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Short-Packet Communications over Multiple-Antenna Rayleigh-Fading Channels

Giuseppe Durisi, Tobias Koch, Johan Östman, Yury Polyanskiy, Wei Yang

arXiv:1412.7512v3cs.IT

TL;DR

Short-packet mission-critical communications require jointly managing reliability, throughput, latency, and channel-estimation overhead, but prior analyses were often asymptotic. The paper derives finite-blocklength, finite-SNR bounds for multiple-antenna Rayleigh block-fading channels and uses them to characterize design tradeoffs. The bounds are tight for short packets, identify rate-maximizing antenna and diversity choices, and show that infinite-blocklength capacity metrics can misestimate throughput.

  • Problem

    Prior results were often asymptotic in packet length or SNR, leaving the reliability–throughput–latency tradeoff and antenna requirements for short-packet mission-critical communications insufficiently characterized.

  • Method

    The paper derives finite-blocklength achievability and converse bounds for maximum coding rate in Rayleigh block-fading MIMO channels with unknown channel realizations at the receiver.

  • Results

    The bounds tightly delimit maximum coding rates for short packets and identify how many transmit antennas and time-frequency diversity branches maximize rate under channel-estimation overhead.

  • Takeaways & Limitations

    Short-packet system design must balance resource-exploitation gains against channel-estimation costs rather than rely solely on ergodic or outage capacity.

Abstract

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Motivated by the current interest in ultra-reliable, low-latency, machine-type communication systems, we investigate the tradeoff between reliability, throughput, and latency in the transmission of information over multiple-antenna Rayleigh block-fading channels. Specifically, we obtain finite-blocklength, finite-SNR upper and lower bounds on the maximum coding rate achievable over such channels for a given constraint on the packet error probability. Numerical evidence suggests that our bounds delimit tightly the maximum coding rate already for short blocklengths (packets of about 100 symbols). Furthermore, our bounds reveal the existence of a tradeoff between the rate gain obtainable by spreading each codeword over all available time-frequency-spatial degrees of freedom, and the rate loss caused by the need of estimating the fading coefficients over these degrees of freedom. In particular, our bounds allow us to determine the optimal number of transmit antennas and the optimal number of time-frequency diversity branches that maximize the rate. Finally, we show that infinite-blocklength performance metrics such as the ergodic capacity and the outage capacity yield inaccurate throughput estimates.

I. INTRODUCTION

The paper studies short-packet communication over multiple-antenna Rayleigh block-fading channels, where reliability, throughput, latency, and channel-estimation overhead must be balanced. It develops finite-blocklength bounds to characterize this tradeoff and guide antenna, diversity, and coding choices.

  • Motivation: Multiple antennas provide spatial degrees of freedom that can reduce error probability through diversity or increase data rate through spatial multiplexing.These two benefits cannot generally be harvested concurrently, creating a diversity–multiplexing tradeoff.
  • Motivation: Mission-critical machine-type communications require short packets with stringent latency, reliability, and availability requirements.Industrial automation applications may require about 100 bits within 100 µs with a 10^-9 packet error rate.
  • Research gap: Most existing results are asymptotic in packet length, SNR, or both, leaving their relevance to mission-critical machine-type communications unclear.The paper addresses this gap with a nonasymptotic analysis that also accounts for channel-estimation overhead.
  • Contributions: The paper derives nonasymptotic achievability and converse bounds on maximum coding rate for given SNR, packet size, and packet reliability in Rayleigh block-fading channels.The model allows codewords to span multiple fading realizations in time and frequency, with channel statistics known but no a priori channel realization knowledge at the receiver.
  • Contributions: Numerical results show the bounds tightly delimit maximum coding rates and identify throughput-maximizing transmit-antenna choices as time-frequency diversity varies.The analysis also compares diversity, multiplexing, and partial antenna deactivation while accounting for channel-estimation overhead.
  • System model: The study evaluates finite-blocklength rate behavior using multiple-antenna fading models, matrix-valued inputs and outputs, and independent Rayleigh fading across coherence intervals.The channel remains constant for nc time-frequency slots, while fading and noise take independent realizations across successive coherence intervals.

III. MAXIMUM CODING RATE

This section defines maximum coding rate for finite-blocklength Rayleigh block-fading channels and relates it to classical asymptotic performance metrics. It motivates nonasymptotic bounds because ergodic capacity, outage capacity, and the DMT have limitations for short packets.

