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Beyond Massive-MIMO: The Potential of Data-Transmission with Large Intelligent Surfaces

Sha Hu, Fredrik Rusek, Ove Edfors

arXiv:1707.02887v1cs.IT

TL;DR

The paper addresses the lack of analysis of information-transfer capabilities for electronically active structures by taking a first look at uplink capabilities with a large intelligent surface. It defines space-normalized capacity and analyzes independent signal dimensions for different terminal deployments, obtaining wavelength-dependent results and one signal dimension per antenna-element in the lattice design.

  • Problem

    Prior analysis had not examined the information-transfer capabilities of electronically active structures.

  • Method

    The paper takes a first look at large intelligent surface uplink information-transfer capabilities and defines space-normalized capacity per area-unit.

  • Results

    2/λ terminals can be spatially multiplexed per m for one-dimensional deployment, while π/λ^2 terminals can be spatially multiplexed per m^2 for two- and three-dimensional deployment.

  • Takeaways & Limitations

    The LIS analysis obtains one independent signal dimension for every spent antenna-element on the LIS.

Abstract

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In this paper, we consider the potential of data-transmission in a system with a massive number of radiating and sensing elements, thought of as a contiguous surface of electromagnetically active material. We refer to this as a large intelligent surface (LIS). The "LIS" is a newly proposed concept, which conceptually goes beyond contemporary massive MIMO technology, that arises from our vision of a future where man-made structures are electronically active with integrated electronics and wireless communication making the entire environment "intelligent". We consider capacities of single-antenna autonomous terminals communicating to the LIS where the entire surface is used as a receiving antenna array. Under the condition that the surface-area is sufficiently large, the received signal after a matched-filtering (MF) operation can be closely approximated by a sinc-function-like intersymbol interference (ISI) channel. We analyze the capacity per square meter (m^2) deployed surface, \hat{C}, that is achievable for a fixed transmit power per volume-unit, \hat{P}. Moreover, we also show that the number of independent signal dimensions per m deployed surface is 2/λfor one-dimensional terminal-deployment, and π/λ^2 per m^2 for two and three dimensional terminal-deployments. Lastly, we consider implementations of the LIS in the form of a grid of conventional antenna elements and show that, the sampling lattice that minimizes the surface-area of the LIS and simultaneously obtains one signal space dimension for every spent antenna is the hexagonal lattice. We extensively discuss the design of the state-of-the-art low-complexity channel shortening (CS) demodulator for data-transmission with the LIS.

I. INTRODUCTION

The paper introduces the large intelligent surface (LIS) as an electronically active extension of massive MIMO and analyzes its uplink information-transfer capabilities. It derives capacity and spatial-dimension results, evaluates antenna-lattice implementations, and discusses low-complexity channel-shortening demodulation.

  • Motivation and concept: The LIS extends massive MIMO by treating electromagnetically active surfaces with integrated electronics as wireless receiving arrays.The vision connects electronically active man-made structures to intelligent environments and future communication systems.
  • Problem and approach: The paper fills an information-transfer gap by taking a first analytical look at LIS uplink capabilities under ideal line-of-sight propagation.The model assumes no scatterers or reflections and isotropic terminal signals.
  • Capacity and spatial dimensions: The normalized capacity limit is P_hat/(2N0) nats/s/Hz/volume-unit as wavelength λ approaches zero under fixed transmit power per volume-unit P_hat.N0 denotes the spatial power spectral density of additive white Gaussian noise.
  • Capacity and spatial dimensions: 2/λ terminals per m of deployed surface can be spatially multiplexed for one-dimensional terminal deployment, versus π/λ^2 terminals per m^2 for two- and three-dimensional deployments.These results concern an infinitely large LIS.
  • Numerical findings: A medium-sized LIS can accommodate around 100 uplink terminals with only a minuscule per-terminal capacity loss compared with one terminal.The paper attributes this behavior to effective interference suppression by the LIS.
  • Implementation and demodulation: The hexagonal lattice minimizes LIS surface area while providing one independent signal dimension per antenna, saving 23% area over a rectangular lattice.The paper also investigates channel shortening as a low-complexity demodulation method, with LMMSE performance close to the optimal receiver near the achievable signal-dimension density.

