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Communicating with Extremely Large-Scale Array/Surface: Unified Modelling and Performance Analysis

Haiquan Lu, Yong Zeng

arXiv:2104.13162v1cs.IT

TL;DR

XL-array communications invalidate conventional sizeless-element and uniform-plane-wave assumptions because signal phase, amplitude, and projected aperture vary across large apertures. The paper develops a unified discrete-array/continuous-surface model and derives closed-form optimal single-user MRC/MRT SNR expressions. The resulting SNR grows with M with diminishing return, while far-field analysis recovers direction-dependent linear scaling and motivates the uniform-power distance criterion.

  • Problem

    Conventional XL-array models treat elements as sizeless points and often assume uniform plane waves, neglecting physically important phase, amplitude, and projected-aperture variations.

  • Method

    The paper proposes a unified 3D model for discrete arrays and continuous surfaces that explicitly represents element physical size, signal phase, amplitude, and projected aperture, then derives closed-form optimal single-user MRC/MRT SNR expressions.

  • Results

    The generic model predicts SNR growth with M with diminishing return governed by collective array properties, while far-field analysis recovers linear scaling that depends on AoA/AoD.

  • Takeaways & Limitations

    Accurate XL-array analysis requires modelling projected-aperture and amplitude variations, and separating near- and far-field behavior with UPD alongside direction-dependent Rayleigh distance.

Abstract

from arXiv · show

Wireless communications with extremely large-scale array (XL-array) correspond to systems whose antenna sizes are so large that conventional modelling assumptions, such as uniform plane wave (UPW) impingement, are longer valid. This paper studies the mathematical modelling and performance analysis of XL-array communications. By deviating from the conventional modelling approach that treats the array elements as sizeless points, we explicitly model their physical area/aperture, which enables a unified modelling for the classical discrete antenna arrays and the emerging continuous surfaces. As such, a generic array/surface model that accurately takes into account the variations of signal phase, amplitude and projected aperture across array elements is proposed. Based on the proposed model, a closed-form expression of the resulting SNR with the optimal single-user MRC/MRT beamforming is derived. The expression reveals that instead of scaling linearly with the antenna number M as in conventional UPW modelling, the SNR with the more generic model increases with M with diminishing return, which is governed by the collective properties of the array, such as the array occupation ratio and the physical sizes of the array along each dimension, while irrespective of the properties of the individual array element. Additionally, we have derived an alternative insightful expression for the optimal SNR in terms of the vertical and horizontal angular spans. Furthermore, we also show that our derived results include the far-field UPW modelling as a special case. One important finding during the study of far-field approximation is the necessity to introduce a new distance criterion to complement the classical Rayleigh distance, termed uniform-power distance (UPD), which concerns the signal amplitude/power variations across array elements, instead of phase variations as for Rayleigh distance.

I. INTRODUCTION

XL-array systems challenge conventional far-field and sizeless-element models because phase, amplitude, and projected-aperture variations become significant. The paper proposes a unified physical model for discrete arrays and continuous surfaces and derives performance insights under optimal beamforming.

  • Motivation: XL-array communications become relevant for B5G/6G as antenna sizes increase beyond conventional massive MIMO scales and may converge toward continuous surfaces.The paper uses XL-array communications as a general term encompassing related large-aperture architectures.
  • Motivation: Large arrays make users more likely to lie outside the far-field region, invalidating uniform plane-wave assumptions and motivating spherical-wave modelling.For D = 4 meters, the Rayleigh distance is 373.3 m at 3.5 GHz and 2986.7 m at 28 GHz.
  • Modelling gap: Sizeless-point models neglect element-dependent projected apertures, potentially producing physically impossible received powers exceeding transmit power.Different arrival angles across a large array cause different effective apertures normal to the incoming wave.
  • Proposed model: The proposed unified 3D model represents discrete arrays and continuous surfaces while accounting for phase, amplitude, physical element size, and projected-aperture variations.The model includes both zenith and azimuth arrival/departure angles and explicitly models each element's aperture.
  • Main results: Under optimal single-user MRC/MRT beamforming, SNR grows with antenna number M with diminishing return rather than the linear scaling predicted by UPW modelling.The scaling is governed by collective properties such as occupation ratio ξ and physical dimensions Ly and Lz, not individual element properties.
  • Far-field and geometry: Far-field approximation recovers linear SNR scaling with M but reveals dependence on signal direction, motivating UPD and direction-dependent Rayleigh distance criteria.The paper also derives angular-span expressions and studies projected-aperture effects for ULA architectures.

