Global Positioning System Reference
In-Depth Information
as:
1
−
τ
T
,
τ
≤
T
()
R
τ
=
c
c
BPSK
−
R
(4.14)
0
,
else
here
()
(
)
S
f
=
T
sinc
2
π
fT
c
BPSK
−
R
c
×
1.023-MHz chipping rate. As will be discussed in Sections 4.3 and 4.5 and Chapter
10, GPS and GALILEO employ frequencies that are multiples of 1.023 MHz.
A BOC signal may be viewed as being the product of a BPSK-R signal with a
square wave subcarrier. The autocorrelation and power spectrum are dependent on
both the chip rate and characteristics of the square wave subcarrier. The number of
square wave half-periods in a spreading symbol is typically selected to be an integer:
The notation BPSK-R(
n
) is often used to denote a BPSK-R signal with
n
T
T
k
=
c
(4.15)
s
where
T
s
1/(2
f
s
) is the half-period of a square wave generated with frequency
f
s
.
When
k
is even, a BOC spreading symbol can be described as:
=
[
]
()
()
(
)
g
t
=
g
t
sgn sin
π
t T
+
ψ
(4.16)
BOC
BPSK
−
R
s
where sgn is the signum function (1 if the argument is positive,
−
1 if the argument is
negative) and
is a selectable phase angle. When
k
is odd, a BOC signal may be
viewed as using two symbols over every two consecutive chip periods—that given in
(4.16) for the first spreading symbol in every pair and its inverse for the second.
Two common values of
ψ
are 0º or 90º, for which the resultant BOC signals are
referred to as
sine phased
or
cosine phased
, respectively.
With a perfect coin-flip spreading sequence, the autocorrelation functions for
cosine- and sine-phased BOC signals resemble saw teeth, piecewise linear functions
between the peak values as shown in Table 4.1. The expression for the
autocorrelation function applies for
k
odd and
k
even when a random code is
assumed. The notation BOC(
m,n
) used in the table is shorthand for a BOC modula-
tion generated using an
m
× 1.023-MHz square wave frequency and an
n
×
1.023-MHz chipping rate. The BOC subscripts
s
and
c
refer to sine-phased and
cosine- phased, respectively.
The power spectral density for a sine-phased BOC modulation is [3]:
ψ
π
f
f
(
)
2
2
T
sinc
π
f T
tan
,
k
even
c
c
2
()
s
S
f
=
(4.17)
(
)
2
cos
(
π
fT
BOC
π
f
f
s
c
2
T
tan
,
k
odd
c
)
2
2
π
fT
s
c
and the power spectral density for a cosine-phased BOC modulation is:
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