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Tri-Domain Multiuser MIMO Precoding Optimization and Channel Estimation with Spatial-EM Reconfigurable Antenna

Yining Li, Ziwei Wan, Zhen Gao, Keke Ying, Lipeng Zhu, Rui Zhang

arXiv:2605.23682v3eess.SPcs.IT

TL;DR

The paper addresses the underexplored integration of EM and spatial antenna reconfiguration, including the resulting channel-estimation and precoding challenges. It proposes movement-aided parametric CE and tri-domain ZF/WMMSE precoding for SEMRA; simulations report improved eCSI accuracy and higher SE than EMRA under matched pilot overhead.

  • Problem

    EMRA and SMA provide distinct reconfiguration capabilities, but their integration remains underexplored and creates coupled channel-estimation and precoding challenges.

  • Method

    SEMRA combines coordinated movement-aided parametric CE with alternating tri-domain optimization over antenna positions, EM pattern weights, and digital precoders.

  • Results

    The proposed CE improves eCSI accuracy, while SEMRA-ZF and SEMRA-WMMSE achieve higher SE than EMRA under the same per-user pilot overhead; SEMRA-WMMSE further improves over SEMRA-ZF in tested loads.

  • Takeaways & Limitations

    A denser movement-synthesized virtual array supports AoD estimation and eCSI assembly at desired transmission positions without additional pilots.

Abstract

from arXiv · show

In this paper, we propose a tri-domain reconfigurable multiuser multiple-input multiple-output (MIMO) communication system that integrates the electromagnetic (EM) reconfigurable antenna (EMRA) with the spatially movable antenna (SMA), termed the spatial-EM reconfigurable antenna (SEMRA). The proposed system offers EM, spatial, and digital domain degrees of freedom (DoFs) for joint channel reconfiguration, yet introduces new challenges in channel estimation (CE) and precoding optimization. Specifically, for multiuser orthogonal frequency division multiplexing (OFDM) downlink, the precoding design is formulated as a tri-domain optimization problem over antenna positions, EM-domain radiation-pattern weights, and digital precoders. We first develop a zero-forcing (ZF)-based baseline algorithm to decouple the design of spatial reconfiguration, and then propose a weighted minimum mean square error (WMMSE)-based tri-domain joint optimization algorithm for further improving the spectral efficiency (SE). Furthermore, we propose a low-overhead movement-aided channel estimation scheme in which coordinated antenna repositioning across pilot slots synthesizes a denser virtual array, enabling more accurate angle-of-departure (AoD) estimation and EM-domain channel state information (eCSI) reconstruction under the same per-user pilot overhead as the EMRA baseline. The resulting parametric representation enables eCSI assembly at desired antenna positions without additional pilots. Simulation results show that the proposed CE scheme improves eCSI estimation accuracy and the proposed SEMRA achieves higher SE than the EMRA baseline under the same pilot overhead.

I. INTRODUCTION

SEMRA integrates electromagnetic radiation-pattern reconfiguration with spatial antenna repositioning, adding physical degrees of freedom to digital precoding. The paper develops movement-aided channel estimation and tri-domain precoding, reporting improved eCSI accuracy and spectral efficiency over EMRA under matched pilot overhead.

  • Motivation and framework: SEMRA combines EMRA and SMA to jointly exploit electromagnetic and spatial reconfiguration in multiuser MIMO.The architecture adds antenna-position variables to radiation-pattern and digital-precoding design.
  • Motivation and framework: EM-domain precoding uses radiation-pattern basis coefficients alongside conventional digital precoding, requiring both sCSI and eCSI.eCSI links propagation geometry to the radiation-pattern basis for joint EM-domain and digital-domain transmission design.
  • Challenges: AoD errors perturb steering and basis matrices used for eCSI assembly, while excessive antenna spacing can introduce spatial aliasing and grating-lobe ambiguity.These effects make channel estimation central to reliable EM-domain precoding.
  • Proposed channel estimation: Movement-aided CE coordinates local antenna repositioning across pilot slots to synthesize a denser virtual array with the same per-user pilot overhead as EMRA.The resulting position-independent parametric model supports eCSI assembly at desired antenna positions without additional pilots.
  • Tri-domain precoding: The proposed precoding framework alternates over antenna positions, EM-domain pattern weights, and digital precoders using ZF and WMMSE designs.ZF provides a lower-complexity baseline, while WMMSE jointly optimizes the tri-domain variables for further SE improvement.
  • Results: Numerical results show improved eCSI accuracy and higher SE than EMRA under the same pilot overhead, with SEMRA-WMMSE outperforming SEMRA-ZF in tested multiuser loads.The SEMRA advantage is especially evident at larger inter-element spacings, where EMRA is more sensitive to spatial aliasing.

