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Joint Pilot and Payload Power Allocation for Massive-MIMO-enabled URLLC IIoT Networks

Hong Ren, Cunhua Pan, Yansha Deng, Maged Elkashlan, Arumugam Nallanathan

arXiv:1912.12438v1eess.SPcs.IT

TL;DR

The paper addresses uplink resource allocation for massive-MIMO-enabled industrial URLLC when finite blocklength and imperfect CSI make conventional Shannon-capacity allocation unsuitable. It derives rate lower bounds for MRC and ZF and jointly optimizes pilot and payload powers with low-complexity iterative algorithms. Simulations show rapid convergence and better performance than existing benchmark algorithms.

  • Problem

    Finite blocklength and decoding-error requirements make conventional Shannon-capacity-based resource allocation unsuitable for critical IIoT URLLC communications.

  • Method

    The paper derives closed-form achievable-rate lower bounds with imperfect CSI for MRC and ZF, then jointly optimizes pilot and payload powers using low-complexity iterative algorithms.

  • Results

    Simulations demonstrate that the proposed algorithms converge rapidly and outperform existing benchmark algorithms.

  • Takeaways & Limitations

    Finite-blocklength achievable-data-rate expressions are important for resource allocation in massive-MIMO industrial URLLC systems.

Abstract

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The Fourth Industrial Revolution (Industrial 4.0) is coming, and this revolution will fundamentally enhance the way the factories manufacture products. The conventional wired lines connecting central controller to robots or actuators will be replaced by wireless communication networks due to its low cost of maintenance and high deployment flexibility. However, some critical industrial applications require ultra-high reliability and low latency communication (URLLC). In this paper, we advocate the adoption of massive multiple-input multiple output (MIMO) to support the wireless transmission for industrial applications as it can provide deterministic communications similar as wired lines thanks to its channel hardening effects. To reduce the latency, the channel blocklength for packet transmission is finite, and suffers from transmission rate degradation and decoding error probability. Thus, conventional resource allocation for massive MIMO transmission based on Shannon capacity assuming the infinite channel blocklength is no longer optimal. We first derive the closed-form expression of lower bound (LB) of achievable uplink data rate for massive MIMO system with imperfect channel state information (CSI) for both maximum-ratio combining (MRC) and zero-forcing (ZF) receivers. Then, we propose novel low-complexity algorithms to solve the achievable data rate maximization problems by jointly optimizing the pilot and payload transmission power for both MRC and ZF. Simulation results confirm the rapid convergence speed and performance advantage over the existing benchmark algorithms.

I. INTRODUCTION

Industrial wireless networks must replace wired connections while meeting ultra-reliability and low-latency requirements under finite blocklength. The paper develops massive-MIMO rate bounds and joint power-allocation algorithms because Shannon-capacity-based allocation is not optimal in this setting.

  • Industrial applications require deterministic communications with ultra-reliability of 1 −10−9 and latency of 1 ms.
  • Existing approaches often adapt upper layers while leaving the physical layer unchanged, but reported standards cannot meet the stringent requirements of critical industrial applications.
  • Finite blocklength causes decoding errors, making Shannon capacity inapplicable and making short-packet rate expressions non-convex in SNR or blocklength.
  • Massive MIMO can support multiple devices simultaneously and provide deterministic communications through spatial degrees of freedom and channel hardening.
  • The paper derives closed-form achievable-rate lower bounds with imperfect CSI for MRC and ZF receivers, treating short-packet effects as a penalty relative to Shannon capacity.
  • It jointly optimizes pilot and payload power under URLLC constraints, using low-complexity algorithms whose simulations show rapid convergence and better performance than benchmark schemes, especially Shannon-capacity-based ones.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system models uplink URLLC transmission from many single-antenna industrial devices to a massive-MIMO controller, using TDD channel estimation with finite pilot and payload resources.

