(v8) Each cell contains at least one node.

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19.3 INFORMATION THEORY AND NETWORK ARCHITECTURES 19.3.1 Network architecture For the optimization of ad hoc and sensor networks we will discuss some performance measure for the architectures shown in Figures 19.8 19.11. We will mainly focus on transm port capacity CT := sup l=1 Rl l , where the supremum is taken over m, and vectors (R1 , R2 , . . . , Rm ) of feasible rates for m source-destination pairs, and l is the distance

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Figure 19.8 A planar network: n nodes located on a two-dimensional plane, with minimum separation distance min .

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Figure 19.9 A regular planar network: n nodes located on a plane at (i, j) with 1 i, j n. (Reproduced by permission of IEEE [33].) between the lth source and its destination. For the planar network from Figure 19.8 we assume [33]: (1) There is a nite set N of n nodes located on a plane. (2) There is a minimum positive separation distance min between nodes, i.e. min := mini= j i j > 0, where i j is the distance between nodes i, j N . (3) Every node has a receiver and a transmitter. At time instants t = 1, 2, . . . , node i N sends X i (t), and receives Yi (t) with Yi (t) =

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where Z i (t), i N , t = 1, 2, . . . are Gaussian independent and identically distributed (i.i.d.) random variables with mean zero and variance 2 . The constant > 0 is referred to as the path loss exponent, while 0 will be called the

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Figure 19.10 A linear network: n nodes located on a line, with minimum separation distance min . (Reproduced by permission of IEEE [33].)

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Figure 19.11 A regular linear network: n nodes located on a line at 1, 2, . . . , n. (Reproduced by permission of IEEE [33].) absorption constant. A positive generally prevails except for transmission in a vacuum, and corresponds to a loss of 20 log10 e decibel per meter. (4) Denote by Pi 0 the power used by node i. Two separate constraints on n {P1 , P2 , . . . , Pn } are studied: total power constraint Ptotal : i=1 Pi Ptotal or individual power constraint Pind : Pi Pind , for i = 1, 2, . . . , n. (5) The network can have several source destination pairs (s , d ), = 1, . . . , m, where s , d are nodes in N with s = d , and (s , d ) = (s j , d j ) for = j. If m = 1, then there is only a single source destination pair, which we will simply denote (s, d). A special case a regular planar network where the n nodes are located at the points (i, j) is for 1 i, j n; see Figure 19.9. This setting will be used mainly to exhibit achievability of some capacities, i.e. inner bounds. Another special case is a linear network where the n nodes are located on a straight line, again with minimum separation distance min ; see Figure 19.10. The main reason for considering linear networks is that the proofs are easier to state and comprehend than in the planar case, and can be generalized to the planar case. Also, the linear case may have some utility for, say, networks of cars on a highway, since its scaling laws are different. A special case of a linear network is a regular linear network where the n nodes are located at the positions 1, 2, . . . , n; see Figure 19.11. This setting will also be used mainly to exhibit achievability results.

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