v2010.10.26 - Convex Optimization

5.4. EDM DEFINITION 427by translated ellipsoids of graduated orientation and eccentricity as inFigure 126.Depicted in Figure 125 is one cell phone x 1 whose signal power isautomatically and repeatedly measured by 6 base stations ◦ nearby. 5.20Those signal power measurements are transmitted from that cell phone tobase station ˇx 2 who decides whether to transfer (hand-off or hand-over)responsibility for that call should the user roam outside its cell. 5.21Due to noise, at least one distance measurement more than the minimumnumber of measurements is required for reliable localization in practice;3 measurements are minimum in two dimensions, 4 in three. 5.22 Existenceof noise precludes measured distance from the input data. We instead assignmeasured distance to a range estimate specified by individual upper andlower bounds: d i1 is the upper bound on distance-square from the cell phoneto i th base station, while d i1 is the lower bound. These bounds become theinput data. Each measurement range is presumed different from the others.Then convex problem (936) takes the form:minimize trGG∈S 7 , X∈R2×7 subject to d i1 ≤ tr(GΦ i1 ) ≤ d i1 , i = 2... 7tr ( )Ge i e T i = ‖ˇx i ‖ 2 , i = 2... 7tr(G(e i e T j + e j e T i )/2) = ˇx T i ˇx j , 2 ≤ i < j = 3... 7whereX(:, 2:7) = [ ˇx 2 ˇx 3 ˇx 4 ˇx 5 ˇx 6 ˇx 7 ][ ] I XX T≽ 0 (946)GΦ ij = (e i − e j )(e i − e j ) T ∈ S N + (889)This semidefinite program realizes the wireless location problem illustrated inFigure 125. Location X ⋆ (:, 1) is taken as solution, although measurement5.20 Cell phone signal power is typically encoded logarithmically with 1-decibel incrementand 64-decibel dynamic range.5.21 Because distance to base station is quite difficult to infer from signal powermeasurements in an urban environment, localization of a particular cell phone • bydistance geometry would be far easier were the whole cellular system instead conceived socell phone x 1 also transmits (to base station ˇx 2 ) its signal power as received by all othercell phones within range.5.22 In Example 4.4.1.2.4, we explore how this convex optimization algorithm fares in theface of measurement noise.