bv_cvxbook_extra_exercises

# Use the result of part b to show that the rank of z

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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 M-matrix.) 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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