  • Maximum coding rate: A channel code spans l coherence intervals, each containing n_c channel uses, so the blocklength is n = l n_c.The code includes an encoder, per-subcodeword power constraints, and a decoder subject to a maximum error-probability constraint.
  • Maximum coding rate: The maximal coding rate R*(l, n_c, ε, ρ) is the largest rate achievable for blocklength n = l n_c, packet error probability ε, and SNR ρ.Its dependence on n_c, l, and the antenna configuration exposes how channel characteristics affect the rate–reliability tension.
  • Relation to previous results: Most prior characterizations are asymptotic in packet length, coherence interval, SNR, or combinations of these limits.The section therefore develops nonasymptotic bounds on the maximum coding rate for short-packet communications.
  • Relation to previous results: Ergodic capacity is the limit as l → ∞, whereas outage capacity is the limit as n_c → ∞ for fixed l, ε, and ρ.These limiting metrics describe different asymptotic regimes of maximum coding rate.
  • Relation to previous results: Ergodic capacity does not depend on packet reliability ε, while outage-capacity analyses omit channel-estimation overhead as n_c becomes large.The resulting limitations are especially relevant when packets are short and channel learning is costly.
  • Relation to previous results: The DMT compactly describes diversity–multiplexing tradeoffs, but its high-SNR formulation and finite-SNR extensions remain asymptotic in blocklength.These restrictions limit its significance for the short-packet scenarios studied here.

A. Output Distribution Induced by USTM Inputs

This section derives a closed-form channel-output density for scaled isotropically distributed inputs with orthonormal columns. The construction also permits using only a selected subset of the available transmit antennas.

  • Output distribution: USTM inputs are scaled isotropically distributed matrices with orthonormal columns, and the analysis permits using only e_m_t of m_t available transmit antennas.The corresponding output density is a key ingredient in the nonasymptotic coding-rate bounds.
  • Output distribution: For n_c ≥ m_t + m_r, the output density is expressed using the ordered nonzero singular values of the received matrix Y.The construction defines q = min{e_m_t, m_r}, p = max{e_m_t, m_r}, and μ = ρ n_c/e_m_t.
  • Output distribution: The density formula uses a p × p real matrix whose entries depend on the squared singular values and a regularized incomplete Gamma function.The regularized incomplete Gamma function is denoted by the expression introduced after the matrix definition.
  • Output distribution: The resulting density expression is reported as easier to compute and more numerically stable than an alternative expression.This comparison is stated as a numerical-evaluation remark rather than a change to the underlying distribution.

B. USTM Dependence-Testing (DT) Lower Bound

This section constructs nonasymptotic lower and upper bounds on maximum coding rate using USTM-induced output distributions. The bounds are formulated through dependence-testing and meta-converse arguments under the channel and power constraints.

  • USTM bounds: The DT lower bound uses the USTM-induced output distribution and ordered eigenvalues of independent Gaussian-matrix products.The eigenvalues enter the bound through the function ψ_e_m_t and its logarithm.
  • USTM bounds: The MC upper bound uses the USTM-induced output distribution as an auxiliary output distribution.The resulting upper bound holds for every blocklength n and every 0 < ε < 1.
  • USTM bounds: For the upper-bound construction, the received-matrix eigenvalues are formed from i.i.d. USTM-distributed outputs, while admissible diagonal covariance matrices satisfy tr{Σ_k} = n_cρ.The covariance matrices are constrained separately for each coherence interval.
  • USTM bounds: The upper bound can be relaxed using a result from finite-blocklength information theory to facilitate numerical evaluation.The relaxed upper bound is the one used in the numerical evaluations reported later.
  • USTM bounds: A converse bound under a less stringent per-codeword power constraint can be obtained by optimizing over the admissible covariance matrices.The main formulation instead uses the per-coherence-interval power constraint.

VI. BOUNDS ON THE CODING RATE FOR ORTHOGONAL SPACE-TIME CODES

This section studies orthogonal space-time codes that use transmit antennas for full spatial diversity and compares their rate bounds with the general bounds. The comparison characterizes the rate penalty of diversity-exploiting transmission.

