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Power Scaling Laws and Near-Field Behaviors of Massive MIMO and Intelligent Reflecting Surfaces

Emil Björnson, Luca Sanguinetti

arXiv:2002.04960v5cs.ITeess.SP

TL;DR

Large-array SNR and power-scaling results have mainly been derived under far-field assumptions, leaving their validity for asymptotically large arrays unresolved. The paper uses a deterministic planar-array model with near-field effects to revisit these laws and comparisons. It finds finite asymptotic limits, an equal-size SNR disadvantage for IRSs, and concave-mirror focusing by optimized IRSs.

  • Problem

    Existing SNR and power-scaling analyses for mMIMO and IRSs assume far-field operation, which cannot describe the asymptotic regime where array size grows without bound.

  • Method

    The paper derives and applies a deterministic closed-form channel-gain model for arbitrary planar arrays that includes distance variation, effective areas, and polarization mismatches.

  • Results

    Equal-sized IRSs cannot achieve higher SNR than the two mMIMO setups; far-field gains eventually taper to finite limits, with channel gain converging to 1/3 for mMIMO and bounded by 1/9 for IRSs.

  • Takeaways & Limitations

    An IRS may need a larger surface to outperform mMIMO, while an optimized IRS can focus signals like a concave mirror rather than acting as an anomalous plane mirror.

  • Takeaways & Limitations

    The asymptotic analysis applies to fixed transmitter, receiver, and array locations in the considered deterministic propagation scenarios.

Abstract

from arXiv · show

The use of large arrays might be the solution to the capacity problems in wireless communications. The signal-to-noise ratio (SNR) grows linearly with the number of array elements $N$ when using Massive MIMO receivers and half-duplex relays. Moreover, intelligent reflecting surfaces (IRSs) have recently attracted attention since these can relay signals to achieve an SNR that grows as $N^2$, which seems like a major benefit. In this paper, we use a deterministic propagation model for a planar array of arbitrary size, to demonstrate that the mentioned SNR behaviors, and associated power scaling laws, only apply in the far-field. They cannot be used to study the regime where $N\to\infty$. We derive an exact channel gain expression that captures three essential near-field behaviors and use it to revisit the power scaling laws. We derive new finite asymptotic SNR limits but also conclude that these are unlikely to be approached in practice. We further prove that an IRS-aided setup cannot achieve a higher SNR than an equal-sized Massive MIMO setup, despite its faster SNR growth. We quantify analytically how much larger the IRS must be to achieve the same SNR. Finally, we show that an optimized IRS does not behave as an "anomalous" mirror but can vastly outperform that benchmark.

I. INTRODUCTION

Large arrays motivate mMIMO and IRS because their far-field SNR scaling suggests substantial power savings. This paper argues that those laws fail for asymptotically large arrays, requiring near-field-aware channel modeling and revised scaling analysis.

  • Motivation: mMIMO uses many active antennas to provide array gains and spatial multiplexing, while passive IRS elements shape additional propagation paths.mMIMO arrays are described as having at least 64 antennas in 5G, whereas IRS elements adjust phase and polarization to reflect signals toward a destination.
  • Motivation: SNR scales as N for optimally beamformed mMIMO and as N^2 for an optimally configured IRS in the far-field.These respective behaviors motivate transmit-power reductions proportional to 1/N and 1/N^2 for maintaining a target SNR.
  • Problem: Far-field power-scaling laws assume approximately equal directions and channel gains across the array, an assumption that breaks down as array size grows.The paper distinguishes this array far-field from the Fraunhofer distance of an individual antenna element.
  • Problem: Near-field analysis requires element-wise modeling because varying incident directions change polarization mismatch across large arrays.The conference version used an approximate model that neglected polarization effects, whereas the present analysis accounts for them in the near-field.
  • Contributions: The paper derives an exact arbitrary-size planar-array channel gain and uses it to establish near- and far-field behavior for mMIMO, relaying, and IRS communications.The analysis revises power-scaling laws, compares equal-sized IRS and mMIMO setups, and derives expressions for the IRS size needed to exceed competing systems.
  • Organization: The paper is organized around propagation preliminaries, planar-array channel modeling, asymptotic limits, and spectral-efficiency analysis for three relay or array setups.It begins with free-space propagation and later defines conventional mMIMO, half-duplex mMIMO relaying, and IRS-aided communications.

