Signal Processing - September 2017 - 43

order O (1/ (Dx) bin). If we assume that the
Table 2. Acquisition complexity as a function of (Dx) bin .
code length in chips is L code and the number of frequency bins to be searched in the
L code * N freq / (Dx) bin
Number of correlators Ncorrs
Oc 1 m
(Dx) bin
acquisition stage is N freq, Table 2 shows
the required number of correlators to
Oc 1 m
c 1 m
Minimum sampling frequency fs
(Dx) bin
(Dx) bin
search the whole time-frequency space and
1
9N fft + 4N fft log 2 (N fft)
Oc
m
MAC operations [17] in an FFTthe minimum possible sampling frequency
(Dx) bin log 2 ((Dx) bin))
based acquisition
N
=
next
2
pow
(
N
)
to acquire the desired time resolution. The
fft
corrs
number of multiply and accumulate
(MAC) operations per one frequency bin
are also showing, assuming a fast Fourier transform (FFT)
Unambiguous solutions
implementation of the acquisition stage with N fft FFT
points. N fft is typically taken as the next power of two highGeneric block diagram for unambiguous processing
er or equal with the number of required correlators. Clearly,
To have a unified view of unambiguous techniques, we present
(Dx) bin plays an important role and defines the tradeoff
them all via a generic block diagram, as shown in Figure 4. The
between a low-complexity acquisition and a high-resolution
general procedure is the following: first, the received signal and
or accurate acquisition.
the reference codes are filtered with various linear or nonlinear filTo increase the speed and decrease the complexity of the
ters (most cases use linear time-invariant or time-variant filters),
acquisition, methods that deal with the notches in the ambiguous
denoted here via RxFilti (filters for the received signal) and CFilti
function have been found to be able to use
(filters for the reference codes) with i = 1 : N
In
addition
to
the
basic
sine
(
D
x
)
a larger value of
and N being the number of (dual) branches,
bin (for example the
same 0.5 chips as used in the legacy GPS
and cosine BOC waveforms, then correlated, possibly averaged coherently
signals). Such methods will be discussed in
and then noncoherently combined or postprothere are several other
the section "Unambiguous Solutions."
cessed. The filter purposes, filter types, numBOC-based modulations
In tracking, the challenges come from
ber N of branches, and noncoherent
typically obtained by
the presence of false peaks, because the
processing in the combiner block differ for
combining sine and
code estimate can easily jump from the
one algorithm to another. The main algocosine BOC modulations
correct peak to a false peak (see Figure
rithms and how they fit into the block diagram
1), especially when there are many false
of Figure 4 are presented next. As shown in
of various orders.
peaks within ±1 chip from the main peak.
Figure 4 and described in detail in the section
It is easy to see that both the number of ambiguities and the
"Principal Dichotomy of Unambiguous Solutions: Wide Main
number of false peaks for a BOC-modulated signal is equal
Lobe Versus Narrow Main Lobe," the unambiguous processing
to 2 (N B + N cos) - 4. The more ambiguities we have, the more
can be based either on widening the main correlation lobe, or on
challenging it is to mitigate them in acquisition and tracking.
preserving the narrow main correlation lobe, while removing
Higher-order BOC-modulated signals are more challengsome or all of the sidelobes of the correlation envelope.
ing than lower-order BOC modulation signals and cosineIn Figure 4, we denote via R i, i = 1, 2, f, N the outputs
BOC modulated signals are more challenging than sine-BOC
of the coherent integration. R i in each branch clearly depend
modulated signals of the same N B order. Also, the nearest
on the receiver and code filters (RxFilt and Cfilt) in that parincorrect peak to the main one is placed at ! 1/N B from the
ticular branch. Most unambiguous algorithms rely on N = 1
main peak, so an increased BOC modulation order implies a
or N = 2 branches, but there are also some unambiguous
closely spaced strong fake peak.
algorithms employing a number of branches proportional to
the BOC modulation order. For the solutions with N = 1 or
N = 2, the combiner is typically a weighted combination of the
Modulation types used in modernized GNSSs
It was previously mentioned that the ambiguities pose a
envelopes or squared envelopes of the R 1 and R 2 correlations
threat in the modernized GNSS signals. To understand better
shown in Figure 4 followed by noncoherent averaging. In some
which GNSS signals suffer of ambiguities, Table 1 shows the
unambiguous algorithms, not only are the envelopes | R 1 | and
| R 2 | used in the weighted combinations, but also the envelope of
signals proposed or already in use in GNSS that will rely on
BOC. The number of ambiguities (also equal to the number
their difference | R 1 - R 2 |. To be able to use this generic model
of false peaks) and the type of GNSS signals using such modof Figure 4 for the vast majority of unambiguous approaches,
ulations are also shown. Details on various GNSS signals and
in some algorithms, the RxFilt and CFilt filters are absent or
bands can be found, e.g., in [1]. All future GNSS modulations
the branch corresponding to them is absent. The ambiguwill have to deal with the ambiguities. Also, as higher BOC
ous or full-BOC processing can also be modeled via the
modulation orders implies more challenging acquisition and
block diagram in Figure 4; for full-BOC, we have two dual
tracking structures, the most challenging modulations in
branches (N = 2), the received signal filters RxFilt 1, RxFilt 2
GNSS are the BeiDou B1-D, B3-A, and B1-P signals and the
are not used, and the reference code filters CFilt 1, CFilt 2 are
Galileo E1-A and E6-A signals.
the BOC modulation filters whose transfer functions are, e.g.,
IEEE SIGNAL PROCESSING MAGAZINE

September 2017

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