In this paper, we study efficient codebook design for limited feedback in extremely large-scale multiple-input-multiple-output (XL-MIMO) frequency division duplexing (FDD) systems. It is worth noting that existing codebook designs for XL-MIMO, such as polar-domain codebook, have not well taken into account user (location) distribution in practice, thereby incurring excessive feedback overhead. To address this issue, we propose in this paper a novel and efficient feedback codebook tailored to user distribution. To this end, we first consider a typical scenario where users are uniformly distributed within a specific polar-region, based on which a sum-rate maximization problem is formulated to jointly optimize angle-range samples and bit allocation among angle/range feedback. This problem is challenging to solve due to the lack of a closed-form expression for the received power in terms of angle and range samples. By leveraging a Voronoi partitioning approach, we show that uniform angle sampling is optimal for received power maximization. For more challenging range sampling design, we obtain a tight lower-bound on the received power and show that geometric sampling, where the ratio between adjacent samples is constant, can maximize the lower bound and thus serves as a high-quality suboptimal solution. We then extend the proposed framework to accommodate more general non-uniform user distribution via an alternating sampling method. Furthermore, theoretical analysis reveals that as the array size increases, the optimal allocation of feedback bits increasingly favors range samples at the expense of angle samples. Finally, numerical results validate the superior rate performance and robustness of the proposed codebook design under various system setups, achieving significant gains over benchmark schemes, including the widely used polar-domain codebook.

翻译：本文研究了在超大规模多输入多输出（XL-MIMO）频分双工（FDD）系统中，针对有限反馈的高效码本设计问题。值得注意的是，现有的XL-MIMO码本设计（例如极域码本）未能充分考虑实际中的用户（位置）分布，从而导致过高的反馈开销。为解决这一问题，本文提出一种新颖且高效的、适应用户分布的反馈码本设计方法。为此，我们首先考虑一个典型场景，即用户在特定极坐标区域内均匀分布，并基于此构建了一个和速率最大化问题，以联合优化角度-距离采样点以及在角度/距离反馈之间的比特分配。由于接收功率关于角度和距离采样点缺乏闭式表达式，该问题求解具有挑战性。通过利用Voronoi划分方法，我们证明了均匀角度采样对于接收功率最大化是最优的。对于更具挑战性的距离采样设计，我们推导了接收功率的一个紧下界，并证明几何采样（即相邻采样点之间的比值为常数）能够最大化该下界，从而成为一个高质量的次优解。随后，我们通过一种交替采样方法扩展了所提框架，以适应更一般的非均匀用户分布。此外，理论分析表明，随着阵列尺寸增大，反馈比特的最优分配越来越倾向于距离采样，而相应减少角度采样所占的比特。最后，数值结果验证了所提码本设计在各种系统配置下具有优越的速率性能和鲁棒性，相较于包括广泛使用的极域码本在内的基准方案，取得了显著的性能增益。