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

The sum-capacity C of a static uplink channel with n single-antenna sources and an n-antenna destination is known to scale linearly in n, if the random channel matrix fulfills the conditions for the Marcenko-Pastur law: if each source transmits at power P/n, there exists a positive c_ο, such that lim_n → _∞ C/n = c_o almost surely. This paper addresses the question to which extent this result carries over to multi-hop networks. Specifically, an L + 1-hop network with n non-cooperative source antennas, n fully cooperative destination antennas, and L relay stages of n_R (cooperative or non-cooperative) relay antennas each is considered. Four relaying strategies are assessed based on the interrelationship be- tween two sequences. For each considered strategy XF, there exists a sequence (c_L^XF)_L=o^∞, such that c_L^XF = lim_n → ∞ R_L^XF/n almost surely, where R_L^XF denotes the supremum of the set of sumrates that are achievable by the strategy over L + 1 hops. This sequence depends on the sequence (P_L)_L=o^∞, where P_L corresponds to the power of the source stage and each of the relay stages in an Z+1-hop network. Results are summarized as follows: · Decode & forward (DF): For n_R = n, c_L^DF is constant with respect to L, if also P_L is constant with respect to L. · Quantize & forward with (CF) and without (QF) Slepian & Wolf compression: For n_R = n, there exists a sequence (P_L)_L=o^∞, such that c_L^QF/CF is positive and constant with respect to L. The corresponding sequence (P_L)_L=o^∞ grows linearly with L for CF and exponentially with - for QF. · Amplify & forward (AF): Fix n_R/n = β and P_L ∝ L. Then, (i) there exists c >; 0, such that lim_L -_∞ c_L^ΛΓ = c, if β ∈ Ω (L^1+ε) and (ii) lim_L → _∞ c_L^AF = 0, if β ∈ O (L^1-ε).