Geoscience Reference
In-Depth Information
Doppler and noise effect on the moments of
y
2
(
t
+
τ
)
ˆ
should be noticed. Instead of
y
2
(
t
+
τ
)
ˆ
,
we work on the real part:
y
2
r
(
t
+
τ
)=
ˆ
s
(
t
)
cos
(
2
π
(
t
+
τ
)
ε
)+
ω
2
(
ˆ
t
+
τ
)
ˆ
,
t
≥
0.
(13)
y
2
r
(
+
τ
)
ˆ
includes the noise and signal, and the signal is also affected by Doppler which
changes the moments of the signal. Therefore, we prefer to obtain MGF of
y
2
r
(
t
+
τ
)
ˆ
t
firstly,
then, the moments are obtained from this MGF by (7).
The noise-free signal in (13) is
ω
2
(
+
τ
)
ˆ
(
+
τ
)
ˆ
independent from the noise
t
, so MGF of
y
2
r
t
is:
M
y
2
r
(
u
)=
M
r
(
u
)
M
(
u
)
,
(14)
ω
2
where
M
r
(
)
u
is MGF of the first term in right side of (13), and:
exp
0.5
ω
2
u
2
.
2
M
ω
2
(
u
)=
σ
(15)
The time varying variance will be comprehensively discussed in the sequel. Here, the problem
is to estimate
M
r
follows a Gaussian mixture distribution in (6). The presence of
the cosine term changes the first term in the right side of (13) to a non-stationary process.
Although the cosine term is time variant, fortunately, it is deterministic.
(
u
)
.
s
(
t
)
Now, we obtain
M
r
(
u
)
:
i
=
1
p
i
exp
μ
s
i
u
+
0.5
σ
s
i
u
2
N
(
)=
⇒
M
s
u
i
=
1
p
i
exp
μ
s
i
u
+
0.5
σ
u
2
.
N
s
i
cos
2
(
)=
(
π
(
+
τ
)
ε
)
M
r
u
;
t
2
t
ˆ
(16)
Both
M
r
(
u
)
and
M
ω
2
(
u
)
are expressed as the series for
u
→
0, then by multiplying these two
series and ordering their terms, MGF of
y
2
r
(
t
+
τ
)
ˆ
is asymptotically obtained in the context of
(7):
M
y
2
r
(
u
) =
M
r
(
u
)
M
ω
2
(
u
)
u
2
m
r
2
2!
u
3
m
r
3
3!
u
4
m
r
4
4!
=(
1
+
um
r
1
+
+
+
+
···
)
u
2
m
u
3
m
u
4
m
ω
2
2
2!
ω
2
3
3!
ω
2
4
4!
×
(
1
+
um
+
+
+
+
···
)
ω
2
1
u
2
(
m
r
2
+
m
ω
2
2
+
2
m
r
1
m
ω
2
1
)
=
+
(
m
r
1
+
)+
1
u
m
ω
2
1
2!
u
3
(
m
r
3
+
m
+
3
m
r
1
m
+
3
m
r
2
m
)
ω
2
3
ω
2
2
ω
2
1
+
3!
u
4
(
m
r
4
+
+
+
+
)
m
6
m
r
2
m
4
m
r
1
m
4
m
r
3
m
ω
2
4
ω
2
2
ω
2
3
ω
2
1
+
+
···
.
(17)
4!
The moments extracted from
M
r
(
u
)
are shown in Table (II). There exists also another problem.
The resulting moments of
y
2
r
(
are time dependent. Since the cosine term is deterministic,
the time average of the moments can be substituted instead. Let's define:
t
+
τ
)
ˆ
T
1
T
cos
i
ζ
i
(
ε
)=
(
π
(
+
τ
)
ε
)
ˆ
2
t
dt
,
(18)
0