III. PLANAR ANTENNA ARRAYS

The planar-array model accounts for geometry, effective antenna area, and polarization in the near-field. It yields a channel-gain expression applicable to arbitrarily large arrays and clarifies when far-field approximations cease to be reliable.

  • Array model: The planar array is modeled as N equally spaced, edge-to-edge antennas with individual area A and total area NA.The assumptions restrict each antenna to A ≤ (λ/4)^2 and, in the stated construction, N to a square grid.
  • Near-field effects: Near-field propagation requires accounting for varying element distances, effective antenna areas, and polarization-mismatch losses.These quantities vary because array elements receive signals from different angular directions and distances.
  • Channel formulation: The general planar-array formulation computes channel gains for arbitrarily located transmitters and receivers across the array elements.It extends prior work by providing a way to calculate the gain associated with each element.
  • Channel formulation: The channel-gain upper bound is tight when each receive antenna is sufficiently small compared with the wavelength, specifically a ≤ λ/4.The lemma assumes negligible phase variation over an antenna area and is used later for IRS analysis.
  • Exact channel gain: Unlike equal-effective-area models, the new channel-gain expression supports near-field operation and asymptotically large planar arrays.The formulation is used to examine both the far-field approximation and the large-array limit.
  • Exact channel gain: The exact expression depends on total area through Nβ_d, so equivalent total array areas have the same modeled behavior across frequency and element sizes.Reducing wavelength shrinks the allowed individual antenna area, requiring more elements to fill the same total area.

B. Far-field Approximation and Large-array Limit

The far-field approximation accurately predicts received-power scaling for practical array sizes, but the exact channel-gain expression is required for near-field and asymptotic analysis. As arrays become very large, finite limits arise from changing distance, effective area, and polarization effects.

  • For relatively small planar arrays, the received power is proportional to N.
  • The model assumes d ≫ λ and therefore excludes the reactive near-field of the transmitting antenna, even when the array itself is in the near-field.
  • 105 antennas are needed before the far-field approximation error exceeds 5%, while 108 antennas approach the upper limit of 1/3 for d = 25 m and λ = 0.1 m.The setup uses A = (λ/4)^2 and f = 3 GHz.
  • The far-field approximation is accurate when 9NA ≤ d^2, equivalently when the distance is approximately three times the array width or height.
  • For d = 25 m, the approximation applies to arrays up to 8.3 × 8.3 m; the permissible antenna count grows quadratically with distance or carrier frequency while area remains constant.
  • Including distance, effective-area, and polarization variations yields a finite asymptotic channel gain, whereas omitting effective-area variation causes divergence and violates energy conservation.Neglecting polarization alone gives a limit of 1/2, but not the correct channel-gain value.

IV. THREE DIFFERENT MIMO SETUPS

The paper compares conventional uplink mMIMO, half-duplex mMIMO relaying, and IRS-aided communication using deterministic LoS planar-array models. Perfect channel state information is assumed, and maximum-ratio processing is used where described.

  • Conventional uplink mMIMO: The conventional mMIMO setup uses a single-antenna source transmitting to a planar receive array with N antennas.
  • Half-duplex mMIMO relay: The half-duplex mMIMO relay receives from the source and retransmits to a single-antenna destination using two equal-duration phases.The relay uses decode-and-forward repetition coding, with no direct link.
  • IRS-aided communication: The IRS-aided setup replaces the planar array with an IRS of N passive elements operating as a full-duplex relay.
  • All three setups use line-of-sight propagation and deterministic channels with perfect channel state information.
  • Maximum-ratio combining maximizes the conventional mMIMO receiver SNR, while maximum-ratio precoding maximizes the relay's second-phase SNR.
  • Half-duplex mMIMO relay: The relay end-to-end spectral efficiency is determined by the minimum SNR across its two phases and includes a one-half pre-log factor.

C. IRS-aided Communication

The mMIMO channel model yields linear SNR growth only in the far-field; as arrays grow, near-field channel gain saturates and any transmit-power scaling to zero drives asymptotic SNR to zero.

