Globally Optimal Movable Antenna-Enabled Multiuser Communication: Discrete Antenna Positioning, Power Consumption, and Imperfect CSI

Movable antennas (MAs) represent a promising paradigm to enhance the spatial degrees of freedom of conventional multi-antenna systems by dynamically adapting the positions of antenna elements within a designated transmit area. In particular, by employing electro-mechanical MA drivers such as stepper motors, the positions of the MA elements can be discretely adjusted to shape a favorable spatial correlation for improving system performance. Although preliminary research has explored beamforming designs for MA-enabled systems, the intricacies of the power consumption and the precise positioning of MA elements are not well understood, yet. Moreover, the assumption of perfect channel state information (CSI) adopted in the current literature is generally impractical due to the significant pilot overhead and the extensive time required for acquiring close-to-perfect CSI. To address these challenges, in this paper, we model the motion of MA elements through discrete steps and quantify the associated power consumption as a function of these movements. Furthermore, by leveraging the properties of the MA channel model, we introduce a novel CSI error model tailored for MA-enabled systems that facilitates robust resource allocation design. In particular, we jointly optimize the beamforming and the MA positions at the base station (BS) for minimization of the total BS power consumption, encompassing both radiated power and MA motion power, while guaranteeing a minimum required signal-to-interference-plus-noise ratio for each user. To this end, novel algorithms exploiting the branch and bound (BnB) method are developed to obtain the globally optimal solution for perfect and imperfect CSI, respectively. Moreover, to support practical real-time implementation, we propose low-complexity suboptimal algorithms with guaranteed convergence by leveraging successive convex approximation (SCA). Our numerical results validate the global optimality of the proposed BnB-based algorithms for both CSI scenarios. Furthermore, we unveil that both proposed SCA-based algorithms approach the optimal performance of the BnB-based algorithms within only a few iterations, thus highlighting their practical advantages. Additionally, we show that compared to the state-of-the-art approach, the proposed low-complexity SCA-based schemes achieve considerable performance gains, especially in high-load systems with a small number of antenna elements.