Digital Signal Processing Reference
In-Depth Information
Receiver-receiver synchronization (
Node A
and
Node B
)
Clock
offset
T
1,1
T
1,
i
T
1,
N
Node P
P
Node A
A
(A)
2,1
(A)
2,
i
(A)
2,
N
T
T
T
}
i
j =
1
}
i
j =
1
}
N
j =
1
}
N
j =
1
(B)
2,1
(A)
2,1
ˆ
T
T
{
(B)
2,
j
{
(A)
2,j
{
(B)
2,
j
{
(A)
(AB)
T
T
T
T
o
2,
j
Node B
B
(B)
2,
i
(B)
2,
N
(B)
2,1
T
T
T
FIgure 13.5
Clock synchronization model of RRS.
X
i
(PA)
denotes the nondeterministic delay components (random portion of delays) and
d
(PA)
denotes the deterministic delay component (propagation delay) from node P to
node A; then
T
2,
i
(A)
can be written as
()
A
()
PA
()
PA
PA
θθ
,,
()
PA
()
(
TTd
=+ +
X
++⋅
T
−,
T
11
)
(13. 26)
2
,
i
1
i
i
o
s
1
i
where
T
1,
i
is the transmission time at the reference node, and θ
(PA)
and θ
(PA)
are the clock
offset and skew of node A with respect to the reference node, respectively. Similarly, we
can decompose the arrival time at node B as
()
B
()
PB
()
PB
PB
θθ
,,
()
PB
()
(
TTd
=+ + ++⋅
X
T
−,
T
11
)
(13. 27)
2
,
i
1
i
i
o
s
1
i
where
d
(PB)
,
X
i
(PB)
, θ
(PB)
, and θ
(PB)
stand for the propagation (fixed) delay, random por-
tion of delays, clock offset, and skew of node B with respect to the reference node,
respectively.
Subtracting (13.27) from (13.26), we obtain
()
A
()
B
BA
θθ
(
BA
)
(
)
(()
PA
()
PB
()
PA
()
PB
TT
−=+⋅ −+
(
TTd
)
−+−
d
XX
i
,
(13. 28)
2
,
i
2
,
i
o
s
1
,
i
1 1
,
i
where θ
(BA)
θ
(PA)
- θ
(PB)
and θ
(BA)
θ
(PA)
- θ
(PB)
are the relative clock offset and skew
between node A and node B at the time they receive the
i
t h
broadcast packet from the
reference node, respectively. Here, we assume these random portions of delays
X
i
(PA)
and
X
i
(PB)
are normal distributed RVs with mean μ and variance σ
2
/2. Indeed, (13.28)
assumes exactly the same form as (13.17). Hence, the same steps can be applied to derive
the joint clock offset and skew estimator for ROS. More specifically, let the noise compo-
nent
z
[
i
]
μ′ +
X
i
(BA)
, where μ′
d
(PA)
-
d
(PB)
and
z
[
i
] ~ N(μ′,σ
2
). Let us also define
x
[
i
]
T
2,
i
(B)
- μ′ and
w
[
i
]
z
[
i
] - μ′. Using similar steps as in ROS, it is straightforward to
show that the same form of the joint clock offset and skew estimator (13.22) can also be
(A)
-
T
2,
i
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