Image Processing Reference
In-Depth Information
Where x is the beat signals array; n is the number of points in the signal array x; * denotes a
complex conjugate. According aforementioned formula, figure 4 plots power spectrum of a
100 Hz sine wave. As expected, we get a very strong peak at a frequency of 100 Hz.
Therefore, we can acquire the frequency corresponding to the maximum power from the
plot of auto power spectrum.
3.2.2 Fine measurement
The beat signals from the ADCs are fed into PC to realize fine measuring too. Fine
measurement includes the cross-correlation and interpolation methods. To illuminate the
cross-correlation method, figure 5 shows a group of simulation data. The simulation signals
of 1.08Hz are digitized at the sampling frequency of 400Hz. The signal can be expressed by
following formula.
f
xn
() sin(2
n
)
(3.2)
0
f
s
Where f indicates the frequency of signal, the
f is sampling frequency, n refers the number
of sample, and
 represents the initial phase. In the figure 5, the frequency of signal can be
expressed:
f

f
f
(1
0.05)
Hz
(3.3)
N
There the
f
refers the integer and
 indicates decimal fraction. In addition, there is the
N
initial phase
s f H  . There are sampled two seconds data in the figure 5, so
we can divide it into data1 and data2 two groups. Data1 and data2 can be expressed
respectively by following formulas:
  and
0
400
0
f

f
N
xn
(
)
sin(2
n
),
n
[0, 399]
(3.4)
1
0
f
s
f

f
N
xn
(
)
sin(2
n
),
n
[400,799]
2
0
f
s
(3.5)
f

f
N
sin(2
n
2
(
f
 
f
)),
n
[0, 399]
0
N
f
s
According the formula (3.5), the green line can be used to instead of the red one in the figure
5 to show the phase difference between data1 and data2. And then the phase difference is
the result that the decimal frequency  of signal is less than 1Hz. Therefore, we can
calculate the phase difference to get  . The cross-correlation method is used to calculate
the phase difference of adjacent two groups data.
The cross-correlation function can be shown by following formula:
N
1
1
1
f

f
N
Rm
()
xnxnm
() (

)
cos(2
m
2(
f

f
)
(3.6)
xx
1
2
N
N
2
f
12
n
0
s
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