Global Positioning System Reference
In-Depth Information
{
}
()
()
j
2
π
f t
c
st
=
Re
s t e
l
(4.3)
where Re{·} denotes the real part of. The in-phase and quadraphase components of
the real signal
s
(
t
) are related to its complex envelope by:
()
()
()
st
=
st
+
s t
(4.4)
l
I
Q
Two signal characteristics of great importance for satellite navigation applica-
tions are the
autocorrelation function
and
power spectral density
. The
autocorrelation function for a lowpass signal with constant power is defined as:
T
1
2
()
() (
)
−
∫
*
R
τ
=
lim
stst
+
τ
dt
(4.5)
l
l
T
T
→∞
T
where * denotes complex conjugation. The power spectral density is defined to be
the Fourier transform of the autocorrelation function:
∞
∫
()
()
Sf
=
R
τ
e
−
j
2
πτ
f
dt
(4.6)
−∞
The power spectral density describes the distribution of power within the signal
with regard to frequency.
It is often convenient to model some portions of a DSSS signal as being random.
For instance, the data symbols and PRN code are often modeled as
coin-flip
sequences
(i.e., they randomly assume values of either
1 with each outcome
occurring with equal probability and with each value being independent of other
values). The autocorrelation function for a DSSS signal with random components is
generally taken to be the average or expected value of (4.5). The power spectral den-
sity remains as defined by (4.6).
As an example, consider a baseband DSSS signal without data employing rect-
angular chips with a perfectly random binary code, as shown in Figure 4.3(a).
The autocorrelation function illustrated in Figure 4.3(b) is described in equation
form as [8]:
+
1or
−
τ
()
RA
τ
=
2
1
−
for
τ
≤
T
(4.7)
c
T
c
=
0
elsewhere
The power spectrum of this signal shown in Figure 4.3(c) (as a function of angu-
lar frequency
ω =
2
π
f
) may be determined using (4.6) to be:
()
(
)
2
2
(4.8)
Sf
=
AT
sinc
π
fT
c
c
sin
x
where sinc(
x
)
=
. What is important about a DSSS signal using a random
x
binary code is that it correlates with itself in one and only one place, and it is
uncorrelated with any other random binary code. Satellite navigation systems
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