Global Positioning System Reference
In-Depth Information
Section 10.7. The E b /N 0 is related to the C/N 0 by the length of the navigation
data t b , which is 20 ms (20
10 3 ). The relation can be written as
×
E b
N 0 =
Ct b
N 0 =
C
N 0 × 20 × 10 3
and
P e = erfc C
10 3
N 0 ×
×
20
(10.2)
This probability of error P e represents the error of 20 ms of data. However,
40 ms of data are needed to determine a phase transition. If a mistake is made in
either the first or the second 20 ms of data, the detection of the phase transition
will be wrong. Thus the probability of making a wrong decision on the phase
transition will be 2 P e .
Figure 10.1 shows the results of 2 P e versus C/N 0 . In this figure the data
points are obtained by the Monte Carlo method. For every data point there are
1000 or 2000 runs, and the probability of error is the total error divided by
the total number of runs. The three points with the highest S / N have 2000
runs because the error rate is low. When the probability is low, a large number
of runs are needed to confirm the result. For example, at C/N 0 = 24 dB the
probability of error is about 1 . 5 × 10 3 , calculated from Equation (10.2). The
Probability of phase error from 40 ms of data
10 0
10 1
10 2
10 3
10 4
10 5
10 6
10 7
14
16
18
20
22
24
26
28
Input C/N 0 in dB/Hz
FIGURE 10.1 Probability of phase transition error from 40 ms of data.
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