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

This article provides capacity expressions for multi-user and multi-cell wireless communication schemes when the transmitters and receivers are equipped with multiple antennas and when the transmission channel has a certain correlation profile. In mathematical terms, this contribution provides novel deterministic equivalents for the Stieltjes and Shannon transforms of a class of large dimensional random matrices. These results are of practical relevance to evaluate the rate performance of communication channels with multiple users, multiple cells and with transmit and receive correlation at all communication pairs. In particular, we analyse the per-antenna achievable rates for these communication systems which, for practical purposes, is a relevant measure of the trade-off between rate performance and operating cost of every antenna. We study specifically the per-antenna rate regions of (i) multi-antenna multiple access channels and broadcast channels, as well as the capacity of (ii) multi-antenna multi-cell communications with inter-cell interference. Theoretical expressions of the per-antenna mutual information are obtained for these models, which extend previous results on multi-user multi-antenna performance without channel correlation to the more realistic Kronecker channel model. From an information theoretic viewpoint, this article provides, for scenario (i), a deterministic approximation of the per-antenna rate achieved in every point of the MAC and BC rate regions, a deterministic approximation of the ergodic per-antenna capacity with optimal precoding matrices in the uplink MAC and an iterative water-filling algorithm to compute the optimal precoders, while, for scenario (ii), this contribution provides deterministic approximations for the mutual information of single-user decoders and the capacity of minimum mean square error (MMSE) decoders. An original feature of this work is that the deterministic equivalents are proven asymptotically exact, as the system dimensions increase, even for strong correlation at both communication sides. The above results are validated by Monte Carlo simulations.

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