Digital Signal Processing Reference
In-Depth Information
o
: Clock offset (time difference)
s
: Clock skew (frequency difference)
Linear clock skew model
T
(P)
3,
N
T
(P)
2,
N
(AP)
(A)
(AP)
i
(
T
+
d
+
X
)
T
(P)
3,
i
s
1,
i
T
(P)
2,
i
(AP)
(A)
(A)
1,1
(
T
-
T
)
T
(P)
3,2
T
(P)
2,2
s
4,
N
(AP)
(A)
(PA)
i
(
T
-
d
-
X
)
T
(P)
3,1
s
4,
i
T
(P)
2,1
Node P
(AP)
o
Node A
T
(A)
1,1
T
(A)
4,1
T
(A)
T
(A)
4,2
T
(A)
1,
i
T
(A)
4,
i
T
(A)
1,
N
T
(A)
4,
N
1,2
(
T
(A)
1,1
= 0)
d
+
X
(AP)
i
d
+
X
(PA)
i
FIgure 13.3
Two-way timing message exchange model that assumes clock offset and skew.
2
2
( )
( )
N
θ
o
−
,
+
T
i
(
T
i
d
,
++
)
1
θ
θ
−
,
+
,
−
T
i
(
T
i
d
)1
+
θ
2
1
s
o
3
4
s
.
∑
−
1
2
+
1
+
θ
1
+
θ
−
N
s
s
,
( )
=
σ
2
i
=
1
f
xy
πσ
2
e
XY
,
Further assuming that the fixed portion of delay
d
is known and θ′
s
1/(1 + θ
s
), then
t he log-likelihood f function (ignoring irrelevant additive and mu ltiplicative constants) for
(θ
o
,θ′
s
), based on observations {
T
1,
i
}
i
=1
, {
T
2,
i
}
i
=1
, {
T
3,
i
}
i
=1
, and {
T
4,
i
}
i
=1
, is given by
N
{
}
.
∑
1
,
( )
=−
2
2
ln
L
θθ
θθ
′ −++
(
T
)
(
Td
1
+ ′ −+−
)
θθ
s
(
TTd
i
)
(
)
(13.8)
os
s
o
2
,
i
,
i
o
3
,
4
,
i
i
=
It has been shown in [25] that the values of θ
o
and θ
s
that maximize the above log-
likelihood function are given, respectively, by
N
N
N
∑∑
1
∑
2
2
(
TT
+
)
(
TT
+ −
)
(
TTQ
+
)
1
,
i
4
,
i
2
,
i
3
,
i
2
,
i
3
,
i
ˆ
i
=
i
=
1
i
=
1
θ
o
ML
=
,
(13.9)
N
N
∑∑
(
TT
+
)
(
T
+
T
,
−
)
2
Q
2
,
i
3
,
i
1
,
i
4
i
i
=
1
i
=
1
N
N
N
N
∑∑
∑
∑
2
2
−
(
TT
+
)
−
2
NTT
(
+
)
(
T
+
T
)
QTT
(
+
)
1
,
i
4
,
i
2
,
i
3
,
i
2
,
i
3
,
i
2
,
i
3
,
i
ˆ
i
=
1
i
=
1
i
=
1
i
=
1
θ
s
ML
=
+
−,
1
(13.10)
N
N
N
N
∑
∑
∑∑
(
TT
+
)
(
TT
+
)
(
TT
+
)
(
TT
+ −
)
2
NQ
1
,
i
4
,
i
1
,
i
4
,
i
2
,
i
3
,
i
1
,
i
4
,
i
i
=
1
i
=
1
i
=
1
i
=
1
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