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Beam Focusing for Near-Field Multi-User MIMO Communications

Haiyang Zhang, Nir Shlezinger, Francesco Guidi, Davide Dardari, Mohammadreza F. Imani, Yonina C. Eldar

arXiv:2105.13087v1eess.SP

TL;DR

The paper studies beam focusing for near-field multi-user MIMO and develops architectures and sum-rate-based solutions for exploiting it. The results indicate that focused beams can support reliable communication for users at identical angles with little interference, while DMAs achieve the most focused beams among the considered architectures.

  • Problem

    Near-field beam focusing offers the possibility of supporting multiple coexisting orthogonal links, even for users at similar angles.

  • Method

    The paper models near-field channels and transmission patterns for fully-digital, hybrid, and DMA architectures, then proposes sum-rate maximization solutions for each.

  • Results

    Users with identical angles can achieve high rates with little interference, and DMAs achieve the most focused beams among the considered architectures.

  • Takeaways & Limitations

    Near-field beam focusing can increase achievable rate and has the potential to decrease co-channel interference.

Abstract

from arXiv · show

Large antenna arrays and high-frequency bands are two key features of future wireless communication systems. The combination of large-scale antennas with high transmission frequencies often results in the communicating devices operating in the near-field (Fresnel) region. In this paper, we study the potential of beam focusing, feasible in near-field operation, in facilitating high-rate multi-user downlink multiple-input multiple-output (MIMO) systems. As the ability to achieve beam focusing is dictated by the transmit antenna, we study near-field signaling considering different antenna structures, including fully-digital architectures, hybrid phase shifter-based precoders, and the emerging dynamic metasurface antenna (DMA) architecture for massive MIMO arrays. We first provide a mathematical model to characterize near-field wireless channels as well as the transmission pattern for the considered antenna architectures. Then, we formulate the beam focusing problem for the goal of maximizing the achievable sum-rate in multi-user networks. We propose efficient solutions based on the sum-rate maximization task for fully-digital, (phase shifters based-) hybrid and DMA architectures. Simulation results show the feasibility of the proposed beam focusing scheme for both single- and multi-user scenarios. In particular, the designed focused beams are such that users residing at the same angular direction can communicate reliably without interfering with each other, which is not achievable using conventional far-field beam steering.

I. INTRODUCTION

High-frequency, large-aperture systems often operate in the near field, where spherical wavefronts enable location-specific beam focusing. The paper studies this capability for multi-user downlink MIMO with fully-digital, hybrid, and DMA architectures.

  • Motivation: Near-field spherical wavefronts enable beam focusing at specific locations rather than only steering toward directions.This may support multiple coexisting orthogonal links for users at similar angles.
  • Research gap: Near-field communication remains insufficiently studied from a communication perspective, especially for practical massive MIMO downlinks.Prior work largely characterized antenna behavior or considered limited point-to-point, single-user, uplink, or ideal-architecture settings.
  • Contributions: The paper studies focused-beam formation and multi-user downlink effects across fully-digital, phase-shifter-based hybrid, and DMA architectures.It quantifies architecture capabilities and their effects on downlink multi-user systems.
  • Approach: The authors formulate near-field multi-user MIMO channels and transmission models, then optimize beam focusing for sum-rate under each antenna architecture.The fully-digital design is used as a baseline for deriving the hybrid setting.
  • Results: Accounting for near-field characteristics yields notable achievable sum-rate gains over designs assuming conventional far-field operation.The resulting beam patterns are fundamentally different from far-field patterns.
  • Results: Simulations show that all proposed architectures concentrate transmissions at desired focal points and support reliable simultaneous communication for users sharing an angular direction.The latter capability is described as unavailable with conventional beam steering.

II. SYSTEM MODEL

The system model describes radiative near-field operation, its distance boundaries, and the resulting need to account for spherical wavefronts. It then frames transmission-pattern design as sum-rate maximization for downlink multi-user MIMO.