3) Non-uniform spherical wave (NUSW) model [28], [38]:

The NUSW model accounts for element-dependent propagation distances and resulting amplitude and phase variations. The paper contrasts this with models that assume uniform power and notes that projected-aperture variation remains unmodelled in the conventional alternatives.

  • NUSW model: The NUSW model assigns each array element an exact propagation distance to model both amplitude and phase variations across the array.These variations become important as array dimensions increase or link distance decreases.
  • Model limitation: Conventional UPW, USW, and NUSW models omit projected-aperture variation, which can lead to received-power predictions exceeding transmit power as array dimensions grow.This limitation motivates the more generic physical model.
  • Signal model: The received uplink signal combines the user transmission, channel response, and additive white Gaussian noise before linear receive beamforming.The receive beamformer v has unit norm, and the resulting SNR is then evaluated.
  • Beamforming: For single-user communication, maximum-ratio combining is the optimal linear receive beamformer.The beamformer is proportional to the array response vector and normalized by its norm.
  • Special case: At end-fire directions, projected apertures vanish and the resulting SNR is zero.The paper excludes these cases when analyzing the non-trivial operating regime.

III. CLOSED-FORM EXPRESSION AND PERFORMANCE ANALYSIS

The paper derives a closed-form optimal SNR for XL-array communication and interprets it through geometric angular spans and asymptotic array limits. The analysis shows saturation behavior and includes continuous surfaces as a special case.

  • Under r ≫d, Theorem 1 gives a closed-form SNR for single-user XL-array communication with optimal MRC/MRT beamforming.The expression is based on the proposed array response model.
  • The SNR depends on collective UPA properties, including occupation ratio ξ and physical dimensions Ly and Lz, rather than individual element size A or separation d.This follows when element separation is much smaller than link distance, allowing the element sum to be approximated by integration.
  • Geometric interpretation: The angular-span formulation expresses SNR through four geometric angles, with larger array dimensions increasing the corresponding spans and SNR for fixed user location.For increasing link distance, the angular spans decrease, producing smaller SNR.
  • Special cases: When A = d^2 and ξ = 1, the XL-array becomes a continuous surface, so the derived results include the continuous-surface result as a special case.
  • Asymptotic behavior: For an infinitely large array or surface, the SNR approaches a constant proportional to the array occupation ratio ξ rather than increasing unbounded as in conventional UPW models.Only the occupied fraction of the total transmitted power is captured; ξ = 1 recovers the continuous-surface result that half the isotropic-source power is captured.

IV. FAR-FIELD APPROXIMATION AND UNIFORM-POWER DISTANCE

The far-field analysis introduces the uniform-power distance (UPD) to complement Rayleigh-distance-based approximation, while retaining projected-aperture effects in the SNR expression.

  • The far-field analysis introduces UPD as a new distance criterion complementing the classical Rayleigh distance.UPD addresses amplitude/power variations across array elements rather than phase variations alone.
  • Under far-field conditions, optimal MRC/MRT SNR increases linearly with antenna number M.This scaling is consistent with conventional results but still depends on the total projected aperture MA sin θ cos φ.
  • Far-field SNR depends on AoA/AoD through the projected-aperture factor sin θ cos φ, even when all elements share a common direction.Highly inclined incident waves need not satisfy sin θ cos φ ≈ 1.
  • The general far-field model and conventional UPW model differ by sin θ cos φ, so ignoring projected aperture generally over-estimates SNR.