  • A. Factory System Model: The central controller serves K industrial devices, such as actuators and robots, over an uplink massive-MIMO link.The controller has M antennas, while each device has one antenna and M ≫ K.
  • A. Factory System Model: Devices transmit emergency measurements or operating states so the controller can process them and provide prompt feedback.The analysis focuses on uplink transmission; downlink solutions can be derived similarly.
  • A. Factory System Model: Each device channel is decomposed into a large-scale gain containing pathloss and shadowing and a small-scale fading vector.The channel matrix is H = [h_1, h_2, · · ·, h_K].
  • B. Channel Estimation in Massive MIMO URLLC: TDD is adopted because devices may not support the processing and feedback burden associated with FDD channel estimation.All devices receive orthogonal pilot resources, and channel reciprocity provides downlink CSI from uplink estimation.
  • B. Channel Estimation in Massive MIMO URLLC: Each transmission frame contains l_p pilot symbols and l_d data symbols, with total length L = l_p + l_d.The corresponding durations are t_p = l_p/B and t_d = l_d/B.
  • B. Channel Estimation in Massive MIMO URLLC: The MMSE channel estimate is paired with an independent estimation error distributed as CN(0, δ_k I_M).The received training noise matrix has independently generated Gaussian elements.

C. Achievable Data Rate for Massive MIMO URRLC

The paper develops finite-blocklength achievable-rate lower bounds for massive-MIMO URLLC with imperfect CSI, using MRC and ZF detection and exploiting channel hardening for tractable allocation.

  • C. Achievable Data Rate for Massive MIMO URRLC: The frame-level rate accounts for pilot symbols and includes a finite-blocklength decoding penalty governed by channel dispersion.As L approaches infinity, the rate approaches the classic Shannon-capacity expression (1 − β) log_2(1 + γ_k).
  • C. Achievable Data Rate for Massive MIMO URRLC: The analysis considers two low-complexity detection schemes: maximum-ratio combining (MRC) and zero-forcing (ZF).The corresponding SINR expressions differ and lead to separate power-allocation treatments.
  • C. Achievable Data Rate for Massive MIMO URRLC: The receiver processes the estimated-channel signal with a linear detector based on the estimated channel matrix, treating estimation-error terms as interference and noise.The resulting SINR γ_k determines the kth device’s instantaneous achievable rate.
  • C. Achievable Data Rate for Massive MIMO URRLC: Massive-MIMO channel hardening motivates deriving the achievable-rate lower bound as a function of large-scale fading rather than small-scale fading.This reduces computational delay because optimized allocations can be reused across consecutive fading blocks until large-scale parameters change.
  • C. Achievable Data Rate for Massive MIMO URRLC: Channel estimation is performed at each coherence interval, whereas power allocation is updated only when large-scale fading gains change.This separation makes the optimization solutions applicable over a larger time scale, which is appealing for URLLC.
  • C. Achievable Data Rate for Massive MIMO URRLC: The lower bounds are valid for any antenna count, and their gap from the actual ergodic rate decreases as the number of antennas grows.The paper therefore uses these bounds for massive-MIMO analysis and optimization.
  • C. Achievable Data Rate for Massive MIMO URRLC: The optimization jointly selects pilot and payload powers, and simulations verify the tightness of the derived lower bounds.The resulting solutions are described as applicable on a large time scale.

D. Problem Formulation

The paper formulates weighted sum-rate maximization under finite-blocklength imperfect-CSI rates, per-device rate requirements, and energy constraints, then treats MRC and ZF separately.

  • D. Problem Formulation: Pilot and data transmission powers are jointly optimized for each device under imperfect CSI and finite blocklength.The objective is a weighted sum-rate maximization problem with polynomial-time algorithms sought.
  • D. Problem Formulation: The weighted sum-rate objective uses device weights to promote fairness, while constraints impose minimum rates and per-device energy limits.The rate expressions differ between MRC and ZF.
  • D. Problem Formulation: Power control with interference is NP-hard even under perfect CSI and becomes more complicated with the paper’s more general setting.This motivates the search for efficient low-complexity optimization algorithms.
  • D. Problem Formulation: Auxiliary variables are introduced to transform the original optimization into an equivalent problem with the same power-allocation solutions and objective value.The reformulated problem is then solved separately for MRC and ZF because their SINR expressions differ.

