Low-Complexity and Power-Efficient Precoding Codebook Design on Sparse Grassmannian

We propose a sparse Grassmannian design for precoding codebooks. Due to their sparse structure, our proposed codebooks achieve low peak-to-average power ratio (PAPR), low complexity of precoder multiplication, and low storage cost, while demonstrating performance comparable to the optimal codebook. Specifically, we introduce a method for constructing codebooks based on Schubert cell decomposition on the Grassmann manifold. Designing an optimal Grassmannian precoding codebook generally requires high computational complexity. In the proposed approach, by exploiting its sparsity, the objective function can be simplified, and the search space can also be significantly reduced compared to state-of-the-art codebooks. Numerical simulations in uplink systems demonstrate that the proposed sparse codebook asymptotically approaches the optimal codebook and outperforms the codebook currently adopted in 5G NR, in terms of achievable rate under uncorrelated Rayleigh fading channels, while maintaining substantially lower PAPR than conventional dense designs. These results confirm that the proposed sparse codebook can be a practical and power-efficient alternative to conventional codebooks for a wide range of uplink transmission scenarios.

Sparse Grassmannian Design for Noncoherent Codes via Schubert Cell Decomposition

In this paper, we propose a method for designing sparse Grassmannian codes for noncoherent multiple-input multiple-output systems. Conventional pairwise error probability formulations under uncorrelated Rayleigh fading channels fail to account for rank deficiency induced by sparse configurations. We revise these formulations to handle such cases in a unified manner. Furthermore, we derive a closed-form metric that effectively maximizes the noncoherent average mutual information (AMI) at a given signal-to-noise ratio. We focus on the fact that the Schubert cell decomposition of the Grassmann manifold provides a mathematically sparse property, and establish design criteria for sparse noncoherent codes based on our analyses. In numerical results, the proposed sparse noncoherent codes outperform conventional methods in terms of both symbol error rate and AMI, and asymptotically approach the performance of the optimal Grassmannian constellations in the high-signal-to-noise ratio regime. Moreover, they reduce the time and space complexity, which does not scale with the number of transmit antennas.