A. Uniform-Power Distance

The uniform-power distance is defined from the weakest-to-strongest element-power ratio, making it direction-dependent and distinct from the phase-based Rayleigh distance.

  • Rayleigh distance considers maximum phase difference across elements, whereas UPD accounts for amplitude/power differences that affect beamformed SNR.With optimally aligned signal phases, amplitude variations determine the resulting SNR.
  • UPD r_UPD(θ, φ) is the minimum distance at which the weakest-to-strongest element-power ratio Γ(r, θ, φ) reaches threshold Γ_th.The ratio Γ(r, θ, φ) increases with link distance.
  • Unlike the classical Rayleigh distance, UPD depends on direction (θ, φ) and generally forms a surface.
  • For the proposed channel power-gain model, the general UPD solution can be obtained in closed form, or numerically for any given array model.The closed-form expression is described as sophisticated and omitted.

B. Direction-Dependent Rayleigh Distance

The direction-dependent Rayleigh distance extends the classical phase-based criterion to account for signal direction and reduces to the classical distance in a special case.

  • The direction-dependent Rayleigh distance reflects how direction (θ, φ) affects phase variations across array elements.
  • r_ddRayl(θ, φ) is defined as the minimum link distance for which the maximum phase error Δφ(r, θ, φ) does not exceed π/8.
  • The generalized Rayleigh distance is generally difficult to solve in closed form but can be obtained numerically and forms a Rayleigh surface.
  • In the special direction case, the resulting expression is consistent with the classical Rayleigh distance.The consistency follows after applying a first-order Taylor approximation.

V. UNIFORM LINEAR ARRAY

For the ULA special case, the generic SNR depends on angular span and angular difference, while distance variation and projected aperture reduce edge-element contributions as angular span grows.

  • V. UNIFORM LINEAR ARRAY: The ULA case sets M_y = 1 and M = M_z, reducing the general closed-form SNR to a simpler expression.
  • V. UNIFORM LINEAR ARRAY: The ULA SNR depends on angular span Δ_span(M) and angular difference Δ_diff(M), which capture distinct geometric relationships to the user.As M → ∞, sin(α_1(M)) + sin(α_2(M)) approaches 2.
  • V. UNIFORM LINEAR ARRAY: The asymptotic ULA result is a constant depending on projected distance r sin θ and projected aperture A cos φ, differing from the UPA asymptotic limit.
  • V. UNIFORM LINEAR ARRAY: For an infinitely long one-dimensional ULA, the captured transmitted-power fraction differs from the two-dimensional UPA because its array aperture is much smaller.
  • V. UNIFORM LINEAR ARRAY: As zenith angle θ increases, the projected aperture of antenna elements increases, and the cosine term in the ULA expression reflects this directional variation.
  • V. UNIFORM LINEAR ARRAY: As angular span increases, normalized path loss and projected aperture decrease, reducing the channel-power contribution of ULA end elements relative to the center.Distance variation has a more significant impact than projected-aperture variation in the plotted scenario.

VI. NUMERICAL RESULTS

The numerical results compare array models for XL-array communications, showing that aperture-aware modelling changes SNR scaling and angular dependence, especially for large or inclined arrays.

  • A. Comparison of Different Array Models: For moderate M, all array models predict SNR increasing linearly with M.
  • A. Comparison of Different Array Models: As M grows, the proposed model approaches a constant SNR, whereas UPW/USW continue increasing linearly and unbounded.
  • A. Comparison of Different Array Models: The UPW/USW and NUSW models generally overestimate the proposed SNR, with larger gaps for inclined wave incidence.
  • A. Comparison of Different Array Models: NUSW also exhibits diminishing returns, but its asymptotic SNR can exceed the physically possible captured-power limit because projected aperture is ignored.
  • A. Comparison of Different Array Models: For a square UPA with M_y=M_z=400, UPW/USW SNR is angle-independent, while proposed and NUSW SNR vary with user zenith angle.
  • A. Comparison of Different Array Models: As θ increases, NUSW SNR decreases but proposed-model SNR increases because the proposed model includes both distance and projected-aperture variations.

