On capacity scaling of (long) MIMO amplify-and-forward multihop networks

Consider a multiple-input-multiple-output-multihop network consisting of n_S noncooperating source antennas, n_D fully cooperating destination antennas, and L clusters of n_R noncooperating relay antennas each. The source signals reach the destination antennas via L + 1 hops by traversing all relay clusters. Relay antennas scale their received signals by a common constant before retransmission (amplify-and-forward). We perform an asymptotic capacity analysis assuming the elements of the (per hop) channel matrices to satisfy the conditions for the Marcenko-Pastur law. Our result: In the limit L rarr infin sum-capacity scales linearly with min{n_S, n_D}, if and only if L scales no faster than linearly with the ratio n_R/ min{n_S, n_D}.