Geoscience Reference
In-Depth Information
n
Moment
Central moment Cumulant
0
1
1
−
1
μ
0
μ
2
2
2
2
2
μ
+
σ
σ
σ
3
2
+
3
μ
3
μσ
0
0
4
2
2
4
4
μ
+
μ
σ
+
σ
σ
4
6
3
3
0
Table 1. Normal distribution moments
and Doppler creates a non-stationary signal, as seen in (5). At first, by using both sensors in
the receiver, the time delay is predicted, then this estimated delay facilitates determination of
the Doppler shift subsequently. The time delay estimation is described here and discussions
about the Doppler estimation are provided in the sequel. It is required to consider the MGF
of the normal distributed variate as the starting ground for the next steps:
exp
2
u
2
,
M
(
u
)=
μ
u
+
0.5
σ
(8)
where
are the mean and variance of normal distribution. The related moments are
depicted in table (I). We suppose the received noise-free signal in the second sensor is denoted
by:
μ
and
σ
(
)=
(
− τ
)
(
ε
)
r
t
s
t
exp
j
2
π
t
.
(9)
2
ω
1
(
)=
(
− τ
)
First, we assume there is no Doppler i.e.
r
t
s
t
, and the noise variances,
σ
and
2
σ
ω
2
, are constant. As mentioned above, we utilize the MGF for the estimation purposes. The
noise terms in both sensors have normal distributions. Since the noise terms in (5) and signal
s
(
t
)
are independent, the difference between MGF of two received signals
y
1
(
t
)
and
y
2
(
t
)
in
(5) is derived from the noise-free terms
s
(
t
)
and
r
(
t
)
. Since
r
(
t
)
is the delayed replica of
s
(
t
)
,
(
)
it includes two blocks. When the second sensor has not sensed the received signal yet,
r
t
ω
2
(
)
merely contains the noise
and its MGF can be calculated by (8), but, as soon as the
transmitted signal arrives at this sensor,
y
2
(
t
)
shows a similar behavior to
y
1
(
)
t
t
. This suitable
observation could be used for the time delay estimation.
So, MGF of signal detected at the first sensor is considered as a reference for our estimation
in the second sensor.
Indeed, the moments of
y
1
(
)
are extractable from this known MGF
by using (7). These moments are employed as the reference for comparing among results
retrieved from the second sensor.
t
In the second sensor, a rectangular running window is
(
)
(
)
implemented on
y
2
step by
step. The window length depends on two parameters. First, it must be long enough to be
trustable in calculating the estimated moments, on the other hand, it should not be so long
that damages the real-time characteristics of estimator. Anyway, there is a trade-off between
these two factors. The window length is considered constant and moves from the beginning
t
and this window helps to extract different segments of
y
2
t
