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Communication Through a Large Reflecting Surface With Phase Errors

Mihai-Alin Badiu, Justin P. Coon

arXiv:1906.10751v1eess.SPcs.IT

TL;DR

The paper addresses communication through an LRS when reflector phases cannot be estimated or configured perfectly. It models generic phase errors, derives an equivalent Nakagami fading channel, and finds that performance remains robust despite uncertainty.

  • Problem

    Perfect phase estimation and high-precision configuration of LRS reflection phases are unfeasible, motivating analysis under generic phase errors.

  • Method

    The paper models reflector-phase deviations with a generic circular distribution and derives a Nakagami-fading representation of the composite LRS channel.

  • Results

    The equivalent channel has average SNR scaling with n2 and diversity order growing with n, while phase uncertainty attenuates both parameters.

  • Takeaways & Limitations

    Numerical results for limited reflector counts agree with theory and show that performance is remarkably robust against phase errors.

Abstract

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Assume the communication between a source and a destination is supported by a large reflecting surface (LRS), which consists of an array of reflector elements with adjustable reflection phases. By knowing the phase shifts induced by the composite propagation channels through the LRS, the phases of the reflectors can be configured such that the signals combine coherently at the destination, which improves the communication performance. However, perfect phase estimation or high-precision configuration of the reflection phases is unfeasible. In this paper, we study the transmission through an LRS with phase errors that have a generic distribution. We show that the LRS-based composite channel is equivalent to a point-to-point Nakagami fading channel. This equivalent representation allows for theoretical analysis of the performance and can help the system designer study the interplay between performance, the distribution of phase errors, and the number of reflectors. Numerical evaluation of the error probability for a limited number of reflectors confirms the theoretical prediction and shows that the performance is remarkably robust against the phase errors.

I. INTRODUCTION

The paper studies large reflecting surfaces that enhance communication by coherently combining reflected waves, while addressing the impracticality of perfect phase knowledge and precise phase configuration.

  • An LRS is modeled as an array of passive reflector elements that induce adjustable phase shifts on reflected signals.
  • Appropriately configured reflector phases make the reflected waves combine coherently at the destination, enhancing communication.
  • Perfect phase estimation and high-precision reflection-phase configuration are infeasible in practice.
  • The paper models generic phase errors and shows that an LRS with n imperfect reflectors is equivalent to point-to-point Nakagami fading.
  • Despite phase uncertainty, the average SNR scales with n2 and diversity order grows with n, while numerical results show robust performance.

II. SYSTEM MODEL

The system consists of a single-antenna source and destination assisted by n reflectors, with phase deviations modeled as random circularly distributed errors around ideal coherent-combining settings.

  • The system contains a single-antenna source, a single-antenna destination, and n reflector elements, with no direct source–destination link.
  • The source-to-reflector and reflector-to-destination fading coefficients are assumed mutually independent across the 2n channels.
  • The received model uses transmitted symbols with unit second moment, normalized complex Gaussian receiver noise, and γ0 as the single-reflector average SNR.
  • For ideal operation, each reflector phase cancels the combined channel phase arg(Hi1) + arg(Hi2) to maximize receiver SNR.
  • Phase errors are represented by random deviations Θi from ideal settings, including imperfect estimation and discrete phase configurations.
  • The phase errors are modeled as identically distributed independent variables with a common characteristic function represented by circular moments.

III. EQUIVALENT SCALAR FADING CHANNEL

The paper represents transmission through a large reflecting surface as an equivalent point-to-point Nakagami fading channel for large numbers of imperfect reflectors. Phase uncertainty attenuates the equivalent channel parameters, while reflector diversity and average SNR retain favorable scaling with system size.

  • Special cases: With no phase errors, the composite coefficient is real and the result recovers the ideal no-error case studied for Rayleigh fading.Zero phase errors imply first and second trigonometric moments equal to one.
  • Special cases: With uniformly distributed phase errors, the equivalent coefficient is zero-mean circularly symmetric complex normal, resembling Rayleigh fading.This case represents complete lack of knowledge about the constituent channel phases.
  • Equivalent channel: For large n and nonzero first trigonometric moment, the magnitude of the composite channel has a Nakagami distribution.This follows the paper’s large-n equivalent-channel theorem.
  • Equivalent channel: The LRS transmission is statistically equivalent to direct transmission over a Nakagami fading channel with n^2 larger receive power than one reflector.The single-reflector SNR γ0 accounts for path loss, shadowing, and reflection loss.
  • Phase-error dependence: The equivalent fading distribution depends on the phase-error distribution through its first two trigonometric moments.For the considered zero-mean symmetric errors, these moments are real and subunitary.
  • Phase-error dependence: Broader phase-error distributions reduce the average SNR through attenuation by the squared first trigonometric moment.The average SNR remains bounded and cannot exceed the transmit signal power divided by receiver noise.

IV. PERFORMANCE ANALYSIS

The performance analysis models phase uncertainty through estimation and quantization error distributions. These models provide the trigonometric moments used to characterize their effects on the equivalent channel.