  • Exact channel model: The exact mMIMO channel gain accounts for varying element distances, polarization mismatches, and effective areas across an arbitrarily sized planar array.The resulting expression depends on the total array area, while frequency determines how many antennas are needed to realize that area.
  • Far-field behavior: In the far-field, the mMIMO SNR is proportional to N, supporting the conventional 1/N transmit-power scaling law.The approximation requires d cos(η) ≫ NA, where NA is the array width or height.
  • Near-field behavior: As N →∞ with constant transmit power, the mMIMO channel gain saturates in the near-field rather than sustaining unbounded far-field SNR growth.The far-field approximation eventually breaks down as the array size increases.
  • Power scaling: Any scaling Ptx = P/N^ρ with ρ > 0 causes asymptotic mMIMO SNR to converge to zero, although such laws remain useful in practical array sizes.For the illustrated setup, ρ = 1 keeps SNR approximately constant for N ≤10^6, while larger N leads to zero SNR whenever ρ > 0.
  • Practical range: For d = 25 m and η = 0, far-field scaling remains accurate up to N ≤10^6; at d = 2.5 m, it remains accurate up to N ≤10^4.These correspond to arrays up to approximately 25×25 m and 2.5×2.5 m, respectively.

B. Half-Duplex mMIMO Relay

The half-duplex mMIMO relay retains linear end-to-end SNR growth and 1/N power scaling in the far-field, but its asymptotic behavior is bounded by near-field channel-gain saturation.

  • Relay setup: The relay geometry places the source and destination relative to a planar transmit-and-receive array, with propagation delays determining the element phases.The source and destination may have distances and angular positions specified separately from the array center.
  • Far-field behavior: In the far-field, the relay end-to-end SNR grows proportionally to N when both source and destination satisfy d cos(η) ≫ NA.The corresponding approximation is obtained by combining the uplink and downlink far-field results.
  • Power scaling: Far-field operation permits reducing both transmit powers, Ptx and Prelay, as 1/N while maintaining the SNR achieved with N = 1.This practical scaling result applies when the far-field approximation remains valid.
  • Asymptotic behavior: As N →∞, the relay’s near-field channel gain approaches a finite limit, so power scaling that drives transmit powers to zero yields zero asymptotic spectral efficiency.The paper states that the total channel gain is upper bounded by one.

C. IRS-aided Communication

The IRS analysis derives an upper bound and compares IRS-aided communication with mMIMO and relaying across near-field and far-field regimes. Although IRS SNR can grow quadratically with N in the far-field, equal-sized mMIMO can achieve higher SNR, and near-field limits constrain asymptotic scaling.

  • IRS SNR bound: The optimized IRS SNR is upper bounded by the corresponding mMIMO SNR multiplied by the IRS-to-destination channel gain.Equality requires the element-wise magnitude vectors of the source-to-IRS and IRS-to-destination channels to be parallel.
  • IRS SNR bound: An equal-sized IRS cannot achieve a higher SNR than the corresponding mMIMO setup with the same transmit power.The comparison follows from the channel-gain factor being below one, or asymptotically 1/3, under energy conservation.
  • Comparison with relaying: At sufficiently high SNR, the IRS-aided setup can outperform a half-duplex mMIMO relay because the relay incurs a 1/2 pre-log factor.This spectral-efficiency advantage can occur even when the IRS SNR is lower.
  • Far-field behavior: In the far-field, IRS SNR grows as N^2 because the SNR is proportional to the square of a sum with N terms.The far-field condition requires both source and destination distances to exceed the array dimensions sufficiently.
  • Far-field behavior: The IRS total channel gain grows as N^2 while the mMIMO receiver gain grows as N in the illustrated setup.The example uses d = 25 m, δ = 2.5 m, λ = 0.1 m, and places the destination near the IRS.
  • Near-field behavior: The far-field IRS approximation remains accurate to roughly 10^4 elements, while the proposed upper bound stays close to the exact curve beyond 10^4 elements.This supports using the bound to characterize larger-array near-field behavior.
  • Near-field behavior: As N →∞, the IRS SNR is asymptotically upper bounded, and conventional power scaling laws yield zero asymptotic spectral efficiency.For practical array sizes, IRS SNR may still grow as N^2, allowing transmit power reduction as 1/N^2 while maintaining constant SNR.

