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From Bayesian Asymptotics to General Large-Scale MIMO Capacity

Sheng Yang, Richard Combes

TL;DR

We present a unified framework that connects Bayesian asymptotics with information theory to analyze the asymptotic capacity of general large-scale MIMO channels, including nonlinearities and hardware impairments. The central result expresses capacity in the large-$n_r$ limit as $C(P)=\frac{d}{2}\log\frac{n_r}{2\pi e}+\log\mathsf{JF}(\lambda^*)+o(1)$, where the (tilted) Jeffreys factor $\mathsf{JF}(\lambda^*)$ is determined by the Fisher information of the single-antenna output and the optimal input tilting $\lambda^*$. The asymptotically optimal input is the Jeffreys prior, and the paper provides a practical recipe for capacity computation, constellation design via a compander-like transform, and a low-complexity receiver that quantizes outputs into a finite number of bins while preserving near-capacity performance. Applications span AWGN with clipping, low-resolution ADCs, energy-detection, imperfect CSIR, and optical Poisson channels, and extensions to non-i.i.d. settings are discussed. Overall, the Fisher information is shown to govern the channel’s asymptotic behavior, enabling scalable capacity analysis and guiding practical transceiver design for next-generation large-scale MIMO systems.

Abstract

We present a unifying framework that bridges Bayesian asymptotics and information theory to analyze the asymptotic Shannon capacity of general large-scale MIMO channels including ones with nonlinearities or imperfect hardware. We derive both an analytic capacity formula and an asymptotically optimal input distribution in the large-antenna regime, each of which depends solely on the single-output channel's Fisher information through a term we call the (tilted) Jeffreys factor. We demonstrate how our method applies broadly to scenarios with clipping, coarse quantization (including 1-bit ADCs), phase noise, fading with imperfect CSI, and even optical Poisson channels. Our asymptotic analysis motivates a practical approach to constellation design via a compander-like transformation. Furthermore, we introduce a low-complexity receiver structure that approximates the log-likelihood by quantizing the channel outputs into finitely many bins, enabling near-capacity performance with computational complexity independent of the output dimension. Numerical results confirm that the proposed method unifies and simplifies many previously intractable MIMO capacity problems and reveals how the Fisher information alone governs the channel's asymptotic behavior.

From Bayesian Asymptotics to General Large-Scale MIMO Capacity

TL;DR

We present a unified framework that connects Bayesian asymptotics with information theory to analyze the asymptotic capacity of general large-scale MIMO channels, including nonlinearities and hardware impairments. The central result expresses capacity in the large- limit as , where the (tilted) Jeffreys factor is determined by the Fisher information of the single-antenna output and the optimal input tilting . The asymptotically optimal input is the Jeffreys prior, and the paper provides a practical recipe for capacity computation, constellation design via a compander-like transform, and a low-complexity receiver that quantizes outputs into a finite number of bins while preserving near-capacity performance. Applications span AWGN with clipping, low-resolution ADCs, energy-detection, imperfect CSIR, and optical Poisson channels, and extensions to non-i.i.d. settings are discussed. Overall, the Fisher information is shown to govern the channel’s asymptotic behavior, enabling scalable capacity analysis and guiding practical transceiver design for next-generation large-scale MIMO systems.

Abstract

We present a unifying framework that bridges Bayesian asymptotics and information theory to analyze the asymptotic Shannon capacity of general large-scale MIMO channels including ones with nonlinearities or imperfect hardware. We derive both an analytic capacity formula and an asymptotically optimal input distribution in the large-antenna regime, each of which depends solely on the single-output channel's Fisher information through a term we call the (tilted) Jeffreys factor. We demonstrate how our method applies broadly to scenarios with clipping, coarse quantization (including 1-bit ADCs), phase noise, fading with imperfect CSI, and even optical Poisson channels. Our asymptotic analysis motivates a practical approach to constellation design via a compander-like transformation. Furthermore, we introduce a low-complexity receiver structure that approximates the log-likelihood by quantizing the channel outputs into finitely many bins, enabling near-capacity performance with computational complexity independent of the output dimension. Numerical results confirm that the proposed method unifies and simplifies many previously intractable MIMO capacity problems and reveals how the Fisher information alone governs the channel's asymptotic behavior.

Paper Structure

This paper contains 29 sections, 9 theorems, 124 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Proposed Method
  3. Channel model
  4. Bayesian asymptotics and the Jeffreys factor
  5. Recipe for asymptotic capacity in large-scale MIMO
  6. A toy application: Real AWGN
  7. Applications to Different Channels
  8. AWGN with clipping
  9. AWGN with low-resolution ADC
  10. Complex AWGN with energy detection
  11. MIMO channels with imperfect CSI
  12. Optical SIMO Poisson channel
  13. 1-bit ADC with dithering
  14. Constellation Design
  15. Jeffreys constellation
  16. ...and 14 more sections

Key Result

Theorem 1

Consider a large-scale MIMO setting where the number of receive antennas ${n_\mathrm{r}}$ grows large. Under mild regularity conditions (see Appendix app:CB_cond), the channel capacity subject to both peak- and average-power constraints satisfies where the $o(1)$ term vanishes when ${n_\mathrm{r}} \to \infty$; $\lambda^*\ge0$ is the smallest $\lambda$ such that the (tilted) Jeffreys prior satisfi

Figures (8)

  • Figure 1: Illustration of the Jeffreys factor in the SIMO real-AWGN case vs. $\lambda$.
  • Figure 2: Jeffreys prior for a clipped SIMO AWGN channel with different clipping thresholds $B=\alpha A$. Average power $P=A^2/9$. Left: Jeffreys prior with different clipping ratios. Right: Jeffreys factor normalized by the peak amplitude $A$ vs. clipping ratio $\alpha$.
  • Figure 3: Jeffreys prior (left) and Jeffreys factor (right) for real SIMO AWGN with $L$-level quantizers and peak amplitude $A$. Average power constraint $P = A^2/9$.
  • Figure 4: Jeffreys prior (left) and Jeffreys factor (right) for a complex SIMO channel with energy detection.
  • Figure 5: MIMO channel with imperfect CSIR, ${n_\mathrm{t}}=4$ transmit antennas, peak-power constraint $A$, average power constraint $P=A^2/9$. Left: optimal distribution on the input radius derived from the Jeffreys prior. Right: The Jeffreys factor.
  • ...and 3 more figures

Theorems & Definitions (21)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 1
  • proof
  • Lemma 2
  • ...and 11 more