Acquisition of Weak Signals - Fundamentals of Global Positioning System Receivers: A Software Approach - page 256

In these three figures it is shown that the threshold obtained is a linear function
of the standard deviation of the noise, which is reasonable. If actual data are used
for acquisition, the first step is to find the standard deviation of the noise. Since
in a GPS system with 2 MHz bandwidth data collection arrangement the signal
is about 19 dB below the noise, the signal strength will not affect much of

the standard deviation of the noise. The standard deviation of the noise can
be measured in the time domain using 1 ms of input data. Once the standard
deviation of noise is determined, the threshold in Figure 10.16 can be multiplied
by the standard deviation and used as the actual threshold.
With 1 false alarm over 100 runs a threshold is determined. This threshold is
used to determine the probability of detection. In the next section we show some
results obtained through simulated signals.

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10.14 PROBABILITY OF DETECTION OF SIMULATED SIGNALS

Simulated data are used to evaluate the probability of detection. Different signal
conditions can be selected to perform the test. The best signal conditions are
defined as ones whose input frequency is coincident with one of the frequency
bin and the initial phase of the C/A code is on a digitized data point. Under this
condition the locally generated signal can be perfectly matched to the input signal.
This condition is used for testing the probability of detection. In order to add
minor uncertainty, the carrier frequency is arbitrarily set randomly 50 Hz within

a frequency bin. From these results the worst signal condition can be predicted.
The noise used has a 0 mean and standard deviation of 1 and the S / N is
varied by adjusting the signal strength. For every S / N value the test is performed
100 times. For each run different noise is generated with 0 mean and standard
deviation of 1. The number of detections is recorded. Figure 10.19 shows the
S / N versus number of noncoherent summations for detecting the signal about 90
times out of 100 runs. This curve is limited to about C/N 0 =

±

25 . 5 dB because
it is difficult to obtain 90 detections when the input signal is weak. A separate
test has been carried out on the weak signals around C/N 0 = 24 dB. This curve

confirms the noncoherent integration gain discussed in Section 10.7. A smaller
number of summations makes more improvement.
For the lower S / N value test, the initial phase of the C/A code of the input
signal is set to zero, but the Doppler frequency is random within ± 5Hz to
provide a lower uncertainty. The results are listed in Table 10.3. In this table
each data point is obtained from only 100 runs.

TABLE 10.3 Signal Detection at C / N 0 around 24 dB

C/N 0 (dB) 25 24.5 24 23.5 23

P d 82 77 74 60 50

False alarm 2 11 4 11 9