Signal Processing - September 2017 - 48

here is for a BOC c (15, 2.5) modulation, as one of the modulations with many ambiguities. In this example, the statistics are
computed over 10,000 random points for single-path Nakagami-m fading channels, corresponding to a rural scenario.
Here we used 4 ms coherent integration and two blocks of
noncoherent integration ^ N c = 4 ms, N nc = 2 h. In Figure 6(a),
we used a 0.5 chip delay step for the acquisition stage, similarly with what is used traditionally in GPS receivers and
ensuring a fast acquisition time. In Figure 6(b), we used a
very small time-bin step of 0.01 chips to cover also the acquisition in narrowband mode, which is much slower, but slightly
more accurate. The results are shown only for the single-path
case, to focus solely on the ambiguities effects. However, similar observations can be drawn from results with multipath
channels. The acquisition performance metric considered here
is the detection probability of the LOS path at 10 -3 false
alarm probability. As a general rule, the best performance in
fast acquisition (i.e., high time-bin case) is achieved with wide
main lobe unambiguous approaches. The full BOC case is
overlapping with the SCM case, and it is also rather close to
the Benedetto algorithms. In particular, for the 0.5 time-bin
step, the dual-sideband BF algorithm gives the best acquisition results. In slow acquisition or narrowband mode (i.e.,
small time-bin step), wide main lobe approaches are still
among the best, but their performance is now very close to
full BOC and SCM approaches, also overlapping in Figure 6.
Also, as a general rule, the majority of unambiguous approaches with narrow main lobe exhibit a rather poor performance
in the acquisition stage. Next, we show that their benefit stays
in providing a lower tracking error variance than the wide
main lobe unambiguous algorithms.
Regarding the tracking performance, there are myriad
possible implementations of an unambiguous tracker. Readers are reminded of the discussion in the section "Generic

2) postcorrelation processing algorithms, where the unambiguous processing is done after the correlation
3) both pre- and postcorrelation processing, where their
ambiguity removal stages are split before and after the
correlation.
The noncoherent integration is not included in the postcorrelation processing because such a processing does not remove
the ambiguities, per se. The classification in Figure 5 strictly
refers to the processing stages involved in removing the ambiguities. While the first classification (see the section "Principal Dichotomy of Unambiguous Solutions: Wide Main Lobe
Versus Narrow Main Lobe") helps a designer to better understand the possible behavior of an unambiguous algorithm in
the presence of multipath (e.g., narrow main lobe algorithms
are likely to better deal with multipath than wide main lobe
algorithms), this second classification helps the designer to
have an at-a-glance estimate of the complexity of each algorithm (e.g., the last category is likely to be more complex than
the first two categories, and the first category is likely to be
less robust to noise than the second category).

Performance and implementation complexity
Once the different techniques have been introduced, it is interesting to elaborate on the performance they provide, since this
is one of the key aspects to be considered when choosing one
technique in front of some other. Another key aspect is the implementation complexity, discussed in the section "Complexity Considerations."

Acquisition and tracking performance comparisons
Figure 6 illustrates the acquisition performance of full BOC
(ambiguous) and 12 representative unambiguous algorithms,
eight of them corresponding to the wide main lobe type and four
of them for the narrow main lobe type. The example shown

(∆t )bin = 0.5 Chips

0.8
0.6

(∆t )bin = 0.01 Chips

Wide
Main Lobe

0.4
Narrow
0.2
Main
Lobe
0
20 25 30 35 40 45 50 55 60
C/N0 (dB-Hz)

Pd at Pfa = 0.001

0.8
0.6

Wide
Main Lobe

0.4

Narrow
Main Lobe

0.2
0
20

35
40
45
C/N0 (dB-Hz)

(a)

(b)
Full BOC
MH, Dual
PUDLL

BF, Single
UAL, Single
SCM
SLC

BF, Dual
MH, Single
UAL, Dual
GRASS
Ben1
Ben2

FIGURE 6. The acquisition performance of one ambiguous and 12 unambiguous algorithms. BOCc(15,2.5) modulation. (a) 0.5 chips time-bin step.
(b) 0.01 chips time-bin step.

IEEE SIGNAL PROCESSING MAGAZINE

September 2017

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