  • System model: The paper models downlink near-field multi-user MIMO channels, transmission patterns, antenna architectures, and sum-rate-maximizing beam-pattern design.The system-model development covers near-field transmission, channel modeling, signal models, and optimization formulation.
  • A. Near-Field Region: The radiative near-field is the Fresnel region between the Fraunhofer distance dF and Fresnel distance dN.Distances beyond dF are treated as far field, while dN marks the minimum distance for reactive-field components from the antenna itself.
  • A. Near-Field Region: For large arrays at mmWave frequencies, the planar-wave approximation can fail and spherical wavefronts must be modeled.The near-field region can extend over practical communication distances.
  • A. Near-Field Region: For an antenna with D = 0.5 meters at 28 GHz, users closer than 47 meters reside in the near field.The passage presents this as an example of the near-field operating range.
  • A. Near-Field Region: The model motivates exploiting spherical wavefronts to generate focused beams and improve communication rates by alleviating multi-user interference.This connects near-field propagation characteristics to downlink performance objectives.

B. Near-Field Channel Model

The paper models near-field multi-user MIMO channels using spherical-wave propagation and position-dependent channel responses, then relates these responses to beam focusing and antenna architecture.

  • B. Near-Field Channel Model: The base station uses a uniform planar array with N = N_d × N_e elements arranged across horizontal and vertical directions.Element coordinates are represented as p_i,l = (x_l, y_i, 0).
  • B. Near-Field Channel Model: For three receivers sharing an angular direction, far-field beam steering produces notable interference, whereas near-field focusing produces minor interference.The figure compares dedicated beams using far-field steering and near-field focusing.
  • B. Near-Field Channel Model: Near-field operation occurs between the Fresnel limit d_N and Fraunhofer distance d_F, where spherical waves enable focused-beam generation.The receivers’ positioning information is assumed known at the base station.
  • B. Near-Field Channel Model: Each user’s received signal combines element emissions weighted by distance-dependent phase, channel gain, radiation profile, and additive white Gaussian noise.The wave number is k = 2π/λ, and the phase term captures propagation distance.
  • B. Near-Field Channel Model: Unlike far-field beam steering, near-field channel-vector diversity permits focusing toward a spatial position rather than only a direction.In the far field, element responses share approximately equal path loss and a constant phase gradient.
  • B. Near-Field Channel Model: The study considers fully-digital, phase-shifter-based hybrid, and dynamic metasurface antenna architectures with different signal-processing capabilities.These architectures determine how the transmitted signal and resulting beam pattern are formed.
  • 1) Fully-digital antenna:: Fully-digital precoding gives each antenna element a dedicated RF chain, providing flexible processing but typically increasing massive-MIMO cost.The achievable user rate is computed while treating interference as noise.

2) Phase shifter based hybrid antenna:

The hybrid and DMA architectures constrain how digital signals reach the antenna elements, requiring distinct signal models for phase-shifter networks and microstrip-fed metasurface elements.

  • 2) Phase shifter based hybrid antenna:: Hybrid antennas use N_RF RF chains, fewer than the N antenna elements, with each RF-chain output connected through a phase-shifter network.The analog matrix Q maps the N_RF-dimensional digital vector to the N antenna elements.
  • 2) Phase shifter based hybrid antenna:: Hybrid analog precoding imposes a unit-modulus constraint on the elements of Q.This constraint contributes to the non-convexity of the hybrid design problem.
  • 2) Phase shifter based hybrid antenna:: The hybrid transmitted precoder is ˜w_m = Qw_m, and its achievable user rate is evaluated using the fully-digital rate expression.The digital vector has dimension N_RF × 1, while Q has dimension N × N_RF.
  • 3) DMA:: DMAs place sub-wavelength-spaced metamaterial radiators on multiple microstrips, enabling dense element packing and externally adjustable element responses.Each microstrip is fed by one RF chain, and all its elements radiate the input signal.
  • 3) DMA:: For a DMA, each element emits s_i,l = h_i,l q_i,l z_i, combining microstrip propagation, tunable element response, and microstrip input.The model assumes frequency-flat element responses and a Lorentzian-constrained phase response.
  • 3) DMA:: The DMA architecture is illustrated through a microstrip physical model and its corresponding mathematical signal-transmission model.The illustration shows multiple elements associated with microstrip-based signal transmission.
  • 3) DMA:: The DMA output is s = HQz, where H captures fixed microstrip propagation and Q contains configurable weights.The equivalent precoder for user m is HQw_m, whose achievable rate is computed using the same rate model.

