Cutset Bounds on the Capacity of MIMO Relay Channels

We analyze the ergodic capacity of multiple-input multiple-output (MIMO) Rayleigh-fading relay channels. We first derive the probability density function of a sum of independent complex central Wishart matrices-called the central hyper-Wishart matrix-and its joint eigenvalue density. We then derive a trace representation for the max-flow min-cut upper bound on the ergodic capacity of general full-duplex MIMO relay channels where each communicating node is equipped with N antennas and has access only to respective receive channel state information. We also establish the Schur monotonicity theorem for this cutset bound as a functional of the signal-to-noise ratios (SNRs) of three communication links. We further characterize the exact ergodic capacity in the regularity SNR regime where the upper and lower bounds coincide.