Signal Processing - September 2017 - 80

deviation of the measurement noise, the geometry, and the
maximum allowed probabilities of false alarm and missed detection. In safety-of-life applications, the integrity of the estimates becomes critical [49]. The main principle behind most
RAIM algorithms is to check for consistency of the navigation
solution. This is accomplished after computing the pseudorange residual errors, obtained upon subtracting an estimation
of the pseudoranges to the observed measure. Note that this
is straightforward in the context of an LS-based conventional
receiver but should be carefully thought about within DPE.
Conversely, since some of the integrity approaches operate
at the position-domain level, DPE seems a natural way for
implementing RAIM as it already operates in the positiondomain [50].

DPE initialization
Since DPE involves an optimization problem that needs to
be solved numerically, some sort of initialization is required.
This is reflected in Figure 1(b). As a rule of thumb, we would
like the initial point to lie in the main lobe of the cost function, such that it can eventually converge. For instance, for the
GPS L1 C/A signal, it can be seen from Figure 2(b) that initial
errors on the order of ±300 m are fine, in the sense of having an
initial point in the influence of the main optimum that allows
to eventually reach convergence. In general, the width of the
main lobe depends on the width of the signal's CAF. This is
due to the nature of the cost function (5), where M correlation
functions are added. Therefore, for signals whose chip rate is
higher, the main lobe is going to be narrower, requiring better
initialization. Similarly, for binary offset carrier-type signals
where the main peak is already narrower given the same chip
rate, initialization might play a role. The ±300 m initial error
requirement is probably the most relaxed case-then this
could go all of the way to a few tens of meters depending on
the particular signals under test. Some further understanding

needs to be gained on how initialization impacts the different
signals or combination of them in multiconstellation schemes.

Results and discussion
A set of representative experiments to highlight the benefits
of direct positioning when compared to legacy two-steps positioning approaches is provided in this section. Here we focus
on synthetic data results, providing comparison against the
MSE lower bound described in the section "DPE" and between
several receiver configurations and GNSS signals. We focus
on open-loop receiver architectures, i.e., the sort of receiver
that processes batches of data. This architecture is particularly
important in cloud-based systems, where chunks of I&Q data
are sent to a server that process them and provides the user
with a PVT solution [51].
Since the DPE's cost function may have multiple optima,
there is a need for an efficient, simple and computationally
affordable optimization. Notice that grid-based method results
unpractical because of the dimensionality of the PVT problem.
When implementing DPE, we used an stochastic optimization
method known as the accelerated random search (ARS) algorithm [52], which is a modification of the pure random search
algorithm that accelerates its convergence. Remarkably, ARS
possesses interesting convergence properties under relatively
mild conditions.
A first set of results considers M = 7 satellites varying its
C/N 0 at the same time, emulating for instance scenarios where
LOSS is obstructed. Particularly, GPS and Galileo systems
are analyzed. The GPS L1 C/A signal and Galileo E1 signal
are considered with a precorrelation bandwidth of 4 MHz. In
the case of Galileo, the E1 signal is considered with a precorrelation bandwidth of 10 MHz. In both cases, noncoherent
integrations are not applied. Conversely, different coherent
integration times are tested. The results can be consulted
in Figure 4 for the GPS signals and in Figure 5 for Galileo.

106

106
MLE
ZZB

105

104
Two−
Steps

DPE

103

RMSE (m)

105

104
Two−
Steps

DPE

103
102

102
101
30

MLE
ZZB

101

C/N0 (dB Hz)

(a)

(b)

FIGURE 4. GPS L1 signal processing with (a) 1-ms and (b) 10-ms coherent integration times. DPE outperforms two-steps, as predicted by the breakdown
points of the bound.

IEEE SIGNAL PROCESSING MAGAZINE

September 2017

Table of Contents for the Digital Edition of Signal Processing - September 2017