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Performance Analysis of Large Intelligent Surfaces (LISs): Asymptotic Data Rate and Channel Hardening Effects

Minchae Jung, Walid Saad, Youngrok Jang, Gyuyeol Kong, Sooyong Choi

arXiv:1810.05667v4cs.IT

TL;DR

The paper addresses performance analysis for large antenna arrays under practical limitations, including overhead, hardware impairments, imperfect channel estimation, and spatially correlated Rician interference. It derives asymptotic uplink-rate and channel-hardening results, finding agreement with simulations and diminishing noise and interference effects as antenna numbers increase.

  • Problem

    Massive antenna arrays face overhead that can seriously degrade performance, motivating analysis under practical limitations.

  • Method

    The paper asymptotically analyzes uplink data rate, derives mutual-information moments and an uplink-rate bound, and models device-specific spatially correlated Rician fading.

  • Results

    The analyses agree with extensive simulations, and channel hardening occurs in the LIS-based system.

  • Takeaways & Limitations

    Noise and interference from channel-estimation errors and the NLOS path become negligible as the number of antennas increases.

  • Takeaways & Limitations

    The analysis explicitly considers hardware impairments, imperfect channel estimation, and interference as practical limitations.

Abstract

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The concept of a large intelligent surface (LIS) has recently emerged as a promising wireless communication paradigm that can exploit the entire surface of man-made structures for transmitting and receiving information. An LIS is expected to go beyond massive multiple-input multiple-output (MIMO) system, insofar as the desired channel can be modeled as a perfect line-of-sight. To understand the fundamental performance benefits, it is imperative to analyze its achievable data rate, under practical LIS environments and limitations. In this paper, an asymptotic analysis of the uplink data rate in an LIS-based large antenna-array system is presented. In particular, the asymptotic LIS rate is derived in a practical wireless environment where the estimated channel on LIS is subject to estimation errors and interference channels are spatially correlated Rician fading channels. Moreover, the occurrence of the channel hardening effect is analyzed and the performance bound is asymptotically derived for the considered LIS system. The analytical asymptotic results are then shown to be in close agreement with the exact mutual information as the numbers of antennas and devices increases without bounds. Moreover, the derived ergodic rates show that noise and interference from estimation errors and the non-line-of-sight path become negligible as the number of antennas increases. Simulation results show that an LIS can achieve a performance that is comparable to conventional massive MIMO with improved reliability and a significantly reduced area for antenna deployment.

I. INTRODUCTION

The paper motivates LISs as dense, surface-integrated antenna arrays that can simplify deployment and channel acquisition while reducing device power and interference. It develops an asymptotic uplink-rate and channel-hardening analysis under practical estimation, hardware, and interference conditions, finding comparable rates to massive MIMO with improved reliability and reduced deployment area.

  • Motivation and potential benefits: LISs can use man-made surfaces for wireless communication, enabling nearby users to transmit at lower power than in massive MIMO.The resulting lower device power can save battery energy and reduce interference within the LIS.
  • Motivation and potential benefits: Dense LIS antenna arrays are feasible because greater-than-half-wavelength spacing is unnecessary for full diversity under highly correlated LOS channels.The paper also associates dense deployment with near-field LOS communication.
  • Motivation and potential benefits: LIS desired channels can simplify channel estimation and feedback compared with massive MIMO systems requiring CSI for hundreds of antennas.The simplification is attributed to the desired channel's strong correlation with the LOS path.
  • Research gap: Prior LIS studies used idealized assumptions including infinite surfaces, perfect channel estimation, LOS interference, or independent Rayleigh fading.These assumptions did not represent practical LIS environments and their limitations.
  • Approach and contributions: The paper analyzes uplink ergodic rate asymptotically using finite LIS units, channel-estimation errors, hardware impairments, and spatially correlated Rician interference channels.Interference channels combine deterministic LOS and stochastic NLOS components determined by the interfering device's distance from the target LIS unit.
  • Approach and contributions: The analysis derives asymptotic rate, variance, and performance bounds to study channel hardening and operating parameters such as optimal LIS-unit size.It also shows that noise, estimation errors, hardware impairments, and NLOS interference become negligible relative to LOS interference, while simulations report comparable rates to massive MIMO, improved reliability, and reduced deployment area.

II. SYSTEM MODEL

The system models an uplink LIS array with finite, device-centered LIS units, realistic resource allocation, LOS desired channels, and spatially correlated Rician interference channels.

  • LIS geometry: Each LIS unit is a finite square subarea of the surface, centered on its corresponding device and containing M antennas on a rectangular lattice.The unit area is limited to 2L × 2L, with antenna spacing ΔL.
  • Resource allocation: Overlapping LIS units can degrade performance, so devices with similar (x,y) coordinates are assigned orthogonal resources through allocation and scheduling.The LIS units are centered according to device locations.
  • Wireless channel model: The desired device-to-unit channel is modeled as a LOS path because the NLOS contribution becomes negligible as M increases.The LOS channel state depends on antenna-specific distances and the signal wavelength.
  • Wireless channel model: Interference channels combine LOS and spatially correlated NLOS components under Rician fading, with correlation modeled for a uniform planar array.The NLOS model uses dominant paths and an independent fast-fading vector.

