(8)
In this case, k1, k2 can be known when S receives three messages among
all that have been relayed after the same initiator.
B1 S
t0
t1
t2
Time Time
B2 B3 B4 B5
X
X
(B1, 4, B2
,
)
(B1, 3, B2,
, B3,
)
(B1, 2, B2,
, B
3,
, B4,
, B
5
,
)
Fig. 4. Range Detection: Signal is Relayed With Loss
Comparing Equations (3)(4) with (7)(8), we can summarize the result of
range detection as
dSA = dSB + k1, (9)
dSC = dSB + k2, (10)
where, A,B,C are the three relay nodes in the same group that convey messages
originated from the same source and are sorted according to the sequence
184 Fang Liu et al.
Chapter 7 TPSS
in which they relay the signal.
Remarks:
(i) All times are estimated locally. For example, the arrival times of the signals
(t1, t2, etc.) are measured at sensor S??™s local timer; the time di?®erences at
relay nodes (??tb,??tc, etc.) are computed based on the beacon??™s local timer.
(ii) For each sensor S, range detection is conducted on each group that contains
messages from the same initiator. Corresponding location computation
is taken in the next step. Averaging all the results computed for S, the final
result is taken as the coordinates of node S.
(iii) For each group, there may exist multiple methods to select the three
messages. Consider a signal travelling along beacons 1 to 4, and assume all
the relayed signals arrive at S finally. We have ds,i = ds,i??’1 + ki??’1, where
ki = v ?· (ti+1 ??’ ti ??’ ??ti+1) ??’ di,i+1, dij(dsj) is the distance between node
i(s) and j, ??ti is the time di?®erence at the relay node i, and ti is the time S
receives the message from beacon i, for i = 2, 3, and 4.
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