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Mutual Information Density of Massive MIMO Systems over Rayleigh-Product Channels

Xin Zhang, Shenghui Song

TL;DR

This work develops a central limit theorem for the mutual information density (MID) of massive MIMO systems over Rayleigh-product channels in a finite-blocklength regime, enabling closed-form Gaussian approximations of MID via its asymptotic mean and variance. The authors derive a CLT under a joint growth regime for the number of scatterers, antennas, and blocklength, and use it to obtain tight upper and lower bounds on the optimal average packet error probability, including high- and low-SNR insights that reveal the impact of rank deficiency. The analysis shows that rank-deficiency worsens FBL performance and that, as the number of scatterers grows, the Rayleigh-product channel behavior converges to the standard Rayleigh channel; the results degenerate to known IBL/MID bounds in corresponding limits. Numerical results validate the accuracy of the MID CLT and FBL bounds, and demonstrate good agreement with LDPC-code performance, highlighting practical implications for latency-constrained MIMO systems with partial CSI and rank-deficient channels.

Abstract

The Rayleigh-product channel model is utilized to characterize the rank deficiency caused by keyhole effects. However, the finite blocklength analysis for Rayleigh product channels is not available in the literature. In this paper, we will characterize the mutual information density (MID) and perform the FBL analysis to reveal the impact of rank-deficiency in Rayleigh-product channels. To this end, we first set up a central limit theorem for the MID over Rayleigh-product MIMO channels in the asymptotic regime where the number of scatterers, number of antennas, and blocklength go to infinity at the same pace. Then, we utilize the CLT to obtain the upper and lower bounds for the packet error probability, whose approximations in the high and low signal to noise ratio regimes are then derived to illustrate the impact of rank deficiency. One interesting observation is that rank-deficiency degrades the performance of MIMO systems with FBL and the fundamental limits of Rayleigh-product channels degenerate to those of the Rayleigh case when the number of scatterers approaches infinity.

Mutual Information Density of Massive MIMO Systems over Rayleigh-Product Channels

TL;DR

This work develops a central limit theorem for the mutual information density (MID) of massive MIMO systems over Rayleigh-product channels in a finite-blocklength regime, enabling closed-form Gaussian approximations of MID via its asymptotic mean and variance. The authors derive a CLT under a joint growth regime for the number of scatterers, antennas, and blocklength, and use it to obtain tight upper and lower bounds on the optimal average packet error probability, including high- and low-SNR insights that reveal the impact of rank deficiency. The analysis shows that rank-deficiency worsens FBL performance and that, as the number of scatterers grows, the Rayleigh-product channel behavior converges to the standard Rayleigh channel; the results degenerate to known IBL/MID bounds in corresponding limits. Numerical results validate the accuracy of the MID CLT and FBL bounds, and demonstrate good agreement with LDPC-code performance, highlighting practical implications for latency-constrained MIMO systems with partial CSI and rank-deficient channels.

Abstract

The Rayleigh-product channel model is utilized to characterize the rank deficiency caused by keyhole effects. However, the finite blocklength analysis for Rayleigh product channels is not available in the literature. In this paper, we will characterize the mutual information density (MID) and perform the FBL analysis to reveal the impact of rank-deficiency in Rayleigh-product channels. To this end, we first set up a central limit theorem for the MID over Rayleigh-product MIMO channels in the asymptotic regime where the number of scatterers, number of antennas, and blocklength go to infinity at the same pace. Then, we utilize the CLT to obtain the upper and lower bounds for the packet error probability, whose approximations in the high and low signal to noise ratio regimes are then derived to illustrate the impact of rank deficiency. One interesting observation is that rank-deficiency degrades the performance of MIMO systems with FBL and the fundamental limits of Rayleigh-product channels degenerate to those of the Rayleigh case when the number of scatterers approaches infinity.
Paper Structure (32 sections, 9 theorems, 162 equations, 6 figures, 1 table)

This paper contains 32 sections, 9 theorems, 162 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Characterizing the MI of MIMO Systems by RMT
  3. Characterizing the MID of MIMO Systems by RMT
  4. Challenges
  5. Contributions
  6. System Model and MID
  7. Mutual Information Density (MID)
  8. MID Characterization
  9. Preliminary Results
  10. CLT for the MID
  11. Bounds for the Optimal Average Error Probability
  12. Performance Metrics
  13. Numerical Results
  14. Approximation Accuracy of Upper and Lower Bounds
  15. Impact of $n$ on the Optimal Average Error Probability
  16. ...and 17 more sections

Key Result

Theorem 1

(EMI of Rayleigh-product channels) Given Assumption A and the channel matrix ${\bold{H}}$ defined in (cha_mod), the per antenna mutual information is given by and there holds true that hoydis2011asymptotic and zhang2022asymptotic where $\overline{C}(\sigma^2)$ is given by Here $\omega$ is the root of the following cubic equation which satisfies $\omega>0$ and $\eta+(\eta-1)\omega>0$.

Figures (6)

  • Figure 1: Comparison with existing works.
  • Figure 2: Lower bound.
  • Figure 3: Upper bound.
  • Figure 4: Bounds with different numbers of the scatterers.
  • Figure 5: Comparison of the bounds with LDPC coding ($L=24$).
  • ...and 1 more figures

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • proof
  • Remark 1
  • Lemma 1
  • Remark 2
  • Theorem 3
  • proof
  • Remark 3
  • Remark 4
  • ...and 18 more