II. RECEIVED SIGNAL MODEL AT LIS FOR MULTIPLE TERMINALS

The LIS received-signal model considers multiple single-antenna terminals under perfect LoS propagation and narrow-band assumptions, then applies matched filtering to obtain a discrete matrix-form signal model.

  • K autonomous single-antenna terminals transmit independent Gaussian data symbols toward a two-dimensional LIS under perfect LoS propagation.The terminals transmit with equal power per Hz, and their symbols are zero-mean, unit-variance, and independent.
  • The model uses an isotropic terminal signal, wavelength λ, symbol period T, and negligible propagation delays relative to T.Negligible transmit-time differences produce no temporal interference in the narrow-band system.
  • The effective channel accounts for spherical propagation, free-space path loss, arrival angle, carrier frequency, and terminal-to-surface distance.The formulation applies to both near-field and far-field scenarios with respect to the LIS.
  • Matched filtering combines spatial and temporal correlation for each transmit signal, yielding discrete received samples with effective noise.The post-filter noise remains zero-mean but becomes colored, and the samples are assembled into a matrix formulation.
  • The resulting model includes received signal power, inter-user interference, and colored noise under equal terminal transmit powers.The paper studies the terminals’ communication capability with the LIS using this received-signal model.

C. Independent Signal Dimensions with the LIS

This section defines capacity and independent signal dimensions per deployed LIS area under fixed transmit power density, examining terminal deployments of different dimensionality and large-surface array gain.

  • The capacity is normalized by deployed LIS length or surface area to measure achievable communication performance per area-unit.The resulting space-normalized capacity has units of nats/s/Hz/area-unit.
  • The LIS can harvest many independent signal dimensions per deployed area-unit while also providing substantial array gain.For an isotropically transmitting terminal, half of the transmitted power reaches an infinitely large LIS, yielding g_k,k = P/2.
  • The analysis considers one-dimensional terminal deployments along a line and two- or three-dimensional deployments with spacings Δx and Δy.For three-dimensional deployments, terminals can be projected onto an xy-plane for analyzing independent signal dimensions.
  • The independent signal dimensions ρ are defined as the high-SNR pre-log factor of the normalized capacity.For one-dimensional deployments, capacity is normalized by LIS length; for two-dimensional deployments, it is normalized by surface area.
  • Fixed transmit power per volume-unit is required when calculating independent signal dimensions, because using fixed terminal power can make normalized capacity unbounded as terminal spacing shrinks.The paper denotes this power density by P-hat.

E. On the Approximation of an Integral

The paper approximates the integral governing inter-terminal coupling with a sinc function, then uses the resulting ISI model to analyze capacity and signal dimensions for one-dimensional deployments.

  • Close approximations are sought for φℓk when ℓ≠k because closed-form solutions appear out of reach.
  • The sinc approximation is justified when λ/z_k is sufficiently small, with λ/z_k ≲1 identified as a practical condition.
  • For λ=0.4 m, the exact integral and approximation are almost aligned, with relatively small approximation errors.
  • The resulting effective channel impulse response g_k,ℓ is real and follows the sinc-based model.
  • The one-dimensional deployment analysis derives capacity for both optimal reception and matched filtering under equally spaced terminals and an infinitely long LIS.
  • 2/λ independent signal dimensions per m are available in one-dimensional terminal deployments, and matched filtering reaches the same asymptotic normalized-capacity slope when 1/θ is an integer or λ is sufficiently small.

B. The Two-Dimensional Case: Terminals on a Plane

For terminals deployed on a two-dimensional plane, the paper derives space-normalized capacity and signal-space dimensionality using spatial spectral analysis of the LIS channel.

  • With the LIS dimensions tending to infinity, ζ=1/2 for all z₀ and capacity does not depend on distance.
  • The received signal after sinc-based matched filtering is analyzed through its two-dimensional spatial power spectral density.
  • Radial symmetry permits the relevant Fourier transform to be evaluated using a degree-zero Hankel transform.
  • As λ→0, the space-normalized capacity converges to P̂/2N₀ nats/s/Hz/m², matching the one-dimensional infinite-bandwidth limit.
  • π independent signal dimensions are obtained for every λ² of deployed surface area, corresponding to π/λ² dimensions per m².

