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Mutual-Coupling-Aware Movable and Fluid Antennas on Holographic Surfaces: A Wavenumber-Domain Circuit-Field Unification

Giovanni Iacovelli, Chandan Kumar Sheemar, Symeon Chatzinotas

arXiv:2609.04968v1eess.SP

TL;DR

The paper addresses the missing wavenumber-domain, coupling-consistent formulation of flexible positioning and port selection for multi-user transmission. It unifies impedance-kernel and circuit models, formulates physically constrained sum-rate optimization, and develops modal relaxation and FFT-based initialization for port placement.

  • Problem

    Existing coupling-aware FAS/MA optimization uses spatial or current domains, while wavenumber-domain treatments address fixed apertures, leaving flexible positioning and port selection un unified for multi-user transmission.

  • Method

    The paper derives a full impedance-kernel model, establishes circuit-field equivalence, represents ports as constant-modulus wavenumber codewords, and optimizes precoders and positions under coupled power and voltage constraints.

  • Results

    The formulation provides an exact wavenumber dichotomy, asymptotic resistive-kernel diagonalization, a modal upper bound, and FFT-based codeword projection for initializing port positions.

  • Takeaways & Limitations

    Coupling must enter the design loop for sub-half-wavelength packing, whereas optimized layouts on unconstrained apertures can self-decouple and recover the classical model.

Abstract

from arXiv · show

Movable and fluid antenna systems turn antenna position into a design variable. At sub-wavelength spacings, however, their behavior is governed by mutual coupling, modeled today by two disjoint traditions: circuit-theoretic impedance matrices with element-level constants, and field-theoretic kernels with norm-type power constraints. This paper unifies the two. Starting from the impedance kernel of a holographic surface, a Poynting-anchored balance identifies its resistive part with ohmic plus radiated power and its reactive part with stored-energy imbalance, and a circuit-field equivalence shows that the multiport impedance matrix is the kernel sampled at the port separations, in a single closed spherical-Hankel form. In the wavenumber domain the resistive kernel asymptotically diagonalizes in the aperture size: visible modes radiate at closed-form prices, evanescent modes only dissipate, and a flexible port becomes a constant-modulus codeword whose coupling is the pullback of the spectral weight. Coupling-aware multi-user sum-rate maximization over precoders and port positions is then formulated under physical power and voltage constraints and solved by weighted-MMSE and projected-gradient steps with closed-form gradients. A modal relaxation upper-bounds every port configuration and seeds the search by FFT-based codeword projection. A half-wavelength corollary and a superdirectivity margin quantify when coupling hurts, and when it helps.

I. INTRODUCTION

Flexible antenna positioning creates a design opportunity but makes independent-port abstractions fragile under simultaneous sub-wavelength radiation. The paper fills the empty intersection between coupling-aware flexible positioning and wavenumber-domain multi-user design with a physically constrained unified model.

  • I. INTRODUCTION: Sub-wavelength multi-port radiation makes independent-port channel abstractions inadequate because electromagnetic coupling becomes part of the system behavior.The issue arises in MIMO-FAS, fluid antenna arrays, and pixel-based implementations.
  • I. INTRODUCTION: Prior coupling-aware work uses circuit-theoretic matrices or field-theoretic operators, while flexible-position formulations and wavenumber-domain treatments remain separated.The literature review identifies distinct circuit-first and field-first cultures.
  • I. INTRODUCTION: The paper targets the previously empty intersection of wavenumber-domain coupling consistency, flexible positioning, and multi-user port selection.It also introduces physically correct coupled power and voltage constraints.
  • I. INTRODUCTION: Its contributions include a full impedance-kernel model, circuit-field equivalence, an exact light-circle dichotomy, positioning-HMIMO equivalence, and coupling-aware sum-rate optimization.The formulation connects resistive power, reactive voltage and stored-energy effects, constant-modulus codewords, and alternating optimization.

B. The impedance kernel and complex power

The impedance kernel is grounded in complex power: its resistive part accounts for ohmic and radiated power, while its reactive part represents stored-energy imbalance. This establishes the physical meaning of the kernel components used later in the design constraints.

