Time Synchronization for Wireless Sensor Networks - Adaptive Signal Processing in Wireless Communications - page 388

and when d  d (AP) = d (PA) is unknown, it is proved in [23] that the ML estimator of θ (AP)
is given by

min U − min V

ˆ

1 ≤≤ iN i 1 ≤≤ i Ni

θ (AP) = . (13. 31)

2

On the other hand, with X i (AP) and X i (PA) in (13.2) and (13.3) being modeled as indepen-
dent and normally distributed RVs with the same mean μ and variance σ 2 /2, the maxi-
mum likelihood (ML) estimate for θ (AP) takes the equation (derived in section 13.3.1)

∑∑

N N

U − V

1 1

i i

N N

ˆ θ (AP) =

i = 1 i = 1

. (13. 32)

2

Notice from (13.30)-(13.32), it is clear that if only one round of message exchange is
performed, the TPSN presented in (13.30) is the ML estimator under both exponential
and Gaussian delay models.

13.4.1.3 Joint Clock Offset and Skew Estimation Based on Two-Way
Message Exchanges
When clock skew exists between two nodes, the clock offset between them will increase
linearly, as shown in Figure 13.3 . In order to establish long-term synchronization, it is
more efficient to estimate jointly the clock offset and skew. In section 13.3.1, we derived
the joint offset and skew ML estimators (see equations (13.9) and (13.10)), when the vari-
able delays X i (PA) and X i (AP) are modeled as independent Gaussian distributed RVs. When
X i (PA) and X i (AP) are exponentially distributed RVs, the likelihood function for joint esti-
mation of the clock offset and skew is very complicated. However, a solution to this
problem has been recently reported in [27].
Notice that the joint offset and skew ML estimators (equations (13.9) and (13.10))
under Gaussian delay assumption are quite complicated. Besides, there is no simple
closed-form solution for the ML joint offset and skew estimation when the delays are
exponentially distributed. For these reasons, a family of robust and simple clock offset
and skew estimators, named maximum likelihood like estimators (MLLEs), has been
proposed in [25].

13.4.1.4 Tiny-Sync and Mini-Sync
Tiny-sync and Mini-sync [28] are two lightweight clock synchronization protocols that
also use the two-way message exchanges. Node A and node P exchange messages just
like in Figure 13.3. The only difference here is that node P replies to node A immediately
after receiving the message, i.e., T 2, i

(P) . Assuming the clocks between node A and
node P are linearly related, from (13.6) and (13.7) we have

(P) = T 3, i

() P () AP

T − θ

2 , i o () A () AP

=++ , TdX

1 , i i

() AP

1 + θ

s