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

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