Global Positioning System Reference
In-Depth Information
number of runs can be found through the law of large numbers ( 6 ) . The law can
be written as
1
4 2
P
{|
x n
p
|
}≥ 1
( 10 . 3 )
In words, Equation (10.3) can be described as follows: x n is the measured
probability of occurrence (in this case it is the probability of error) and p is the
expected probability (or the expected probability of error). When C/N 0 =
24 dB,
p = 1 . 5 × 10 3 . The symbol ε represents the expected error. For example, if
the expected error is 100% of p ,then ε = 100% × p = 1 . 5 × 10 3 . P can be
considered as the confidence level, which is also a probability of occurrence, and
n is the number of runs required.
The next step is to find the total number of runs required. The discussion
above can be summarized as follows: To obtain a confidence of 80% so that
the measured probability of error x n is within 100% of p (or ε = 1 . 5 × 10 3 ),
Equation (10.3) is written as
1
4 2
1
P {| x n p | }= 0 . 8 = 1
= 1
or
4 n
×
2 . 25
×
10 6
10 6
9 × 0 . 2 =
10 5
n
=
5 . 56
×
(10.4)
It is impractical to run the simulation this many times. By this equation, the 1000
runs only provide a confidence of 0.8 with ε
= 2 . 5 × 10 2 ,which
is about the fifth point from the top in Figure 10.1. The remaining five points have
an even smaller probability of error, and therefore should provide less confidence.
However, because the data points and the curve are close, Equation (10.2) will
be used to find the probability of error. The actual method of determining the
navigation phase transition is presented in Section 10.17.
Now let us find the average time it takes for a mistake in navigation data to
be produced. Let us refer this time as the false alarm time. The false alarm time
can be obtained every 20 ms from 20 × 10 3 / 2 P e because 2 P e is the probability,
which is used to obtain Figure 10.1. Figure 10.2 shows the results.
As mentioned in Chapter 5, it takes 30 seconds to collect five subframes and
18 seconds to collect the first three subframes that contain the ephemeris data
to calculate the user position. It appears that it requires about 18 seconds of
data without navigation data error, which in turn requires that C / N 0 = 24 . 3dB,
as shown in Figure 10.2. When C / N 0 = 23 dB, the average false alarm time is
about 4.2 seconds, and when C / N 0 =
= 1 . 2 p and p
25 dB, the average false alarm time is about
53.2 seconds. The navigation data do not change rapidly from page (containing
5 subframes; Section 5.5) to page. If a navigation data error is made, the parity-
check will find the bit error, and the data will be ignored. The mistake may be
corrected from multiple numbers of subframes (the same subframe number on
different pages.) For simplicity, let us choose C / N 0 =
39 dB
referenced to 2 MHz bandwidth) as the sensitivity goal where it corresponds to
24 dB ( S / N
=−
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