In this paper, we investigate the ergodic sum rate (ESR) capacity achieving uplink (UL) transmit design for massive multiple-input multiple-output (MIMO) low-earth-orbit (LEO) satellite communications with statistical channel state information at the user terminals (UTs). The UL massive MIMO LEO satellite channel model with uniform planar array configurations at the satellite and UTs is presented. We prove that the rank of each UT's optimal transmit covariance matrix does not exceed that of its channel correlation matrix at the UT side, which reveals the maximum number of independent data streams transmitted from each UT to the satellite. We then prove that the transmit covariance matrix design can be transformed into the lower-dimensional matrix design without loss of optimality. We also obtain a necessary and sufficient condition when single data stream transmission from each UT to the satellite can achieve the ESR capacity. A conditional gradient (CG) method is developed to compute the ESR capacity achieving transmit covariance matrices. Furthermore, to avoid the exhaustive sample average, we utilize an asymptotic expression of the ESR and devise a simplified CG method to compute the transmit covariance matrices, which can approximate the ESR capacity. Simulations demonstrate the effectiveness of the proposed approaches.

翻译：本文研究了在用户终端（UTs）仅知统计信道状态信息条件下，面向大规模多输入多输出（MIMO）低地球轨道（LEO）卫星通信的上行（UL）遍历和速率（ESR）容量可达发射设计。首先，建立了卫星与用户终端均配置均匀平面阵列的UL大规模MIMO LEO卫星信道模型。我们证明每个UT最优发射协方差矩阵的秩不超过其UT端信道相关矩阵的秩，这揭示了每个UT向卫星传输的最大独立数据流数量。进而证明发射协方差矩阵设计可在不损失最优性的情况下转化为低维矩阵设计。此外，推导了当每个UT向卫星传输单数据流即可实现ESR容量时的充要条件。为计算ESR容量可达的发射协方差矩阵，提出了一种条件梯度（CG）方法。更进一步，为避免穷举式样本平均，利用ESR的渐近表达式，设计了一种简化CG方法来计算可近似ESR容量的发射协方差矩阵。仿真验证了所提方法的有效性。