Scalable Long-Term Beamforming for Massive Multi-User MIMO
Fully digital massive multiple-input multiple-output (MIMO) systems with large numbers (1000+) of antennas offer capacity gains from spatial multiplexing and beamforming, but receivers that scale to these array dimensions face challenges in both channel estimation overhead and digital computation. Long-term beamforming addresses both by projecting the data onto a low-dimensional subspace that can be tracked at a slow time scale from the long-term channel parameters. In this setting, we show how to compute, in closed form, the projection matrix that maximizes a capacity upper bound, using a matrix inverse square root; the same projection is shown to maximize the mean post-projection signal-to-interference-plus-noise ratio (SINR) exactly. Computationally efficient methods are then presented for the matrix computation, realizable with matrix-matrix multiplies and hence amenable to systolic array implementations in hardware. Bounds on the SINR degradation are derived, and ray tracing simulations in a realistic rural uplink setting show a small loss relative to instantaneous minimum mean-square error (MMSE) beamforming when the covariance is accurately estimated. The efficient Gram-domain form of the instantaneous MMSE receiver applies the maximum-ratio combining reduction before an inverse whose dimension is the total number of streams. Against this baseline, the method estimates and refreshes beamforming coefficients three orders of magnitude less often and decouples the real-time path across users. With the conjugate-gradient solve and a rank-one projection, its total arithmetic cost is 2% higher at the ten-user operating point and lower above a stream-dimension crossover that we characterize.