III. WEIGHTED SUM DATA RATE FOR MRC

The MRC weighted-sum-rate problem is difficult because finite-blocklength penalties and non-convex constraints complicate direct optimization. The paper develops tight logarithmic lower-bound approximations that enable iterative geometric-programming solutions.

  • III. WEIGHTED SUM DATA RATE FOR MRC: G(x) is concave, and the proposed log-function approximation F(x) is tight at the expansion point.The paper compares F(x) with a linear approximation and reports higher accuracy for F(x) across the tested region.
  • III. WEIGHTED SUM DATA RATE FOR MRC: Finite blocklength introduces the decoding-error penalty akG(χk), making the MRC weighted-sum-rate optimization difficult.The problem also contains non-convex constraints and is not generally a difference-of-convex program.
  • III. WEIGHTED SUM DATA RATE FOR MRC: The log-function approximation converts the original optimization into a geometric program with an obtainable optimal solution.A logarithmic change of variables equivalently transforms the geometric program into a convex optimization problem solvable by interior-point methods.
  • III. WEIGHTED SUM DATA RATE FOR MRC: The achievable-rate lower bound for ln(1 + χk) is tight at the chosen approximation point and is combined with the approximation of G(χk).These bounds are used to maximize a lower bound on the objective rather than the original objective directly.
  • III. WEIGHTED SUM DATA RATE FOR MRC: Algorithm 1 iteratively solves the approximated problem until the objective change falls below the error tolerance.The algorithm is initialized with a feasible power allocation and uses CVX to solve each geometric-program subproblem.

1) Initialization of Algorithm 1:

Algorithm 1 requires a carefully selected feasible initial power allocation because energy-feasible allocations may violate minimum SINR requirements. An auxiliary feasibility problem provides an alternative initialization procedure.

  • 1) Initialization of Algorithm 1:: Random power allocations satisfying per-device energy constraints may still violate minimum SINR requirements.The initialization therefore must be chosen carefully before Algorithm 1 starts.
  • 1) Initialization of Algorithm 1:: An auxiliary variable ϕ is introduced to construct an alternative optimization problem for finding an initial allocation.The auxiliary problem is always feasible because it admits at least ϕ = 0 with an associated allocation.
  • 1) Initialization of Algorithm 1:: The original problem is feasible if the optimal auxiliary value satisfies ϕ ≥ 1.The paper assumes Problem (24) is always feasible and that the optimal ϕ in Problem (37) is no smaller than one.
  • 1) Initialization of Algorithm 1:: Algorithm 1 produces a nondecreasing objective sequence with an upper bound imposed by the devices’ energy constraints.Consequently, the algorithm is guaranteed to converge.
  • 1) Initialization of Algorithm 1:: Because the original problem is non-convex, Algorithm 1 generally cannot obtain the globally optimal solution.The algorithm can converge to a KKT point or local optimal solution, and its limit depends on the initial input.

A. Algorithm Design

For ZF, the paper reformulates the SINR constraints and uses local monomial approximations to obtain geometric-program subproblems. The resulting iterative Algorithm 2 converges to a feasible solution.

  • A. Algorithm Design: ZF requires separate derivations because its SINR expression differs from the MRC case.The ZF SINR is reformulated before constructing the optimization algorithm.
  • A. Algorithm Design: The ZF SINR numerator is a posynomial, so its constraint cannot be directly transformed into the required geometric-program format.The paper introduces Theorem 3 to address this mismatch.
  • A. Algorithm Design: Theorem 3 replaces polynomial terms with best local monomial approximations at each iteration.The approximation makes the left-hand side a posynomial and the right-hand side a monomial, satisfying geometric-program conditions.
  • A. Algorithm Design: Algorithm 2 solves a geometric program in each iteration using CVX and updates the approximation parameters and power allocation.It begins from a feasible allocation and terminates when the objective change is below the error tolerance.
  • A. Algorithm Design: Algorithm 2 is guaranteed to converge and converges to a feasible solution of Problem (26).The convergence proof shows that the previous iteration’s solution remains feasible for the next approximated problem.

