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
$c_{0}$
, 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 $L$.