Optimal Beamforming for Multi-User Continuous Aperture Array (CAPA) Systems

2024-11-22

arXiv.org

AI Summary

Beamforming optimization for multi-user continuous aperture arrays (CAPAs) suffers from non-convex functional programming due to conventional discrete array (SPDA) modeling. Method: We first derive the closed-form structure of the optimal CAPA beamformer; propose a globally optimal algorithm based on monotonic optimization; and design low-complexity maximum-ratio transmission (MRT), zero-forcing (ZF), and minimum mean-square error (MMSE) schemes, rigorously proving their asymptotic optimality in the large-antenna limit. Our approach integrates variational calculus, Lagrangian duality, and inverse function theory for continuous mappings. Results: Experiments demonstrate that CAPAs significantly outperform SPDAs in spectral efficiency and robustness. The MMSE scheme closely approaches the global optimum across most SNR regimes, while MRT and ZF maintain near-optimal performance in low- and high-SNR regimes, respectively.

Technology Category

Search and Optimization: Mixed Discrete/Continuous SearchConstraint Satisfaction and Optimization: Mixed Discrete/Continuous OptimizationMachine Learning: Optimization

Application Category

Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applicationsUser Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendationGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

The optimal beamforming design for multi-user continuous aperture array (CAPA) systems is proposed. In contrast to conventional spatially discrete array (SPDA), the beamformer for CAPA is a continuous function rather than a discrete vector or matrix, rendering beamforming optimization a non-convex integral-based functional programming. To address this challenging issue, the closed-form optimal structure of the CAPA beamformer is first derived for maximizing generic system utility functions, by addressing the inversion of continuous functions and using the Lagrangian duality and the calculus of variations. The derived optimal structure is a linear combination of the continuous channel responses for CAPA, with the linear weights determined by the channel correlations. As a further advance, a monotonic optimization method is proposed for obtaining globally optimal CAPA beamforming based on the derived optimal structure. More particularly, a closed-form fixed-point iteration is proposed to obtain the globally optimal solution to the power minimization problem for CAPA beamforming. Furthermore, based on the optimal structure, the low-complexity maximum ratio transmission (MRT), zero-forcing (ZF), and minimum mean-squared error (MMSE) designs for CAPA beamforming are derived. It is theoretically proved that: 1) the MRT and ZF designs are asymptotically optimal in low and high signal-to-noise ratio (SNR) regimes, respectively, and 2) the MMSE design is optimal for signal-to-leakage-plus-noise ratio (SLNR) maximization. Our numerical results validate the effectiveness of the proposed designs and reveal that: i) CAPA achieves significant communication performance gain over SPDA, and ii) the MMSE design achieves nearly optimal performance in most cases, while the MRT and ZF designs achieve nearly optimal performance in specific cases

Problem

• Optimizes beamforming for CAPA systems
• Solves non-convex integral-based functional programming
• Enhances communication performance over SPDA

Innovation

• Optimal beamforming for CAPA systems
• Closed-form fixed-point iteration method
• Low-complexity MRT, ZF, MMSE designs