V. SIMULATION RESULTS

Simulations evaluate lower-bound tightness, convergence, and performance under finite-blocklength URLLC settings. The proposed joint pilot-and-payload allocation converges rapidly and is especially advantageous for ZF at low energy limits and larger device counts.

  • V. SIMULATION RESULTS: The derived data-rate lower bounds are tight for both MRC and ZF across the tested numbers of transmit antennas.The ZF curves are almost overlapped, supporting use of the bounds for optimization instead of directly optimizing the expectation expression.
  • V. SIMULATION RESULTS: 2 or 3 iterations are sufficient for both proposed algorithms to converge across various energy limits.This result demonstrates the low complexity of the proposed algorithms.
  • V. SIMULATION RESULTS: Shannon-capacity-based conventional allocation is more likely to violate rate requirements at small energy limits and cannot guarantee URLLC QoS there.At high energy, its penalty term can become negligible and its performance can approach the proposed algorithm.
  • V. SIMULATION RESULTS: Jointly optimizing pilot and payload power outperforms fixed pilot-power allocation, with the gain especially evident for ZF at low energy limits.Joint allocation can shift power toward channel estimation when energy is scarce, while gains become marginal at high energy.
  • V. SIMULATION RESULTS: As the number of devices increases, average weighted sum rate rises for all schemes except the conventional algorithm.For ZF, the proposed algorithm’s gain over fixed pilot power increases with device count, whereas MRC shows similar performance between the two.
  • V. SIMULATION RESULTS: Smaller blocklength creates a larger gap between the proposed algorithm and the Shannon-capacity upper bound.The gap decreases as blocklength increases, and the proposed algorithm outperforms fixed pilot power for both MRC and ZF.

VI. CONCLUSIONS

The paper studies uplink massive-MIMO resource allocation for critical IIoT communications under finite channel blocklength. It derives achievable-rate bounds and jointly optimizes pilot and payload powers with low-complexity algorithms that converge rapidly and outperform benchmarks.

  • The study targets critical IIoT scenarios where multiple robots or actuators transmit URLLC signals simultaneously under finite channel blocklength.
  • Closed-form achievable data-rate lower bounds are derived for imperfect CSI with both MRC and ZF receivers under short-packet transmission.
  • The weighted sum-rate problem jointly optimizes pilot and payload power subject to energy, minimum-rate, and decoding-error requirements.
  • Low-complexity iterative algorithms solve the non-convex optimization problem and converge rapidly.
  • The proposed algorithm outperforms existing benchmark algorithms, especially schemes based on conventional Shannon capacity.

APPENDIX A PROOF OF THEOREM 1

The appendix derives the MRC and ZF achievable-rate expressions through instantaneous-SINR transformations, distributional identities, and substitutions. It also establishes concavity and monotonicity properties used in the analysis.

  • For MRC, the proof begins from the instantaneous SINR and evaluates expectations using independent random variables and a complex Wishart identity.
  • Substitution of the derived expectation into the earlier SINR expression yields the MRC quantity used in the theorem.
  • For ZF, the relation AH ˆH = I_K makes the desired coefficient one and the cross-user coefficients zero, simplifying the instantaneous SINR.
  • The ZF derivation then represents the channel columns through independent complex-normal variables and applies the associated expectation identity.
  • The appendix proves concavity by combining concavity of T(x) and √x, then establishes the required inequality through derivative and monotonicity arguments.

APPENDIX E PROOF OF THEOREM 3

The appendix proves Theorem 3 using derivative relations, an auxiliary lemma, and inequalities applied across all device variables. The argument concludes by combining the resulting inequalities.

  • The proof starts by expressing partial derivatives of W(˘x) and Y(˘x) with respect to each x_i.
  • Substituting the specified values of x, λ, and τ_i establishes the second equality in equation (44).
  • A lemma provides an inequality for any non-negative reference vector ˘x, with equality only when x equals ˘x.
  • Applying the lemma separately to each x_i and multiplying the resulting K inequalities completes the proof.
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