Quasi-Static Multiple-Antenna Fading Channels at Finite Blocklength (1311.2012v2)

Abstract: This paper investigates the maximal achievable rate for a given blocklength and error probability over quasi-static multiple-input multiple-output (MIMO) fading channels, with and without channel state information (CSI) at the transmitter and/or the receiver. The principal finding is that outage capacity, despite being an asymptotic quantity, is a sharp proxy for the finite-blocklength fundamental limits of slow-fading channels. Specifically, the channel dispersion is shown to be zero regardless of whether the fading realizations are available at both transmitter and receiver, at only one of them, or at neither of them. These results follow from analytically tractable converse and achievability bounds. Numerical evaluation of these bounds verifies that zero dispersion may indeed imply fast convergence to the outage capacity as the blocklength increases. In the example of a particular $1 \times 2$ single-input multiple-output (SIMO) Rician fading channel, the blocklength required to achieve $90\%$ of capacity is about an order of magnitude smaller compared to the blocklength required for an AWGN channel with the same capacity. For this specific scenario, the coding/decoding schemes adopted in the LTE-Advanced standard are benchmarked against the finite-blocklength achievability and converse bounds.

• The paper establishes that outage capacity precisely predicts performance in quasi-static MIMO fading channels even at finite blocklength.
• The paper demonstrates that channel dispersion is zero, ensuring rapid convergence to outage capacity regardless of CSI availability.
• Numerical evaluations reveal that required blocklengths for 90% capacity in Rician fading models are significantly lower than in AWGN channels, guiding LTE-Advanced design.

Quasi-Static Multiple-Antenna Fading Channels at Finite Blocklength

This paper provides a thorough analysis of the achievable rates over quasi-static MIMO fading channels with finite blocklength, a matter of particular interest due to the latency constraints in modern communication systems. The authors consider scenarios both with and without CSIT and CSIR, unveiling insights into the outage capacity as a reliable performance measure under these conditions.

Key Findings

1. Outage Capacity and Finite Blocklength: The research establishes that the outage capacity can accurately predict the performance of slow-fading channels even at finite blocklength. This defies the general expectation which assumes such capacity measures are only asymptotically valid.
2. Zero Channel Dispersion: Uniquely, the channel dispersion is shown to be zero, indicating rapid convergence to the outage capacity as blocklength increases. This result holds regardless of the availability of CSI at either the transmitter or receiver.
3. Numerical Evaluations: The evaluation involves comparative analysis with additive white Gaussian noise (AWGN) channels, reinforcing that for a Rician fading model with specific parameters, the required blocklength to achieve 90% capacity is considerably smaller than that needed for corresponding AWGN channels.
4. Implications for LTE-Advanced: The research’s bounds serve as benchmarks against which the performance of LTE-Advanced coding schemes are evaluated, showing that these practical codes achieve approximately 85% of the normal approximation rate, offering insights into code design and performance tuning under delay constraints.

Theoretical Contributions

• Achievability and Converse Bounds: The paper presents sharp finite-blocklength achievability bounds using innovative geometric arguments revolving around angles in high-dimensional vector spaces. The converse bounds are derived using refined techniques involving water-filling strategies.
• Normal Approximation: A normal approximation to the achievable rate was derived and demonstrated to be closely aligned with both the achievability and converse bounds for extended blocklengths.
• SIMO and Isotropic Codes: The paper specifically addresses the case of SIMO channels and isotropic code design, offering a more nuanced understanding of rate constraints in these specific configurations.

Proof Techniques

1. Meta-Converse Theorem: This foundational result guides the bounding process for converse results in diverse scenarios addressed by the paper.
2. Cramer-Esseen Type Central Limit Theorem: This theorem is employed to approximate the finite blocklength error probabilities, offering insights into the rapid convergence behavior of achievable rates.
3. Kappa-Beta Bound: The achievability bound employs the kappa-beta bound on transformed channel models, specifically leveraging the geometry of subspaces.

Practical Implications

The findings are instrumental for designing communication strategies in systems where short packet transmission is paramount, such as in IoT and ultra-reliable low-latency communications (URLLC). The insights into zero dispersion emphasize the significant reduction in rate loss encountered due to finite blocklength, thus impacting how these systems will be optimized.

Conclusion

The paper substantiates that the outage capacity stands as a meaningful performance metric for MIMO channels even under stringent blocklength constraints. Through detailed theoretical developments and comprehensive numerical evaluations, it provides valuable tools and benchmarks for modern communication standards and protocols, with anticipated impacts on future technological advancements.