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Analog Beamforming Enabled Multicasting: Finite-Alphabet Inputs and Statistical CSI

Yanjun Wu, Zhong Xie, Zhuochen Xie, Chongjun Ouyang, Xuwen Liang

TL;DR

This work analyzes the average multicast rate (AMR) of a multicast channel employing analog beamforming with a single RF chain and finite-alphabet inputs under statistical CSI. It derives AMR expressions for both non-cooperative and cooperative multicasting, and provides high-SNR asymptotics showing the AMR converges to $\log_2 M$, with diversity order 1 for non-cooperative and $K$ for cooperative transmissions; the beamformer influences AMR primarily through the array gain terms $\sum_{k=1}^K 1/(\mathbf{f}^H \mathbf{R}_k \mathbf{f})$ (non-cooperative) and related cooperative equivalents. To maximize AMR, the paper proposes efficient beamforming algorithms: a manifold-optimization-based conjugate gradient method (RM-CGD) to maximize array gain and a genetic algorithm (GA) for the cooperative design, along with a simpler RM-CGD surrogate that maximizes $\sum_k \log(\mathbf{f}^H \mathbf{R}_k \mathbf{f})$. Numerical results validate the theory, show rapid convergence, and demonstrate that cooperation increases both AMR and diversity at the expense of feedback overhead. Overall, the findings offer concrete, scalable strategies to boost multicast performance in practical analog-beamforming systems with finite constellations and statistical CSI.

Abstract

The average multicast rate (AMR) is analyzed in a multicast channel utilizing analog beamforming with finite-alphabet inputs, considering statistical channel state information (CSI). New expressions for the AMR are derived for non-cooperative and cooperative multicasting scenarios. Asymptotic analyses are conducted in the high signal-to-noise ratio regime to derive the array gain and diversity order. It is proved that the analog beamformer influences the AMR through its array gain, leading to the proposal of efficient beamforming algorithms aimed at maximizing the array gain to enhance the AMR.

Analog Beamforming Enabled Multicasting: Finite-Alphabet Inputs and Statistical CSI

TL;DR

This work analyzes the average multicast rate (AMR) of a multicast channel employing analog beamforming with a single RF chain and finite-alphabet inputs under statistical CSI. It derives AMR expressions for both non-cooperative and cooperative multicasting, and provides high-SNR asymptotics showing the AMR converges to , with diversity order 1 for non-cooperative and for cooperative transmissions; the beamformer influences AMR primarily through the array gain terms (non-cooperative) and related cooperative equivalents. To maximize AMR, the paper proposes efficient beamforming algorithms: a manifold-optimization-based conjugate gradient method (RM-CGD) to maximize array gain and a genetic algorithm (GA) for the cooperative design, along with a simpler RM-CGD surrogate that maximizes . Numerical results validate the theory, show rapid convergence, and demonstrate that cooperation increases both AMR and diversity at the expense of feedback overhead. Overall, the findings offer concrete, scalable strategies to boost multicast performance in practical analog-beamforming systems with finite constellations and statistical CSI.

Abstract

The average multicast rate (AMR) is analyzed in a multicast channel utilizing analog beamforming with finite-alphabet inputs, considering statistical channel state information (CSI). New expressions for the AMR are derived for non-cooperative and cooperative multicasting scenarios. Asymptotic analyses are conducted in the high signal-to-noise ratio regime to derive the array gain and diversity order. It is proved that the analog beamformer influences the AMR through its array gain, leading to the proposal of efficient beamforming algorithms aimed at maximizing the array gain to enhance the AMR.
Paper Structure (17 sections, 3 theorems, 28 equations, 2 figures, 2 algorithms)

This paper contains 17 sections, 3 theorems, 28 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. System Model
  3. Transmission Model
  4. Channel Model
  5. Non-Cooperative Scenario
  6. Performance Evaluation
  7. Multicast Rate
  8. Channel Statistics and AMR
  9. Asymptotic Analysis
  10. Analog Beamforming Design
  11. Cooperative Scenario
  12. Performance Evaluation
  13. Channel Statistics
  14. Asymptotic Analysis
  15. Analog Beamforming Design
  16. ...and 2 more sections

Key Result

Lemma 1

If $f\left(x\right)$ is ${\mathcal{O}}\left(x^m\right)$ as $x\rightarrow0^{+}$ and ${\mathcal{O}}\left(x^n\right)$ as $x\rightarrow+\infty$, then $\left|{\mathcal{M}}\left[f\left(x\right);t\right]\right|<\infty$ when $-m<t<-n$.

Figures (2)

  • Figure 1: The non-cooperative ( \ref{['fig2a']}) and cooperative ( \ref{['fig2b']}) AMR versus the number of iterations.
  • Figure 2: The AMR versus the SNR. The analytical results are calculated by \ref{['AMI_Nooperative']} and \ref{['AMI_Cooperative']}. The asymptotic results are calculated by \ref{['Asym_AMI_Nooperative']} and \ref{['Asym_AMI_Cooperative']}.

Theorems & Definitions (8)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5