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Unformatted text preview: 12.3 Power control for sum rate maximization in interference channel. We consider the optimization
log 1 +
Aij pj + vi
n pi = 1 subject to
i=1 pi ≥ 0, i = 1, . . . , n with variables p ∈ Rn . The problem data are the matrix A ∈ Rn×n and the vector v ∈ Rn .
We assume A and v are componentwise nonnegative (Aij ≥ 0 and vi ≥ 0), and that the diagonal
elements of A are equal to one. If the oﬀ-diagonal elements of A are zero (A = I ), the problem
has a simple solution, given by the waterﬁlling method. We are interested in the case where the
oﬀ-diagonal elements are nonzero.
We can give the following interpretation of the problem, which is not needed below. The variables
in the problem are the transmission powers in a communications system. We limit the total power
to one (for simplicity; we could have used any other number). The ith term in the objective is 94 the Shannon capacity of the ith channel; the fraction in the argument of the log is the signal to
interference plus noise ratio.
We can express the problem as
j =1 Bij pj n log maximi...
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- Fall '13
- The Aeneid