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Movable Antenna Arrays with Imperfect Channel State Information in Rich Scattering Environments

Yizhen Zhao, Amna Irshad, Emil Bjornson

arXiv:2608.29182v1eess.SP

TL;DR

Fixed antenna arrays cannot adapt to changing propagation, and prior movable-antenna studies have not fully connected imperfect CSI, precoding, and placement optimization. This paper combines MMSE estimation, estimated-CSI ZF precoding, and PSO-based placement under sum-rate and fairness objectives. Gains are strongest in sparse near-LoS channels, diminish with channel richness, and require precoder-geometry compatibility for fairness benefits to appear.

Abstract

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The growing demand for high spectral efficiency in 6G and beyond has driven research into adaptive antenna architectures capable of exploiting the spatial structure of multipath propagation channels. Conventional base station arrays are deployed with fixed element positions and cannot adapt to the instantaneous spatial structure of the propagation channel. In contrast, movable antenna (MA) systems enable dynamic reconfiguration of antenna positions, allowing the array geometry to track the channel characteristics of the current user set. While prior MA studies have demonstrated significant gains under perfect or statistical channel state information (CSI) assumptions, the interplay between imperfect instantaneous CSI, antenna placement optimization, and precoder design has received limited attention. This paper addresses this gap by proposing a practical end-to-end framework encompassing uplink pilot transmission, MMSE channel estimation under a clustered multipath model, and downlink ZF precoding designed from estimated CSI. Antenna positions are optimized via particle swarm optimization under two objectives: sum-rate maximization and max-min fairness. Simulation results show that MA gains are most pronounced under sparse, near-LoS propagation and diminish as channel richness increases. Furthermore, we reveal a fundamental coupling between array geometry and precoder design: a fairness-oriented antenna geometry encodes spatial fairness information that is only recoverable when evaluated with a compatible power allocation strategy. A mismatched precoder can completely mask the geometric advantage, leading to misleading conclusions about the robustness of antenna placement to the choice of optimization objective. These findings provide practically relevant guidance for the design and evaluation of movable antenna systems under realistic operating conditions.

I. Introduction

Movable antennas address the inability of fixed arrays to adapt to changing users and propagation, but imperfect CSI complicates placement and precoder evaluation. The paper proposes an end-to-end framework and shows that gains depend on channel structure and precoder compatibility.

Motivation

  • Fixed array geometries cannot adapt after deployment to time-varying user distributions or propagation environments.Source: §I · p. 1
  • Movable antennas reposition elements within a bounded aperture to align with dominant propagation directions, exploit spatial degrees of freedom, and reduce channel correlation.Source: §I · p. 1

Research gap

  • Imperfect pilot-based channel estimates degrade precoding quality and the reliability of antenna-placement objectives.Source: §I · p. 1
  • Prior work did not evaluate the complete chain of pilot transmission, MMSE estimation, ZF precoding, and PSO placement while explicitly accounting for imperfect CSI.Source: §I · p. 2

Contributions

  • The proposed framework combines MMSE estimation, precoding, and PSO-based placement optimization using ergodic rates under imperfect CSI.Source: Contributions · p. 2
  • Movable-antenna gains are significant in sparse directional channels but diminish in rich scattering, while fairness gains require compatible power allocation.Source: Contributions · p. 2

II. System Model

The system is a multi-user downlink in which a base station with movable antennas serves single-antenna users through clustered multipath channels. Each channel comprises clusters distributed around its line-of-sight direction with limited angular spread.

System model

  • A base station equipped with N movable antennas serves K single-antenna users in a time-division duplexing downlink.Source: §II · p. 2

Clustered channel model

  • Each user channel contains Ncl scattering clusters whose directions are distributed around the line-of-sight direction with limited angular spread.Source: §II · p. 2
  • The azimuth and elevation angles ϕk,i and θk,i specify the direction of the i-th cluster for user k.Source: A. Clustered Channel Model · p. 2

Array response

Channel representation

  • The channel model uses complex fading coefficients for each cluster, with cluster powers determining the corresponding spatial covariance matrix.Source: A. Clustered Channel Model · p. 2

B. Pilot-Based MMSE Channel Estimation

During uplink training, users transmit orthogonal pilots, and the base station applies MMSE estimation to obtain channel estimates while retaining an estimation-error covariance.

