On Capacity Scaling of Multi-Antenna Multi-Hop Networks: The Significance of the Relaying Strategy in the “Long Network Limit”
Jörg Wagner, Armin Wittneben · IEEE Transactions on Information Theory · 2012
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$c_{0}$, such that$\lim_{n \ura{} \infty}C/n=c_{0}$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$n_{\cal R}$(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${\rm XF}$, there exists a sequence$(c_{L}^{\rm XF})_{L=0}^{\infty}$, such that$c_{L}^{\rm XF}=\lim_{n \ura{} \infty}R^{\rm XF}_{L}/n$almost surely, where$R^{\rm XF}_{L}$denotes the supremum of the set of sum-rates that are achievable by the strategy over$L\!+\!1$hops. This sequence depends on the sequence$(P_{L})_{L=0}^{\infty}$, where$P_{L}$corresponds to the power of the source stage and each of the relay stages in an$L\!+\!1$-hop network. Results are summarized as follows:Decode & forward (DF): For$n_{\cal R}=n$,$c^{\rm DF}_{L}$is constant with respect to$L$, if also$P_{L}$is constant with respect to$L$.