B. Uplink Data Rate

The uplink rate is formulated from the received SINR after matched-filter detection, incorporating transmit signals, interference, noise, hardware impairments, and channel-estimation errors.

  • Rate formulation: The instantaneous uplink data rate is R_k = log(1 + γ_k), where γ_k is the instantaneous SINR at LIS unit k.The rate is evaluated for the signal received from all devices at the target LIS unit.
  • Received signal: The received signal includes the desired uplink signal, interference from other devices, thermal noise, and residual hardware-impairment noise.Transmit signals are modeled as independent zero-mean unit-variance Gaussian variables, while ρ_k and ρ_j denote uplink transmit SNRs.
  • Receiver model: The receiver uses matched filtering with f_k = ĥ_kk, reducing to the true desired channel h_kk under perfect channel estimation.Under imperfect CSI, the estimation error is modeled independently of h_kk and n_k.
  • Desired-signal power: The LIS desired-signal power is deterministic because it is formed from the squared sum of LOS channel gains across antennas.This differs from conventional massive MIMO, where NLOS fading makes the desired-signal power non-deterministic at the base station.
  • Asymptotic analysis: The rate analysis examines mutual-information moments as both the number of antennas M and the number of devices K increase without bounds.The SINR expression includes estimation-error and hardware-impairment contributions through the effective received-noise terms.

III. ASYMPTOTIC RATE ANALYSIS

The asymptotic analysis exploits the finite LIS geometry to characterize rate behavior as antennas and devices grow, including deterministic signal power and channel-hardening consequences.

  • System scaling: The paper analyzes an LIS-based large antenna-array system with densely distributed antennas, where increasing M can reduce ΔL within a fixed LIS-unit area.The analysis targets systems with many connected IoT devices.
  • System scaling: LIS communication imposes no constraint on the relationship between M and K, unlike conventional massive MIMO models that keep M/K constant.The contrast is stated as a system-level distinction in the asymptotic analysis.
  • Desired-signal asymptotics: The desired signal power S_k converges to a constant value p̄_k determined by the device distance z_k and the LIS-unit size L.This power is also related to the energy captured over the finite surface region.
  • Desired-signal asymptotics: Within a constrained physical area, the LIS can capture more total energy as M increases because denser antenna spacing increases the number of LOS paths and spatial channels.The paper distinguishes this setting from prior far-field analyses based on perfect NLOS channels.
  • Rate moments: Corollary 1 expresses the mean and variance of the uplink rate through the random variable I_k, making its asymptotic moments central to the rate analysis.The mean and variance of R_k are determined exclusively by I_k in the corollary.

A. Asymptotic Analysis of Rk

The paper derives asymptotic moments for intermediate SINR variables and the uplink rate, yielding deterministic, location- and correlation-dependent performance estimates.

  • Derivation: The asymptotic derivation first analyzes the moments of I_k, then derives covariances among X_k, Y_jk, and Z_k.These quantities are used to obtain asymptotic moments of the SINR and rate.
  • Asymptotic SINR: The asymptotic SINR mean and variance are deterministic values determined by device locations and channel-correlation matrices.The resulting quantities are denoted by μ̄_γk and σ̄²_γk.
  • Performance evaluation: The asymptotic data rate can therefore be evaluated from deterministic quantities such as device locations and correlation matrices.The paper connects these estimates to ergodic rate, reliability, and scheduling-diversity evaluation without extensive simulations.
  • Performance evaluation: Theorem 1 moments closely agree with the mutual-information moments of an actual LIS system as the numbers of antennas and devices increase.This agreement supports the asymptotic approximation in the large-system regime.

IV. CHANNEL HARDENING EFFECT AND PERFORMANCE BOUND

The section establishes channel hardening for LIS systems and derives an asymptotic uplink-rate performance bound. As antenna count grows, normalized interference converges to a constant, rate variance vanishes, and fading becomes negligible.

  • The channel hardening effect describes shrinking mutual-information fluctuations as the number of antennas grows.
  • The analysis verifies channel hardening and derives the performance bound of the ergodic uplink rate.
  • The normalized interference term ¯I_k/M^2 converges to a constant without variance as M increases.
  • Theorem 2 states that the asymptotic variance of R_k goes to zero as M →∞, while the asymptotic mean determines the uplink-rate bound.
  • As M increases, LIS fading behaves as a static channel and its impact on the uplink data rate becomes negligible.
  • The analysis notes a scope boundary: convergence may fail if as many interference-dominant devices as antennas transmit toward a target LIS unit.

