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Continuous Aperture Array (CAPA)-Based Multi-Group Multicast Communications

Mengyu Qian, Xidong Mu, Li You, Michail Matthaiou

TL;DR

This work analyzes a CAPA-based downlink multi-group multicast system and targets energy-efficient operation under per-group QoS constraints. It develops two optimization pipelines within Dinkelbach's fractional programming framework: a CoV-based BCD algorithm that jointly optimizes the CAPA current patterns, per-group rates, and auxiliary variables, and a low-complexity ZF-based CAPA scheme using representative users for inter-group interference suppression. A key theoretical result is that the optimal CAPA beamformer lies in the span of all users' channels, and the ZF design yields a closed-form beamformer with reduced complexity. Numerical results show CAPA substantially improves EE over SPDAs for moderate apertures, but very large apertures or wide intra-group user spreads can degrade performance, highlighting practical design trade-offs and the value of CAPA in multicast scenarios.

Abstract

A continuous aperture array (CAPA)-based multi-group multicast communication system is investigated. An integral-based CAPA multi-group multicast beamforming design is formulated for the maximization of the system energy efficiency (EE), subject to a minimum multicast SE constraint of each user group and a total transmit power constraint. To address this non-econvex fractional programming problem, the Dinkelbach's method is employed. Within the Dinkelbach's framework, the non-convex group-wise multicast spectral efficiency (SE) constraint is first equivalently transformed into a tractable form with auxiliary variables. Then, an efficient block coordinate descent (BCD)-based algorithm is developed to solve the reformulated problem. The CAPA beamforming design subproblem can be optimally solved via the Lagrangian dual method and the calculus of variations (CoV) theory. It reveals that the optimal CAPA beamformer should be a combination of all the groups' user channels. To further reduce the computational complexity, a low-complexity zero-forcing (ZF)-based approach is proposed. The closed-form ZF CAPA beamformer is derived using each group's most representative user channel to mitigate the inter-group interference while ensuring the intra-group multicast performance. Then, the beamforming design subproblem in the BCD-based algorithm becomes a convex power allocation subproblem, which can be efficiently solved. Numerical results demonstrate that 1) the CAPA can significantly improve the EE compared to conventional spatially discrete arrays (SPDAs); 2) due to the enhanced spatial resolutions, increasing the aperture size of CAPA is not always beneficial for EE enhancement in multicast scenarios; and 3) wider user distributions of each group cause a significant EE degradation of CAPA compared to SPDA.

Continuous Aperture Array (CAPA)-Based Multi-Group Multicast Communications

TL;DR

This work analyzes a CAPA-based downlink multi-group multicast system and targets energy-efficient operation under per-group QoS constraints. It develops two optimization pipelines within Dinkelbach's fractional programming framework: a CoV-based BCD algorithm that jointly optimizes the CAPA current patterns, per-group rates, and auxiliary variables, and a low-complexity ZF-based CAPA scheme using representative users for inter-group interference suppression. A key theoretical result is that the optimal CAPA beamformer lies in the span of all users' channels, and the ZF design yields a closed-form beamformer with reduced complexity. Numerical results show CAPA substantially improves EE over SPDAs for moderate apertures, but very large apertures or wide intra-group user spreads can degrade performance, highlighting practical design trade-offs and the value of CAPA in multicast scenarios.

Abstract

A continuous aperture array (CAPA)-based multi-group multicast communication system is investigated. An integral-based CAPA multi-group multicast beamforming design is formulated for the maximization of the system energy efficiency (EE), subject to a minimum multicast SE constraint of each user group and a total transmit power constraint. To address this non-econvex fractional programming problem, the Dinkelbach's method is employed. Within the Dinkelbach's framework, the non-convex group-wise multicast spectral efficiency (SE) constraint is first equivalently transformed into a tractable form with auxiliary variables. Then, an efficient block coordinate descent (BCD)-based algorithm is developed to solve the reformulated problem. The CAPA beamforming design subproblem can be optimally solved via the Lagrangian dual method and the calculus of variations (CoV) theory. It reveals that the optimal CAPA beamformer should be a combination of all the groups' user channels. To further reduce the computational complexity, a low-complexity zero-forcing (ZF)-based approach is proposed. The closed-form ZF CAPA beamformer is derived using each group's most representative user channel to mitigate the inter-group interference while ensuring the intra-group multicast performance. Then, the beamforming design subproblem in the BCD-based algorithm becomes a convex power allocation subproblem, which can be efficiently solved. Numerical results demonstrate that 1) the CAPA can significantly improve the EE compared to conventional spatially discrete arrays (SPDAs); 2) due to the enhanced spatial resolutions, increasing the aperture size of CAPA is not always beneficial for EE enhancement in multicast scenarios; and 3) wider user distributions of each group cause a significant EE degradation of CAPA compared to SPDA.
Paper Structure (36 sections, 5 theorems, 64 equations, 6 figures, 1 table, 4 algorithms)

This paper contains 36 sections, 5 theorems, 64 equations, 6 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Prior Works
  3. Motivations and Contributions
  4. Organization and Notations
  5. System Model and Problem Formulation
  6. System Model
  7. Transmit Signal
  8. Receive Signal
  9. System EE
  10. Problem Formulation
  11. Dinkelbach's Framework
  12. CoV-based BCD Algorithm
  13. Problem Reformulation
  14. BCD Algorithm
  15. Subproblem with respect to $\mathbf{J}$ and $\mathbf{r}$
  16. ...and 21 more sections

Key Result

Lemma 1

For given $\{h_{g,k}(\mathbf{s})\}_{\forall g \in \mathcal{G}, \forall k \in \mathcal{K}_g}$, the problem CP_with_r_gk is equivalent to the following one: where $y(\mu_{g,k},\mathbf{J})$ is given by y_def at the top of the next page, while $\boldsymbol\mu = \{\mu_{g,k}\}_{\forall g \in \mathcal{G}, \forall k \in \mathcal{K}_g}$ are auxiliary variables. For each $\mu_{g,k}$, its optimal value is

Figures (6)

  • Figure 1: CAPA-based downlink multi-group multicast communication system.
  • Figure 2: Convergence of the proposed CAPA algorithm, with $G = 3$ and $K_g = 3$.
  • Figure 3: EE versus the aperture size, with $G = 3$ and $K_g = 3$ in multicast and $G = 3$ and $K_g = 1$ in unicast.
  • Figure 4: EE versus the users' spread radius, with $G = 3$ and $K_g = 2$.
  • Figure 5: EE versus the number of users.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof