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Intelligent Reflecting Surface-aided URLLC in a Factory Automation Scenario

Hong Ren, Kezhi Wang, Cunhua Pan

arXiv:2103.09323v3cs.NI

TL;DR

Wireless factory automation requires URLLC analysis beyond conventional Shannon capacity, while obstacles motivate IRS-assisted links. The paper analytically derives ADR and ADEP under finite blocklength across seven channel and deployment cases, including a roughly 0.5-bit-per-channel gap relative to Shannon capacity.

  • Problem

    Wireless factory automation faces stringent URLLC requirements, while short channel-code blocklengths make conventional Shannon capacity inapplicable.

  • Method

    The paper analytically derives ADR and ADEP for IRS-aided factory-automation URLLC under finite blocklength across seven channel and deployment cases.

  • Results

    A roughly 0.5 bit per channel gap exists between Shannon capacity and the finite-blocklength result in the direct-channel case.

  • Takeaways & Limitations

    The derived analytical results cover IRS-aided URLLC performance across Rayleigh, Nakagami-m, Rician, correlated-channel, phase-alignment, direct-link, and multiple-IRS cases.

  • Takeaways & Limitations

    The analysis assumes only one transmit antenna and notes that limited factory space may constrain the transmission distance between the IRS and devices.

Abstract

from arXiv · show

Different from conventional wired line connections, industrial control through wireless transmission is widely regarded as a promising solution due to its reduced cost, increased long-term reliability, and enhanced reliability. However, mission-critical applications impose stringent quality of service (QoS) requirements that entail ultra-reliability low-latency communications (URLLC). The primary feature of URLLC is that the blocklength of channel codes is short, and the conventional Shannon's Capacity is not applicable. In this paper, we consider the URLLC in a factory automation (FA) scenario. Due to densely deployed equipment in FA, wireless signal are easily blocked by the obstacles. To address this issue, we propose to deploy intelligent reflecting surface (IRS) to create an alternative transmission link, which can enhance the transmission reliability. In this paper, we focus on the performance analysis for IRS-aided URLLC-enabled communications in a FA scenario. Both the average data rate (ADR) and the average decoding error probability (ADEP) are derived under finite channel blocklength for seven cases: 1) Rayleigh fading channel; 2) With direct channel link; 3) Nakagami-m fading channel; 4) Imperfect phase alignment; 5) Multiple-IRS case; 6) Rician fading channel; 7) Correlated channels. Extensive numerical results are provided to verify the accuracy of our derived results.

I. INTRODUCTION

Factory automation requires wireless communications with millisecond-scale latency and packet error probabilities around 10^-6–10^-9, but dense obstacles can block signals. The paper addresses this challenge by analyzing IRS-aided URLLC under finite blocklength across multiple channel and deployment conditions.

  • Dense machinery, moving objects, and pillars block wireless signals, degrading reliability and motivating an IRS-created alternative transmission link.An IRS uses independently phase-controllable reflecting elements to constructively combine reflected waves with the direct signal.
  • Existing IRS studies largely emphasize transmission design, while analytical performance for IRS-aided systems remains limited.
  • Shannon capacity is unsuitable for short-packet URLLC because finite blocklength prevents decoding error probability from vanishing and can overestimate performance.Using Shannon capacity directly can underestimate delay outage probability and fail to guarantee QoS requirements.
  • The paper derives ADR and ADEP expressions under finite channel blocklength for Rayleigh, direct-link, Nakagami-m, imperfect-alignment, multiple-IRS, Rician, and correlated-channel cases.The derived results are supported by numerical validation and include a roughly fixed gap between Shannon capacity and finite-blocklength ADR.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system models IRS-assisted short-packet communication from a single-antenna controller to a remote factory device when metallic machinery blocks the direct link. Channel-state information supports IRS phase-shift design and performance analysis.

  • A. System Model: With fixed transmit power P, the received signal contains unit-power data and zero-mean noise with variance σ^2, yielding an instantaneous SNR.
  • A. System Model: Metallic machinery can severely weaken or eliminate the controller–device direct link, motivating an IRS-mounted alternative path.
  • A. System Model: The factory scenario uses one antenna at both controller and device, while the IRS contains N reflecting phase shifters.
  • A. System Model: The controller-to-IRS and IRS-to-device channels are represented by N-dimensional complex vectors h and g.
  • A. System Model: Under the baseline model, h and g have independent zero-mean complex Gaussian elements with variances α and β, respectively.
  • A. System Model: When multiple devices are present, OFDM can allocate one sub-carrier per device to preserve reliability while retaining the derivations.
  • A. System Model: The transmitter is assumed to know instantaneous and distributional channel information for IRS phase design and performance analysis.
  • A. System Model: The IRS reflection matrix uses unit-modulus phase shifts φ_n ∈ [0, 2π] for its N elements.

B. Short Packet Transmission Theory

The paper formulates URLLC using finite-blocklength coding because factory delay constraints prevent asymptotic Shannon-capacity assumptions. It then derives approximations for rate and decoding reliability under IRS phase optimization.

