Signal Processing - September 2017 - 49

DS-SS Receiver Chain and Code-Tracker Classification" and
Figure 3, where we showed that there are many code trackers
that can be used with any DS-SS system. When the ambiguities are also taken into account, any of the code trackers
can be in fact combined with one of the unambiguous methods discussed in the section "Unambiguous Solutions"-this
is how we obtained the overall unambiguous code-tracker
structure, as shown in Figure 2. The tracking loop goal is to
converge to the correct peak even if the acquisition stage converged to a wrong peak from the BOC-modulated envelope.
Thus, there is a significant number of possible code-tracking
combinations, and very few of them have been actually studied in the current literature.
Figure 7(a) illustrates an example of code-tracking performance when full BOC and four unambiguous algorithms
(two with wide main lobe and two with narrow main lobe)
are combined with NCORR and HRC, respectively, as two
of the most used code trackers. This example is based on
a BOC(1, 1)-modulated signal, an early-late spacing of
0.3 chips, a code loop bandwidth of 1 Hz, 20 ms of total
integration time (10 ms coherent integration and two blocks
of noncoherent integration), and a double-sided bandwidth
of 40.92 MHz. The statistics were computed over 12,000
random points for Nakagami-m fading channels. The tracking performance metric shown in Figure 7(a) is the standard
deviation of the delay error in the absence of multipaths; the
standard deviation is also compared with the Cramer-Rao

Tracking Performance

30
Multipath Error in Code Tracking (m)

103

Wide Main Lobe

Std Err (m)

102

Narrow
Main Lobe

101

100

10-1
20

lower bound and with the theoretical NCORR performance
as derived in the literature [35], [36]. Figure 7(b) shows a
different metric: the multipath error envelope metric in the
presence of multipaths. The multipath power was 3 dB lower
than the LOS power in the multipath error envelope curve.
For clarity of the curves, we have selected the BOC(1, 1)
case in Figure 7(a) and (b), but similar observations hold
for other BOC-based modulations encountered in GNSS.
As seen in Figure 7 (and it has also been observed from
an extensive search in the literature and additional simulations that we have run), the lowest tracking error variance
is achieved with the basic NCORR structure and narrow
main lobe unambiguous acquisition algorithms. The tracking error variance among the different narrow main lobe
unambiguous trackers is rather similar and also close to
the ambiguous error tracking variance. Thus, looking only
at the code error variance, one could misleadingly draw
the conclusion that a full BOC ambiguous approach with
basic NCORR is the best in tracking. However, the tracking performance metrics are not only the variance of the
code-tracking error, but also the probability to slide away
from the main peak into a false peak, the probability to
jump back to the correct peak (LOS peak) when starting
the tracking from an incorrect delay, the mean time to lose
lock while in tracking, the multipath robustness, and the
complexity of the tracking stage, including the unambiguous processing part.

20
10

-10
Narrow Main
Lobe Algorithms

-20
-30

C/N0 (dB-Hz)
Full BOC, NCORR
UAL Dual, NCORR
GRASS, NCORR
PUDLL, NCORR
Benedetto 1, NCORR
Full BOC, HRC

Wide Main
Lobe
Algorithms

0.2

0.4

0.6

0.8

1.2

Multipath Spacing in the Channel (Chips)
UAL Dual, HRC
GRASS, HRC
PUDLL, HRC
Benedetto 1, HRC
CRLB
NCORR,th

Full BOC
UAL, Dual
GRASS
PUDLL
Benedetto 1

(a)

(b)

FIGURE 7. (a) The tracking error standard deviation of four unambiguous algorithms combined with two types of code trackers. (b) Errors due to multipaths in the absence of noise. One LOS and one NLOS spaced as shown on the x-axis. BOC(1, 1) modulation.

IEEE SIGNAL PROCESSING MAGAZINE

September 2017

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