Digital Signal Processing Reference
In-Depth Information
navigation data symbol, the received GPS signal in the absence of noise and interference
is given by
{
}
st Ag t
d
()
= −
(
γ
)exp[
j
ω φ
c
+
],
(14 .8)
where
A
is the signal amplitude,
g
(
t
) is the ±1 valued C/A code, γ is the time delay of the
original signal, ω
c
is the carrier frequency, and ϕ is the carrier phase. After downconver-
sion and phase tracking, the baseband signal is expressed as
ˆ
)]
stAg t
db
()
= −
(
γ
)exp[
j
(
φ φ ,
−
(14 .9)
where ϕ
ˆ
is the estimated carrier phase. The code replica, generated by the receiver, is
therefore
ˆ
)
st gt
D
()
= γ ,
(
(14.10)
where γ
ˆ
is the estimated time delay prior to processing through the GPS correlation
loops. The received data vector
x
(
k
) is the superposition of the GPS signals, interferers,
and AWGN noise, and can be expressed as
x
()
k
=
as
() ()
θ
k
+
ai
() ()
ϕ
k
+
n
(),
k
(14 .11)
D
D
I
M
where the 2
N
-by-1 vector
n
(
k
) represents the AWGN noise at the 2
N
elements. The SNR
of the GPS is in the vicinity of -20 dB, and the JNR is typically 20 dB. It is noted that
with RHCP signals,
a
iV
= -
j a
iH
(
i
= 1, 2, …,
N
). In this case, assuming all the antennas are
omnidirectional, the steering vectors
a
(θ
i
) and
a
(φ
i
)can be expressed as
H
d
d
j
2
π
λ
(
N
−
1
)sin
θ
j
2
π
λ
(
N
−
1
)sin
θ
i
a
()
θ
= − …
1
j
e
−
je
,
(14 .12)
i
H
d
d
j
2
π
λ
(
N
−
1
)sin
ϕ
j
2
π
λ
(
N
−
1
)sin
ϕ
i
i
a
()
ϕ
= − …
1
j
e
−
je
,
(14 .13)
i
where
λ
is the wavelength. For a horizontally polarized interference,
a
iV
= 0 (
i
= 1, 2,
…,
N
), whereas for a vertically polarized interference,
a
iH
= 0 (
i
= 1, 2, …,
N
), and for
LHCP interference,
a
iV
=
j a
1
H
(
i
= 1, 2, …,
N
). Let
r
represent the data spatial correlation
matrix,
H
[()()],
rxx
=
Ek k
(14.14)
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