Global Positioning System Reference
In-Depth Information
(a)
(b)
FIGURE 6.7. Output from parallel frequency space search acquisition. The figure only
includes the first 500 chip shifts and the frequency band from 5-15 MHz. (a) PRN 19 is
not visible so no significant peaks are present in the spectrum. (b) PRN 21 is visible so a
significant peak is present in the spectrum. The peak is situated at code phase 359 chips
and frequency 9.548 MHz.
The goal of the acquisition is to perform a correlation with the incoming signal
and a PRN code. Instead of multiplying the input signal with a PRN code with
1023 different code phases as done in the serial search acquisition method, it is
more convenient to make a circular cross correlation between the input and the
PRN code without shifted code phase. In the following, a method of performing
circular correlation through Fourier transforms will be described; see Oppenheim
& Schäfer (1999), page 746, and Tsui (2000), page 140.
The discrete Fourier transforms of the finite length sequences
x
(
)
(
)
n
and
y
n
both with length
N
are computed as
N
−
1
N
−
1
e
−
j
2
π
kn
/
N
e
−
j
2
π
kn
/
N
X
(
k
)
=
x
(
n
)
and
Y
(
k
)
=
y
(
n
)
.
(6.4)
n
=
0
n
=
0
The circular cross-correlation sequence between two finite length sequences
x
(
n
)
and
y
(
n
)
both with length
N
and with periodic repetition is computed as
N
−
1
N
−
1
1
N
1
N
z
(
n
)
=
x
(
m
)
y
(
m
+
n
)
=
x
(
−
m
)
y
(
m
−
n
).
(6.5)
m
=
0
m
=
0
1
In the following we will omit the scaling factor
N
.
The discrete
N
-point Fourier transform of
z
(
n
)
can be expressed as
N
−
1
N
−
1
e
−
j
2
π
kn
/
N
Z
(
k
)
=
x
(
−
m
)
y
(
m
−
n
)
n
=
m
=
0
0
N
−
1
N
−
1
e
j
2
π
km
/
N
e
−
j
2
π
k
(
m
+
n
)/
N
X
∗
(
=
x
(
m
)
y
(
m
+
n
)
=
k
)
Y
(
k
),
(6.6)
m
=
0
n
=
0
where
X
∗
(
k
)
is the complex conjugate of
X
(
k
)
.
Search WWH ::
Custom Search