Digital Signal Processing Reference
In-Depth Information
,
∑
N
T
=
T
N
1
1
i
i
=
1
and
∑
N
T
=
T
N
4
4
,
i
i
=
1
stand for the average of time observations. Therefore,
2
2
2
2
∂
ln
L
θθσ
θ
, ′,
∂∂
ln
L
, ′,
∂
′
θθσ
θ
θ
os
os
−
E
−
E
2
∂
o
o
s
()
=
I
θ
,
∂
2
ln
L
θθσ
θθ
, ′,
2
∂
2
ln
L
θ
,
θσ
′′ ,
2
os
o
s
−
E
−
E
∂ ′
∂ ′
θ
2
(13.11)
so
s
2
2
N
′
N TT
+
s
1
4
2
=
.
1
∑
N
( )
+
( )
+
2
2
2
σ
NT T
+
Td Td
σ
2
1
4
1
,
i
4
,
i
2
θ
′
i
=
1
s
The CRB can be obtained by taking the [
i
,
i
[i,i]th element of the inverse of the Fisher infor-
mation matrix (i.e., var (θ
ˆ
i
) ≥ [
I
-1
(θ)]
ii
), and the inverse
I
-1
(θ) is given by
−
T
+
T
P
1
4
2
2
2
2
PNTT
− +
θ
′
NPNT T
2
− +
,
(13.12)
2
σ
1
4
s
1
4
I
−
1
()
θ
=
2
−+
TT
θ
s
2
2
′
1
4
2
2
2
PN
−
TTT
+
2
PNTT
− +
1
4
1
4
where
∑
( )
+
( )
+
1
N
2
2
2
2σ .
P
T
dTd
1
,
i
4
,
i
i
=
Consequently, the CRBs of the joint clock offset and skew estimator are given by
2
σθ
(
1
+
)
2
P
var(
ˆ
,
(13.13)
ML
θ
)
≥
s
o
2
22
NPNT T
− +
1
4
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