D. Problem Formulation

The paper formulates near-field multi-user precoding as achievable-sum-rate maximization under architecture-specific constraints, then develops optimization procedures for fully-digital, hybrid, and DMA systems.

  • D. Problem Formulation: D. Problem Formulation: The target is to maximize achievable sum-rate for users with similar directions but different distances from the base station.The objective represents the total reliably conveyed bits per channel use under transmit power P_max.
  • D. Problem Formulation: D. Problem Formulation: The feasible precoding set captures architecture-specific constraints for fully-digital, hybrid, and DMA transmitters.Hybrid designs impose unit modulus, while DMA designs impose Lorentzian constraints on non-zero Q entries.
  • D. Problem Formulation: D. Problem Formulation: Near-field effects enter through the equivalent channel vectors rather than an explicit near-field term in the optimization objective.The formulation resembles far-field sum-rate problems while retaining near-field channel information.
  • D. Problem Formulation: D. Problem Formulation: Maximizing near-field sum-rate yields focused beams that mitigate interference between users sharing an angular direction.The objective does not explicitly constrain beam shape, yet the resulting beams become focused.
  • A. Fully-Digital Beam Focusing: A. Fully-Digital Beam Focusing: The fully-digital feasible set is unconstrained, and the reformulated problem introduces auxiliary variables while preserving the global optimum.The resulting block-coordinate procedure is concave in each variable block when the other two blocks are fixed.
  • A. Fully-Digital Beam Focusing: A. Fully-Digital Beam Focusing: Alternating optimization solves the reformulated fully-digital problem, with transmit-power control handled through a Lagrangian multiplier.The multiplier can be selected using methods such as bisection.
  • A. Fully-Digital Beam Focusing: A. Fully-Digital Beam Focusing: The method yields focused beams that allow users at the same angle to coexist with minimal cross-interference.For M > 1, the original problem is non-convex; weighted sum-MSE methods provide convergence to a stationary point.

B. Beam Focusing via Phase-Shifters Based Hybrid Precoding

The hybrid beam-focusing design alternates between digital and analog precoding, approximating a fully-digital solution while respecting unit-modulus phase-shifter constraints.

  • B. Beam Focusing via Phase-Shifters Based Hybrid Precoding: The hybrid problem is non-convex because its optimization variables are coupled and the analog precoder has unit-modulus constraints.Alternating optimization separates updates of the digital and analog components.
  • B. Beam Focusing via Phase-Shifters Based Hybrid Precoding: The hybrid design seeks a phase-shifter-based mapping whose effective precoder is close to the fully-digital solution in Frobenius norm.The unconstrained target ˜W_opt is obtained from the fully-digital design.
  • B. Beam Focusing via Phase-Shifters Based Hybrid Precoding: For fixed Q, the digital precoder W minimizing the surrogate objective is obtained as a least-squares solution.This update is stated in Lemma 2.
  • B. Beam Focusing via Phase-Shifters Based Hybrid Precoding: For fixed W, vectorizing Q converts the analog update into optimization over a product of L complex circles.The resulting search space is a Riemannian submanifold of C^L.
  • B. Beam Focusing via Phase-Shifters Based Hybrid Precoding: The analog update uses a Riemannian conjugate-gradient method with tangent-space gradients, Armijo steps, and retraction.Retraction maps the update back onto the complex-circle manifold.
  • B. Beam Focusing via Phase-Shifters Based Hybrid Precoding: The overall hybrid procedure initializes Q and W, obtains a fully-digital target, alternates analog and digital updates, and outputs Q_j and W_j.The digital update uses Lemma 2 after each analog update.

