Sparse Grassmannian Design for Noncoherent Codes via Schubert Cell Decomposition

TL;DR

This paper introduces a novel method for designing sparse Grassmannian codes for noncoherent MIMO systems, improving error rates and mutual information while reducing complexity through Schubert cell decomposition.

Contribution

It develops a unified error probability formulation and a closed-form metric for sparse noncoherent code design based on Schubert cell decomposition.

Findings

Outperforms conventional codes in error rate and mutual information

Approaches optimal Grassmannian performance at high SNR

Reduces computational complexity independent of transmit antennas

Abstract

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.