A manifold-based codebook design scheme for near-field LoS channel in XL-MIMO systems

In this paper, we propose a manifold-based codebook design scheme for near-field line-of-sight (LoS) channel in extremely large-scale multiple-input–multiple output (XL-MIMO) systems, where the user is equipped with multiple antennas. The LoS XL-MIMO channel model has been developed, considering the spatial distribution characteristics of multiple receiving antennas. The proposed scheme employs manifold optimization to design the codebook, resulting in a higher achievable rate compared to the conventional polar-domain codebook. Specifically, to establish the codebook, the angle and distance parameters in space are quantified by minimizing the incoherence between two channel matrices of different user locations. Since the near-field LoS channel matrix is no longer rank one, multiple codewords are designed for each quantified point to cover the LoS paths. Furthermore, the joint design of codewords at the BS and user side is formulated as an optimization problem with a constant modulus constraint, which defines a geometric structure in the form of a complex circle manifold. To solve the constant modulus optimization problem, we decouple it into two independent sub-problems. The Riemannian gradient descent algorithm on the complex circle manifold is applied to efficiently solve such problems. Numerical results demonstrate the superiority of our proposed scheme in improving the achievable rate.