Digital Signal Processing Reference
In-Depth Information
2
2
2
⋅
σθ
1
+
2
2
≥
∂
∂ ′
θ
θ
′
− +
σθ
(13.14)
var(
ˆ
s
PNTT
ML
θ
)
s
s
s
=
.
s
2
2
2
PNTT
2
− +
1
4
1
4
In fact, finding the joint MLE of clock skew requires quite a number of computations
as in (13.10), and the fixed portion of delays
d
must be known (or estimated), which
might not be applicable for wireless sensor networks consisting of low-end terminals.
In practice, it requires an additional estimation procedure, which might deteriorate the
robustness of the joint MLE. To overcome this limitation, a family of robust and simple
clock offset and skew estimators that do not require prior knowledge of
d
have been
proposed in [25].
13.3.2 Receiver-Only Synchronization
Due to the power constraint, the communication range of a sensor is strictly limited
to a (radio-geometrical) circle whose radius depends on the transmission power (see
Figure 13.1
). In this figure, every node within the checked area (e.g., node B) can receive
messages from both node P and node A. Suppose that node P is a parent (or reference)
node, and node P and node A perform a pairwise synchronization using two-way tim-
ing message exchanges [17]. Then, all the nodes in the common coverage region of node
P and node A (checked region) can receive a series of synchronization messages contain-
ing the information about the time stamps of the pairwise synchronization. Using this
information, node B can also be synchronized to the parent node, node P, with no extra
timing message transmissions. This approach is called receiver-only synchronization
(ROS). In general, all the sensor nodes lying within the checked area can be synchro-
nized by only receiving timing messages using ROS. Here, node P and node A can be
regarded as super nodes since they provide synchronization beacons for all the nodes
located in their vicinity.
In Figure 13.1, consider an arbitrary node, say node B, in the checked region. While
node P and node A exchange time messages, node B can overhear these time messages.
Hence, node B is capable of observing a set of time readings ({
T
2,
i
(
B
)
}
i=1
) at its local clock
when it receives packets from node A, as depicted in
Figure 13.2
. Besides, node B can
also receive the information about a set of time stamps {
T
2,
i
(
P
)
}
i=1
obtained by receiving the
packets transmitted by node P. Considering the effects of both clock offset and skew, the
reception time at node P in the
i
t h
uplink message
T
2,
i
(
P
)
is given by
()
P
()
A
()
AP
()
AP
()
A
(()
A
AP AP
++,
()
()
TT
= ++⋅ −
θθ
(
TT
)
d
(13.15)
2
,
i
1
,
i
o
s
1
,
i
1 1
,
i
where θ
(AP)
stands for the relative clock skew between node A and node P. Likewise, the
reception time at node B in the
i
th
uplink message
T
2,
i
(
B
)
can be represented by
()
B
()
A
()
AB
()
AB
()
A
(()
A
AB AB
+ + ,
()
()
TT
= ++⋅ −
θθ
(
TT
)
d
(13.16)
2
,
i
1
,
i
o
s
1
,
i
1 1
,
i
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