Signal Processing - September 2017 - 42

the sign of a sine carrier), and cosine BOC modulation,
which we will refer to as BOCc (created by taking the signum of a cosine carrier):
s BOC (t) = sign ^sin (N B rtfc)h, 0 # t # 1/fc
s BOC c (t) = sign ^cos (N B rtfc)h, 0 # t # 1/fc,

(1)

where fc is the code chip rate. If we consider the reference chip
rate of a C/A GPS code fref = 1.023 MHz, the typical notations for BOC and BOC c modulations are BOC (m, n)
and BOC c (m, n), respectively, with m = N B fc 2fref and
n = fc fref . It is noted that there can be many BOC or BOC c
waveforms of the same order N B . For example, BOC(1, 1) and
BOC(2, 2) modulations have exactly the same modulation order
NB = 2, but different chip rates: 1.023 MHz and 2.046 MHz,
respectively. The relationship between the BOC modulation
order and the m, n parameters of a BOC modulation is
N B = 2m/n.
As an example, let us assume that a three-chip sequence
[1, 1, -1] is transmitted via a BOC-modulated signal of
order N B = 2. The BOC-modulated signal will basically
"split" each chip into two alternating subchips 1, -1, and
the resulting sequence to be transmitted will be [1, -1, 1,
-1, -1, 1]. In the BOCc case, the resulting signal looks
like splitting each subchip further into two sub-subchips
with alternating sign [16]. The BOCc-modulated signal of
order NB = 2 of the aforementioned chip sequence will be
[1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1], where the corresponding
duration of each digit will be half compared to the BOC case.
It is as if the BOCc modulation acts as a double BOC modulation [16], and that is why we can model both sine and cosine
BOC modulations with an additional parameter called N cos ,
which is equal to one for BOC and equal to two for BOCc signals. In addition to the basic sine and cosine BOC waveforms,

there are several other BOC-based modulations typically
obtained by combining sine and cosine BOC modulations of
various orders. For example, the multiplexed BOC (MBOC)
modulation used in Galileo and modernized GPS is obtained
as a combination of two BOC signals of orders N B = 2 and
N B = 12, respectively, and it has two main variants: a composite BOC (CBOC), relying on weighted multiplexing, and
a time MBOC (TMBOC), relying on time multiplexing. The
alternate BOC (AltBOC) modulation is obtained as a combination of a BOC with a BOC c of the same orders. More
details on various BOC modulation classes and their equivalent models can be found, e.g., in [1], [2], [5], and [16].

Ambiguity-related challenges
The notches or ambiguities are very challenging in the
acquisition process because, for correctly acquiring a correlation peak, the time distance between two consecutive correlations, also called the search step in time (Dx) bin, has to
be sufficiently small to not miss a correlation peak, but at
the same time, it has to be sufficiently high to ensure a fast
acquisition process. For example, in BPSK-modulated codes
in GPS, where the main lobe width is two chips, a time-bin
step of 0.5 chips is typically used [2], [3], [6]. However, in a
BOC-modulated case, the main lobe of the correlation envelope (see an example in Figure 1) has a width close to 1/N B,
which means that a time-bin step higher than this value can
significantly increase the misdetection probability in the
acquisition stage. This approximation of the main lobe
width is more exact as the N B increases; the exact main
lobe width values are shown in Table 1. To minimize the
misdetection probability in the acquisition, it is good to
choose a small time-bin step: (Dx) bin # 1/ (2N B). On the
other hand, the acquisition time and complexity are inversely proportional to the search step (a larger time step means a
faster acquisition), meaning an acquisition complexity of the

Table 1. A list of signals proposed or already in use for GNSS that are vulnerable to ambiguities.

42

Modulation Type

Ambiguous
(Yes/No)

Number of ambiguities
within one chip

Main lobe width
[chips]

Where used [1], [2]

CBOC (+ )

Yes

2

0.70

Galileo (E1-B)

CBOC (- )

Yes

2

0.69

Galileo (E1-C)

TMBOC

Yes

2

0.70

GPS(L1C-p), BeiDou (B1-C)

TMBOC

Yes

2

0.70

GPS(L1C-p), BeiDou (B1-C)

AltBOC (15, 10)

Yes

4

0.33

Galileo(E5)

BOC (1, 1)

Yes

2

0.67

GPS(L1C-d), Glonass (L1OC-p, L1OCM), BeiDou (B1-C)

BOC (5, 2.5)

Yes

6

0.28

Glonass (L1SC)

BOC (10, 5)

Yes

6

0.28

GPS(M-code)

BOC (14, 2)

Yes

12

0.07

BeiDou (B1-D, B1-P)

BOC c (10, 5)

Yes

8

0.22

Galileo (E6-A)

BOC c (15, 2.5)

Yes

24

0.08

Galileo (E1-A), BeiDou (B3-A)

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September 2017

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