Global Positioning System Reference
In-Depth Information
Doppler. Under these assumptions, the correlation
between the reference code of the
modulation type MOD (for example, BPSK or CBOC(-)) and the received MOD-modulated
signal via an
L
-path channel can be written as:
R
(
τ
)
rx
L
l
=
1
α
l
e
j
θ
l
R
(
τ
)=
R
MOD
(
τ
−
τ
l
)
(5)
rx
where
(
τ
)
is the auto-correlation function of a signal with the modulation type MOD. The analytical
expressions for MEEs become complicated in the presence of more than two paths due to the
complexity of channel interactions. Therefore, an alternative Monte-Carlo simulation-based
approach is presented here, in accordance with Bhuiyan & Lohan (2010), for multipath error
analysis in more than one path scenarios (i.e, for
L
α
l
,
θ
l
,
τ
l
are the amplitude, phase, and delay, respectively, of the
l
-th path; and
R
MOD
≥
2). First, a sufficient number of
random realizations,
N
random
are generated (i.e., in the simulation, we choose
N
random
equals
to 1000), and then we look at the absolute mean error for each path delay over
N
random
points.
The objective is to analyze the multipath performance of some of the proposed advanced
techniques along with some conventional DLLs in the presence of more than two channel
paths, which may occur in urban or indoor scenarios.
The following assumptions are made while running the simulation for generating the RAE
curves Hein et al. (2006). According to Hein et al. (2006), RAE is computed from the area
enclosed within the multipath error and averaged over the range of the multipath delays
from zero to the plotted delay values. In the simulation, the channel follows a decaying Power
Delay Profile (PDP), which can be expressed by the equation:
α
l
=
α
l
exp
−
μ
(
τ
l
−
τ
1
)
,
(6)
where
(
τ
l
−
τ
1
)
=
0 for
l
>
1,
μ
is the PDP coefficient (assumed to be uniformly distributed
in the interval
[
0.05; 0.2
]
, when the path delays are expressed in samples). The channel path
phases
and the number of channel paths
L
is uniformly distributed between 2 and
L
max
, where
L
max
is set to 5 in the simulation. A
constant successive path spacing
x
ct
is chosen in the range
θ
l
are uniformly distributed in the interval
[
0; 2
π
]
[
]
chips with a step of 0.0417
chips (which defines the multipath delay axis in the RAE curves). It is worth to mention here
that the number of paths is reduced to only one LOS path when
x
ct
0; 1.167
=
0.
The successive
τ
l
=
path delays can be found using the formula
lx
ct
in chips. Therefore, for each channel
realization (which is a combination of amplitudes
α
L
, phases
θ
=
θ
1
,...,
θ
L
,
fixed path spacings, and the number of channel paths L), a certain LOS delay is estimated
τ
1
α
=
α
1
,...,
,
θ
(
α
,
L
)
from the zero crossing of the discriminator function (i.e.,
D
(
τ
)=
0), when searched
in the linear range of
D
in case of conventional DLLs, or directly from the auto-correlation
function in case of advanced multi-correlator based techniques. The estimation error due to
multipath is ˆ
(
τ
)
,
θ
(
α
)
− τ
1
, where
τ
1
is the true LOS path delay. The RAE curves are generated
in accordance with Hein et al. (2006). RAE is actually computed from the area enclosed within
the multipath error and averaged over the range of the multipath delays from zero to the
plotted delay values. Therefore, in order to generate the RAE curves, the Absolute Mean
Error (AME) is computed for all
N
random
random points via eqn. 7:
τ
1
,
L
τ
1
)
−
τ
1
)
θ
AME
(
x
ct
)=
mean
(
(
α
,
,
L
,
(7)
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