C. The Three-Dimensional Case: Terminals in a Sphere

The three-dimensional terminal-deployment case has the same signal-space dimensionality per surface area as the two-dimensional case, because propagation preserves the relevant spectral domain.

  • The two-dimensional analysis supplies the spectral-domain result used to study three-dimensional terminal deployments.
  • The domain of S_x₀,y₀,z₀(f₁,f₂) is independent of the terminal distance z₀ from the wall.
  • π/λ² signal-space dimensions per m² are available at an intermediate plane P.
  • The number of dimensions in the three-dimensional volume is unchanged relative to the two-dimensional case.
  • Sampling implementation: The sampled LIS is constrained by the bandlimited support D(λ^-1) ∩ V(S^-T), and sufficiently dense sampling avoids aliasing loss.
  • Sampling implementation: The attainable dimensions per spent antenna reach ρ_ant=1 when V(S^-T)⊂D(λ^-1).

S |V (S)|

The paper formulates LIS antenna sampling as a lattice-design problem and identifies the scaled hexagonal lattice as the area-minimizing solution under one dimension per antenna.

  • The lattice generator solving the area-minimization problem is the scaled hexagonal lattice generator.
  • The hexagonal lattice achieves ρ_ant=1 with an antenna density determined by its fundamental cell.
  • The rectangular lattice must satisfy the same reciprocal-lattice containment constraint to achieve ρ_ant=1.
  • Hexagonal sampling requires only a fraction 4√3/3 of the surface area required by rectangular sampling, saving 23% of surface area.
  • The hexagonal efficiency gain exceeds the normal packing-efficiency comparison because each spent antenna is constrained to harvest one signal-space dimension.

V. CHANNEL SHORTENING DEMODULATOR DESIGN FOR DATA-TRANSMISSION WITH LIS

This section develops a low-complexity channel-shortening (CS) demodulator for LIS uplink data transmission, controlling detection complexity through the interfering depth ν. CS shortens the ISI channel and approaches optimal detection with substantially lower complexity when ν is small.

  • Complexity-performance trade-off: The LMMSE receiver incurs a loss relative to capacity, whereas CS can approach optimal demodulation with much lower complexity for small ν.The paper motivates CS as an intermediate-complexity alternative between optimal BCJR detection and LMMSE equalization.
  • CS demodulator design: CS transforms an arbitrary channel matrix into a band-limited effective channel with interfering depth ν among terminals.The demodulator shortens the ISI channel matrix G into an effective channel H and constrains Φ to its middle 2ν+1 diagonals.
  • CS demodulator design: The CS demodulator comprises channel shortening followed by a BCJR detector with |X|^ν states.The optimal BCJR detector instead has |X|^(K−1) states, so small ν reduces the state-space complexity.
  • Parameter optimization: The parameters W and Φ are optimized by maximizing the achievable information rate under the band-limited structure imposed on Φ.The resulting design includes a closed-form expression for the optimal achievable information rate and prefiltering parameters.
  • Complexity-performance trade-off: ν=0 and ν=K−1 recover the LMMSE and BCJR demodulators, respectively, spanning the complexity spectrum between linear and optimal detection.The CS framework therefore provides a tunable family of receivers rather than a single fixed-complexity detector.

VI. NUMERICAL RESULTS

The numerical results evaluate LIS capacity, independent signal dimensions, and CS achievable information rates under one-, two-, and three-dimensional terminal deployments. They show convergence and saturation of normalized capacity with increasing terminal density, while LIS reception suppresses interference and preserves robust performance.