  • For D bits transmitted over M channel uses, the coding rate is R = D/M and finite-blocklength analysis determines the decoding error probability.
  • Factory delay requirements constrain channel blocklength, so decoding error cannot be assumed to vanish as in asymptotic Shannon-capacity analysis.
  • The achievable rate approximation uses blocklength M, target error probability ε, inverse Q-function Q^-1(·), and channel dispersion V(γ) = 1 − (1 + γ)^−2.
  • With CSI at the BS, the IRS phase shifts are selected to maximize instantaneous SNR γ.
  • The random SNR is approximated by a Gamma distribution whose parameters are obtained from the first and second moments of the IRS-assisted channel variable X.
  • The average data rate is approximated by replacing channel dispersion with its high-SNR approximation V(γ) ≈ 1 and using a generalized hypergeometric-function expression.
  • The average decoding error probability is derived for fixed packet size D and blocklength M by averaging the finite-blocklength error expression over the SNR distribution.

IV. EXTENSIONS TO MORE GENERAL CASES

The analysis extends the baseline IRS-assisted URLLC model to direct links, Nakagami-m fading, and imperfect phase alignment. These variants retain Gamma-based SNR approximations for deriving performance metrics.

  • A direct BS-to-device link with coefficient h_0 following CN(0, η) is incorporated into the IRS-assisted model.
  • For the direct-link extension, the resulting SNR is approximated by a Gamma distribution with parameters determined by the moments of Y.
  • Under Nakagami-m fading, the controller-to-IRS and IRS-to-device amplitudes use parameters m_1, m_2 and average channel power gains α, β.
  • For Nakagami-m fading, phase alignment produces an SNR proportional to Z, whose distribution is approximated by a Gamma law.
  • Imperfect phase alignment is modeled by uniform errors ω_n ∼ U(−Δ, Δ), where Δ characterizes the phase-error level.
  • For imperfect alignment, the SNR is proportional to G and is likewise approximated by a Gamma distribution.
  • When Δ = 0, the imperfect-alignment model reduces to the perfect-CSI case.

D. Multiple-IRS Case

The multiple-IRS extension models geographically separated IRSs with uncorrelated links and CSI-based phase optimization, while also treating Rician fading. Both cases use Gamma approximations for the relevant SNR variables.

  • Multiple IRSs are assumed sufficiently separated that their channels are uncorrelated, which can increase diversity.
  • For the multiple-IRS model, h_i and g_i denote controller-to-IRS and IRS-to-user channels, with elements distributed as CN(0, α_i) and CN(0, β_i).
  • CSI at the controller determines the phase shifts φ_i,n that maximize instantaneous SNR across the IRS links.
  • The multiple-IRS channel variable U and its SNR γ = ρU are approximated using a Gamma distribution with moment-based parameters.
  • The Rician extension models the channel amplitudes with Rician fading and uses the line-of-sight power ratio and common scale parameter in its characterization.
  • For Rician fading, phase shifting yields an SNR proportional to V, whose distribution is approximated by a Gamma law using the moments of ξ_n.

F. Correlated Channels

The correlated-channel analysis models the IRS geometry and channel covariance, approximates the resulting SNR distribution with a Gamma distribution, and evaluates finite-blocklength ADR and ADEP against simulations. Across the reported cases, the derived results generally agree with simulations in the relevant low-error regime, while high-SNR approximation gaps can appear.

  • Correlated-channel model: The correlated-channel model specifies covariance matrices for the BS–IRS and IRS–device links and fixes the IRS phase-shift matrix before approximating the SNR distribution.The correlated links use covariance matrices Rci and Rid, with Rci = Rid = R in the stated special case; the SNR γ is approximated as Gamma distributed.
  • IRS geometry: The IRS is modeled as a rectangular surface with N = NHNV reflecting elements arranged in NH rows and NV columns, with element dimensions dH × dV.Element locations are defined through horizontal and vertical indices using modulus and truncation operations.
  • Numerical results: The derived ADR results closely match simulations across SNR values, while larger N increases ADR through higher passive beamforming gain and short-packet ADR remains roughly 0.5 bits per channel use below Shannon capacity.The numerical curves average over 10000 randomly generated channels, and the comparison uses Shannon capacity as an upper bound.
  • Nakagami-m fading: Under Nakagami-m fading, increasing N from 50 to 100 provides a 2 bits per channel use gain, while Shannon capacity overestimates short-packet performance.The Nakagami parameters are set to m1 = 2 and m2 = 2, and high-SNR approximation errors remain visible for ADEP.

D. Imperfect Phase Alignment

The analysis examines imperfect IRS phase alignment and shows that phase errors reduce ADR and increase ADEP, while larger IRSs improve ADR. Derived results remain consistent with simulations across the considered settings.

