Image Processing Reference
In-Depth Information
N
1
B
0
(3.20)
m
0
In view of the Eq. (3.20), although the B isn't strictly zero, their sum is equal to zero. We all
known that on the right-hand side of Eq.(3.14) is the sum of cross-correlation function.
Applying the Eq. (3.20) to (3.14) term by term, we obtain that the Eq.(3.14) strictly hold. Now
we have the knowledge that the term C doesn't effect on the measurement results and we
just need to discuss the term B as follows. Eq. (3.12) can be given by
N
1
1
1
R
(0)
R
(
m
)
cos(

)
B
(3.21)
ij
ij
ij
N
2
m
0
Let the error terms that are caused by the white Gaussian noise and the quantization noise
be represented by
BR
R
and
BR
R
respectively. So B can be
1
xg
gx
2
xl
lx
ij
i
(1)
j
ij
i
(1)
j
ij i
(1)
j
ij
i
(1)
j
expressed by
.
Here, quantization noise is generally caused by the nonlinear transmission of AD converter.
To analysis the noise, AD conversion usual is regarded as a nonlinear mapping from the
continuous amplitude to quantization amplitude. The error that is caused by the nonlinear
mapping can be calculated by using either the random statistical approach or nonlinear
determinate approach. The random statistical approach means that the results of AD
conversion are expressed with the sum of sampling amplitude and random noise, and it is
the major approach to calculate the error at present.
We assume that
BB B
1
2
2
 '.
gt is Gaussian random variable of mean '0'and standard deviation '
()
In the view of Eq.(3.15) and (3.17), we have obtained the standard deviation as follow:
2
2
g
2
 
(3.22)
B
N
1
Assume that the AD converter is round-off uniformly quantizer and using quantization
step . Then ()
lt is uniformly distributed in the range

/2
and its mean value is zero
2
and standard deviation is
(
/ )
. We have
2
2
12
2
(3.23)
B
N
2
For
B and
B
are uncorrelated, then
2
2
2

2
12
g
2
2
2
 
(3.24)
B
B
B
NN
1
2
The mean square value of 1 cos(
  on the right-hand side of formula (3.21) will be
)
B
ij
2
calculated by the following formula to evaluate the influence of noise on measurement
initial phase difference.
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