  • B. The impedance kernel and complex power: A norm-only current constraint is physically agnostic because supplied power depends on nonlocal electromagnetic fields acting against the current.The field-based power budget replaces a simple current-norm interpretation.
  • B. The impedance kernel and complex power: The complex power delivered by the generators separates into real power and imaginary energy exchange through the impedance kernel.The kernel maps current density to the voltage density required at each surface point.
  • B. The impedance kernel and complex power: The resistive kernel combines ohmic dissipation with radiation, whereas the reactive kernel contributes no average power and tracks stored electric-minus-magnetic energy.The reactive balance is 2ω(We − Wm).
  • B. The impedance kernel and complex power: The radiation kernel admits a plane-wave decomposition, linking radiative power to far-field flux and the transverse current spectrum.This provides the physical anchor for the later spectral representation.

C. Port model and the circuit-field equivalence

The paper derives a circuit-field equivalence in which the multiport impedance matrix is determined by matching complex power and samples the electromagnetic kernel at port separations. This makes coupling enter both physical power and voltage feasibility constraints.

  • C. Port model and the circuit-field equivalence: Matching complex power for every excitation uniquely determines the circuit multiport impedance matrix from the field kernel.Its entries recover induced-EMF mutual impedance.
  • C. Port model and the circuit-field equivalence: The real and imaginary parts of the port matrix inherit the resistive and reactive kernels, including the stored-energy identity at port level.The port-level reactive quadratic form equals 2ω(We − Wm).
  • C. Port model and the circuit-field equivalence: As κ∆→0, mutual impedance entries converge to a common closed-form kernel sampled at port separations, while the radiation self-resistance approaches the Hertzian value.The mutual approximation applies off the diagonal for separated translated element profiles.
  • C. Port model and the circuit-field equivalence: Self-reactance and ohmic self-resistance remain profile-dependent element properties rather than universal point-source values.These diagonal terms diverge under the point-profile limit according to the element profile.
  • C. Port model and the circuit-field equivalence: For multiple active ports, coupling makes both feasible power and voltage sets position-dependent through Cε(U) and Z(U).The reactive kernel particularly affects voltage feasibility at small spacings.

III. WAVENUMBER-DOMAIN REPRESENTATION AND THE FAS-HMIMO EQUIVALENCE

In the wavenumber domain, the impedance kernel separates exactly at the light circle: visible modes govern radiation while evanescent modes govern reactive behavior. Flexible antenna positions therefore become constant-modulus spectral codewords whose nonuniform power prices retain coupling.

  • III. WAVENUMBER-DOMAIN REPRESENTATION AND THE FAS-HMIMO EQUIVALENCE: The planar Fourier transform of the impedance kernel exposes separate radiation and reaction behavior for tangential aperture currents.The scalar polarized kernel is the relevant spectral object.
  • III. WAVENUMBER-DOMAIN REPRESENTATION AND THE FAS-HMIMO EQUIVALENCE: The light circle ∥k̄∥=κ exactly separates the resistive and reactive spectra.The resistive spectrum occupies the visible disc, while the reactive spectrum occupies the evanescent region.
  • III. WAVENUMBER-DOMAIN REPRESENTATION AND THE FAS-HMIMO EQUIVALENCE: The resistive spectrum contains the ohmic floor plus an obliquity-and-polarization weight on visible modes, whereas the reactive spectrum is entirely evanescent.Reactive behavior changes sign across the polarization cone and grows with wavenumber magnitude.
  • III. WAVENUMBER-DOMAIN REPRESENTATION AND THE FAS-HMIMO EQUIVALENCE: The spectral split follows from the Weyl plane-wave expansion and the fact that kz is real inside the light disc but imaginary outside it.The transverse polarization factor supplies the obliquity weighting.
  • III. WAVENUMBER-DOMAIN REPRESENTATION AND THE FAS-HMIMO EQUIVALENCE: Only visible spectral components radiate, while the ohmic floor taxes every component through the non-flat per-wavenumber power price.This makes average power directly computable from the resistive spectral representation.

B. Fourier modes and the modal Gram

The Fourier representation separates radiative and reactive behavior: visible modes asymptotically diagonalize the resistive Gram, while sharply truncated modes have divergent reactive energy. Port positions then appear as constant-modulus wavenumber codewords whose coupling is inherited from the spectral kernel.

