Acquisition of Weak Signals - Fundamentals of Global Positioning System Receivers: A Software Approach - page 233

error function is not currently available. In order to use the Matlab program,
the erfc − 1 ( y ) must be expressed in terms of erf − 1 ( y ).
The error function is defined as

x

2
√ π e − t 2 dt y = erf (x) = or

0

= erf − 1 (y)

x (10.8)

The complimentary error function is defined as

z = erfc(w) = 1 − erf (w) or

erf (w) = 1 − z (10.9)

The inverse of these two functions can be written as

erfc − 1 (z)

w =

erf − 1 ( 1

w = − z) (10.10)

Thus,

erfc − 1 (z) erf − 1 ( 1 − = z) ( 10 . 11 )

Substituting this relation into Equation (10.7) yields

[ erf − 1 ( 1 erf − 1 ( 1 2 P d ) ] 2

D c ( 1 ) = − 2 P fa ) − − ( 10 . 12 )

The corresponding noncoherent integration loss is

log 1

+ √ 1

dB

9 . 2 n/D c ( 1 )
1 + √ 1 + 9 . 2 /D c ( 1 )

+

L(n) = 10 × ( 10 . 13 )

10 − 7

where n is number of noncoherent integrations. Given P d = 0 . 9and P fa =

21 can be obtained from
Equation (10.12). If n is given the noncoherent integration loss, L ( n )canbe
obtained from Equation (10.13). When n is a large value, the integration loss is
approaching L(n) = 5 × log( n ).
The noncoherent integration gain G i can be written as

as discussed in Section 10.4, the value of D c ( 1 ) ≈

G i (n) = G c (n) − L(n) = 10 log (n) − L(n) ( 10 . 14 )

Figures 10.3 and 10.4 show the noncoherent integration loss and the noncoherent
integration gain, respectively. It shows that when the noncoherent integration
number n is small, the gain is relatively large. For example, when the noncoherent
integration number is 2, the noncoherent gain G i ( 2 )

≈

2 . 7 dB, which is only

0.3 dB from the coherent gain of 3 dB. When n = 100, G i ( 100 ) ≈ 14 . 6dB,

which is about 5.4 dB lower than the coherent gain of 20 dB.