Satellite Signal Acquisition, Tracking, and Data Demodulation - Understanding GPS: Principles and Applications - page 228

false alarm, P fa , as was described for the Tong detector. Note that the M of N algo-
rithm in Figure 5.39 contains a superior reference noise source technique compared
to that of the Tong algorithm in Figure 5.37. This requires a preplanned receiver
design that provides the same PRN code for each channel that is not used by (does
not correlate with) any SV in the constellation—such as G1(t) for C/A code and PRN
38 for P(Y) code—and one component of a spare complex correlator (shown as Q N
component in Figure 5.39).
The overall probability of false alarm in N trials is [17]:

N
n

N
n

N M − 1

( (

)

Nn −

)

Nn −

∑ ∑

n n

P = P 1 − P = 1 − P 1 − P

FA fa fa fa fa

(5.48)

nM = n = 0

( )

=− 1 BM − 1; NP ,

fa

where B ( k;N,p ) is the cumulative pdf.
The overall probability of detection in N trials is [17]:

N
n

N

( )

Nn −

( )

= ∑

n

P = P 1 − P = 1 − BMNP − 1; , (5.49)

D d d d

nM

Figure 5.40 illustrates the M of N probability of detection versus C / N into the
detector for N 10 −6 . By inspection of Figure = 8 and M = 3, 4, 5, and 6 when P FA = 1 ×

5.40, it is clear that M 5 is the optimum value. The data were generated by com-

puting P fa given M , N, and P FA using the following equation [17]:

=

( )

− 1

PBMNP
fa

= − 11 ;, − (5.50)

FA

100%

90%

80% 3of8

5 of 8 (Best)

70%

60%

4of8

50%

40%

6of8

30%

20%

10%

0%

8 12

6 10

0 2 4

−

C/N (dB) into envelope detector with P = 10

FA

Figure 5.40 Probability of detection for M of N search detector.