C. DMA-Based Beam Focusing

The DMA beam-focusing design addresses a non-convex sum-rate problem by first relaxing Lorentzian element constraints and then projecting the solution back onto the feasible set. This approach yields accurate focused beams and effective single-user DMA weights.

  • Problem formulation: DMA weight configuration is formulated to maximize the near-field multi-user sum-rate, but joint optimization of DMA weights and digital precoding is non-convex.The design first examines the single-user case before extending to multiple users with an alternating algorithm.
  • Problem formulation: Lorentzian constraints couple each element’s amplitude and phase, making the original optimization difficult to solve.The feasible Lorentzian responses lie on a circle in the upper half of the complex plane.
  • Single-user design: The method relaxes Lorentzian weights to constant-amplitude, arbitrary-phase weights, solves the relaxed problem, and projects the result back onto the Lorentzian set.The projection maps phase-only weights to Lorentzian-constrained weights as illustrated in Fig. 5.
  • Single-user design: The relaxed single-user problem remains non-convex but admits a closed-form solution for the coupled design variables.The resulting phase profile combines near-field focusing with compensation for microstrip transmission delay, synchronizing the signals.
  • Single-user design: The projected DMA weights produce accurate focused beams and provide an effective solution for the single-user case.The paper reports that numerical results verify the accuracy of the relaxation and projection approach.

2) Multi-user case:

For multiple users, DMA precoding is optimized through alternating updates of the digital precoder and DMA weights. The weight update is reduced to element-wise scalar optimization solved numerically.

  • Multi-user optimization: The multi-user DMA problem optimizes the DMA matrix and digital precoding vectors in an alternating manner.The procedure iterates between the two variable blocks until convergence.
  • DMA-weight update: With fixed digital precoders, the DMA-weight subproblem is reformulated into an equivalent optimization problem.The reformulation uses auxiliary variables and removes zero elements from the relevant vectors.
  • DMA-weight update: The relaxed multi-user formulation has the same optimal value as the corresponding equivalent problem, and its solution can be recovered from that problem.Recovery follows the structural relation specified for the DMA matrix.
  • DMA-weight update: Alternating optimization updates each nonzero DMA element separately while keeping the remaining elements fixed.Each resulting subproblem is a single-variable optimization over the element phase.
  • Algorithm: The single-variable phase problems are solved with numerical techniques such as one-dimensional search, forming Algorithm 3.Algorithm 3 alternates digital-precoder and DMA-weight updates for M > 1.

D. Discussion

The discussion compares antenna architectures for near-field sum-rate optimization, emphasizing the interference-management benefits of beam focusing and the scalability–performance trade-offs of DMAs. It also identifies frequency-selective and fully connected DMA extensions as future directions.

  • Scope of beam design: The study does not explicitly design transmission beams, although numerical results show that focused beams are generated.Near-field focusing is implicitly represented through the channel-related vectors in the objective.
  • Communication benefits: Near-field beam focusing enables users with identical angles to achieve high rates with little interference.This multi-user separation is a consequence of optimizing the achievable sum-rate in the near-field setting.
  • Architecture trade-offs: The fully-digital architecture is expected to achieve the largest sum-rates for a fixed UPA with fixed element placement.Its flexibility comes at the cost of assigning a dedicated RF chain to each antenna element.
  • Architecture trade-offs: DMAs are the most scalable architecture in cost and power efficiency, but their Lorentzian elements make design more challenging because gain and phase are coupled.The paper addresses this challenge by optimizing elements separately in an alternating procedure.
  • Beam focusing: For a given antenna aperture, sub-wavelength DMA spacing allows more elements and can produce the most focused single-user beams among the considered architectures.The paper states that numerical results demonstrate this single-user focusing advantage.
  • Future directions: Frequency-selective DMA responses could enable frequency-variant beam-focusing patterns for high-rate wideband communications with many users.These capabilities are unavailable in conventional phase-shifter-based hybrid architectures.
  • Limitations and extensions: The partially connected one-dimensional microstrip DMA architecture incurs performance loss relative to fully connected analog combiners in multi-user scenarios.Two-dimensional waveguides could provide a fully connected model, but the paper leaves this extension for future work.

