Geoscience Reference
In-Depth Information
n
Moment
01
i
1
1
p
i
μ
s
i
∑
=
i
s
i
+
σ
s
i
cos
2
2
1
p
i
(
μ
(
2
π
(
t
+
τ
)
ε
))
ˆ
∑
=
i
s
i
+
s
i
cos
2
3
1
p
i
(
μ
3
μ
s
i
σ
(
2
π
(
t
+
τ
)
ε
))
ˆ
∑
=
4
∑
N
i
s
i
s
i
σ
s
i
cos
2
s
i
cos
4
1
p
i
(
μ
+
6
μ
(
2
π
(
t
+
τ
)
ε
)+
ˆ
3
σ
(
2
π
(
t
+
τ
)
ε
))
ˆ
=
(
)
Table 2. Moments extracted from
M
r
u
;
t
ζ
i
(
ε
)
where
T
is the observation time.
Note that for dependency of
on
ε
, the moments of
y
2
r
(
+
τ
)
t
ˆ
are dependent on
ε
too. Finally, for obtaining the time-independent moments of
,
m
y
2
r
,
i
, it suffices that all “cos
i
y
2
r
(
+
τ
)
(
π
(
+
τ
)
ε
)
t
ˆ
2
t
ˆ
” terms in the time-dependent moments
ζ
i
(
ε
)
to be substituted by
. The final moments are depicted in table (III).
Since now, the moments were obtained analytically, it means we only calculated the right side
of equation (4). On the other hand, the moments of the observed signal in the second receiver
can be calculated statistically by:
T
1
T
y
i
2
r
(
m
i
=
+
τ
)
t
ˆ
dt
.
(19)
0
Now the left side of the equation (4) is also obtained. Both of these two procedures must yield
same results. Thus,
should be selected in such a way that this equality holds. To do this,
MSE criterion is used again:
ε
i
=
1
m
y
2
r
,
i
−
m
i
L
2
=
MSE
.
(20)
(
+
Similar to the previous section,
L
is considered as 4. So Doppler of the received signal
y
2
r
t
τ
)
ˆ
is estimated:
i
=
1
m
y
2
r
,
i
−
m
i
L
2
ˆ
ε
=
arg min
ε
.
(21)
4.3 Noise power estimation
The noise power estimation is similar to Doppler estimation. Indeed, these two estimations
are done simultaneously. It could be seen that the moments do not merely depend on Doppler.