  • One-dimensional deployments: When 1/θ is an integer, terminals do not interfere and optimal and MF receivers have identical normalized capacities.For other values, the MF receiver is inferior to the optimal receiver.
  • One-dimensional deployments: As terminal spacing decreases, space-normalized capacity approaches the optimal-receiver limit and saturates at Δx=λ/2 in the one-dimensional case.The MF receiver also converges but remains suboptimal.
  • Two-dimensional deployments: For two-dimensional deployments, space-normalized capacity reaches a limit and saturates at Δs=λ^2/π with the optimal receiver.The MF receiver converges as well but remains inferior to the optimal receiver.
  • Three-dimensional deployments: As three-dimensional terminal spacing decreases, space-normalized capacity increases under fixed transmit power per volume-unit.The simulations compare fixed per-terminal power with fixed power density in a finite room.
  • Three-dimensional deployments: With 32 to 320 terminals, capacity per terminal remains fairly flat despite increased interference, demonstrating LIS interference suppression.The result is reported for randomly located terminals and comparisons between optimal and MF receivers.

C. Capacities with CS Demodulator for LIS

This section evaluates CS demodulators in finite-LIS scenarios and compares their sum-rates and capacity losses with optimal and LMMSE receivers. Small memory depth can closely approach optimal performance, while larger depth reduces the gap at higher complexity.

  • Finite-LIS effects: At 640 terminals, LMMSE incurs approximately 14% capacity loss relative to the optimal receiver because the LIS has limited size.An infinitely large LIS would provide π/λ^2=13 independent signal dimensions per m^2 of deployed surface area.
  • CS complexity-performance trade-off: Setting ν=K/2 reduces the CS performance gap to the optimal receiver to less than 5%.This improvement comes with large ν, which makes the CS demodulator overly complex unless a low-complexity modulation scheme is used.
  • Finite-LIS effects: The LMMSE demodulator performs reasonably well because the LIS efficiently suppresses interference.The paper relates this behavior to the interference-suppression results in Figs. 8–10.
  • Conclusions: The paper concludes that LISs maintain robust performance as terminal numbers increase and are highly effective at suppressing interference.It presents LIS data transmission as a research direction beyond massive MIMO.

APPENDIX A: ARGUMENTATIONS OF PROPERTY 1

The appendix develops a sinc-function approximation for the LIS channel response using Fourier-transform arguments and numerical comparisons. The approximation is accurate over a stated wavelength range but relies on a small-λ condition.

  • Fourier-transform derivation: The Fourier transform of the channel response is derived as a rectangular function, whose inverse transform yields a sinc-function form.The derivation uses Fourier cosine transforms and properties of modified Bessel functions.
  • Approximation construction: The convolution involving the modified Bessel function K0 lacks a tractable closed-form expression, motivating an approximation.The appendix lower-bounds the amplitude of K0 with a rectangular function before applying the approximation.
  • Numerical validation: The sinc-function approximation of Φ(f) works well for λ up to 1 m.Numerical integrals and sinc-based approximations are compared for several wavelength values.
  • Approximation conditions: The approximation treats Φc(f) as a Dirac delta-function when λ is small, subject to a bandwidth condition.The bandwidth of K0 must be larger than that of Φc(f), so λ should not be too small.
  • Numerical validation: For λ=[0.1, 0.5, 1, 2] m, numerical capacities are [1.9269, 1.6383, 1.4044, 1.1334] nats/s/Hz, while sinc approximations are [1.9053, 1.6178, 1.3794, 1.0876].The sinc-based values are close and slightly smaller than the numerical capacities.

APPENDIX B: PROOF OF PROPERTY 2

The appendix derives the capacity and interference expressions through auxiliary-parameter substitutions and Parseval-based calculations, then proves that the relevant lattice is hexagonal.

  • Capacity derivation: The capacity is split into two parts according to whether G(f) is folded by β or β+1 times.The two parts use amplitudes βθζP and (β+1)θζP, with integration-interval lengths θ−˜θ and ˜θ, respectively.
  • Capacity derivation: The resulting capacity expression is transformed using the definitions of α and β to obtain the capacity stated in Property 3.
  • Interference derivation: Parseval’s identity and the same auxiliary-function argument yield the interference power expression stated in (31).
  • Lattice geometry: The lattice proof reduces the two-dimensional case to a centrally symmetric hexagonal cell, possibly degenerating into a rectangle, inscribed in a circle.The proof first assumes λ=1 and uses scaling for arbitrary λ.
  • Lattice geometry: The lattice generated by S−T has maximal cell area when its defining triangle is equilateral, making both that lattice and the lattice generated by S hexagonal.
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