Acquisition of Weak Signals - Fundamentals of Global Positioning System Receivers: A Software Approach - page 264

In this equation, A 2 is considered as signal amplitude and 2 An + n 2 is noise.
The corresponding (S/N) sq in dB can be considered as the energy ratio as
S
N

10 log 10 log

A 4
<( 2 An

A 4
3 σ n + 4 A 2 σ n

sqdB =

=

n 2 ) 2 > +

= 10 log A 2
σ n

− 10 log 4 + 3 × 10 − 10 log (A 2 /σ n )/ 10 (10.30)

S
N

db −

10 log 4 10 − ((S/N) db / 10 ) = + 3 ×

In the operation above, <> is the expectation value, and the noise distribution
n is assumed to be Gaussian. The relation between the input (S/N) dB and the
(S/N) sqdB is shown in Figure 10.21. This curve indicates that when the input
S / N is low, squaring will more than double the input S / N , such as from

−

40 dB

to approximate to 84.78 dB. When the input S / N is high, the output S / N is

decreased by approximately 6 dB, such as from 60 to 53.98 dB.
The application of this relation can be illustrated with an example. If the
C/N 0 = 24 dB, 1 ms of processed data has a S/N =− 6 dB (24 - 30) because
1 ms has a nominal bandwidth of 1 kHz. Two ms of processed data have a
S/N =− 3 dB (24 - 27), and 4 ms of processed data have a S/N = 0 dB. The
corresponding squared output S / N 's from 1, 2, and 4 ms are approximately − 18,
− 12.99, and − 8.45 dB from Equation (10.30) or Figure 10.21.

−

60

40

20

0

− 20

− 40

− 60

− 80

− 100
− 40

− 30 − 20 − 10 0 10

Input S/N in dB

20 30 40 50 60

FIGURE 10.21 Input S / N versus output S / N of squared signal.