Signal Processing - September 2017 - 78

mainly considers open-loop configurations, although DPE
can be additionally formulated in a closed-loop scheme yielding to other possible receiver architectures [33], [38]. Thirdly,
computing DPE's solution to PVT involves the optimization
of a multidimensional, nonconvex function. This is typically a
cumbersome operation. In the literature there are some implementations reported, including grid-based search, stochastic
optimization algorithms, or relaxations to the cost function to
be optimized [22], [39], [40].
There are several open issues regarding the implementation
of DPE in a real receiver. Besides the aforementioned possible
receiver architectures, there are some practical aspects that
should be accounted for to fully exploit the benefits that DPE
brings. Next, we point out the main research directions and
suggests some avenues to explore.

Atmospheric perturbations
The propagation of GNSS signals through the atmosphere affects
its reception. In particular, the uppermost and the lower parts
of the atmosphere cause nonnegligible effects, ionosphere and
troposphere, respectively. Since atmospheric errors can be on
the order of tenths of meters if not properly corrected, correction of ionospheric- and tropospheric-induced delays in DPE's
equations is of paramount importance.
The free electrons in the ionosphere (due to gases ionized
by solar radiation) cause the GNSS signal to propagate through
a dispersive medium. In that case, the propagation velocity is
no longer the vacuum speed of light but a function of the carrier frequency (?1/f 2c ). This is seen as a delay in the signal
modulation (i.e., pseudorange is delayed) and as an advancement of the phase of the signal in an identical amount. There
are three main alternatives to combat the effect of ionosphere
in observable measurements:
1) There exist parametric models of such delays, with the
parameters being broadcasted in the navigation message by
the satellites. For instance, conventional single-frequency
GPS receivers make use of the so-called Klobuchar model
[41]. Other ionospheric models can be found in the literature
such as NeQuick [42], which is the model adopted by the
Galileo system. Incorporating the information provided by
these models in a DPE receiver is straightforward: when computing x (c), the corresponding cDI i terms need to be computed from the model and accounted for in the calculations.
2) Another solution is to rely on a real-time correction network, which broadcasts ionospheric corrections from a
number of reference stations. This is referred to as differential operation. In satellite-based augmentation systems
(SBASs) like the wide-area augmentation system (WAAS)
or the European Geostationary Navigation Overlay Service
(EGNOS), the correction for the ionospheric delay is
achieved by transmitting a grid of vertical ionospheric delay
values and performing some kind of interpolation [43].
Likewise, when using ionospheric models, the use of SBAS
corrections can be readily incorporated in DPE's algorithm.
3) Dual-frequency GNSS receivers can directly solve for the
delay, taking advantage of its dependency with the carrier
78

frequency. Basically, dual-frequency receivers can eliminate the effect through a linear combination of code or carrier measurements (observables). Since DPE is not
necessarily producing observables, this approach does not
appear as suitable to this type of receivers even when having dual-frequency capabilities.
Tropospheric delay is a function of the elevation and altitude
of the receiver. It is dependent on many factors such as atmospheric pressure, temperature, and relative humidity. Unlike ionospheric delay, tropospheric delay is not frequency dependent
(because it is not a dispersive medium at GNSS frequencies),
and therefore it cannot be eliminated through linear combinations of observations on different frequency bands. Thus, it
becomes the dominant source of atmospheric error in multiplefrequency receivers. In contrast to ionosphere-induced delays,
the tropospheric path delay lengthens the propagation time
equally for code and carrier signal components. The mitigation
of such effect is typically achieved by differential techniques
and interpolation, due to its high spatial correlation. In terms
of mitigating the effect of the DTi term in DPE receivers, the
aforementioned method can be easily incorporated when computing the cost function (5).
In summary, either when atmospheric models or SBAS
information is considered, the terms DTi and DI i in (3) can
be compensated for by knowing the approximate position of
the receiver and time. This provides an easy way for reducing
atmospheric-induced errors without increasing the computational complexity.

Coarse time positioning
DPE aims to overcome the conventional approach in the most
challenging scenarios, where the received signals are dramatically degraded. In such scenarios, GNSS signals might not be
strong enough to extract the navigation message, and, hence,
neither the time-of-week (TOW) information nor the ephemerides might be available, thus completely denying PVT. Since
this information is required for a DPE receiver to start operation, the receiver might need then some external assistance that
can provide information such approximate user position, ephemerides, Doppler shift, time, satellite clock corrections, and
numerous other parameters essential to determining a user's
PVT in such complex environments. These parameters allow
for the computation of the atmospheric corrections, accounting for the ionospheric and tropospheric delays, which, as discussed previously, can be critical [44].
Likewise, the absence of TOW information is also crucial when it comes to estimating the PVT. The TOW is, in
essence, the transmission time of the received signal from the
satellite under study. Together with a valid set of ephemerides, the transmission time enables determination of the satellite position. If the accuracy of the time assistance is on the
order of seconds, the TOW needs to be estimated in a process referred to as coarse time positioning [45]. TOW has an
impact on how position and velocity are computed for a given
satellite, p i _ p i (TOW) and v i _ v i (TOW), respectively. In
the context of DPE, this implies that, when TOW is unknown,

IEEE SIGNAL PROCESSING MAGAZINE

September 2017

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