Global Positioning System Reference
In-Depth Information
2. Generate a new set of frequency components
X
1
(
k
) from
X
(
k
)as
X
N
k
N
2
−
X
1
(k)
=
2
+
for
k
=
0
∼
1
X
k
N
2
+
N
2
∼
X
1
(k)
=
−
for
k
=
N
−
1
(6.15)
In Figure 6.8, the lower input frequency is converted into a higher output
frequency and the higher frequency is converted into a lower frequency.
This operation puts the two separate bands in Figure 6.8 into the correct
frequency range as the input signals. Or one can consider that it shifts
the center of the input signal from zero to
f
s
/2. If the IF is not at zero
frequency a different shift is required. If the IF of the I-Q channels is at
f
s
/2, no shift is required and this step can be omitted, because the input
signal will not split into two separate bands.
3. Generate additional frequency components for
X
1
(
k
)as
X
1
(N)
=
0
k)
∗
X
1
(N
+
k)
=
X
1
(N
−
for
k
=
1
∼
N
−
1
(6.16)
Including the results from Equation (6.15) there are total 2
N
frequency
components from
k
=
0
∼
2
N
−
1.
4. The final step is to find the new data in the time domain through inverse
FFT as
2
N
−
1
1
2
N
jπnk
N
x
1
(n)
=
X
1
(k)e
(
6
.
17
)
=
k
0
The
N
points of complex data generate 2
N
points of real data and they
contain the same amount of information. These 2
N
points cover the same
time period; therefore, the equivalent sampling frequency is doubled or
f
s
1
=
2
f
s
.
Actual complex data are collected from satellites with I-Q channels of zero IF
from Xetron Corporation. The sampling frequency is 3.2 MHz and the Nyquist
bandwidth is also 3.2 MHz. The operations from Equations (6.14) to (6.17) are
used to change these data to real data with IF
1
.
6 MHz. The number of real
data is double the number of complex data and the equivalent sampling frequency
is 6.4 MHz. These data are processed and the user position has been found.
=
6.15 EFFECT OF SAMPLING FREQUENCY ACCURACY
Although the sampling frequency discussed in this chapter is given a certain math-
ematical value, the actual frequency used in the laboratory usually has limited
accuracy. The effect of this inaccuracy will be discussed as follows.
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