all critical eavesdropper positions (Li et al., 30 Jun 2025).

Secure multiuser beamforming generalizes these ideas to multiple legitimate users and multiple cooperating Eves. One recent framework formulates the sum secrecy rate under the Gaussian wiretap-coding theorem and solves the joint digital-beamforming and MA-placement problem by fractional programming plus block coordinate descent. Each update is either closed-form or a low-complexity one-dimensional or bisection search. The reported sum secrecy gain over the FPA baseline is up to $[0,D]$37–$[0,D]$38 across SNR and Eve-count settings, with convergence typically within $[0,D]$39 outer iterations (Cheng et al., 9 Jan 2026).

7. Hardware realizations, control overhead, and performance limits

A central practical issue for MA arrays is actuator complexity. The element-wise single-layer MA model maximizes flexibility but scales poorly. TL-MA addresses this by splitting motion into subarray translation and intra-subarray fine-tuning. Numerically, TL-MA reduces the sum-displacement of MA motors by about $[0,D]$40 relative to element-wise single-layer MA while maintaining comparable rate performance; the reported best operating region occurs around $[0,D]$41–$[0,D]$42 subarrays, balancing large-scale and fine-tuning degrees of freedom (Yao et al., 19 Nov 2025). CL-MA addresses the same issue more aggressively: because only $[0,D]$43 motors control an $[0,D]$44 grid, hardware scales linearly rather than with the number of ports. In uplink SDMA, instantaneous CL-MA yields more than $[0,D]$45 dB power savings over a dense UPA at $[0,D]$46 users and $[0,D]$47 dB over a sparse UPA, while the statistical CL-MA design is about $[0,D]$48 dB worse than the instantaneous design but avoids about $[0,D]$49 repositionings (Zhu et al., 6 May 2025).

The actuation technology itself imposes a second layer of constraints. Mechanically driven MAs, such as servo motors or MEMS actuators, provide continuous 2D or 3D displacement but incur movement latency in the $[0,D]$50s–ms range as well as energy and wiring complexity. Electronically driven MAs, such as multi-mode patch antennas with switchable phase centers, can reconfigure in the ns–$[0,D]$51s range but only through discrete position quantization and at the cost of added RF-circuit complexity (Ma et al., 31 Aug 2025). This motivates predictive control. In UAV communications, a Transformer-enhanced LSTM is trained on secrecy-optimal MA trajectories generated by PSO. For prediction horizon $[0,D]$52, the reported NMSE reduction exceeds $[0,D]$53 relative to the best baseline, the position-accuracy gain is $[0,D]$54 at threshold $[0,D]$55 m, the inference latency is $[0,D]$56 ms, and predictive control improves secrecy rate by $[0,D]$57–$[0,D]$58 over reactive control under latency (Yu et al., 14 Aug 2025).

Theoretical analyses show that MA benefits are not monotone in all physical parameters. In continuous moving arrays with best-position selection, reduced inter-element spacing can improve high-SNR performance because correlated elements can align simultaneously with a strong fading hotspot. The reported ordering is

$[0,D]$59

and new upper-tail CDF and level-crossing-rate expressions are derived for both correlated and uncorrelated elements (Siriwardana et al., 16 May 2026). By contrast, in the wideband multi-user MIMO study, rich scattering, large EVM, and wide UL–DL frequency separation weaken or erase the MA advantage (Irshad et al., 23 Mar 2026). These results indicate that mobility and compactness can be beneficial in some channel-statistical regimes but are not universally optimal.

Mutual coupling creates a further dual aspect. In standard array design it is usually detrimental, yet closely packed coupled MA arrays can exploit it for directivity enhancement. For $[0,D]$60, the GS–GD optimized design outperforms the uncoupled half-wavelength ULA by at least $[0,D]$61–$[0,D]$62 across directions and approaches the exhaustive-search optimum closely; at endfire, the directivity tends toward $[0,D]$63 (Xu et al., 17 Mar 2026). At the same time, the same work notes that superdirective designs require extremely precise current control and are sensitive to element tolerances and loss (Xu et al., 17 Mar 2026). This controversy is characteristic of the broader MA literature: the same positional freedom that enables larger apertures, lower CRBs, or higher secrecy can also amplify hardware sensitivity, calibration burden, and control latency.

Future directions identified in the literature include six-dimensional movable antennas that jointly optimize 3D position and 3D orientation, movable-element intelligent reflecting surfaces, extremely-large MA deployments, MA-aided secure sensing, and robust position design under imperfect CSI (Ma et al., 31 Aug 2025). A plausible implication is that MA arrays are best understood not as a single array topology but as a general aperture-reconfiguration paradigm whose utility depends on the joint geometry of propagation, hardware, and control timescales.