A Proof of a Single-Letter Capacity Formula for MIMO Gauss-Markov Rayleigh Fading Channels

Over the past decades, the problem of communication over finite-state Markov channels (FSMCs) has been investigated in many works and the capacity of FSMCs has been studied in closed form under the assumption of the availability of partial/complete channel state information at the sender and/or the receiver. In our work, we focus on infinite-state Markov channels by investigating the problem of message transmission over time-varying single-user multiple-input multiple-output (MIMO) Gauss-Markov Rayleigh fading channels, as an example of MIMO ergodic Rayleigh fading channels, with average power constraint and with complete channel state information available at the receiver side (CSIR). We prove a single-letter formula for the channel capacity and in particular the formula pointed out by Telatar for the channel capacity of MIMO ergodic Rayleigh fading channels for the case when the Gaussian noise is uncorrelated across antennas.