C. EM-Domain CSI Representation

The channel representation expands each transmit radiation pattern in orthonormal basis functions, separating position-independent path information from antenna-position and EM-weight effects. This enables the formulation of a coupled sum-SE optimization over spatial positions, EM weights, and digital precoders.

  • EM-domain representation: Transmit radiation patterns are represented as weighted combinations of K real-valued orthonormal basis functions.The basis functions are constructed from real spherical harmonics, and each antenna has a coefficient vector α_m.
  • eCSI construction: The eCSI representation maps path geometry and equivalent path gains through the basis-function matrix Ω_u into per-antenna channel coefficients.The resulting vectors are stacked across the M antennas for use in the effective channel.
  • EM-domain representation: The EM precoding matrix Λ is block diagonal, with each block containing one antenna's pattern-weight vector.The coefficients satisfy the per-antenna unit-norm constraint ∥α_m∥2 = 1.
  • SEMRA model: SEMRA jointly optimizes per-antenna EM pattern weights and antenna positions, unlike the EMRA with TFA architecture.Antenna positions enter the channel through the BS-side steering matrix, while EM weights affect the equivalent channel through Λ.
  • Problem formulation: The sum-SE maximization jointly varies digital precoders, EM-domain weights, and antenna positions through the effective channel vectors.Coupled variables and unit-norm constraints make the resulting problem highly nonconvex.

III. TRI-DOMAIN ALTERNATING OPTIMIZATION: ZF-BASED BASELINE

The SEMRA ZF baseline extends EM-digital alternating precoding with a spatial antenna-position block. It alternates digital, EM, and spatial updates while maintaining feasible EM structure through masked-manifold operations.

  • SEMRA extends EM-digital alternating ZF design by adding a spatial-domain antenna-position block.
  • Baseline role: SEMRA-ZF provides a lower-complexity reference that makes the added spatial block comparable with EM and digital baseline designs, but does not optimize global user power allocation.
  • Digital-domain optimization: The ZF procedure updates digital precoders from normalized channel directions for M ≥ U.
  • EM-domain optimization: The EM precoder is optimized as a block-diagonal matrix with per-antenna unit-norm constraints on a masked oblique manifold.
  • EM-domain optimization: Column-wise normalization retracts EM updates while preserving the block-diagonal sparsity imposed by Hadamard masking.
  • Alternating implementation: A single manifold-gradient step is used for the EM block in each outer iteration to keep per-iteration cost moderate.

B. Spatial-Domain Antenna Position Optimization

The spatial block optimizes antenna positions using gradients of the spectral-efficiency objective derived from the position-dependent eCSI model. Projected position trials are accepted only when they do not reduce sum SE.

  • Gradient construction: The spatial gradient is derived from the parametric eCSI model and depends on position-dependent channel terms rather than explicit AoA or UE-side path parameters.
  • Position update: A projected-gradient step updates each antenna position with the EM precoder and digital precoders fixed.
  • Feasibility projection: Componentwise projection enforces the planar constraint and clips each coordinate to its feasible box region.
  • Acceptance rule: The algorithm retains a feasible position trial only if it does not decrease the current sum SE; otherwise, it preserves the current positions.
  • WMMSE extension: The WMMSE extension retains the EM and spatial update structure while replacing normalized ZF with auxiliary, weighted-MSE, and digital-precoder updates.

A. WMMSE Problem Transformation

The WMMSE transformation introduces receive equalizers and weighted MSE terms for each user-subcarrier pair while retaining the effective channel dependence on antenna positions and EM weights.

  • Each user-subcarrier pair uses a scalar receive equalizer βu,g to form an estimated transmitted symbol.
  • The resulting MSE expression separates signal distortion from multiuser interference.
  • The effective channel hu,g depends on antenna positions and EM-domain weights, although those dependencies are omitted for notational brevity.

2) WMMSE Equivalence:

WMMSE equivalence converts sum-SE maximization into alternating updates over auxiliary variables and digital precoders under a total-power constraint. The auxiliary and digital subproblems admit closed-form solutions, while EM and spatial blocks remain nonconvex.

  • WMMSE reformulation: WMMSE equivalence reformulates sum-SE maximization using MMSE receivers βu,g and positive MSE weights ρu,g for each user-subcarrier pair.
  • WMMSE equivalence: After optimizing βu,g and ρu,g, minimizing the WMMSE objective is equivalent to maximizing spectral efficiency.
  • Nonconvex blocks: The EM- and spatial-domain blocks remain nonconvex and retain the update structure developed elsewhere in the algorithm.
  • Block updates: With positions and EM precoders fixed, block coordinate descent updates auxiliary variables and digital precoders through convex subproblems with closed-form solutions.
  • Digital precoder update: The digital-precoder subproblem is solved by a Lagrangian method under the global total-power constraint.
  • Digital precoder update: The optimal digital update can redistribute power across users and subcarriers, unlike normalized ZF, with the multiplier found by bisection when needed.

