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On Noncoherent Multiple-Antenna Rayleigh Block-Fading Channels at Finite Blocklength

Chao Qi, Tobias Koch

TL;DR

This work addresses the finite-blocklength performance of noncoherent MIMO Rayleigh block-fading channels, deriving a high-SNR normal approximation for the maximum coding rate under blocklength and reliability constraints. The authors extend high-SNR dispersion analysis to the MIMO noncoherent setting by leveraging unitary space-time modulation and asymptotic capacity/dispersions at high SNR, complemented by a rigorous DT-based lower bound and MC-based upper bound. The resulting approximation, valid as both the number of coherence intervals and the SNR grow, clarifies the interplay between diversity, multiplexing, and channel-estimation cost at finite blocklength and provides practical guidance on the optimal number of active transmit antennas. These insights offer a tractable proxy for system design in short-packet wireless protocols and set the stage for analytical exploration of diverse system parameters in finite-blocklength regimes.

Abstract

This paper investigates the maximum coding rate at which data can be transmitted over a noncoherent, multiple-input, multiple-output (MIMO) Rayleigh block-fading channel using an error-correcting code of a given blocklength with a block-error probability not exceeding a given value. A high-SNR normal approximation is derived that becomes accurate as the signal-to-noise ratio (SNR) and the number of coherence intervals over which we code tend to infinity. The obtained normal approximation complements the nonasymptotic bounds that have appeared in the literature, but whose evaluation is computationally demanding. It further lays the theoretical foundation for an analytical analysis of the fundamental tradeoff between diversity, multiplexing, and channel-estimation cost at finite blocklength and finite SNR.

On Noncoherent Multiple-Antenna Rayleigh Block-Fading Channels at Finite Blocklength

TL;DR

This work addresses the finite-blocklength performance of noncoherent MIMO Rayleigh block-fading channels, deriving a high-SNR normal approximation for the maximum coding rate under blocklength and reliability constraints. The authors extend high-SNR dispersion analysis to the MIMO noncoherent setting by leveraging unitary space-time modulation and asymptotic capacity/dispersions at high SNR, complemented by a rigorous DT-based lower bound and MC-based upper bound. The resulting approximation, valid as both the number of coherence intervals and the SNR grow, clarifies the interplay between diversity, multiplexing, and channel-estimation cost at finite blocklength and provides practical guidance on the optimal number of active transmit antennas. These insights offer a tractable proxy for system design in short-packet wireless protocols and set the stage for analytical exploration of diverse system parameters in finite-blocklength regimes.

Abstract

This paper investigates the maximum coding rate at which data can be transmitted over a noncoherent, multiple-input, multiple-output (MIMO) Rayleigh block-fading channel using an error-correcting code of a given blocklength with a block-error probability not exceeding a given value. A high-SNR normal approximation is derived that becomes accurate as the signal-to-noise ratio (SNR) and the number of coherence intervals over which we code tend to infinity. The obtained normal approximation complements the nonasymptotic bounds that have appeared in the literature, but whose evaluation is computationally demanding. It further lays the theoretical foundation for an analytical analysis of the fundamental tradeoff between diversity, multiplexing, and channel-estimation cost at finite blocklength and finite SNR.

Paper Structure

This paper contains 38 sections, 17 theorems, 208 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. System Model
  3. Main Results
  4. High-SNR Normal Approximation
  5. Numerical Results
  6. Engineering Wisdom: Optimal Number of Active Transmit Antennas
  7. Proof of Theorem \ref{['Thm_R']}
  8. Lower Bound on $R^{\ast}(L, T, \epsilon, \rho)$
  9. Upper Bound on $R^{\ast}(L, T, \epsilon, \rho)$
  10. Upper bound on the cardinality of $\mathcal{C}_1$
  11. Upper bound on the cardinality of $\mathcal{C}_2$
  12. Upper bound on the cardinality of $\mathcal{C}$
  13. Conclusion
  14. Proof of Lemma \ref{['DT_Lem_1']}
  15. The USTM Information Density $i^{(U)}(\mathbf{X};\mathbf{Y})$
  16. ...and 23 more sections

Key Result

Theorem 1

Assume that $T\geq n_r+n_t$, $n_r\geq n_t$, and $0<\epsilon<\frac{1}{2}$. Then, at high SNR, where In Equ_Thm_R, $K_{\tilde{I}}(T,\rho)$ and $K_{\tilde{V}}(T,\rho)$ are functions of $T$ and $\rho$ that satisfy and $K_L(L,T,\rho)$ is a function of $L$, $T$, and $\rho$ that satisfies for some $A$, $L_0$, and $\rho_0$ independent of $L$ and $\rho$.

Figures (7)

  • Figure 1: $R^{*}(L,T,\epsilon,\rho)$ as a function of $T$ for a fixed $n=TL$.
  • Figure 2: $R^{\ast}(L, T,\epsilon, \rho)$ as a function of $n$ for a fixed $T$.
  • Figure 3: $R^{\ast}(L, T,\epsilon, \rho)$ as a function of $\epsilon$.
  • Figure 4: $R^{\ast}(L, T,\epsilon, \rho)$ as a function of $\rho$.
  • Figure 5: The minimum error probability $\epsilon$ as a function of the SNR for $R=4$, $T=24$, and $L=7$.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Theorem 1
  • Lemma 1: Dependence-Testing Bound
  • Lemma 2
  • Lemma 3
  • Lemma 4: Polyanskiy-Poor-Verdú---Lemma 47
  • Lemma 5: Berry-Esseen Theorem
  • Lemma 6: Meta-Converse Bound
  • Lemma 7: Wishart matrices
  • Lemma 8
  • Lemma 9
  • ...and 7 more