Acquisition of Weak Signals - Fundamentals of Global Positioning System Receivers: A Software Approach - page 230

P fa = 10 − 11 . Let us use P d = 0 . 9 with P fa = 10 − 7 in the following discussion.
The false alarm time T fa defined in the previous section is

1
P fa B

T fa = ( 10 . 5 )

where B is the bandwidth of the receiver. For the C/A code receiver the knoll-
to-knoll bandwidth is 2.046 MHz. If the receiver has a bandwidth of 2.5 MHz
the corresponding T fa =

10 − 7 )).
Let us discuss the probability of detection and false alarm time from a slightly
different approach by using the detection of a GPS signal as an example. Suppose
that the input signal is digitized at 5 MHz; thus in each ms there are 5000 data
points. Each chip time of the C/A code is about 0 . 9775

×

10 6

×

4sec(1/(2 . 5

s (1 ms/1023). There
are about 5 data points in a chip time. Suppose that in the acquisition method the
correlation is calculated 2500 times. Each time the C/A code moves slightly less
than half the chip time or 2 data points. The result is 2500 correlation outputs per
ms. The chance to have one false alarm in one ms is 2 . 5

µ

10 − 7 ) .
In other words, there is one false alarm in about every 4 sec or T fa = 4sec,
which has the same value of the previous estimation. It should be noted that
T fa = 4 sec is for this special case. It there are 5000 correlation outputs per
ms, the chance of having one false alarm in one ms is 5 × 10 − 4 ( 5000 × 10 − 7 ) .
Under this condition, the false alarm time is about T fa = 2 sec. However, the
probability of detect will also change. For this discussion, let us use T fa = 4sec.
For 1 ms acquisition, the search covers 2500 points in the time domain and 20
components in the frequency domain. The 20 frequency components are required
to cover 20 kHz in 1 kHz steps. Thus each ms of data generates 50,000 (20 ×
2500) correlation outputs. With a probability of false alarm of 10 − 7 the false
alarm time is 200 ms ( 1 /( 50 , 000 × 10 − 7 )) . Statistically this result means that
for every 200 ms of data processed, there is one mistake.
The overall situation can be summarized as follows: With a S / N

10 − 4 ( 2500 × ×

= 14 dB and
performing acquisition on 1 ms of data, a certain threshold can provide 90% prob-
ability of detection and 1 mistake in every 200 ms. This is an acceptable result.
From this discussion one can conclude that with S / N = 14 dB, a satisfactory

result should be obtained. In later sections of this chapter, the goal is to achieve
S / N

= 14 dB from the combination of coherent and noncoherent integrations.

10.5 COHERENT INTEGRATION GAIN

By stripping the C/A code from the input data, the remaining signal becomes a
continuous wave (cw). Once the input signal becomes a cw signal, fast Fourier
transform (FFT) can be used to find the frequency, and this operation is referred
to as coherent integration. Coherent integration is the first step in any acquisition
method to find the GPS signal. The coherent signal processing gain can be found
from the corresponding bandwidth, which is related to the data length.