C. EM-Domain Radiation Pattern Optimization

The EM-domain block optimizes radiation-pattern weights within the WMMSE-based tri-domain alternating procedure, using manifold updates and refreshed channel-dependent quantities.

  • C. EM-Domain Radiation Pattern Optimization: The EM-domain subproblem optimizes the real-valued pattern weights α_m, equivalently represented by a block-diagonal matrix Λ.The masked-oblique manifold framework is adapted to the WMMSE objective and its Euclidean gradient.
  • C. EM-Domain Radiation Pattern Optimization: The EM update extracts per-antenna weights from Λ before the digital refresh, while initialization uses globally normalized MRT under the total-power constraint.The same alternating procedure also updates positions through projected trials and retains a trial only when the sum SE does not decrease.
  • C. EM-Domain Radiation Pattern Optimization: Algorithm 2 alternates digital initialization, channel updates, EM updates, digital refresh, position updates, and objective evaluation.The procedure takes an eCSI model, feasible position sets, and a power budget as inputs, and outputs digital precoders, EM weights, and antenna positions.

1) Spatial Gradient:

The spatial block reuses the decomposition developed for the earlier spatial optimization to derive the position-gradient update in the WMMSE procedure.

  • 1) Spatial Gradient:: The WMMSE spatial-gradient derivation repeats the decomposition from Section III-B and uses the position-gradient expression for the spatial block.The supplied passage identifies this as a reused derivation rather than introducing a separate spatial formulation.

2) Position Update with Projection:

The position block forms projected antenna-position trials, while the movement-aided CE procedure uses coordinated positions to create a denser virtual array for parametric channel estimation.

  • 2) Position Update with Projection:: The spatial block computes one projected-gradient position trial over all antenna positions in each outer iteration.The feasible trial is retained only if it does not decrease the current sum SE; otherwise, the current positions are preserved.
  • V. LOW-OVERHEAD MOVEMENT-AIDED CE SCHEME: SEMRA repositions all antennas across pilot slots, producing NsM distinct spatial samples while separating CE observation positions from later transmission positions.The position-invariant parameterization allows eCSI assembly at optimized data positions without additional pilot overhead.
  • V. LOW-OVERHEAD MOVEMENT-AIDED CE SCHEME: For the CE procedure, each reference antenna position is surrounded by Ns = 4 offsets of ±d/4 along the y- and z-axes, subject to feasibility constraints.The construction requires Dmax ≥ d/4 and is jointly compatible with the non-overlapping condition when Dsep < d/2.
  • V. LOW-OVERHEAD MOVEMENT-AIDED CE SCHEME: With dmin = d/4, the 4M observations form a uniform 2My × 2Mz virtual UPA with spacing dv = d/2 for 2D-ESPRIT AoD estimation.Delay estimation continues to rely mainly on frequency-domain shift invariance.
  • 1) Delay Estimation and sCSI Recovery:: The pilot observations are normalized and stacked across position-pattern pairs into a delay-steering data matrix, whose multipath order is estimated by MDL before 1D-ESPRIT and least-squares recovery.Users use disjoint uniformly spaced pilot-subcarrier subsets, and the recovered full-band sCSI is obtained at the CE observation positions.

2) AoD Estimation on the Virtual Array:

The virtual-array AoD stage applies 2D-ESPRIT to spatial observations, maps spatial frequencies to AoD angles, and reconstructs eCSI at arbitrary target positions from the estimated parametric model.

  • 2) AoD Estimation on the Virtual Array:: Virtual-array observations share a canonical UPA steering structure, enabling re-indexing, spatial smoothing, and 2D-ESPRIT across training blocks.The spatial frequencies returned by 2D-ESPRIT are related to the two AoD angles through the virtual-array spacing constant.
  • 2) AoD Estimation on the Virtual Array:: The AoD inversion recovers principal-branch angles over the front half-space, with unique mapping guaranteed when dv ≤ λ/2.For dv > λ/2, the same principal-branch inversion is retained but global uniqueness is not claimed.
  • 2) AoD Estimation on the Virtual Array:: Estimated AoDs determine the direction vectors, position-dependent steering matrix, and EM basis-function matrix used by the parametric channel model.The steering matrix is assembled at any desired antenna position, whereas the basis functions depend on the estimated angles.
  • 2) AoD Estimation on the Virtual Array:: The equivalent path-gain vector is estimated from the virtual-array observations, after which eCSI can be assembled at target positions through the reconstructed parametric model.Only the target-position steering matrix depends on the target position, so CE and transmission positions need not coincide.

