De nition of feasible rate vectors in .NET

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19.3.2 De nition of feasible rate vectors
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De nition 1 Consider a wireless network with multiple source destination pairs (s , d ), = 1, . . . , m, with s = d , and (s , d ) = (s j , d j ) for = j. Let S := {s , = 1, . . . , m} denote the set of source nodes. The number of nodes in S may be less than m, since we allow a node
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NETWORK INFORMATION THEORY
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to have originating traf c for several destinations. Then a [(2TR1 , . . . , 2TRm ), T, Pe(T ) ] code with total power constraint Ptotal consists of the following: (1) m independent random variables W (transmitted words TRl bits long) with P(Wl = kl ) = 1/2T Rl , for any k {1, 2, . . . , 2TR }, = 1, . . . , m. For any i S, let Wi := R {W : s = i} and Ri :=
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(2) Functions f i,t : R {1, 2, . . . , 2T Ri } R, t = 1, 2, . . . , T, for the source nodes i S and f j,t : Rt 1 R, t = 2, . . . , T , for all the other nodes j S, such / that
X i (t) = f i,t (Yi (1), . . . , Yi (t 1), Wi ), t = 1, 2, . . . , T X j (1) = 0, X j (t) = f j,t (Y j (1), . . . , Y j (t 1)), t = 2, 3, . . . , T such that the following total power constraint holds: 1 T (3) m decoding functions g : RT {1, 2, . . . , |W d |} {1, 2, . . . , 2TR } for the destination nodes of the m source-destination pairs {(s , d ), = 1, . . . , m}, where |W d | is the number of different values W d can take. Note that Wd may be empty. (4) The average probability of error: Pe(T ) := Prob (W1 , W2 , . . . , Wm ) = (W1 , W2 , . . . , Wm ) where W := g (YdT , Wd ), with YdT := Yd (1), Yd (2), . . . , Yd (T ) . (19.48)
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(19.47)
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De nition 2 A rate vector (R1 , . . . , Rm ) is said to be feasible for the m source-destination pairs (s , d ), = 1, . . . , m, with total power constraint Ptotal , if there exists a sequence of [(2TR1 , . . . , 2TRm ), T, Pe(T ) ] codes satisfying the total power constraint Ptotal , such that Pe(T ) 0 as T . The preceding de nitions are presented with total power constraint Ptotal . If an individual power constraint Pind is placed on each node, then Equation (19.47) should be modi ed: 1 T
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(19.49)
and correspondingly modify the rest of the de nitions to de ne the set of feasible rate vectors under an individual power constraint.
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INFORMATION THEORY AND NETWORK ARCHITECTURES
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19.3.3 The transport capacity The capacity region, is the closure of the set of all such feasible vector rates. As in Section 19.2 we will, focus mainly on the distance-weighted sum of rates. De nition 3 As in Section 19.2, the network s transport capacity CT is
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CT :=
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(R1 ,...,Rm ) feasible
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where := s d is the distance between s and d , and R := Rs d . In the following, due to limited space, a number of results from information theory will be presented without the formal proof. For more details the reader is referred to Xie and Kumar [33]. 19.3.4 Upper bounds under high attenuation (r1) The transport capacity is bounded by the network s total transmission power in media with > 0 or > 3. For a single link (s, d), the rate R is bounded by the received power at d. In wireless networks, owing to mutual interference, the transport capacity is upper-bounded by the total transmitted power Ptotal used by the entire network. In any planar network, with either positive absorption, i.e. > 0, or with path loss exponent > 3 CT where c1 ( , , min ) Ptotal 2 (19.50)
2 +7 log ee min /2 (2 e min /2 ) 2 , if > 0 2 min 2 +1 (1 e min /2 ) c1 ( , , min ) := (19.51) 22 +5 (3 8) log e , if = 0 and > 3 ( 2)2 ( 3) min 2 1
(r2) The transport capacity follows an O(n) scaling law under the individual power constraint, in media with > 0 or > 3. Consider any planar network under the individual power constraint Pind . Suppose that either there is some absorption in the medium, i.e. > 0, or there is no absorption at all but the path loss exponent > 3. Then its transport capacity is upper-bounded as follows: c1 ( , , min )Pind n 2 where c1 ( , , min ) is given by Equation (19.51). As in the previous section we use notation: CT f = O(g) if lim sup [ f (n)/g(n)] < +