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.

翻译：本文提出了一种针对非相干多输入多输出系统的稀疏格拉斯曼码设计方法。在非相关瑞利衰落信道下，传统的成对错误概率公式未能考虑由稀疏配置引起的秩亏缺问题。我们修正了这些公式，以统一的方式处理此类情况。此外，我们推导了一种闭式度量，该度量能在给定信噪比下有效最大化非相干平均互信息。我们聚焦于格拉斯曼流形的舒伯特胞腔分解所提供的数学稀疏特性，并基于我们的分析建立了稀疏非相干码的设计准则。数值结果表明，所提出的稀疏非相干码在符号错误率和平均互信息方面均优于传统方法，并在高信噪比区域渐近逼近最优格拉斯曼星座的性能。此外，它们降低了时间和空间复杂度，且该复杂度不随发射天线数量的增加而增长。