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19.3.6 The low-attenuation regime In this scenario no absorption, i.e. = 0, and small path loss exponent are assumed. In this case coherent relaying with interference subtraction (CRIS) is considered an interesting strategy for information transmission in the following scenarios. For a source destination pair (s, d), the nodes are divided into groups, with the rst group containing only the source, and the last group containing only the destination d. Call the higher numbered groups downstream groups, although they need not actually be closer to the destination. Nodes
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in group i, for 1 i k 1, dedicate a portion Pik of their power to coherently transmit for the bene t of node k and its downstream nodes. Each node k employs interference subtraction during decoding to subtract out the known portion of its received signal being transmitted by its downstream nodes. (r8i) If there is no absorption, i.e. = 0, and the path loss exponent < 3/2, then even with a xed total power Ptotal , any arbitrarily large transport capacity can be supported by CRIS in a regular planar network with a large enough number of nodes n. (r8ii) If = 0 and < 1, then even with a xed total power Ptotal , CRIS can support a xed rate Rmin > 0 for any single source destination pair in any regular planar network, irrespective of the distance between them. A similar result exists for the regular linear networks. (r9i) If = 0 and < 1, then even with a xed total power Ptotal , any arbitrarily large transport capacity can be supported by CRIS in a regular linear network with a large enough number of nodes n. (r9ii) If = 0 and < 1/2, then even with a xed total power Ptotal , CRIS can support a xed rate Rmin > 0 for any single source destination pair in any regular linear network, irrespective of the distance between them. A superlinear (n ) scaling law with 1 < < 2 is feasible for some linear networks when = 0 and < 1. (r10) For = 0 and individual power constraint Pind for every 0.5 < < 1, and 1 < < 1/ , there is a family of linear networks for which the transport capacity is C T = (n ). This order optimal transport capacity is attained in these networks by CRIS. 19.3.7 The Gaussian multiple-relay channel The results for the low-attenuation regime rely on the following results for the Gaussian multiple-relay channel. An example of a four-node network with two parallel relays as shown in Figure 19.12. Consider a network of n nodes with i j the attenuation from node i to node j and i.i.d. additive N (0, 2 ) noise at each receiver. Each node has an upper bound on the power available to it, which may differ from node to node. Suppose there is a single source-destination pair (s, d). We call this the Gaussian multiple relay channel. The rst result addresses the case where each relaying group consists of only one node. The strategy used is CRIS. Consider the Gaussian multiple-relay channel with coherent multistage relaying and interference subtraction. Consider M + 1 nodes, sequentially denoted
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Figure 19.12 A four-node network with two parallel relays [33].
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by 0, 1, . . . , M, with 0 as the source, M as the destination, and the other M 1 nodes serving as M 1 stages of relaying. (r11) Any rate R satisfying the following inequality is achievable from 0 to M: 2 j k 1 1 R < min S 2 i j Pik 1 j M k=1 i=0 where Pik 0 satis es
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