Acquisition of Weak Signals - Fundamentals of Global Positioning System Receivers: A Software Approach - page 247

where 10 − 3 represents 1 ms. Since usually f d f L 1 , the approximation can be
used. This equation applies to both positive and negative Doppler frequencies.
For a 5 kHz Doppler shift the time change for 1 ms is 10 − 3 = 3 . 17 × 10 − 9

sec or 3.17 ns, where t can be obtained from Equation (10.21). This time change
is smaller than the sampling time of 200 ns; thus the same C/A code can be
used for acquisition without regenerating it. The digitized C/A code, however,
changes slightly for different Doppler frequencies. For example, if two sets of
5000 points data are generated with Doppler frequencies of 0 and 5 kHz for
satellite 1, only 6 out of 5000 points have different values. Thus use of the same
C/A for acquisition can be justified.
For 10 ms coherent integration, the corresponding time change due to 5 kHz
Doppler frequency shift is 31.7 ns ( 10

− t

×

3 . 17 ) . If 20 noncoherent integrations
are performed, the total length of time is 200 ms. A 5 kHz Doppler frequency
shift can cause a time change of 634 ns ( 200

3 . 17 ) , which corresponds to
three sampling intervals of 600 ns. If this time is not adjusted for the nonco-
herent integration, the peaks will no longer line up. Figure 10.11 shows that the
Doppler frequency is uncompensated. In this set of data, satellite 5 has a Doppler
frequency of

×

2900 Hz. Ten ms coherent integration is performed. In order to
exaggerate this effect, a long data record (2 seconds) is processed. The amplitude
from Figure 10.11a is about 8 . 7 × 10 6 . Figure 10.11b shows a close-up version,
where the peak is spread out. Figure 10.12 shows the compensated results. The
amplitude from Figure 10.12a is about 2 . 3 × 10 7 . Figure 10.12b shows a close-
up, where the peak is sharp. For this long time noncoherent integration, the peaks
can be improved by about 8.4 dB (20 × log ( 23 / 8 . 7 ) ) with this method.
The time shift can be incorporated in the acquisition as follows ( 14 ) . The time
shift for each block of data can be computed as

−

f d T
f L 1

T d = and

S d = exp ( − j 2 πk n T d ) (10.22)

where T d is the required time shift, T is the time delay between the current data
block and the first data block for the noncoherent integration, S d is the frequency
domain equivalent of T d ,and k n is the frequency component of the FFT kernel
function. In obtaining the correlation results, the delay S d is used in the frequency
component of the locally generated signals.

10.13 THRESHOLD DETERMINATION FOR GAUSSIAN NOISE ( 15-17 )

Once the procedures above are accomplished, the acquisition program can pro-
duce a reasonable peak at the correct initial phase of the C/A code and the
carrier frequency. At this stage, a decision must be made to determine whether
the peak represents a true signal. To accomplish this goal, a threshold must be
set. If the peak is higher than the threshold, it is a signal. Otherwise, it is not.