VI. SIMULATION RESULTS

The simulations evaluate channel-estimation accuracy under SNRC and antenna-spacing sweeps, alongside spectral-efficiency comparisons under fixed transmit-power and stated CSI conditions. SEMRA improves ESPRIT-based eCSI estimation and is more robust than EMRA at larger spacings.

  • Simulation setup: The evaluation varies one horizontal-axis quantity at a time while retaining the remaining default simulation settings.The comparisons include EMRA, SEMRA-ZF, SEMRA-WMMSE, and TFA/SMA baselines under their stated CSI conditions.
  • Evaluation scope: The fixed-power evaluation reports communication-rate gains, while actuator energy, movement latency, and calibration overhead are outside the present SE metric.EMRA and SEMRA are evaluated with the same per-user pilot overhead NpatNsJ = 360.
  • Channel estimation versus SNRC: At d = λ/2, NMSE-S curves nearly overlap for EMRA and SEMRA, while ESPRIT reaches about −34.1 dB and OMP saturates near −17.5 dB at 30 dB SNRC.NMSE-S is dominated by delay estimation and sCSI recovery rather than spatial sampling density.
  • Channel estimation versus SNRC: At 30 dB SNRC, SEMRA+ESPRIT reaches about −35.5 dB NMSE-E with estimated sCSI versus about −22.9 dB for EMRA+ESPRIT.The corresponding perfect-sCSI values are about −39.8 dB and −23.4 dB, respectively.
  • Channel estimation versus spacing: At SNRC = 20 dB, EMRA+ESPRIT worsens from about −21.1 dB at d = 0.8λ to about 1.0 dB at d = 1.5λ, whereas SEMRA+ESPRIT remains near −32 dB from 0.9λ to 1.5λ.The denser virtual array mitigates ambiguity and grating-lobe effects at larger spacings while retaining the same pilot overhead.

C. Spectral Efficiency

Across SNR, antenna spacing, user load, displacement, and iteration sweeps, SEMRA consistently improves sum SE over EMRA, with WMMSE outperforming ZF. The gains persist under estimated eCSI, while SEMRA remains more robust to wide-spacing estimation effects.

  • SE versus SNRC: SEMRA-WMMSE > SEMRA-ZF > EMRA across the tested SNRC range, with SEMRA perfect-versus-estimated gaps within about 0.50 bps/Hz at SNRC = 20 dB.The smaller SEMRA gaps are consistent with lower eCSI NMSE, while spatial reconfiguration and stronger digital precoding provide additional gains.
  • SE versus SNRP: At SNRP = 20 dB, SEMRA-WMMSE remains above EMRA and fixed-pattern TFA/SMA baselines under estimated eCSI, with a high-SNRP WMMSE-over-ZF gap of about 1.70 bps/Hz.All curves increase approximately linearly with SNRP, and the SEMRA advantage persists across the full range.
  • SE versus Antenna Spacing: At d = 1.5λ, estimated-eCSI SE reaches about 17.00 bps/Hz for SEMRA-ZF and 18.60 bps/Hz for SEMRA-WMMSE, whereas EMRA falls to about 6.00 bps/Hz.EMRA peaks at about 15.40 bps/Hz near d ≈ 0.8λ before grating-lobe and ambiguity effects dominate; SEMRA increases monotonically and stays near its perfect-eCSI benchmarks.
  • SE versus Number of Users: At U = 4 and U = 5, SEMRA-ZF gains 0.69 and 0.79 bps/Hz over EMRA, while SEMRA-WMMSE gains 3.60 and 4.64 bps/Hz.Both SEMRA variants retain positive paired gains across U = 1, . . . , 7, with WMMSE gains becoming more pronounced as user load increases.
  • SE versus Maximum Antenna Displacement: Across the feasible displacement range, endpoint gains over the Dmax = 0 reference range from 0.13 to 0.62 bps/Hz.Because d is fixed, this sweep isolates additional position freedom from reference-aperture enlargement; perfect-eCSI curves remain above estimated-eCSI curves.
  • Convergence versus Outer Iteration Index: The first 10 iterations capture 99.3% and 96.4% of the improvements observed by iteration 20, leaving only 0.04 and 0.25 bps/Hz of additional gain.SEMRA-ZF plateaus earlier, whereas SEMRA-WMMSE refines more gradually toward higher SE; these results support Nmax = 10.
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