  • Modal resistive behavior: Fourier modes asymptotically diagonalize the resistive Gram for modes separated from the light-circle rim.The result applies as the aperture dimensions grow while retaining a fixed margin from the visible-disc boundary.
  • Modal resistive behavior: Visible modes radiate at individual spectral prices combining the ohmic floor with a broadside-minimal weight that diverges integrably toward the rim.Evanescent modes satisfy [BR]nn → Zs, so they dissipate ohmically without radiating.
  • Reactive behavior: Sharply truncated Fourier modes have logarithmically divergent reactive forms because aperture-edge discontinuities impose infinite stored energy.The paper therefore performs reactive bookkeeping at port level or uses edge-compatible modal families that vanish at the rim.
  • Positioning-HMIMO equivalence: Each port contributes a fixed taper multiplied by a constant-modulus phase codeword, so moving a port is spectral modulation and selecting ports is codebook restriction.The resulting HMIMO manifold contains continuous position codewords, while candidate grids produce finite codebooks.
  • Positioning-HMIMO equivalence: Mutual coupling between multiple ports is the pullback of the wavenumber kernel under the position-to-codeword map, yielding a non-additive quadratic form.The same mode-domain and port-domain complex-power evaluations coincide by Parseval.
  • Half-wavelength corollary: At half-wavelength spacing, the scalar model reduces to uncoupled MIMO, but polarized coupling remains first-order and predominantly reactive, with χ ≈0.43 versus ρ ≈0.15.This residual reactive coupling motivates retaining voltage constraints even away from the superdirective regime.
  • Coupling implications: For one active port, power is position-independent, whereas with A ≥ 2 the feasible power and voltage sets become position-dependent through coupling.Consequently, multi-port position optimization must account for coupling.

IV. COUPLING-AWARE SUM-RATE MAXIMIZATION

The paper formulates multi-user sum-rate maximization over port positions and feed-current precoders under coupled physical constraints. The resulting problem is non-convex because positions affect both channels and impedance kernels, so the design uses alternating optimization blocks.

  • Problem formulation: The transmitter serves K streams through per-user precoders in feed-current units, with the aggregate current formed by their superposition.The formulation uses the flexible-position antenna's coupled channel model.
  • Problem formulation: The master problem maximizes the weighted sum rate over positions and precoders subject to coupled power, voltage, and hardware-exclusion constraints.The normalized impedance and minimum-distance constraints enter the joint optimization.
  • Wavenumber formulation: In the wavenumber representation, the optimization selects K precoded superpositions of A constant-modulus codewords from the admissible position manifold.The spectral weights price each port configuration through the coupling-aware model.
  • Optimization structure: The problem is non-convex because the sum rate, trigonometric position dependence, and position-dependent constraints interact multiplicatively.Both the received channels and feasible set move with port positions.

A. Coupling whitening and the precoder block

The precoder block whitens the coupling matrix so the multi-user problem becomes a canonical weighted sum-rate optimization, while physical power and voltage constraints remain explicit. Its spectral analysis exposes superdirectivity gains from low-radiation eigenmodes and the associated efficiency and stored-energy costs.

  • Coupling whitening: Whitening with Cε(U) makes the coupling matrix positive definite and transforms the precoder variables into a standard MU-MISO weighted sum-rate problem.The ohmic floor ε shifts the positive-semidefinite radiation spectrum, enabling the whitening transformation.
  • WMMSE precoder updates: WMMSE block-coordinate updates solve the whitened precoder subproblem, with bisection selecting the multiplier for the active power constraint.Each pass is monotonically non-decreasing in weighted sum rate; voltage caps can be handled as second-order-cone constraints or through penalties.
  • Physical constraints: Voltage constraints preserve a convex whitened subproblem, while current-unit precoders are recovered by applying the inverse square root of the coupling matrix.The voltage formulation accounts for coupling-dependent drive requirements across ports.
  • Superdirectivity margin: Under Pem ≤ P, beamforming achieves P hHCε^-1h, whereas an agnostic norm constraint prices ohmic dissipation rather than radiation.This distinguishes physical dissipated-power control from a coupling-agnostic current-norm constraint.
  • Superdirectivity margin: The modal gain βi/(ε + λi) exceeds one for channel-weighted modes with λi < 1 − ε and saturates at βi/ε as λi shrinks.Low-radiation eigenmodes can yield gains beyond the port count, but approaching the bound requires vanishing efficiency and increased stored energy.

B. The position block: analytic gradients

The position block uses analytic field-derived gradients for projected port updates and couples them to spectral codeword representations. A relaxed modal problem provides both an upper bound for every port geometry and an FFT-based initialization for the nonconvex search.

