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Rate-Splitting with Hybrid Messages: DoF Analysis of the Two-User MIMO Broadcast Channel with Imperfect CSIT

Tong Zhang, Yufan Zhuang, Gaojie Chen, Shuai Wang, Bojie Lv, Rui Wang, Pei Xiao

TL;DR

This work describes the optimal DoF region for the two-user multiple-input multiple-input multiple-output (MIMO) broadcast channel (BC) with imperfect CSIT and hybrid messages, and provides a hybrid message-aware rate-splitting scheme.

Abstract

Most of the existing research on degrees-of-freedom (DoF) with imperfect channel state information at the transmitter (CSIT) assume the messages are private, which may not reflect reality as the two receivers can request the same content. To overcome this limitation, we therefore consider the hybrid unicast and multicast messages. In particular, we characterize the optimal DoF region for the two-user multiple-input multiple-output (MIMO) broadcast channel (BC) with imperfect CSIT and hybrid messages. For the converse, we establish a three-step procedure to exploit the utmost possible relaxation. For the achievability, since the DoF region is with specific three-dimensional structure regarding antenna configurations and CSIT qualities, we verify the existence or non-existence of corner point candidates via the feature of antenna configurations and CSIT qualities categorization and provide a hybrid message-aware rate-splitting scheme. Besides, we show that to achieve the strictly positive corner points, it is unnecessary to split the unicast messages into private and common parts. This implies adding a multicast message may mitigate the rate-splitting complexity.

Rate-Splitting with Hybrid Messages: DoF Analysis of the Two-User MIMO Broadcast Channel with Imperfect CSIT

TL;DR

This work describes the optimal DoF region for the two-user multiple-input multiple-input multiple-output (MIMO) broadcast channel (BC) with imperfect CSIT and hybrid messages, and provides a hybrid message-aware rate-splitting scheme.

Abstract

Most of the existing research on degrees-of-freedom (DoF) with imperfect channel state information at the transmitter (CSIT) assume the messages are private, which may not reflect reality as the two receivers can request the same content. To overcome this limitation, we therefore consider the hybrid unicast and multicast messages. In particular, we characterize the optimal DoF region for the two-user multiple-input multiple-output (MIMO) broadcast channel (BC) with imperfect CSIT and hybrid messages. For the converse, we establish a three-step procedure to exploit the utmost possible relaxation. For the achievability, since the DoF region is with specific three-dimensional structure regarding antenna configurations and CSIT qualities, we verify the existence or non-existence of corner point candidates via the feature of antenna configurations and CSIT qualities categorization and provide a hybrid message-aware rate-splitting scheme. Besides, we show that to achieve the strictly positive corner points, it is unnecessary to split the unicast messages into private and common parts. This implies adding a multicast message may mitigate the rate-splitting complexity.
Paper Structure (28 sections, 6 theorems, 55 equations, 6 figures, 1 table)

This paper contains 28 sections, 6 theorems, 55 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Organization & Notation
  5. System Model
  6. Main Results and Discussion
  7. Achievabiliy Proof of Theorem 1: Proposed System and Scheme Design
  8. Proposed Hybrid Message-Aware Rate-Splitting System
  9. A Unified Hybrid Message-Aware Rate-Splitting Scheme
  10. GENERAL CASE-1 (If $\frac{N_2-N_1+(M-N_2)\alpha_{2}}{M-N_1} \leq \alpha_{1}$)
  11. GENERAL CASE-2 (If $\max\left\{1-\alpha_2,\frac{N_2-N_1}{M-N_1}\right\} \leq \alpha_{1} \leq \frac{N_2-N_1+(M-N_2)\alpha_{2}}{M-N_1}$)
  12. GENERAL CASE-3 (If $1-\alpha_{2} \leq \alpha_{1} \leq \frac{N_2-N_1}{M-N_1}$)
  13. GENERAL CASE-4 (If $\alpha_{1} \leq 1-\alpha_{2}$)
  14. An Example by Two-User $(4, 2, 3)$ MIMO BC
  15. CASE-1 (If $\frac{1+\alpha_{2}}{2} \leq \alpha_{1}$)
  16. ...and 13 more sections

Key Result

Theorem 1

For the two-user $(M,N_1,N_2)$ MIMO BC with hybrid messages and imperfect CSIT, the DoF region, denoted by $\mathcal{D}$, is given below. where $\alpha_0$ is given in hao2017achievable and copied on the top of this page.

Figures (6)

  • Figure 1: The two-user $(M,N_1,N_2)$ MIMO BC with hybrid messages.
  • Figure 2: Proposed v.s. competing schemes with or without rate-splitting.
  • Figure 3: a) $M=2$, $N_1=N_2=1$, $\alpha_1=\alpha_2=0.5$, b) $M=2$, $N_1=N_2=1$, $\alpha_1=0.25, \alpha_2=0.75$, c) $M=4$, $N_1=2$, $N_2=3$, $\alpha_1=\alpha_2=0.5$, and d) $M=4$, $N_1=2$, $N_2=3$, $\alpha_1=0.25, \alpha_2=0.75$. The hull of the DoF region with delayed CSIT in the presence of errors is blackened.
  • Figure 4: a) $\alpha_{1} \geq \frac{N_2-N_1+(M-N_2)\alpha_{2}}{M-N_1}$, b) $\max\left\{1-\alpha_2,\frac{N_2-N_1}{M-N_1}\right\} \leq \alpha_{1} \leq \frac{N_2-N_1+(M-N_2)\alpha_{2}}{M-N_1}$, c) $1-\alpha_{2} \leq \alpha_{1} \leq \frac{N_2-N_1}{M-N_1}$, and d) $\alpha_{1} \leq 1-\alpha_{2}$ with only $\mathcal{P}_{234}$ exists.
  • Figure 5: System diagram of this $(4,2,3)$ hybrid message two-user MIMO BC.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Theorem 1: DoF Region
  • Remark 1: What Do Hybrid Messages Change?
  • Remark 2: Sum-DoF
  • Theorem 2: Delayed CSIT with Errors
  • Remark 3: Comparison
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4