Image Processing Reference
In-Depth Information
quantization noise of channel i and generates by ADC, n is a positive integer and its value is
in the range 1~ .
Formula (3.8) could be transformed into following normalized expression (3.9) to deduce
conveniently.
f
f
b
i
vn
() sin(2
n
)
gn
()
ln
()
(3.9)
i
i
i
i
N
To realize one time frequency measurement, sampling beat-frequency signal must be
continuous operated at least two seconds. For example, the j-th measurement frequency of
channel i will analyze the j second
ij
vn
and j+1 second
()
v
(
()
n
data from DAQ.
ij
The cross-correlation between
ij
vn
and
(
)
v
(
()
n
have been used by following formula:
ij
N
1
1
Rm
()
vnv
()
(
n m
)
ij
ij
i j
( )
N
n
0
N
1
1
[
xn gn ln
(
)
(
)
(
)]
[
x
(
n m g
)
(
n m l
)
(
n m
)]
ij
ij
ij
i j
( )
i j
( )
i j
( )
(3.10)
N
n
0
1
cos(
m
)
R
R
R
R
R
R
ij
ij
xg
xl
gx
gg
gl
lx
2
ij
i
(1)
j
ij i
(1)
j
ij
i
(1)
j
ij
i
(1)
j
ij i
(1)
j
ij
i
(1)
j
R
R
l
g
l l
ij
ij
(1)
jij
(1)
Formula (3.10) could be split into three parts; with the first part is cross-correlation function
between signals
xn
:
()
1
cos(
A
m
)
(3.11)
ij
ij
2
the second part is the cross-correlation function between noise and signal;
BR
R
R
R
(3.12)
xg
xl
gx
lx
ij
i
(1)
j
ij i
(1)
j
ij
i
(1)
j
ij
i
(1)
j
the third part is the cross-correlation function between noise and noise:
C
R
R
R
R
(3.13)
gg
gl
lg
ll
ij
i
(1)
j
ij i
(1)
j
ij
i
(1)
j
ij i
(1)
j
According to the property of correlation function, if two circular signals are correlated then
it will result in a period signal with the same period as the original signal. Therefore, the C
can be denoted average
i
Rm
over m:
(
)
N
1
1
C
R
()
m
(3.14)
ij
N
m
0
The term
BR
R
R
R
of cross-correlation can't be ignored.
xg
xl
gx
lx
ij
i
(1)
j
ij i
(1)
j
ij
i
(1)
j
ij
i
(1)
j
Because the term B isn't strictly zero. We will discuss the effect of ignoring B and C on
measurement precision in following section.
![](Applications-of-Digital-Signal-Processing_images/img-gen10159.png)
![](Applications-of-Digital-Signal-Processing_images/img-gen10160.png)
![](Applications-of-Digital-Signal-Processing_images/img-gen10161.png)
![](Applications-of-Digital-Signal-Processing_images/img-gen10162.png)
![](Applications-of-Digital-Signal-Processing_images/img-gen10163.png)