The cost of uncorrelation and non-cooperation in MIMO channels

We investigate the sum-capacity loss for using uncorrelated Gaussian inputs over multiple-input multiple-output (MIMO) power-constrained linear additive-noise channels in multi-user configurations. We show that the sum-capacity loss is bounded by a universal constant which depends only on the total number of input and output dimensions of the channel, but is independent of the channel matrix, the noise distribution and the number of users. Specifically, for a multiple-access channel with a total number of nt transmit antennas and base-station with nr receive antennas, the sum-capacity loss is at most C* = min{1/2, nr/2nt log2(1 + nt/nr)} bit per input dimension (or 1 bit per transmit antenna per second per Hertz). If we restrict attention to Gaussian noises, then the capacity loss is upper bounded by CG* = min{0.265, 0.265nr/nt log2(nt/nr)}, and this bound is tight for certain channel matrices and noise spectra. We show also that the same bounds hold for the sum-capacity loss of uncorrelated Gaussian input over linear MIMO broadcast channels, input distribution being interpreted either in terms of the equivalent point-to-point channel with Sato condition, or as the output distribution of a "dirty-paper" transmitter. One implication of these results is the limited value of coherence and water-filling in spatial transmission. Another implication is the limited capacity loss in multi-user configurations relative to the fully cooperative (point-to-point) channel