VI. HOW LARGE IRS IS NEEDED TO ACHIEVE THE SAME SNR?

The section quantifies how many IRS elements are needed to match the spectral efficiency of mMIMO receivers and half-duplex mMIMO relays. IRS scaling can become more favorable at high rates, but the IRS remains larger than the mMIMO array in the example.

  • Analytical comparison: Far-field conditions determine when the IRS provides higher spectral efficiency than an mMIMO receiver or a half-duplex mMIMO relay.The thresholds follow from comparing the far-field expressions for the three setups.
  • Numerical comparison: The IRS needs more than 100 elements before its spectral efficiency is clearly above zero in the illustrated example.After this point, its required element count grows more gradually with spectral efficiency than the relay and mMIMO cases because its SNR grows as N^2.
  • Numerical comparison: Only above 4.4 bit/s/Hz does the IRS require fewer elements than the half-duplex relay in this example.The crossover depends on the stated simulation parameters and equal transmit powers Ptx = Prelay.
  • Numerical comparison: The IRS must always be larger than the mMIMO array to deliver the same spectral efficiency in the example.NmMIMO = 100 delivers 3 bit/s/Hz, whereas approximately NIRS = 4000 elements are needed for the same rate.
  • Physical-size implications: At 3 GHz, the example corresponds to a 0.25 × 0.25 m mMIMO receiver and a 1.6 × 1.6 m IRS for the same 3 bit/s/Hz rate.The physical-size difference reduces asymptotically but does not vanish.
  • Physical-size implications: IRS physical thinness, integration potential, and possible cost and energy advantages may offset its larger area, but its cost and energy consumption remain unquantified.Real-time channel estimation and reconfigurability may dominate implementation cost and energy use.

VII. GEOMETRIC INTERPRETATION OF OPTIMIZED IRS

An optimized IRS is geometrically equivalent to a concave mirror that focuses signals at the destination, rather than a plane mirror redirecting them by angle alone. In the near-field, this optimized focusing can substantially outperform the mirror-mimicking benchmark.

  • Geometric interpretation: The optimized configuration differs from the anomalous-mirror model because its asymptotic distance dependence uses (d + δ)^2 rather than separate factors for d and δ.This distinction shows that SNR optimization does not produce plane-mirror behavior in the large-array near-field.
  • Geometric interpretation: An optimized IRS synthesizes scattering from a concave mirror whose curvature focuses the incoming wave at the destination.The phase shifts determine the SNR-maximizing curvature without physically changing the IRS shape.
  • Numerical comparison: At N = 10^4, the optimized IRS has a 500-times better channel gain than the mirror limit in (54).The mirror-mimicking gain begins converging near N = 360, while the optimized gain continues increasing.
  • Numerical comparison: The mirror analogy is approximately valid for small IRSs in the far-field, where focusing on a distant point resembles reflecting toward the same angular direction.This approximation becomes unreliable in the near-field as the array grows.
  • Geometric interpretation: A mirror-mimicking IRS can use only a limited array area; beyond the area in (55), additional elements effectively scatter signals in other directions.The useful area depends on wavelength and the source and destination geometry.

A. Reconfigurability Under Mobility

Under destination mobility, optimal IRS focusing requires reconfiguration for each destination distance, whereas fixed-focus and mirror-mimicking configurations avoid reconfiguration at the cost of lower or distance-dependent gain.

  • A. Reconfigurability Under Mobility: The SNR-maximizing IRS uses the destination distance δ and requires a different phase configuration as the destination moves.The mirror-mimicking alternative uses only the angle ω and therefore need not be reconfigured along a constant-angle trajectory.
  • A. Reconfigurability Under Mobility: For N = 10^4, the optimal configuration has the highest channel gain across δ ∈ [1, 100] m, while fixed-focus curves intersect it only at their chosen focus distances.The setup uses d = 25 m, A = (λ/4)^2, and λ = 0.1 m.
  • A. Reconfigurability Under Mobility: A 5 m focus provides array gain mainly when the destination is nearby, whereas mirror-mimicking performs better at larger distances.A 25 m focus gives gain larger than or approximately equal to mirror-mimicking over the reported range.

