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

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2 Author(s)
Wagner, J. ; Commun. Technol. Lab., ETH Zurich, Zurich, Switzerland ; Wittneben, A.

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 cο, such that limn C/n = co 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 nR (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 (cLXF)L=o, such that cLXF = limn → ∞ RLXF/n almost surely, where RLXF denotes the supremum of the set of sumrates that are achievable by the strategy over L + 1 hops. This sequence depends on the sequence (PL)L=o, where PL 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 nR = n, cLDF is constant with respect to L, if also PL is constant with respect to L. · Quantize & forward with (CF) and without (QF) Slepian & Wolf compression: For nR = n, there exists a sequence (PL)L=o, such that cLQF/CF is positive and constant with respect to L. The corresponding sequence (PL)L=o grows linearly with L for CF and exponentially with - for QF. · Amplify & forward (AF): Fix nR/n = β and PL ∝ L. Then, (i) there exists c >; 0, such that limL - cLΛΓ = c, if β ∈ Ω (L1+ε) and (ii) limL cLAF = 0, if β ∈ O (L1-ε).

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Information Theory, IEEE Transactions on  (Volume:58 ,  Issue: 4 )