Signal Processing - September 2017 - 76

In high-precision receivers, it is usual to consider carrier
phase measurements, as opposed to code delay, to compute the
PVT solution. Notice that DPE, in its current formulation, can
recover carrier phase measurements. For the PVT solution that
has been estimated with DPE, one can obtain straightforwardly the corresponding pseudoranges and Doppler shifts for each
channel. With these values, the CAF for a particular channel
can be computed to obtain a complex scalar value. The argument of this value yields the phase estimate. Note that the CAF
for each channel is already computed in (5), and, hence, the
phase can be obtained without any additional operation.
DPE shares some similarities with the so-called vector
tracking-loop (VTL) architectures [26]-[32], since both are
making use of the relation between the receiver's PVT and
observables. In VTLs, pseudoranges and pseudorange-rates
predictions are used typically to feed an extended Kalman
filter with the navigation states (PVT). The filter generates
observables predictions, which are provided to the tracking
loops to produce new (often enhanced) observables. With
the new observables, the navigation states are updated. Note
that this process can be seen as providing aid to the tracking
loops according to the navigation states, but the tracking loops
run independently per channel. Therefore, the computation
observables (prior to its adjustment by the navigation filter) is
performed at each channel independently in practice, in a similar way as a conventional receiver with a DLL/PLL, the only
difference being that the filtering comes from the navigation
states and, hence, yields better results since the loops operating points are corrected with the navigation filter. In DPE, the
process works in a slightly different way. Since the problem is
solved directly in the PVT domain, all the signals from the different satellites are processed jointly (5). In the DPE approach,
no pseudoranges or pseudorange-rates are explicitly estimated
since they are implicitly contained in the estimated parameters.
Therefore, the estimated parameters are computed using jointly signals from all satellites.
In this article, we focus more on open-loop processing with
DPE. Note that VTL is a technique designed only for a tracking stage, reusing most of the existing closed-loop circuitry,
and, hence, it has no implementation in open loop to the best
of our knowledge. A closed-loop DPE could be implemented
on tracking the DPE cost function in PVT. This has already
been investigated in [33], but stills remains an interesting and
promising research topic.
From a theoretical perspective, it was shown in [34] that
direct localization outperforms two-steps solutions in terms of
achievable mean-square error (MSE). In the context of positioning (i.e., self-localization), more results exist where the Cramér-Rao lower bounds on PVT estimation are derived for both
positioning approaches, highlighting the potential benefits of
adopting DPE [35], [36]. More recently, [37] provided additional results regarding MSE performance bounds that yield a
better understanding of DPE and its potential. These bounds
are based on the Ziv-Zakai methodology and allow us to determine, e.g., the signal-to-noise ratio (SNR) at which both DPE
and two-steps approaches are able to operate before breaking
76

down (i.e., entering the large-error region). It turns out from
the Ziv-Zakai bound (ZZB) analysis that, under certain scenarios, there is an increase in sensitivity for DPE on the order
of 10 log 10 (M) dB, with M being the total number of used
satellites. Although this effect was already observed in earlier
works, derivation of the new bounds allows analytic interpretation of the enhanced performance of DPE. Particularly, the
lower MSE bound when estimating c is
ZZB (c) = R c 2Q e

M

M

i=1

i=1

/ SNR i o + I -1 (c) C 3/2 e /

SNR i ,
2 o
(6)

where R c is the a priori covariance matrix of c, Q is the
Q-function, and C a (x) is the incomplete gamma function. The
SNR per satellite is defined as SNR i = ^a 2i / ^ N 0 /2hh, assuming
unitary energy for s i (t). I (c) represents the Fisher information
matrix (FIM) of c, whose inverse yields to the Cramér-Rao
bound (CRB). Whereas at high SNRs, the bound converges to
the CRB, at low SNRs, it is dominated by the a priori covariance matrix R c . More precisely, I (c) = P T I (x) P , with
<
P = ^1/ch^u <1 , f, u 



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

Signal Processing - September 2017 - Cover1

Signal Processing - September 2017 - Cover2

Signal Processing - September 2017 - 1

Signal Processing - September 2017 - 2

Signal Processing - September 2017 - 3

Signal Processing - September 2017 - 4

Signal Processing - September 2017 - 5

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Signal Processing - September 2017 - Cover3

Signal Processing - September 2017 - Cover4


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