Geoscience Reference
In-Depth Information
n Moment
01
i
1
1 p i μ s i
=
i
s i + σ
s i ζ 2 ( ε )) + σ
2
ω
2
1 p i ( μ
=
p i μ
N
N
s i
s i ζ 2
2
ω
3
+
3
μ s i σ
( ε )
+
3
σ
p i μ s i
=
=
i
1
i
1
p i
N
i
N
4
s i +
s i σ
s i ζ 2 ( ε ) +
s i ζ 4 ( ε )
4
ω +
2
ω
s i + σ
s i ζ 2 ( ε ))
μ
6
μ
3
σ
+
3
σ
6
σ
p i ( μ
=
1
i
=
1
Table 3. Final moments extracted from M y 2 r (
t
+
τ )
ˆ
They depend onto the noise power as well. So, in (20), MSE includes two parameters, the
noise power and Doppler of the received signal, and should be minimized according to both
of them:
i = 1 m y 2 r , i m i
L
2
2
ω 2
(
)=
ε
ˆ
, ˆ
σ
arg min
ε
.
(22)
2
ω 2
,
σ
Now it is the time to discuss about the variable variance of the noise. This means that in
(14) the noise variance is considered unknown. We can estimate the noise variance given
N 1 signal-free samples which are at hand occasionally.
2
ω 2
σ
So,
becomes a random variate.
Since the noise
is assumed Gaussian, the N 1 -sample based estimated variance is
chi-square distributed with N 1 degrees of freedom:
ω 2
(
t
+
τ )
ˆ
N 1
i = 1 ω
1
N 1
2
ω 2 =
2 i ,
2
ω 2 χ
2 N 1 .
ˆ
σ
σ
ˆ
(23)
2
Hence, the average MGF of the noise over
σ
is obtained in (14) as:
1
M ω 2 (
)=
u
ω 2 u 2 / N 1 )
2
N 1
(
1
σ
ˆ
2
ω 2 u 2
ω 2 u 4
4
=
+
+(
+
1/4 N 1 )
+ ···
1
0.5 ˆ
σ
0.125
σ
ˆ
.
(24)
2
ω 2 , the procedure presented for Doppler
estimation in the previous part does not change, only MGF and the moments of the normal
ˆ
In this non-stationary noise scenario due to
σ
 
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