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signal f k is 50kHz. Decomposing x 1 ( n ) and x 2 ( n ) in the scale of j =5, the range of
bandwidth of
k a is 0~1562.5Hz, that is very suitable for 200-1500Hz chirp signal.
(3) Signal de-noise and reconstruction. The
(5)
k a include the whole available
information of signal, and the noise is mainly set to 0. Reconstruct ()
()
j
j
k a with the
Mallat reconstruction algorithm (inverse of Mallat decomposition) to generate
de-noise signals x 1 ( n ) and x 2 ( n ).
The cross-correlation of x 1 ( n ) and x 2 ( n ) is expressed as
[
]
(
)(
)
R
(
m Exnx nm
)
=
' ( )
' (
+
)
=
E sn nn
'( )
+
' ( )
α
snmD n nm
'(
+
)
+
' (
+
)
xx
'
'
1
2
1
2
(6)
12
=⋅
α
RmD
(
− +⋅
)
α
R mDR mR m
(
− +
)
(
)
+
(
) .
ss
''
ns
' '
sn
''
nn
' '
1
2
1
2
4.3 Triple Correlation with Wavelet Transform (WT-TC)
The triple correlation with wavelet transform is proposed for improving the stability
and accuracy of time-delay estimation in stored grain.
The auto-correlation [7] of de-noise signal x 1 ( n ) is expressed as
[
]
(
)(
)
RmEx
(
)
=
' ( )
nx
' (
nmEs
+
)
=
'( )
nnns
+
' ( )
'(
nmnnm
+
)
+
' (
+
)
xx
'
'
1
1
1
1
(7)
11
=
RmRmRmR m
(
)
+
(
)
+
(
)
+
(
) .
ss
''
n s
' '
sn
''
n n
' '
1
1
1
1
s R i can be considered as the available signal, and
the others can be considered as the noise. For simplifying the calculation, new
expression is shown
'' ()
In the equation (6) and (7),
ym
()
=
R
()
m
=
R
()
m
+
ω
() ,
m y m
()
=
R
()
m
=
α
R
(
m D
)
+
ω
() .
m
(8)
1
xx
'
'
ss
'
'
1
2
xx
'
'
ss
'
'
2
11
12
From equation (8), it can be seen that, the available signal of y 2 ( m ) is the delay and
attenuation of the available signal of y 1 ( m ), and the time-delay of them is also equal to
D . The cross-correlation of y 1 ( m ) and y 2 ( m ) can be expressed as
[
]
(
)(
)
R
()
τ
=
E y m y
() (
m
+
τ
)
=
E
R
()
m
+
ω
()
m
α
R
(
m
D
+
τ
)
+
ω
(
m
+
τ
)
yy
1
2
ss
' '
1
ss
' '
2
(9)
12
=⋅
ατ
RDR DR
(
− +⋅
)
α τ
(
− +
)
( )
τ
+
R
( ) .
τ
RR
ω
R
R
ω
ω ω
1
2
1
2
R
() max[
D
=
R
( ]
m
is the time-delay of x 1 ( m ) and
x 2 ( m ) estimated by triple correlation with wavelet transform.
In the equation (9), τ= D ￿
RR
RR
5 Experiment of Sound Travel-Time Estimation in Stored Grain
During the experiment, MIC1 is 0.08m away from speaker, the distance l between
MIC1 and MIC2 is 1.2m, and that is 1.8m between MIC1 and MIC3. Fill the tank
with soybeans at three different times, make the depth h of microphone in soybeans
be 0.3m, 0.4m and 0.5m respectively, and totally get six sound paths of different
locations and distances. The sampling signals received simultaneously by MIC1,
MIC2 and MIC3 can be estimated the sound travel-time of each sound path by CC,
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