B. UPD and Direction-Dependent Rayleigh Distance

This section evaluates UPD alongside classical and direction-dependent Rayleigh distances, showing that power and phase variations impose distinct, direction-sensitive far-field criteria.

  • B. UPD and Direction-Dependent Rayleigh Distance: UPD is evaluated for an M=Mz=64 ULA while varying θ at fixed φ=0, using both the proposed and NUSW models with Γ_th=90%.
  • B. UPD and Direction-Dependent Rayleigh Distance: The classical Rayleigh distance is conservative for far-field approximation from the phase-modelling perspective.
  • B. UPD and Direction-Dependent Rayleigh Distance: UPD is minimized at θ=π/2 and increases significantly away from that direction, indicating that inclined users are more likely to remain in the near field under UPD.
  • B. UPD and Direction-Dependent Rayleigh Distance: UPD and direction-dependent Rayleigh distance exhibit opposite angular trends, demonstrating the need to model both signal-power and phase variations.
  • C. Uniform Linear Array: For ULA, proposed and NUSW SNR predictions diverge beyond a defined 95% critical-point threshold, with larger deviations for inclined directions.
  • C. Uniform Linear Array: NUSW generally overestimates the proposed-model SNR because it ignores projected aperture, even for small numbers of array elements.
  • Conclusion: The paper concludes that unified aperture-aware modelling covers discrete arrays and continuous surfaces, yielding a closed-form single-user MRC/MRT SNR expression.

APPENDIX A PROOF OF THEOREM 1

Theorem 1 is proved by converting the SNR-related expression into a double integral, evaluating it across geometric cases, and substituting the result into the theorem’s SNR formula.

  • APPENDIX A PROOF OF THEOREM 1: The proof defines f(y,z), partitions the array aperture into M_yM_z equal subrectangles, and approximates the function within each subrectangle.
  • APPENDIX A PROOF OF THEOREM 1: The resulting Riemann-sum approximation is expressed as a double integral and evaluated using standard integral formulas.
  • APPENDIX A PROOF OF THEOREM 1: The alternative angular expression is derived by considering whether the user projection lies within or outside the relevant y-axis segment.
  • APPENDIX A PROOF OF THEOREM 1: For the remaining geometric cases, the proof changes the signs of tan η_1 or tan η_2 and obtains equivalent expressions before combining all cases.

APPENDIX C PROOF OF LEMMA 5

Lemma 5 is proved through small-parameter approximations, first-order Taylor expansions, and combination of the resulting terms.

  • APPENDIX C PROOF OF LEMMA 5: The proof applies arctan x≈x under a small-parameter condition and rewrites denominators using auxiliary variables.
  • APPENDIX C PROOF OF LEMMA 5: First-order Taylor approximations are applied to the denominator terms for small variables, producing an approximation for the full expression.
  • APPENDIX C PROOF OF LEMMA 5: The approximated terms are simplified using array-dimension relations, combined, and used to complete the proof of Lemma 5.

APPENDIX D PROOF OF LEMMA 6

The proof specializes the preceding expression, expands its bracketed terms through a small-ε function, and identifies two resulting terms using Fig. 5 before concluding Lemma 6.

  • The proof substitutes My = 1 and M = Mz into equation (12) as its initial specialization.
  • The first bracketed term is expressed as a function h(ε) and analyzed for small ε.
  • The same procedure is applied to the other three terms inside the bracket.
  • Fig. 5 identifies two terms in the bracket as the sines of α1(M) and α2(M), respectively.
  • The proof concludes after these substitutions, expansions, and identifications.
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