Geoscience Reference
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of the signal to the end. Besides the length, the overlap between adjacent frames is another
item that is determined according to the required accuracy and tolerable complexity in the
time delay estimation. At the beginning of signal block, the windowed signal includes only
the noise part of
y
2
(
t
)
, because of delay
τ
, so it exhibits different moments in comparison
with
y
1
(
t
)
. While the first point of window reaches the onset of delayed signal
s
(
t
−
τ
)
,
the estimated moments become similar to the moments of
y
1
(
. Mean square error (MSE)
criterion is applied for observing the measure of this similarity. At first, we observe large MSE
values, but, the window progression leads to a decrease in MSE and after the
t
)
seconds delay
point, we get a small amount for MSE nearly equal to zero and will remain constant up to the
end of observation time.
τ
(
)
Now, Doppler is considered and
r
t
is obtained from (9).
Doppler changes the constant
amount of MSE which had happened after
seconds. It means that after the delay point,
Doppler increases MSE
g
radually, but this phenomenon is not an annoying event in time
delay estimation, even it helps to find the time delay, because this increasing in MSE takes
place from the delay point, so it causes the delay point to be the point which has minimum
value for MSE.
τ
In figure (1), the Doppler effect on the MSE behavior is showed for three different SNRs. Time
delay is equal to 300 microseconds. In SNR=+10dB, the result is clear. In two other SNRs, the
minimum point is almost matched well with the actual amount of delay, i.e. 300.
We assume the windowed signal in the
k
-th step of window moving is denoted by
y
2
k
and the
i
-th moment of this windowed signal is presented as
m
y
2
k
,
i
. Therefore, the
k
-th window whose
related moments
m
y
2
k
,
i
are the most similar to those of
y
1
(
t
)
,
m
y
1
,
i
, can be estimated by:
i
=
1
m
y
1
,
i
−
m
y
2
k
,
i
L
2
k
=
arg min
k
,
(10)
where in here,
L
is considered 4, and it would reveal a desirable result [Fukunaga et al., 1983].
In fact, when
L
=4, we use 4 moments of signal. So we have 4 equations that are applied to
determine the unknown parameter. Although there is only one unknown parameter, but the
noise signal does not let us find the parameter by only one equation. But the use of four
equations is enough. Note that if more accuracy is needed,
L
can be considered larger. So, the
delay point,
ˆ
, is the first point of
k
-th window.
τ
Despite the presence of Doppler, the proposed moment method estimates the time delay
precisely. Consequently, this method can consider the time delay and Doppler simultaneously,
and thus, is able to estimate the joint time delay and Doppler accurately.
4.2 Doppler estimation
In this section, we can consider the estimated delay ˆ
τ
as the time origin for the received signal
in the second sensor:
y
2
(
+
τ
)=
(
+
τ
)+
ω
2
(
+
τ
)
≥
t
ˆ
r
t
ˆ
t
ˆ
,
t
0.
(11)
According to (9) and (11), we have:
y
2
(
t
+
τ
)=
ˆ
s
(
t
)
exp
(
j
2
π
(
t
+
τ
)
ε
)+
ω
2
(
ˆ
t
+
τ
)
ˆ
,
t
≥
0.
(12)