  • Bounds for orthogonal space-time codes: Orthogonal space-time codes are analyzed as inner codes combined with an outer code defined as in the general channel-code construction.Their purpose is to provide transmit diversity and improve reliability.
  • Bounds for orthogonal space-time codes: The resulting bounds concern transmission strategies that use the available transmit antennas to provide full spatial diversity.They are compared with the general bounds developed for unrestricted antenna usage.
  • Bounds for orthogonal space-time codes: The comparison with the general bounds identifies the rate penalty incurred by employing diversity-exploiting transmission strategies.This frames orthogonal space-time coding as a reliability-oriented design whose throughput cost can be evaluated.

A. 2 × 2 Case: Alamouti

The 2 × 2 analysis evaluates finite-blocklength Alamouti transmission without prior receiver CSI by deriving the induced output distribution and using it to bound the maximum coding rate. The construction uses an Alamouti input matrix whose first column follows a hyperspherical isotropic distribution, while the resulting matrix is not USTM.

  • A. 2 × 2 Case: Alamouti: The 2 × 2 scheme uses an Alamouti space-time inner code and is analyzed when channel state information is unavailable a priori at the receiver.The analysis first derives the scheme’s output distribution, then applies a DT lower bound and an MC upper bound on the maximum coding rate.
  • A. 2 × 2 Case: Alamouti: The coherence interval n_c is assumed even, and the input matrix is constructed from an n_c-dimensional vector satisfying ||a_k||2 = ρn_c/2.The mapping from the input vector to its paired output vector follows the Alamouti rule.
  • A. 2 × 2 Case: Alamouti: The first column A is uniformly distributed over the hypersphere of radius ρn_c/2, corresponding to USTM for a single transmit antenna.This distribution is used to obtain the channel-output pdf for the 2 × 2 construction.
  • A. 2 × 2 Case: Alamouti: For m_t = m_r = 2 and even n_c ≥ 4, the channel-output pdf is expressed using the first and third largest eigenvalues of a scaled 4 × 4 Gram matrix.The relevant eigenvalues are denoted Σ1 and Σ3, with realizations σ1 and σ3.
  • A. 2 × 2 Case: Alamouti: Although A is isotropically distributed, the resulting matrix X is not distributed according to USTM.This distinguishes the Alamouti-induced input distribution from the USTM distribution used elsewhere in the analysis.
  • A. 2 × 2 Case: Alamouti: Treating Alamouti as part of the channel yields lower and upper bounds on its achievable maximum coding rate based on the closed-form output density.The displayed bounds are given for the cases n_c > 4 and n_c = 4.

1) DT lower bound:

The DT lower bound is developed from the channel-output distribution induced by Gaussian matrix variables and the Alamouti mapping. The construction tracks the first and third largest eigenvalues of a scaled Gram matrix, whose positive distinctness holds almost surely.

  • 1) DT lower bound:: The achievability bound is based on the DT bound.This provides the lower bound on the maximum coding rate for the Alamouti-related construction.
  • 1) DT lower bound:: The construction uses independent complex Gaussian n_c × 2 matrices with i.i.d. CN(0, 1) entries.These matrices enter the random-matrix representation used to formulate the bound.
  • 1) DT lower bound:: The associated matrix expression includes the diagonal factors 1 + ρn_c/2, 1 + ρn_c/2, 1, ..., 1.These factors appear in the bound’s random-matrix construction.
  • 1) DT lower bound:: The bound uses Σ_k,1 and Σ_k,3, the first and third largest eigenvalues of the scaled Gram matrix formed from the random matrices.Their realizations are positive and distinct almost surely.

2) MC upper bound:

The numerical results show that the finite-blocklength bounds characterize rate optimization across diversity branches and antennas, while asymptotic capacity metrics can be inaccurate because they miss channel-estimation overhead.