IV. NUMERICAL EVALUATIONS

Numerical evaluations compare near-field beam focusing with beam steering across fully-digital, hybrid, and DMA architectures. Focusing improves rates near target points, while antenna aperture and element spacing materially affect performance.

  • Single-User Scenario: Across all three antenna architectures, near-field focusing increases signal strength and achievable rate near the focusing point.The improvement is observed when the user is located near z = 0.1dF.
  • Single-User Scenario: At the same angular direction but a different distance, focusing reduces observed radiation and rates relative to far-field beam steering.This demonstrates spatial selectivity along the distance dimension.
  • Single-User Scenario: DMA achieves higher rates than fully-digital and hybrid antennas at the near-field focusing point.The paper attributes this to DMA placing more antenna elements within the same area through smaller element spacing.
  • Single-User Scenario: Hybrid antennas achieve the same performance as fully-digital antennas in the evaluated single-user setting.The result is consistent with far-field findings under sufficient RF-chain resources.
  • Single-User Scenario: As antenna length L increases, achievable rate substantially improves for all three architectures.Larger arrays both expand the near-field region and enhance signal strength at the near-field focusing point.
  • Single-User Scenario: DMA performance depends on element spacing: 0.2λ spacing exceeds fully-digital and hybrid antennas, whereas 0.5λ spacing performs worse.The difference is associated with antenna densification and the resulting superdirectivity effect.

B. Multi-User Scenario

The multi-user evaluations test whether near-field focusing can distinguish users sharing an angular direction. Focused beams peak near their assigned focal points with limited interference, unlike conventional far-field steering.

  • Multi-User Scenario: For both users, peak achievable rates occur around their corresponding focal points across all studied antenna architectures.The focused beams support reliable communication with minimal interference-related degradation.
  • Multi-User Scenario: In the near-field two-user scenario, fully-digital and hybrid antennas achieve higher rates than the DMA architecture.The evaluated DMA uses one-dimensional microstrips, producing a partially-connected analog combiner and performance loss relative to fully-connected combiners.
  • Multi-User Scenario: Normalized signal-power maps show maxima near each user’s focusing point and minima near the other user’s focusing point.This verifies enhanced desired signal strength and reduced co-channel interference even for users sharing an angular direction.
  • Multi-User Scenario: For small M, all three architectures show monotonically increasing achievable sum-rate as users are added.The experiments successively add randomly located users in the near-field xz-plane.
  • Multi-User Scenario: As M increases, fully-digital and DMA sum-rate growth slows, while hybrid sum-rate decreases when M ≥8.The slowdown is attributed to co-channel interference, and the hybrid limitation to its fixed number of RF chains.

APPENDIX

The appendix derives optimal digital precoding and weighting designs by reformulating the objectives and decomposing the resulting maximization into tractable subproblems.

  • For any fixed weighting matrix Q, maximum-power maximal-ratio transmission is the optimal digital precoding vector.
  • Replacing Q with F yields the corresponding optimization problem for the digital precoder.
  • The non-convex constraint on Q is removed by rewriting the objective function.
  • The weighting maximization decomposes into Nd identically structured subproblems, each designing one microstrip's weighting coefficients.
  • The phase-design solution follows from the triangle inequality and assigns ψ* according to βiρi,l for each i and l.
  • The derivation uses vectorization identities and removes zero elements of q because they do not affect the objective.
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