Signal Processing - September 2017 - 82

MLE
ZZB

RMSE (m)

104

Two-Steps

103

DPE

102
101
100
24

32 34 36 38
C/N0 (dB Hz)

FIGURE 8. Hybrid processing of four GPS L1 signals and three Galileo
E1 signals with 4-ms coherent integration time. M = 7 satellites varying
power at the same time.

of weighting in the LS algorithm used in two-steps hybridization. This implicitly assumes that both signals provide the same
accuracy for a given integration interval. However, Galileo E1
and GPS L1 signals have different mean-square bandwidth and,
hence, should be weighted differently [36]. The DPE algorithm
however, implicitly takes into account this information. This is a
clear advantage of the DPE algorithm when it comes to the design
of hybrid receivers.
A third set of experiments also involves M = 7 satellites,
although in this case the C/N 0 of only four of them is varied
while keeping it to 40 dB-Hz for the remaining three. The
GPS L1 C/A signal is considered in the experiments. Similarly, the MSE of both positioning approaches is compared
against the new bound in (6) with close results, especially in
the high C/N 0 region. A coherent integration time of 1 ms

106

MLE
ZZB

M=4

Two−Steps
RMSE (m)

RMSE (m)

105
10

is considered. Note that the C/N 0 of the fixed satellites
(40 dB-Hz) is not high enough for the LS algorithm to obtain a
reliable position (see Figure 4). Therefore, these satellites are
not used to compute PVT based on the two-steps approach.
The results in Figure 9 show the performance of the DPE
and LS approaches with the four satellites that have decaying C/N 0 ^ M = 4h and also the performance of the DPE when
the other three satellites with fixed 40 dB-Hz are included (a
total of M = 7 ). These results show the robustness of DPE
to strong shadowing and even low coverage scenarios, outperforming conventional two-steps solutions. The breakdown
point for the two-steps approach appears at much larger C/N 0
values compared to DPE, which can still make use of those
weak satellites. The longer the integration periods, the better
the performance of the methods. Notice that, in this case, the
gain is larger than 10 log 10 (7) . 8.5 dB, this is because this
rule of thumb only holds when all satellites have the same
power (as in previous experiments).
A fourth experiment with noncoherent integrations is
shown in Figure 10, where the same scenario as in Figure 4 is
considered. In this case, the receiver is configured with a coherent integration of 1 ms and with 5 ms of noncoherent integration times. The performance improvement is clear in this case
as well. Notice that the optimal estimator is the one performing coherent integrations, and thus the bound is not necessarily
tight when performing noncoherent integrations. If arbitrarily
large integrations could be performed (e.g., in static receivers),
the performance of the PVT solution could be largely improved.
Finally, a proof-of-concept experiment with real data is provided in Figure 11. The real data were recorded at Northeastern University's main campus (Boston, Massachusetts) with a
general-purpose radio-frequency front end at 4 megasamples/s,
emitting an L1 C/A signal. The C/N 0 for these satellites was
fairly high, always above 45 dB-Hz for the entire 2-min-long
experiment. The two receiver implementations are compared (i.e., two-steps and DPE) in Figure 11. In both cases,

103

M=7

102
101
20

104

Two-Steps

103
102
101

DPE
25

35
40
C/N0 (dB Hz)

remaining three fixed at 40 dB-Hz.

100
30

FIGURE 9. M = 7 GPS L1 signals, four of them varying their C/N0 and the

MLE
ZZB

DPE
32

38 40 42 44
C/N0 (dB Hz)

FIGURE 10. GPS L1 signal processing with 1-ms coherent and 5-ms
noncoherent integration times. M = 7 satellites varying power at the
same time.

IEEE SIGNAL PROCESSING MAGAZINE

September 2017

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