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Unformatted text preview: 12.3 Power control for sum rate maximization in interference channel. We consider the optimization
problem
n
pi
log 1 +
maximize
Aij pj + vi
j =i
i=1
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 oﬀdiagonal 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
oﬀdiagonal 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
n
j =1 Bij pj n log maximi...
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 Fall '13
 F.Borrelli
 The Aeneid

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