Digital Signal Processing Reference
In-Depth Information
Similarly, since the time differences between beacons are proportional to
N
and by far
greater than the variance of delays, the following relationship can be obtained from the
lower bound for the clock skew estimator derived in [25]:
σ
2
ε
2
σ
=
s
,
2
1
, ≥.
N
2
ε
2
sN
,
(
N
−
)
Therefore, for
N
≥ 2, τ
(
N
)
max
can be rewritten as
2
σ
2
ε
σ
−
o
,
1
()
(
N
ε
τ
=−
N
1
)
N
,
N
≥ .
2
(13.4 2)
max
2
σ
ε
s
,
2
Note that ε
s
,1
can be obtained by the specifications of the crystal oscillator, and ε
o
,1
and
ε
s
,2
can be determined by simple experimental tests. Therefore, the maximum resyn-
chronization period is proportional to the number of beacons, and performing clock
skew estimation will significantly increase τ
(
N
)
max
since σ
ε
s
,1
σ
ε
s
,2
.
13.5.3.3 Number of Beacons Required for Each Pairwise Synchronization
The goal of AMTS is to minimize the average number of message exchanges (
—
). Hence,
from (13.41), finding the optimal number of beacons (
N
) resumes to solving the follow-
ing optimization problem:
ˆ
N
=
argmin
M
,
(13.43)
N
with
2
B
N
=
1
2
2
σ
ε
ε
−
()
1
τ
+
o
,
1
sync
2
σ
2
BN
ε
s
,
1
M
=
=
,
2
B
()
N
()
N
τ τ
+
N
≥
2
sync
max
2
σ
ε
()
N
2
o
N
,
1
σ
−
τ
ε
sync
+
−
N
N
1
N
2
σ
ε
s
,
2
where τ
(
N
)
sync
denotes the synchronization time with
N
beacons and will be estimated at
the reference node for different
N
values when the network is first established. Once
N
is
estimated from (13.43), τ
(
N
)
max
can be obtained from (13.42).
Simulation results in [40] show that AMTS requires far less timing messages than
TPSN when there exist multiple numbers of beacon transmissions. Moreover, the gap
between the average number of required timing messages between AMTS and TPSN
significantly increases as
N
increases, and thus AMTP is by far more energy efficient than
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