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Unformatted text preview: is the vector with
all components 1). Show how to construct a feasible power allocation x from z .
(b) Show how to ﬁnd the largest possible SNIR, i.e., how to maximize α subject to the existence
of a feasible power allocation.
To solve this problem you may need the following:
Hint. Let T ∈ Rn×n be a matrix with nonnegative elements, and s ∈ R. Then the following are
equivalent:
(a) s > ρ(T ), where ρ(T ) = maxi λi (T ) is the spectral radius of T . (b) sI − T is nonsingular and the matrix (sI − T )−1 has nonnegative elements.
(c) there exists an x 0 with (sI − T )x ≻ 0. (For such s, the matrix sI − T is called a nonsingular Mmatrix.) Remark. This problem gives an analytic solution to a very special form of transmitter power
allocation problem. Speciﬁcally, there are exactly as many transmitters as receivers, and no power
limits on the transmitters. One consequence is that the receiver noises βi play no role at all in the
solution — just crank up all the transmitters to overpower the noises!
98 12.8 Optimizing...
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 Fall '13
 F.Borrelli
 The Aeneid

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