Signal Processing - March 2016 - 52

ACCurACY BOuNDS fOr Df-L
attenuation Tn , ^ x h, and standard deviation Tv , ^ x h, respectively [11]. According to the diffraction model (5), and the related analytical maps Tn , (x) = E { 6TE 2dB (x; ,; {)@ and Tv , ^ x h =
2
E { 8^TE 2dB ^x; ,; {h - Tn , ^ x hh B obtained as in "DiffractionBased Modeling of Human Body Shadowing," by computing
all derivative terms of (S2), it is possible to obtain the F ^ x h
matrix and, finally, the CRB for the complete L-link network. To
demonstrate the effects of multiple links on the localization
accuracy, the maps shown in Figure 6(b) show the lower
bound to the root mean square error of the location estimate
RMSE = E 8 xt - x 2B $ tr 6F -1^ x h@ for L = 1, 8, 16, 24
links. As expected, the localization accuracy is higher for a
target near the terminals and along the LOS paths. For targets
located in-between, the reduced sensitivity could be counterbalanced by increasing the number of links [11].

The Cramer-Rao bound (CRB) provides a useful benchmark
for assessing DF-L performances. The covariance matrix for
any unbiased estimator xt of the target position x is lower
bounded as Cov ^ xt h = E 6^x - xt h^x - xt hT@ $ F -1 ^ x h, where
F ^ x h is the 2 × 2 Fisher information matrix (FIM). According
to the Gaussian model s~N ^n ^ x h, Q ^ x hh in (5), the FIM
term (i, j) with i, j ! " 1, 2 ,, is
Fi, j = [F ^ x h] i, j =

/ ,L = 1

1

v 2, ^ x h

=

2n , ^ x h 2 n , ^ x h
2v , ^ x h 2v , ^ x h
G,
+2
2x i
2x j
2x i
2x j
(S2)

where the gradient functions 2n , ^ x h /2x i = 2Dn , ^ x h /2x i and
2v , ^ x h /2x i = 2Dv , ^ x h /2x i, i ! {1, 2}, embody the information on the target location provided by the perturbation maps of

(a)
Experimental (Bars)
Log-Normal µ0, = -46.5 dBm, σ0, = 0.8 dB
Log-Normal µ1, = -53 dBm, σ1, = 2.4 dB

Experimental (Bars)
Log-Normal µ0, = -42.6 dBm, σ0, = 0.7 dB
Log-Normal µ1, = -52.3 dBm, σ1, = 2.1 dB
x∉
0.5
0.5 m
8

10

12

7

0.2

0.5

13
x = [x, y]

6

x∈

0.4
0.3
0.2

14

0.1

x2

0
-65 -60 -55 -50 -45 -40 -35 -30
RSS s (dBm)
RSS Mean ∆µ (x)

Outdoor

0.6
11

x∈

0.1

Indoor

9

Probability

Probability

0.4
0.3

x∉

0.7

5

4

3

2

x1

1

0
-65 -60 -55 -50 -45 -40 -35 -30
RSS s (dBm)
RSS Standard Deviation ∆σ (x)

0-11 dB 0-12 dB 0-10 dB 0-9 dB 0-14 dB 0-12 dB 0-18 dB

0-6 dB 0-7 dB 0-7 dB

0-7 dB 0-7 dB

0-5 dB

0-8 dB

4-12 dB 5-9 dB 0-9 dB 0-14 dB 0-9 dB

1-10 dB 0-14 dB

0-7 dB 1-5 dB 0-6 dB

0-4 dB 0-5 dB

0-5 dB

0-7 dB

0-5 dB 0-4 dB 0-4 dB

0-6 dB 0-7 dB

0-6 dB 3-12 dB

2-5 dB 0-5 dB 0-3 dB

0-3 dB 1-4 dB

1-5 dB

1-4 dB

0-5 dB 1-9 dB 0-5 dB

0-5 dB 1-7 dB

0-5 dB

0-4 dB 0-7 dB 0-3 dB

0-6 dB 1-6 dB

1-5 dB

0-4 dB

0-6 dB

(b)

figure 5. (a) Log-normal RSS modeling for two links (top left and top right) and for target inside (red) and outside (blue) the sensitivity area X , for link
,. (b) Experimental RSS mean and standard deviation maps versus the target location x for the highlighted network layout (top center) and some links
in indoor and outdoor scenarios.

52

IEEE SIgnal ProcESSIng MagazInE

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March 2016

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Table of Contents for the Digital Edition of Signal Processing - March 2016