  • Error models: Phase estimation errors use a zero-mean von Mises distribution whose concentration parameter κ represents estimation accuracy.Its characteristic function is expressed using modified Bessel functions.
  • Error models: Quantization errors arise when only 2^q discrete phases are available and are modeled uniformly over [−2^-qπ, 2^-qπ].The resulting first trigonometric moment is given explicitly in the paper.
  • Error models: Independent estimation and quantization errors can be combined because their resulting trigonometric moments equal the products of the respective moments.The paper states that the combined case is therefore directly tractable.

A. Distribution of the SNR

For large n, the instantaneous SNR n2γ0|H|2 is gamma distributed, with simulations agreeing well with the theoretical approximation even at moderate reflector counts.

  • A. Distribution of the SNR: For large n, the instantaneous SNR n2γ0|H|2 has a gamma-distributed model.The result follows from the gamma distribution of |H|2 and the scaling factor n2γ0.
  • A. Distribution of the SNR: Monte Carlo histograms for n = 16 and n = 256 are compared with the gamma pdf under Rician, Rayleigh, and von Mises fading conditions.The histograms use 105 trials, while the solid curves represent the gamma pdf (13).
  • A. Distribution of the SNR: The theoretical approximation agrees very well with Monte Carlo estimates even for moderate n, and the pdf approaches a Gaussian pdf as n increases.For large n, the pdf concentrates around the mean γ̄.

B. Average error probability

The average error probability for BPSK with n = 32 is well predicted by the equivalent Nakagami channel under estimation and quantization errors, although approximation accuracy depends on n.

  • B. Average error probability: For n = 32, simulations show good agreement with the theoretical Nakagami-channel error probability under estimation and quantization errors.The comparison uses BPSK and the same fading models as the preceding example.
  • B. Average error probability: Theoretical curves correspond to the Nakagami channel, marked points to LRS Monte Carlo simulations, and dashed curves to ideal zero-error performance.The figure compares phase-estimation errors on the left with quantization errors on the right.
  • B. Average error probability: For phase estimation, κ = 2 and κ = 8 are ordered by increasing performance; for quantization, q = 1, 2, and 3 bits are similarly ordered.The figure reports average probability of error for n = 32.
  • B. Average error probability: At high average SNR γ̄, the error probability is written as Pe(γ0) = (Gcγ0)−Gd, with diversity gain Gd = m and coding gain Gc = 1a4γ0.These gains are defined relative to the average single-reflector SNR γ0.
  • B. Average error probability: The relationships can guide reflector-count selection for a target diversity or coding gain, though coding-gain sizing is more complicated.The diversity gain can be established from (12), whereas coding-gain sizing depends more complicatedly on the phase-error distribution.

V. CONCLUSION

The composite LRS channel with phase errors is equivalent to point-to-point Nakagami fading, with phase uncertainty attenuating gains while preserving their scaling with reflector count.

  • V. CONCLUSION: The composite channel is equivalent to a point-to-point Nakagami fading channel whose parameters depend on the first two trigonometric moments of phase errors.The constituent fading distributions affect the equivalent channel through their average magnitudes.
  • V. CONCLUSION: Despite phase errors, average SNR grows with n2 and diversity order grows linearly with n, while phase uncertainty attenuates both gains.The scaling is stated relative to the single-reflector SNR.
  • V. CONCLUSION: Numerical results for a limited number of reflectors agree well with theory and show remarkable robustness against phase errors.

APPENDIX A PROOF OF LEMMA 1

The appendix models the reflector contributions as i.i.d. complex variables and uses the central limit theorem to approximate the composite coefficient by a complex normal distribution.

  • APPENDIX A PROOF OF LEMMA 1: The variables |Hi1||Hi2|ejΘi are i.i.d. with common mean µ = a2ϕ1 and variance ν = 1−a4|ϕ1|2.
  • APPENDIX A PROOF OF LEMMA 1: The phase-error assumptions make the trigonometric moments ϕ1 and ϕ2 real, and consequently µ and ρ real.
  • APPENDIX A PROOF OF LEMMA 1: By the central limit theorem, H is approximated for large n by CN(µ, ν/n, ρ/n).Writing H as U + jV, the proof then uses the joint Gaussian characterization of its real and imaginary parts.
  • APPENDIX A PROOF OF LEMMA 1: The real and imaginary components U and V are independent because they are jointly Gaussian with zero covariance.

APPENDIX B PROOF OF THEOREM 1

The proof derives the distribution of the composite channel magnitude by decomposing its squared magnitude into independent components and approximating the resulting cumulant generating function for large n.

  • The squared magnitude |H|² is decomposed into U² and V², with U²/σ² non-central chi-squared and V² gamma distributed.
  • Because U and V are independent, the cumulant generating function of |H|² characterizes the sum of the component distributions.
  • For large n, the first term of the cumulant-generating-function expansion is approximated using a Maclaurin series, with g(t) = O(n^-2).
  • The resulting expression corresponds to the cumulant generating function of a sum of three independent gamma variables.
  • The proof further simplifies this distribution, yielding a gamma approximation for |H|² with shape parameter µ2 and O(n^-1) error.
  • A variable transformation then gives a Nakagami distribution for |H|, while a second-order Gaussian approximation is less accurate for moderate n.
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