Pilot transmission

MMSE estimation

Estimation error

C. Downlink Precoding and Achievable Ergodic Sum Rate

The base station designs linear precoding from estimated channels and allocates power by water-filling, while the achievable ergodic rate accounts for residual interference caused by estimation errors.

Downlink precoding

Power allocation

Imperfect CSI

Achievable rate

III. Problem Formulation

The paper formulates antenna placement as utility maximization over user rates under imperfect CSI and mechanical deployment constraints. The framework supports both sum-rate and max-min fairness objectives.

  • Antenna positions are optimized to maximize a network utility function of the user rates.The utility is evaluated through rates whose dependence on antenna positions arises through the steering vectors.Source: §III · p. 3
  • Each antenna remains within its bounded feasible region, while pairwise spacing is constrained to at least λ/2.The spacing constraint is intended to limit mutual coupling effects.Source: §III · p. 3
  • Imperfect CSI enters the rate expression through the channel-estimation error covariance C_k.This distinguishes the formulation from ideal-CSI antenna-placement problems.Source: §III · p. 3
  • The formulation supports sum-rate and max-min fairness objectives within one optimization framework.The utility is monotonically increasing in each user-rate argument.Source: §III · p. 3
  • The placement problem is non-convex because rates depend nonlinearly on antenna positions and spacing constraints are combinatorial.Each candidate geometry also requires channel estimation and precoding, making gradient-based methods impractical.Source: §III · p. 3

IV. Proposed PSO-Based Solution

The proposed solution uses PSO to search over constrained antenna configurations, evaluating each candidate with estimated-channel rates under the selected utility objective.

  • PSO is used because the placement problem is non-convex, high-dimensional, constrained, and lacks convenient gradient information.Its population-based search can escape local optima through cooperative candidate interactions.Source: §IV · p. 3
  • Each particle represents a candidate antenna configuration T^(p).The particle fitness is the utility U(R_1, ..., R_K) computed from achievable user rates.Source: A. PSO Formulation · p. 3
  • Fitness is evaluated using MMSE-estimated channels rather than true channels, matching the CSI available at the base station.This embeds imperfect-CSI conditions directly into antenna-placement optimization.Source: A. PSO Formulation · p. 3
  • Particle velocities and positions are updated using inertia, cognitive and social coefficients, and random scalings.The cognitive term uses each particle’s personal best, while the social term uses the swarm’s global best.Source: A. PSO Formulation · p. 3

B. Constraint Handling

Constraint handling combines a penalty for spacing violations with exact box bounds, steering PSO toward feasible antenna configurations.

  • Spacing violations are incorporated into the PSO fitness through a penalty term.The penalty reduces fitness according to the maximum violation across antenna pairs.Source: B. Constraint Handling · p. 4
  • The penalty weight is α = 1, and positive-part violations determine the fitness reduction.Particles with greater spacing violations receive larger fitness reductions.Source: B. Constraint Handling · p. 4
  • Box constraints are enforced exactly through PSO variable bounds, keeping every antenna within its feasible region C_n.Unlike spacing constraints, these deployment-region limits are not handled only through a soft penalty.Source: B. Constraint Handling · p. 4

C. PSO Algorithm

The PSO algorithm embeds MMSE estimation and ZF precoding inside every fitness evaluation, producing a realistic but computationally expensive offline optimization procedure.

