  A multi-hop relaying system is analyzed where data sent by a multi-antenna
source is relayed by successive multi-antenna relays until it reaches a
multi-antenna destination. Assuming correlated fading at each hop, each relay
receives a faded version of the signal from the previous level, performs linear
precoding and retransmits it to the next level. Using free probability theory
and assuming that the noise power at relaying levels-- but not at destination--
is negligible, the closed-form expression of the asymptotic instantaneous
end-to-end mutual information is derived as the number of antennas at all
levels grows large. The so-obtained deterministic expression is independent
from the channel realizations while depending only on channel statistics.
Moreover, it also serves as the asymptotic value of the average end-to-end
mutual information. The optimal singular vectors of the precoding matrices that
maximize the average mutual information with finite number of antennas at all
levels are also provided. It turns out that the optimal precoding singular
vectors are aligned to the eigenvectors of the channel correlation matrices.
Thus they can be determined using only the known channel statistics. As the
optimal precoding singular vectors are independent from the system size, they
are also optimal in the asymptotic regime.
