Ergodic Sum Rate Capacity Achieving Transmit Design for Massive MIMO LEO Satellite Uplink Transmission

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.