v2009.01.01 - Convex Optimization

5.4. EDM DEFINITION 375
by translated ellipsoids of graduated orientation and eccentricity as in
Figure 102.
Depicted in Figure 101 is one cell phone x 1 whose signal power is
automatically and repeatedly measured by 6 base stations ◦ nearby. 5.17
Those signal power measurements are transmitted from that cell phone to
base station ˇx 2 who decides whether to transfer (hand-off or hand-over)
responsibility for that call should the user roam outside its cell. 5.18
Due to noise, at least one distance measurement more than the minimum
number of measurements is required for reliable localization in practice;
3 measurements are minimum in two dimensions, 4 in three. 5.19 Existence
of noise precludes measured distance from the input data. We instead assign
measured distance to a range estimate specified by individual upper and
lower bounds: d i1 is the upper bound on distance-square from the cell phone
to i th base station, while d i1 is the lower bound. These bounds become the
input data. Each measurement range is presumed different from the others.
Then convex problem (841) takes the form:
minimize trG
G∈S 7 , X∈R2×7 subject to d i1 ≤ tr(GΦ i1 ) ≤ d i1 , i = 2... 7
tr ( )
Ge i e T i = ‖ˇx i ‖ 2 , i = 2... 7
tr(G(e i e T j + e j e T i )/2) = ˇx T i ˇx j , 2 ≤ i < j = 3... 7
X(:, 2:7) = [ ˇx 2 ˇx 3 ˇx 4 ˇx 5 ˇx 6 ˇx 7 ]
[ ] I X
X T
≽ 0 (851)
G
where
Φ ij = (e i − e j )(e i − e j ) T ∈ S N + (796)
5.17 Cell phone signal power is typically encoded logarithmically with 1-decibel increment
and 64-decibel dynamic range.
5.18 Because distance to base station is quite difficult to infer from signal power
measurements in an urban environment, localization of a particular cell phone • by
distance geometry would be far easier were the whole cellular system instead conceived so
cell phone x 1 also transmits (to base station ˇx 2 ) its signal power as received by all other
cell phones within range.
5.19 In Example 4.4.1.2.2, we explore how this convex optimization algorithm fares in the
face of measurement noise.