The Ergodic Capacity of The MIMO Wire-Tap Channel

We investigate the ergodic secrecy capacity of a Multiple-Input Multiple-Output (MIMO) wire-tap channel, where the instantaneous Channel State Information (CSI) is available at both the legitimate receiver and the eavesdropper, but not at the source. We derive the perfectsecrecy capacity under a general ergodic fading process (with a continuous probability density function) and show that a circularly symmetric Gaussian input is optimal. Our results show that even with no CSI at the transmitter, the fading is beneficial too, and that reliable communication with perfect-secrecy is still possible. When specialized to a Gaussian channel with Rayleigh fading, we further show that a circularly symmetric Gaussian input with diagonal covariance is optimal. Our results show, for instance, that equal power allocation over the transmit antennas is not generally capacity-achieving.