  • Projected position updates: Analytic gradients update port positions using closed-form field expressions, Armijo backtracking, feasibility rescaling, and pairwise push-out constraints.The resulting ascent avoids numerical differentiation of impedance matrices and Sylvester-based computations.
  • Analytic gradients: Moving one port changes both the rate and the drive voltage of neighboring ports, so voltage-aware gradients include coupling-induced neighbor perturbations.The voltage-gradient formulation reduces to the corresponding precoder expression when the position and voltage variables are identified.
  • Spectral implementation: The algorithm represents channels and gradients as codeword inner products over visible spectral modes, allowing iterations to use measured port spectra without reevaluating the continuous channel.Once the spectral coefficients are available, the optimization loop needs neither a geometry model nor a propagation model.
  • Modal relaxation: Relaxing the port-manifold constraint produces a holographic problem whose optimum upper-bounds every A-port configuration and measures the cost of port sparsity.The relaxation preserves the received functional and physical power while dropping the voltage constraint C2.
  • FFT initialization: The relaxed solution seeds port placement through matching pursuit, whose codeword correlations are evaluated by inverse FFT with cost O(A N log N).For candidate grids, the same spectral machinery supports coupling-aware greedy selection with rank-one updates of Cε^-1.

D. Algorithm, convergence, and complexity

The paper combines a modal relaxation, matching-pursuit initialization, weighted-MMSE precoding, and projected-gradient position updates for coupling-aware sum-rate optimization. Numerical results show that modal pricing is accurate at realistic apertures, spectral initialization is effective, and coupling awareness matters most when port density forces sub-half-wavelength spacing.

  • Algorithm and convergence: The alternating algorithm fixes positions for weighted-MMSE precoding, updates positions by projected-gradient ascent, and repeats until the gain falls below tolerance.Both blocks are monotone under their stated update rules, so the sum-rate sequence is non-decreasing and bounded above.
  • Scenario and baselines: The simulation uses a 6λ × 6λ surface with 137 visible modes, three users, exact dyadic-Green-function channels, and ports of effective support approximately λ/5.Users lie in the Fresnel zone, with ε = 0.05, dmin = 0.15λ, and a 48 × 48 candidate grid at approximately λ/8 spacing.
  • Algorithm and convergence: The spectral initialization starts 1 to 2 bit above the half-wavelength start and reaches the best random-restart level in one deterministic run.The result supports using the modal relaxation to seed the nonconvex position search without multistart.
  • Modal relaxation: At S = 8λ, interior modal diagonalization errors are already within a few percent of their limits, while near-rim modes converge more slowly under the S^-1/2 guide.The relative error decreases monotonically with aperture, validating the diagonal pricing used by the modal relaxation.
  • Pullback accuracy: The finite-aperture pullback error follows a 1/S envelope but can be strongly non-monotone, with near-cancellations reaching 10^-4 because periodic images interfere deterministically.Larger elements suppress the image coupling by shading the rim, while the half-wavelength geometry can place axial images on kernel nulls.
  • Multi-user performance: The modal upper bound is flat in port count, the sparsity gap is about 19 bit at A = 2 and 13 to 15 bit at A = 8 at higher SNR, while MP remains within 1 to 2 bit of AO.The results support solving the modal problem once, projecting by FFT-based matching pursuit, and then precoding.
  • Coupling-aware performance: Coupling awareness is negligible at A = 4 with spacings around λ but becomes decisive at A = 16, where the placement pad forces approximately 0.35λ spacing and both kernels are large.The superdirectivity study also reports an endfire margin of 3.8 at d = 0.3λ with efficiency 0.6, whereas broadside remains below one.

APPENDIX A PROOF OF LEMMA 1

The proof derives the impedance-kernel representation from Maxwell fields and connects its real and imaginary quadratic forms to radiated power, reactive energy, and ohmic dissipation. Taking the far-field limit isolates radiated power from reactive circulation.

  • Green-kernel form: The dyadic Green kernel is derived from scalar spherical-wave derivatives and expressed using spherical Hankel functions.The derivation collects powers of (κR)^-1 and uses recurrences for h0, h1, and h2.
  • Poynting identity: The Poynting identity converts the source-field interaction into a surface-flux expression over a ball enclosing the aperture.The proof applies Maxwell’s equations, the divergence theorem, and the sheet-current delta property.
  • Far-field separation: The limit r0 →∞ makes the leading flux real and outward, while reactive circulation vanishes and the stored-energy difference converges.At finite radius the flux is complex; the far-field limit separates escaping power from near-field reactive effects.
  • Radiative quadratic form: The radiative quadratic form reduces to a transverse Fourier-spectrum norm, establishing the kernel’s radiated-power representation.The transverse projector and Parseval-type factorization produce the norm of (I3 − ˆkˆkT) times the transformed current.
  • Ohmic contribution: The complete balance adds sheet ohmic dissipation to the radiative and reactive terms, yielding the proposition’s power decomposition.For a thin conductive sheet, the impressed internal field is Z_sj and its quadratic form supplies the ohmic term.

