On Capacity Scaling of Multi-Antenna Multi-Hop Networks: The Significance of the Relaying Strategy in the "Long Network Limit"

The sum-capacity of a static uplink channel with single-antenna sources and an -antenna destination is known to scale linearly in , if the random channel matrix fulfills the conditions for the Marcenko-Pastur law: if each source transmits at power , there exists a positive , such that almost surely. This paper addresses the question to which extent this result carries over to multi-hop networks. Specifically, an -hop network with non-cooperative source antennas, fully cooperative destination antennas, and relay stages of (cooperative or non-cooperative) relay antennas each is considered. Four relaying strategies are assessed based on the interrelationship between two sequences. For each considered strategy , there exists a sequence , such that almost surely, where denotes the supremum of the set of sum-rates that are achievable by the strategy over hops. This sequence depends on the sequence , where corresponds to the power of the source stage and each of the relay stages in an -hop network. Results are summarized as follows: • Decode & forward (DF): For , is constant with respect to , if also is constant with respect to . • Quantize & forward with (CF) and without (QF) Slepian & Wolf compression: For , there exists a sequence , such that is positive and constant with respect to . The corresponding sequence grows linearly with for CF and exponentially with for QF. • Amplify & forward (AF): Fix and . Then, (i) there exists , such that , if and (ii) , if .