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EE263 homework problems Lecture 2 – Linear functions and ...

EE263 Autumn 2007-08 Prof. S. BoydEE263 homework problemsLecture 2 Linear functions and simple power control algorithm for a wireless some background. We considera network ofntransmitter/receiver pairs. Transmitteritransmits at power levelpi(whichis positive). The path gain from transmitterjto receiveriisGij(which are all nonnegative,andGiiare positive). The signal power at receiveriis given bysi=Giipi. The noise plusinterference power at receiveriis given byqi= +Xj6=iGijpjwhere >0 is the self-noise power of the receivers (assumed to be the same for all receivers).Thesignal to interference plus noise ratio(SINR) at receiveriis defined asSi=si/qi. Forsignal reception to occur, the SINR must exceed some threshold value (which is often inthe range 3 10). Variouspower control algorithmsare used to adjust the powerspitoensure thatSi (so that each receiver can receive the signal transmitted byits associatedtransmitter).

2.4 Representing linear functions as matrix multiplication. Suppose that f: Rn −→ Rm is linear. Show that there is a matrix A∈ Rm×n such that for all x∈ Rn, f(x) = Ax. (Explicitly describe how you get the coefficients Aij from f, and then verify that f(x) = Axfor any x∈ Rn.) Is the matrix Athat represents funique?

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