Global Positioning System Reference
In-Depth Information
Section 6.3.1 describes different models and characteristics of multipath and
shadowing. Section 6.3.2 relates the effect of multipath on signal tracking accuracy
for situations involving different signal modulations, different precorrelation
bandwidths, and different early-late spacings in the code tracking discriminator.
Section 6.2.3 discusses some specialized techniques for multipath mitigation.
6.3.1 Multipath Characteristics and Models
The simplest model of multipath is a set of discrete reflected signals having larger
delays and different amplitudes and carrier phases from the direct path. If the signal
with no multipath is described in analytic signal form as
()
(
)
(
)
−
j
φ
j
2
π
f
t
−
τ
st
=
α
x t
−
τ
e
e
(6.41)
c
0
0
0
0
where
x
(
t
) is the complex envelope of the transmitted signal,
τ
0
is the time for the sig-
nal to propagate from satellite to receiver, and
f
c
is the carrier frequency in hertz,
then a simple model for the complex envelope of a received signal with multipath
(neglecting noise and interference) after frequency down conversion (neglecting any
intentional IF) is
N
( )
(
)
(
)
∑
−
j
φ
−
j
2
π
f
τ
−
j
φ
j
2
π
f t
r t
=
α
e
x t
−
τ
e
+
α
e
x t
−
τ
e
(6.42)
0
c
0
n
n
0
0
n
n
n
1
where there are
N
multipaths,
α
0
is the received amplitude of the direct path and the
α
n
are the received amplitudes of the multipath returns,
τ
0
is the propagation delay of
the direct path, the
τ
n
are the propagation delays of the multipath returns,
φ
0
is the
received carrier phase of the direct path, the
φ
n
are the received carrier phases of the
multipath returns, and the
f
n
are the received frequencies of the multipath returns
relative to the carrier frequency.
In general, each of the parameters in (6.42) is time varying due to motion of the
satellites and the receiver, as well as motion of objects that produce the multipath.
This time variation is not shown explicitly in (6.42) because it complicates the nota-
tion. However, it is accounted for in some of the multipath models discussed next.
The expression (6.42) can be rewritten using parameters that relate the
multipaths to the direct path:
N
~
~
~
~
()
(
)
∑
(
)
−
j
φ
−
j
φ
rt
=
α
e
xt
−
τ
+
α
e
xt
−
τ
−
τ
(6.43)
0
n
n
n
0
0
0
n
1
~
~
where
ααα
n
=
/
is the multipath-to-direct ratio (MDR) of amplitudes,
τ
n
=
is the excess delay of the multipath returns, and the
~
n
0
ττ
n
φ
n
are the received car-
rier phases of the different signal components. The multipath profile producing
(6.43) can be portrayed graphically as a power-delay profile (PDP) by plotting the
points {(
~
−
0
,
~
1
.
The expression (6.43) implies that the received carrier frequencies of the
multipaths are equal to the received carrier frequency of the direct path. This repre-
sentation may not be adequate when relative motion between satellites, scatterers,
τα
n
2
)}
N
n
n
=
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