  • Every candidate geometry is evaluated through the same estimated-channel MMSE-and-ZF pipeline used to compute the deployment utility.This keeps antenna optimization aligned with the system’s realistic CSI conditions.Source: C. PSO Algorithm · p. 4
  • MMSE channel estimation dominates per-evaluation complexity at O(KM^3), compared with ZF null-space computation at O(K^2M^2).The dominance follows from M ≥ K.Source: D. Computational Complexity · p. 4
  • The full PSO loop has complexity O(KM^3 + K^2M^2) across N_MC realizations, particles, and iterations.The expression aggregates the repeated fitness-evaluation costs over the Monte Carlo and swarm loops.Source: D. Computational Complexity · p. 4
  • Approximately 7 × 10^10 operations are required for the parameter values in Table I.The resulting computational burden makes the optimization suited to offline deployment when large-scale channel statistics change.Source: D. Computational Complexity · p. 4

V. Numerical Results

The numerical evaluation uses PSO-based antenna placement under imperfect CSI, with MMSE estimation and ZF precoding evaluated across movable and fixed array geometries.

  • The evaluation uses Monte Carlo simulations over 100 user drops, with 50 independent channel realizations per drop and K = 10 users.Source: §V · p. 4
  • PSO optimizes antenna layouts by building clustered channels, estimating them with MMSE, applying ZF precoding with water-filling, and evaluating user-rate utility.Source: §V · p. 4
  • Each antenna moves within a dedicated 5λ × 5λ cell, while the total deployment aperture spans 20λ × 20λ.Source: §V · p. 4
  • Estimated-channel evaluation designs ZF from the MMSE estimate and includes C_k in the SINR, representing realistic imperfect-CSI operation.Source: §V · pp. 4–5
  • True-channel evaluation uses genie-aided ZF with C_k = 0, and the dashed–solid CDF gap quantifies performance loss from imperfect CSI for each geometry.Source: §V · p. 5
  • The comparison includes sum-rate and max–min movable arrays alongside sparse 20λ/3 fixed arrays and conventional λ/2 fixed planar arrays.Source: §V · p. 5

A. Impact of Channel Structure

Movable antennas provide their strongest gains in sparse, near-LoS channels, while increasing channel richness reduces the spatial selectivity available to antenna repositioning.

A. Impact of Channel Structure

  • For Ncl = 1, MA-PSO achieves the best sum-rate performance, followed by F-USPA and then F-HwUPA.The sparse channel is near-LoS and highly directional, with approximately rank-one spatial covariance that provides strong user separation.Source: A. Impact of Channel Structure · p. 5
  • For Ncl = 10, increased angular diversity and covariance rank cause user-channel overlap, reducing the spatial selectivity that MA repositioning exploits.Source: A. Impact of Channel Structure · p. 5

B. Cross-Objective Performance and the role of the precoder

In sparse channels, the apparent fairness advantage of max–min antenna placement depends on the precoder used for evaluation. Water-filling can mask geometry-level fairness, whereas equal-power ZF reveals it.

B. Cross-Objective Performance and the role of the precoder

  • Under water-filling, MA-MM does not outperform MA-SR on minimum ergodic user rate despite being optimized for fairness.The power allocation favors stronger effective channels and suppresses the benefit of geometry that favors weaker users.Source: B. Cross-Objective Performance and the… · p. 5
  • Under equal-power ZF, the fixed MA-MM geometry achieves a clearly higher minimum-rate CDF than MA-SR in the sparse channel case.Keeping the PSO-derived geometries fixed isolates the effect of changing the precoder.Source: B. Cross-Objective Performance and the… · p. 5, Figure 3 · p. 6
  • The framework reports estimated- and true-channel performance to quantify CSI imperfection, while its gains depend strongly on the propagation environment.Movable antennas yield substantial improvements in sparse near-LoS channels, but gains largely vanish in rich-scattering environments.Source: §VI · pp. 5–6
  • A precoder–geometry mismatch can completely mask the spatial advantage of a fairness-oriented antenna arrangement.Source: §VI · p. 6
  • Future work includes joint pilot-sequence and antenna-position optimization, wideband channels, and adaptive positioning for time-varying user distributions.Source: §VI · p. 6
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