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 the 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 the 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 the more challenging range sampling design, we obtain a tight lower bound on the received power and show thatgeometricsampling, 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.

In this paper, we study efficient limited feedback design for near-field frequency division duplex (FDD) systems, where an extremely large-scale array (XL-array) serves multiple single-antenna users. Although the existing polar-domain codebook caters to the near-field spherical wavefronts, it is designed based on inter-codeword coherence minimization for the entire space. Thus, it suffers considerable performance loss in the achievable rate when the users are randomly located in a specific region. To address this issue, we propose in this paper a new and efficient codebook for limited feedback in near-field FDD systems to facilitate the codebook-based analog beamforming at the BS, while the digital beamforming is designed according to the feedback CSI based on zero-forcing (ZF) technique. Specifically, an optimization problem is formulated to maximize the expected beamforming gain (or equivalent received signal power) at the individual user by optimizing the angle and range sampling in the near-field codebook, given a finite number of feedback bits. Interestingly, we show that when the users are randomly and uniformly located in a polar subspace, the angle domain should be uniformly sampled, while the range samples should be sampled in a new geometric form, with the ratio of adjacent range samples being a constant. Finally, numerical results are presented to validate the significant rate performance gain of the proposed near-field codebook over the existing polar-domain codebook.