  • Imperfect Phase Alignment: With zero phase error, the ADR matches the corresponding perfect-alignment result, supporting the accuracy of the derivation.Derived results also follow the simulation trends for different phase errors and IRS sizes.
  • Imperfect Phase Alignment: Higher phase alignment error lowers ADR, whereas increasing the number of reflecting elements increases ADR.These trends hold across the considered values of phase error and N.
  • Imperfect Phase Alignment: The ADEP analysis considers two IRS sizes and phase alignment errors of Δ = 0, 0.8, and 1.6.The corresponding ADR analysis uses Δ = 0, 1.2, and 2.
  • Imperfect Phase Alignment: At N = 50 and SNR = -30 dB, increasing phase alignment error raises ADEP from 10^-5 to 10^-1.The result demonstrates the importance of accurate channel state information.

E. Multiple-IRS Case

The multiple-IRS analysis shows that adding IRSs and reflecting elements improves ADR and ADEP, while Rician fading benefits from line-of-sight links. Approximation differences arise at large γ.

  • Multiple-IRS Case: ADR increases with SNR and IRS element count, and the derived results generally follow the simulation trends.For N = 40, some differences remain because of approximation error at large γ.
  • Multiple-IRS Case: Increasing IRSs from I = 1 to I = 2 at SNR = −12 dB raises ADR from 6 to 8 bits per channel use.More IRSs provide increased reflecting beamforming and diversity gains.
  • Rician Fading Channel: At N = 50 and SNR = −10 dB, Rician fading achieves 9 bits per channel use versus 6.8 bits per channel use for Rayleigh fading.The comparison shows the advantage of line-of-sight links in IRS-aided communications.
  • Rician Fading Channel: For N = 50, the Rician ADEP can reach 10^-6, with derived results highly accurate when SNR ≤ −37 dB.Differences from simulation appear when SNR ≥ −38 dB because of approximation error at large γ.

G. Correlated Channels

Correlated reflecting-element channels substantially degrade ADR and affect ADEP relative to uncorrelated channels. The paper also derives finite-blocklength IRS-aided URLLC results across seven channel and system cases.

  • Correlated Channels: Correlated channels produce much lower ADR than uncorrelated channels, showing that correlation significantly affects IRS performance.Larger element spacing is recommended to reduce coupling and correlation.
  • Correlated Channels: The correlated-channel ADEP is significantly lower than without correlation, and larger element spacing helps eliminate mutual correlation.The ADEP decreases with SNR in the correlated-channel analysis.
  • Conclusions: The study deploys an IRS to support URLLC in factory automation and uses analytical performance evaluation with numerical validation.The analysis targets ADR and ADEP under the considered channel conditions.
  • Conclusions: The paper derives approximate or accurate closed-form ADR and ADEP expressions for seven IRS-aided URLLC cases.The cases include Rayleigh, direct-link, Nakagami-m, imperfect-alignment, multiple-IRS, Rician, and correlated channels.
  • Conclusions: The conclusions note that instantaneous-CSI-based phase shifts entail high channel-estimation overhead, motivating statistical-CSI designs.Statistical CSI varies more slowly and can reduce estimation overhead.
  • Conclusions: The study assumes one transmit antenna and may require new analysis for multiple antennas and near-field propagation in compact factories.The conventional uniform-plane-wave model may be inaccurate in the near-field regime.

APPENDIX A PROOF OF LEMMA 1

The appendix derives moments and approximations used to obtain the average data rate and decoding-error expressions for the IRS-aided channel model.

  • Proof of Lemma 1: The derivation defines ξ_n as the product of the magnitudes of the two cascaded channel coefficients under Rayleigh fading.The moments of these variables are then used to characterize the aggregate channel quantity.
  • Proof of Lemma 1: The first and second moments of the aggregate random variable are computed using independence among the cascaded channel terms.These moments support subsequent approximations of the average data rate.
  • Proof of Lemma 1: The average data rate is approximated by replacing the channel distribution with an approximate probability density and simplifying its resulting terms.The appendix also uses a logarithmic Gamma function and an approximation for channel dispersion.
  • Proof of Lemma 1: The average decoding-error probability is approximated by linearizing the Q-function around a selected operating point.The linearized expression is substituted into the finite-blocklength error calculation.

APPENDIX E PROOF OF LEMMA 6

Appendix E derives the first and second moments needed for Lemma 6 by introducing auxiliary random variables and applying previously established equalities. The derivation also characterizes a normalized Gaussian variable under conditioning on h.

  • Moment derivation: The appendix calculates the first and second moments of G using auxiliary quantities such as ξ_n = |g_n||h_n| and prior equalities.The first-moment derivation defines ξ_n and invokes equations (A.1) and (A.2), while the second moment uses equations (A.1)–(A.4).
  • Moment derivation: The moment calculations include expectations of products of cosine terms involving phase differences.One required expectation is expressed using sin(∆), cos(∆), sin²(∆), and cos(3∆).
  • Auxiliary variables: The appendix separately derives moments for ξ_i and U from equations (F.1)–(F.8).These intermediate moments support the subsequent calculation of the moments of U.
  • Distributional characterization: Conditioned on h, the relevant variable is circularly symmetric Gaussian, and normalization gives κ ∼ CN(0, 1).The final equality in this part uses a cited prior result.
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