SIMULATION PARAMETERS

The asymptotic analysis identifies which impairments remain relevant as the antenna count grows. LOS interference remains influential, whereas noise, estimation-error interference, and NLOS interference become negligible.

  • Theorem 2 gives an asymptotic rate with vanishing variance, supporting a nearly deterministic uplink data rate.
  • LOS interference is the only interference component that affects the asymptotic ergodic rate.
  • Noise, estimation-error interference, and NLOS-path interference become negligible compared with LOS interference as M increases.
  • If all interference is generated through the NLOS path, the asymptotic bound goes to infinity as M increases.
  • The approximation gap from the Taylor expansions goes to zero as M →∞.

V. SIMULATION RESULTS AND ANALYSES

Simulations across interference types, device deployments, and antenna-array comparisons closely match the asymptotic analysis. LIS rates converge predictably, harden with increasing antennas, and can exceed massive MIMO at practical array sizes while using less physical area.

  • Asymptotic mean values from Theorem 1 closely match simulations across the tested antenna range for LOS and NLOS interference.
  • With LOS interference, the ergodic rate converges to the asymptotic bound as M increases; with NLOS interference, it grows without bound.
  • Only LOS interference affects the asymptotic ergodic rate in the compared LOS- and NLOS-only scenarios.
  • For randomly located devices, the ergodic rate converges to the asymptotic bound, while increasing K decreases rate because interference power increases.
  • The rate variance converges to zero as M increases, demonstrating channel hardening; asymptotic variances approach simulations as K increases.
  • At K = 30 and M = 100, LIS achieves about a 2-fold higher ergodic rate than massive MIMO with ULA.
  • At K = 30 and M = 3600, LIS and massive MIMO achieve almost equal ergodic rates, while LIS occupies 0.25 m2 versus roughly 9 m2 for a 60 × 60 array.
  • LIS rate variance converges to zero, whereas massive-MIMO variance converges to a constant, yielding improved reliability and lower latency for LIS.

VI. CONCLUSIONS

The paper asymptotically analyzes uplink performance for practical LIS systems with hardware impairments, imperfect channel estimation, and spatially correlated Rician interference. It derives rate bounds and shows channel hardening, analytical accuracy, and diminishing impairment effects as system dimensions grow.

  • The analysis considers a large LIS divided into smaller units with limited physical area, under hardware impairments, imperfect channel estimation, and device-specific spatially correlated Rician interference.
  • The paper derives asymptotic moments of mutual information and an asymptotic bound for the uplink data rate.
  • The analyses accurately determine LIS performance analytically without extensive simulations and demonstrate channel hardening in the LIS-based system.
  • Hardware impairments, noise, and interference from channel-estimation errors and the NLOS path become negligible as M increases.
  • The ergodic rate and rate variance converge to the derived asymptotic bound and zero, respectively, as M and K increase.
  • The simulations show close agreement with the asymptotic analyses, while LIS communication is reported as reliable and space-intensive beyond massive MIMO systems.

APPENDIX A

Appendix A develops an asymptotic approximation for a random-variable component of the LIS analysis. It uses the law of large numbers and Lyapunov’s central limit theorem under independence and Gaussian assumptions.

  • The law of large numbers is used to approximate the relevant random variable for large M.
  • Lyapunov’s central limit theorem yields an asymptotically complex Gaussian distribution for the normalized random variable.
  • Under independence across antenna elements, the component variables form a sequence of independent random variables with zero mean and unit variance.
  • The derivation obtains asymptotic means and variances using the large-M approximation and the Lyapunov central limit theorem.

APPENDIX C

Appendix C derives means, variances, and distributions for random variables used in the LIS rate analysis. The derivation accounts for Gaussian components, covariance structure, and deterministic device-location and correlation effects.

  • Several components are modeled as complex Gaussian random variables, including terms formed from independent Gaussian variables.
  • The resulting intermediate expressions support the asymptotic analysis of the LIS rate variables.
  • Deterministic values in the derivation depend on device locations and correlation matrices.
  • The appendix includes covariance terms for dependent variables sharing a common random variable.

APPENDIX D

Appendix D analyzes the asymptotic variance of the mutual-information terms underlying the uplink rate. It identifies the dominant scaling contributions and the role of line-of-sight interference and cross-user covariance.

  • The appendix derives the mean and variance of the uplink mutual-information-related quantities from Gaussian random variables and their independence properties.
  • As K increases, the sum of cross-user covariances becomes dominant, while covariances involving X_k, the summed Y_jk terms, and Z_k become negligible.
  • The asymptotic value of ω_ijk is the covariance between Y_ik and Y_jk for distinct users.
  • The first term of ω_ijk scales as O(M3), while the second and last terms scale as O(M2) as M approaches infinity.
  • The line-of-sight component of the interference channel exclusively produces the deterministic asymptotic term ω̄_ijk.
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