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For the network setting in (r11), Theorem 3.1 in Gupta and Kumar [36] shows that a rate R0 is achievable if there exist some {R1 , R2 , . . . , R M 1 } such that R M 1 and
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From the above, recursively for m = M 2, M 1, . . . , 0, it is easy to prove that Rm <
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For m = 0, this inequality is exactly (r11), showing a higher achievable rate. The right-hand side (RHS) in (r11) can be maximized over the choice of order of the M 1 intermediate nodes. The relaying can also be done by groups, and the next result addresses this. As above, maximization can be done over the assignment of nodes to the groups. Consider again the Gaussian multiple-relay channel using coherent multistage relaying with interference subtraction. Consider any M + 1 groups of nodes sequentially denoted by N0 , N1 , . . . N M with N0 = {s} as the source, N M = {d} as the destination, and the other M 1 groups as M 1 stages of relay. Let n i be the number of nodes in group Ni , i {0, 1, . . . , M}. Let the power constraint for each node in group Ni be Pi /n i 0. (r12) Any rate R satisfying the following inequality is achievable from s to d: 2 j k 1 1 R < min S 2 N N Pik /n i n i 1 j M k=1 i=0 i j
M where Pik 0 satis es k=i+1 Pik Pi , and N x N j := min{ k : k N i, N j}, i, j {0, 1, . . . , M}.
As pointed out earlier, more on the results (r1) (r12) can be found in Xie and Kumar [33].
19.4 COOPERATIVE TRANSMISSION IN WIRELESS MULTIHOP AD HOC NETWORKS The technique discussed in this section allows us to transmit reliably to far destinations that the individual nodes are not able to reach without rapidly consuming their own battery resources, even when using multihop links discussed so far. The results are of interest in both ad hoc and sensor networks. The key idea is to have the nodes simply echo the source s (leader) transmission operating as active scatterers while using adaptive receivers that acquire the equivalent network signatures corresponding to the echoed symbols. The active nodes in the network operate either as regenerative or nonregenerative relays. The intuition is that each of the waveforms will be enhanced by the accumulation of power due to the aggregate transmission of all the nodes while, if kept properly under control, the random errors or the receiver noise that propagate together with the useful signals will cause limited deterioration in the performance. The avalanche of signals triggered by the network leaders forms the so-called opportunistic large array (OLA). In contrast to Sections 19.2 and 19.3, we are interested in this section in a method that utilizes the network as a distributed modem, where one or few sources are effectively transmitting data and all the other users are operating as repeaters. A fresh look into the concept of repeaters as a form of cooperative transmission came recently from References [37 40]. We will assume that, in a network of N nodes transmitting over a shared medium, each node is part of a multiple stage relay of a single source transmitting toward a remote receiver whose position is unknown to all the nodes. If no node in the network is powerful enough to communicate reliably with the remote receiver, the problem is referred to as the reach-back problem. Coordination among nodes in a large network is an extremely dif cult task. In a cooperative transmission mechanism for which cooperation is obtained in a distributed fashion the source (leader) transmits a pulse with complex envelope pm (t) out of an M-ary set of waveforms. The resulting signal at the ith receiver is ri (t) = si,m (t) + n i (t) where si,m (t) is the network-generated signature of the mth symbol. If N nodes echo exactly the same symbol,
si,m (t) =