  • Control signaling: The achievability and converse bounds tightly delimit the maximum coding rate for short packets and identify rate-maximizing antenna choices.For the 2×2 case, the bounds are tight and reveal an optimal coherence interval; for the 4×4 case, the MC and DT bounds also closely characterize the rate.
  • Control signaling: The optimal coherence interval balances channel-estimation cost against the number of available time-frequency diversity branches.When nc is below the optimum, estimation dominates; when nc exceeds it, limited time-frequency diversity becomes the bottleneck.
  • Asymptotic comparisons: Outage capacity overestimates the maximum coding rate by a factor of two at nc = 4, while the ergodic-capacity lower bound overestimates it by a factor of four at nc = 168.Outage capacity is accurate only in the quasi-static regime and misses estimation overhead; the ergodic approximation improves as nc decreases.
  • Control signaling: For the 2×2 system, both transmit antennas maximize the achievability bound for 1 ≤ l ≤ 21, whereas one antenna is optimal for l > 21.The same switching behavior appears for the ergodic-capacity lower bound and the MC upper bound.
  • 2×2 system: The 2×2 Alamouti scheme is nearly optimal for small l, but increasing l makes a single active antenna preferable because estimation cost outweighs added spatial degrees of freedom.The DT lower-bound gap to the Alamouti converse grows with l.
  • 4×4 system: For the 4×4 system, four antennas are optimal when 1 ≤ l < 12, three when l = 12, and two when 12 < l ≤ 21.Using all four antennas for spatial diversity is suboptimal even when the number of time-frequency branches is limited.
  • Ultra-reliable communication: At error probability ϵ = 10^-5, the gap between optimal and orthogonal space-time schemes becomes smaller because stronger reliability requirements favor transmit diversity.This behavior is reported for both 2×2 and 4×4 configurations.

VIII. CONCLUSIONS

The paper develops finite-blocklength bounds for optimizing antennas and time-frequency diversity under channel-estimation costs, while showing that infinite-blocklength metrics misestimate short-packet rates.

  • VIII. CONCLUSIONS: The bounds determine rate-maximizing numbers of time-frequency diversity branches and transmit antennas for fixed packet size.The optimal choice balances resource-exploitation gains against channel-estimation costs.
  • VIII. CONCLUSIONS: For l > 7, the computationally evaluated upper bound restricts the supremum over {Σ_k} to matrices of the form given in (51).
  • VIII. CONCLUSIONS: The optimal antenna configuration indicates whether available antennas should provide transmit diversity or spatial multiplexing.
  • VIII. CONCLUSIONS: Numerical results show that outage and ergodic capacity provide inaccurate maximum-coding-rate estimates for short packets.These metrics also fail to capture the tradeoff among reliability, throughput, latency, and channel-estimation overhead.

APPENDIX A PROOF OF THEOREM 1

The proof of Theorem 1 uses channel symmetry and information-density distributions to derive an achievability bound, then optimizes over effectively used transmit antennas.

  • APPENDIX A PROOF OF THEOREM 1: The transmitter uses emt antennas, producing an emt × mr MIMO Rayleigh block-fading channel.
  • APPENDIX A PROOF OF THEOREM 1: Channel outputs are independent across blocks, allowing the information density to decompose across the l block-fading branches.
  • APPENDIX A PROOF OF THEOREM 1: Unitary invariance makes the information-density distribution independent of Uk, so all Uk can be fixed to a common matrix without loss of generality.
  • APPENDIX A PROOF OF THEOREM 1: The dependence-testing theorem yields a code, and algebra shows its information density has the distribution of S_k, emt.
  • APPENDIX A PROOF OF THEOREM 1: Minimizing over effectively used transmit antennas and solving the resulting inequality for rate yields the achievability bound.

APPENDIX B PROOF OF THEOREM 2

The proof of Theorem 2 applies the meta-converse with an auxiliary USTM-induced output distribution, reducing the optimization to diagonal transmit covariance eigenvalues.

  • APPENDIX B PROOF OF THEOREM 2: The meta-converse theorem for maximal error probability supplies an upper-bound framework for fixed effectively used transmit antennas.
  • APPENDIX B PROOF OF THEOREM 2: The auxiliary distribution uses the USTM-induced output density fY.
  • APPENDIX B PROOF OF THEOREM 2: The converse optimizes over codewords satisfying the power constraint and uses β1−ϵ together with information density.
  • APPENDIX B PROOF OF THEOREM 2: For each codeword, Σk is diagonal with nonnegative entries equal to the eigenvalues of Xk^H Xk.
  • APPENDIX B PROOF OF THEOREM 2: Unitary invariance makes β1−ϵ depend on the codeword only through {Σk}, replacing the codeword infimum with an infimum over these matrices.
  • APPENDIX B PROOF OF THEOREM 2: Under the auxiliary output law, the information density is represented through sums of the variables T_k, emt, leading to the converse bound after minimizing over emt.
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