Manifold-based predictive precoding for the time-varying channel using differential geometry

For limited feedback precoding systems with the feedback delay, the accurate channel state information at the transmitter is crucial for the system performance. We first model the one-dimensional subspaces spanned by the direction vector of MISO time-varying channel as points on the Grassmannian manifold $$G_{n,1}$$Gn,1, and propose the single-stream predictive beamforming algorithm on the Grassmannian manifold $$G_{n,1}$$Gn,1. Next, we extend the algorithm to the spatial multiplexing MIMO time-varying channel and propose multiple-stream predictive precoding algorithm by modeling the column spaces spanned by principal right singular matrix of the channels on the Grassmannian manifold $$G_{n,p}$$Gn,p. More specifically, by exploiting the differential geometric properties of the Grassmannian manifold, we propose an adaptive tracking codebook on the tangent space of Grassmannian manifold to improve the quantization accuracy. The computer simulation results show that the proposed precoding scheme exhibits promised system performance over previous precoding scheme for both transit beamforming MISO system and spatial multiplexing MIMO system.