APPENDIX C PROOF OF PROPOSITION 2

The proof establishes circuit-field equivalence by sampling the impedance kernel with translated port profiles, then characterizes its visible and evanescent spectral parts. It also shows why finite port profiles regularize reactive self-coupling.

  • Circuit-field equivalence: Translated real port profiles induce a complex-symmetric impedance matrix whose entries are kernel correlations at port separations.Translation covariance and reciprocity produce the correlation form and uniqueness of the matrix representation.
  • Self terms: The radiative self-term converges to the Hertzian radiation resistance, while finite-profile reactive self-coupling remains singular as the profile shrinks.The radiative limit gives R_r, whereas the reactive term scales as (κ∆)^-3 for profile width ∆.
  • Scope of the model: A universal point-antenna limit is unavailable because element-specific ohmic and reactive constants depend on the finite profile and circuit model.The bound on |ξ|^2 and the finite-profile regularization prevent assigning universal point values to X_A and R_Ω.
  • Wavenumber dichotomy: The planar transform separates the kernel pointwise: the visible disc contributes a real radiative term, while the evanescent region contributes a purely imaginary reactive term plus ohmic dissipation.Outside the light circle, k_z = |k_z| makes the interaction reactive; the ohmic delta contributes a flat resistive floor.
  • Consistency check: The visible-spectrum inversion recovers the in-plane Green-kernel form and matches the self-term transverse moment at zero separation.Polar integration and angular moments reproduce the restricted kernel and p^H Im{G(0)}p.

APPENDIX E PROOF OF LEMMA 2

The proof shows that large-aperture Fourier modes asymptotically diagonalize the resistive kernel by concentrating their spectral measures near their modal wavenumbers. Visible modes inherit the spectral weight, whereas evanescent modes approach the ohmic floor.

  • Spectral measures: Parseval converts each modulated aperture indicator into a unit-mass spectral measure centered at its modal wavenumber.The Dirichlet-kernel transform supplies the measures used to analyze diagonal and off-diagonal terms.
  • Reactive interpretation: Distributional treatment is required for the planar reactive transform, but profile and Dirichlet-kernel decay keeps the used quadratic forms well defined.The growing distributional component encodes the R^-3 singularity while the evaluated forms remain finite under the stated decay conditions.
  • Diagonal convergence: The diagonal error is controlled by the spectral-weight variation, with tail, bulk, and light-circle rim regions handled separately.The proof uses tail-mass bounds, continuity in the bulk, and Hölder estimates near the rim singularity.
  • Visible modes: For visible modes separated from the light-circle boundary, the resistive diagonal converges to the radiative weight evaluated at the modal wavenumber.The concentration argument yields the weight value in the visible disc as aperture size grows.
  • Evanescent modes: For evanescent modes, the spectral weight vanishes outside the disc, so the resistive diagonal converges to the ohmic sheet term Z_s.The smooth-region contribution is absent when the modal wavenumber lies outside the light circle.

APPENDIX F PROOF OF LEMMA 3

The proof differentiates the coupling-aware channel and rate expressions with respect to port positions. It uses the Green-kernel Jacobian and pairwise voltage dependence to obtain closed-form gradients.

  • Channel gradient: Differentiating the scalar distance and directional terms yields the position Jacobian of the channel kernel.The chain rule handles x = κτ and c = p^T τ/τ, while the directional Jacobian supplies the transverse projector.
  • Voltage gradient: A port-position derivative of the received voltage includes both the port’s own coupling through neighbors and each neighbor’s voltage through that port.Evenness of the coupling kernel combines the two contributions into a pairwise gradient form.
  • Rate gradient: The rate gradient is obtained from the desired-signal and interference terms using ∇|z|^2 = 2 Re{z*∇z}.Only the position-dependent channel entry contributes directly for the differentiated port, after which common factors are collected.
  • Channel gradient: The channel gradient follows by differentiating the Green-kernel factor F with respect to the planar port position.The displacement Jacobian is −Π^T, and the resulting derivatives are substituted into the closed-form kernel expression.
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