Global Positioning System Reference
In-Depth Information
r
X
(τ )
(
0
,
1
)
τ
−
T
T
FIGURE 1.5. Autocorrelation function
r
X
(τ )
for random sequence of pulses with ampli-
tude
±
1.
The ACF for the sample function
x
(
t
)
of a process
X
(
t
)
consisting of a random
sequence of pulses with amplitude
±
1 and with equal probability for the outcome
+
1and
−
1 is [see, e.g., Forssell (1991) or Haykin (2000)]
1
−
|
τ
T
,
for
|
τ
|≤
T
,
r
X
(τ )
=
(1.15)
0
,
otherwise.
The ACF is plotted in Figure 1.5. It follows that the power spectral density is
T
1
e
−
j
ωτ
d
T
sinc
2
ω
−
|
τ
|
T
T
2
S
X
(ω)
=
τ
=
,
(1.16)
−
T
which is plotted in Figure 1.6. The power spectral density of
X
possesses a
main lope bounded by well-defined spectral nulls. Accordingly, the null-to-null
bandwidth provides a simple measure for the bandwidth of
X
(
t
)
(
t
)
.
Note that the power spectral density
S
X
(ω)
of a random sequence of pulses with
amplitude
±
1 differs from the energy spectral density
E
f
(ω)
,givenin(1.11),ofa
single rectangular pulse by only a scalar factor
T
.
1.2
Sampling
A crucial signal processing operation in a GPS or Galileo software-defined re-
ceiver is
sampling
. In the following we briefly review the sampling process.
Consider the signal
x
. Suppose that we sample this signal at a uniform rate—
say once every
T
s
seconds. Then we obtain an infinite sequence of samples, and
we denote this sequence by
(
t
)
,where
n
takes on all integer values. The
quantity
T
s
is called the
sampling period
, and its reciprocal
f
s
{
x
(
nT
s
)
}
=
1
/
T
s
the
sam-
pling rate
.
The sampling operation is mathematically described by
∞
x
δ
(
t
)
=
x
(
nT
s
)δ(
t
−
nT
s
),
n
=−∞
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