A deterministic equivalent for the capacity analysis of correlated multi-user MIMO channels

This paper provides the analysis of capacity expressions in multi-user and multi-cell systems when the transmitters and receivers have a large number of correlated antennas. Our main contribution mathematically translates into a deterministic equivalent of the Shannon transform of a class of large dimensional random matrices. This class of large matrices is used in this contribution to model (i) multi-antenna multiple access (MAC) and broadcast channels (BC) with transmit and receive channel correlation, (ii) multiple-input multiple-output (MIMO) communications with inter-cell interference and channel correlation both at the base stations and at the receiver. These models extend the classical results on multi-user MIMO capacities in independent and identically distributed (i.i.d.) Gaussian channels to the more realistic Gaussian channels with separable variance profile. On an information theoretical viewpoint, this article provides: in scenario (i), an asymptotic description of the MAC and BC rate regions as a function of the transmit and receive correlation matrices and independently of the random channel realizations; an asymptotic expression of the capacity maximizing covariance matrices in the uplink MAC and downlink BC; a water-filling algorithm which, upon convergence, is proved to converge to the capacity achieving power allocation at the transmitters both in MAC and BC. In scenario (ii), the article provides: an expression of the single-user decoding capacity and minimum mean square error (MMSE) decoding capacity when interference is treated as Gaussian noise with a known variance profile; for single-user decoding, the capacity achieving antenna power allocation policy at the transmitter.