VIII. EXTENSION TO GENERAL PROPAGATION SETUPS

The paper’s conclusions extend across propagation setups but remain grounded in a deterministic free-space line-of-sight model. Near-field saturation invalidates unbounded far-field scaling, while optimized IRS focusing differs fundamentally from plane-mirror reflection.

  • VIII. Extension to general propagation setups: The analysis is derived for free-space line-of-sight propagation, where near-field behavior occurs when array dimensions are comparable to propagation distances.Additional scattered paths and partial line-of-sight blockage are outside the directly derived setting.
  • VIII. Extension to general propagation setups: In the far-field, total channel gain grows as N for both mMIMO setups and as N^2 for IRS-aided communication.These classical behaviors remain accurate for many practical deployments with thousands of elements.
  • VIII. Extension to general propagation setups: As N → ∞ in the near-field, channel gain converges to 1/3 for mMIMO and is upper bounded by 1/9 for IRS.Near-field behavior begins when array width or height is comparable to, or larger than, the distance to the terminals.
  • VIII. Extension to general propagation setups: Any power-scaling law that drives transmit power asymptotically to zero also drives asymptotic spectral efficiency to zero.This follows from the finite near-field channel-gain limits.
  • VIII. Extension to general propagation setups: For equal N, IRS always has lower SNR than the two mMIMO setups, but a larger IRS can exceed their SNR after an analytically characterized breaking point.The IRS reflection loss explains why its far-field N^2 growth does not imply superiority at equal size.
  • VIII. Extension to general propagation setups: An optimized IRS synthesizes a concave focusing mirror and can achieve near-field SNRs an order of magnitude above the plane-mirror limit.Its SNR contains the product of the source-to-IRS and IRS-to-destination channel gains.
  • VIII. Extension to general propagation setups: The asymptotic results rely on a deterministic channel model valid only for fixed transmitter, receiver, and array locations.Classical stochastic models do not capture the essential near-field propagation properties needed for this analysis.

APPENDIX A PROOF OF LEMMA 1

The appendix derives a planar-array channel-gain bound from electromagnetic propagation, retaining distance variation, effective-area reduction, polarization loss, and free-space path loss. A closed-form upper bound is then obtained and numerically validated for small elements.

  • APPENDIX A PROOF OF LEMMA 1: The proof first computes each antenna element’s channel gain using electromagnetic arguments, then derives a closed-form upper bound.The bound is obtained after evaluating the required spatial integral.
  • APPENDIX A PROOF OF LEMMA 1: The model uses the electric field and Green function generated by an elementary transmitting surface to form the complex channel to a receive point.The derivation specializes to excitation in the Y direction before defining the channel for an antenna element.
  • APPENDIX A PROOF OF LEMMA 1: For a ≤ λ/4, ζpt,pn tightly upper-bounds |hn(pt)|^2 with relative error below 1 dB in the reported numerical evaluation.The evaluation uses d = 10 m, f = 3 GHz, and receive offsets xn = 0, 5, and 10 m.
  • APPENDIX A PROOF OF LEMMA 1: The channel-gain integral explicitly includes distance-dependent effective area, polarization loss, and free-space path loss.A change of variables reduces the integral before closed-form evaluation.

APPENDIX B PROOF OF COROLLARY 3

The proof of Corollary 3 simplifies the relevant expression through small-argument approximations and trigonometric identities, yielding a compact form proportional to Nβd.

  • For small x, the proof uses tan⁻¹(x) ≈ x after approximating B + 1 ≈ 1 and 2B + 1 ≈ 1.
  • It approximates the denominator using √(1 + x) ≈ 1 + x/2 for x ≈ 0.
  • Using 1 + tan²(η) = 1/cos²(η), the resulting expression is approximately Bπ(1 + tan²(η))^3/2 = Nβd cos(η) cos³(η) = ζd,η.
  • The derivation combines two fractions into one and applies the approximation B tan(η)/(1 + tan²(η)) ≈ 1.
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