Digital Signal Processing Reference
In-Depth Information
where θ
(AB)
and θ
(AB)
stand for the relative clock offset and skew between node A and
node B, and
d
(AB)
and
X
i
(AB)
denote the fixed and random portions of timing delays in the
message transmission from node A to node B, respectively. Here,
X
i
(AB)
is assumed to be
a normal distributed RV with mean μ and variance σ
2
/2.
The linear regression technique can be applied to synchronize node B and compen-
sate the effects of the relative clock skew between node P and node B. Subtracting (13.16)
from (13.15) gives
()
P
()
B
()
BP
()
BP
()
A
(()
A
()
AP
()
AB
()
AP
()
AB
TT
−=+⋅ −
θθ
(
TT
)
+ − +
d
d
XX
i
− .
(13.17)
2
,
i
2
,
i
o
s
1
,
i
1 1
,
i
Since
d
(AB)
and
d
(AP)
are fixed values and
X
i
(AB)
and
X
i
(AP)
are normal distributed RVs, the
noise component can be defined by
z
[
i
]
μ′ +
X
i
(AP)
-
X
i
(AB)
, where μ′
d
(AP)
-
d
(AB)
and
z
[
i
] ~ N(μ′,σ
2
). Let
x
[
i
]
T
2,
i
(B)
- μ′ and
w
[
i
]
z
[
i
] - μ′, then the set of observed data
can be written in matrix notation as follows:
(P)
-
T
2,
i
xH w
=+,
θ
where
x
= [
x
[1]
x
[2]
…
x
[
N
]]
T
,
w
= [
w
[1]
w
[2]
…
w
[
N
]]
T
, θ = [θ
(BP)
θ
(BP)
]
T
, and
T
1
1
1
H
=
.
()
A
()
A
()
A
()
A
0
TT
−
TT
−
12
,
11
,
1
,
N
1 1
,
Note that the noise vector
w
~ N(0,σ
2
I
) and the matrix
H
is the observation matrix
whose dimension is
N
× 2. From [26, theorem 3.2, p. 44], the minimum variance unbi-
ased (MVU) estimator for the relative clock offset and skew is given by θ
ˆ
=
g
(
x
), where
g
(
x
) satisfies
∂
ln
p
x
()
()(())
;
θ
=
I
θ
gx
− .
θ
(13.18)
∂
θ
Since the noise vector
w
is zero mean and Gaussian distributed, from the results in [26,
p. 85], the derivative of the log-likelihood function can be written as
T
∂
ln
p
()
x
;
θ
HH
HH Hx
T
−
1
T
=
[(
)
− ,
θ
]
(13.19)
∂
θ
2
σ
where
H
T
H
is assumed to be invertible. Therefore, comparing (13.18) with (13.9) yields
ˆ
−
HH
T
1
T
θ=
(
)
Hx
,
(13. 20)
T
HH
I
()
θ
=
,
(13. 21)
2
σ
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