Geometric Programming for Communication Systems (Foundations by Mung Chiang

By Mung Chiang

Lately Geometric Programming has been utilized to check various difficulties within the research and layout of conversation structures from info idea and queuing idea to sign processing and community protocols. Geometric Programming for communique structures starts its complete therapy of the topic via delivering an in-depth educational at the conception, algorithms, and modeling equipment of Geometric Programming. It then supplies a scientific survey of the functions of Geometric Programming to the research of verbal exchange platforms. It collects in a single position a variety of released ends up in this region, that are at present scattered in numerous books and plenty of learn papers, in addition to up to now unpublished effects. Geometric Programming for communique structures is meant for researchers and scholars who desire to have a accomplished start line for knowing the idea and purposes of geometric programming in verbal exchange structures.

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N }, output Y ∈ Y = {1, 2, . . , M }, and channel law Pij = Prob{Y = j|X = i}, i = 1, 2, . . , N, j = 1, 2, . . , M . The channel law forms a channel matrix P ∈ RN ×M , where the (i, j) entry of P is Pij ≥ 0 with P1 = 1. A distribution p ∈ RN ×1 on the input, together with a given channel matrix P, induces a distribution q ∈ RM ×1 on the output by q = PT p, and a joint distribution Q ∈ RN ×M on the input output pair by Qij = pi Pij . We also associate with each input alphabet symbol i an input cost si ≥ 0, forming a cost vector s ∈ RN ×1 .

N j=1 zj w ≥ 1, zj ≥ 0, j = 1, 2, . . 5) where P(i) is the ith row of P. 4) means the following: • Weak duality. 4) produces an upper bound on channel capacity with input cost: log M j=1 exp(αj + γS) ≥ C(S). • Strong duality. 4) is C(S). 1. 1. 3) is the following GP (in convex form) minimize log subject to Pα variables α. 6) The constant parameters are P. An equivalent version of the Lagrange dual problem is the following GP (in standard form): minimize subject to variables M j=1 zj Pij M j=1 zj (i) ≥ e−H(P ) , i = 1, 2, .

22) 46 Geometric Programming Appropriate choice of the stepsize α(t) > 0 leads to convergence of the dual algorithm [16]. 20) involving only local variables, upon receiving the updated dual variables {γji , j : i ∈ I(j)} (note that {γij , j ∈ I(i)} are local dual variables). 22). The amount of message passing overhead in the above distributed algorithm can be substantially reduced using the structures of the coupling variables. 1 through distributed GP-based power control. 1 Information Theory Materials in this subsection are in part based on recent publications [30, 33, 82, 84], and some problems have been studied in the 1980s and 1990s [120, 121, 122].

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