means better tracking accuracy. However, when
using a traditional narrow early-minus-late (NEML)
tracking loop [19] with the early-late sep aration ¢,the
discriminator characteristic curve of sine-BOC(m,n)
signal has a smaller linear domain than the one of
the BPSK signal with rectangular symbols and the
same code rate (denoted as BPSK-R(n)). Besides, the
discriminator characteristic curve of a sine-BOC(m,n)
signal has (4m=n ¡2) stable false lock points which
are due to the side peaks of the autocorrelation
function. It is then possible to have the loop locking
on one of the side peaks, which would result in
intolerable biased measurements. This problem is
reputed as an ambiguity problem for BOC modulated
signal tracking.
Several solutions have been proposed to solve
the ambiguity problem for sine-BOC modulated
signals. Among them, there are three representative
ways which are respectively referred to as BPSK-like
technique, bump-jumping technique, and side-peaks
cancellation techniques.
The earliest BPSK-like idea was described in
[7]. As its name implies, the BPSK-like technique
considers the received sine-BOC signal as the
sum of two BPSK signals with carrier frequency
symmetrically positioned on each side of the BOC
carrier frequency. Thus each sidelobe is treated
independently as a BPSK signal, which provides an
unambiguous correlation function and a wider S-curve
steady domain. However, this technique is complex
to implement, and it completely removes all of the
advantages of BOC signal tracking since it causes the
rms bandwidth of the received signal to approach that
of the BPSK signal.
Another technique, presented in [8], is referred
to as bump-jumping technique. This technique uses
an early-minus-late discriminator which is similar
to the traditional BPSK tracking methods. This
technique also uses an extra two correlators which
are referred to as very early (VE) and very late
(VL) correlators. These two correlators are located
at the nearest two side peaks to the main peak. By
comparing the magnitudes of the VE, VL, and prompt
(P) correlators’ outputs, this technique checks whether
the loop is locked on the main peak. If the outputs
of VE or VL correlators are consistently higher than
that of the P correlator, then the loop will jump in the
appropriate direction. When locked on the main peak,
this technique has high tracking accuracy. However,
since it is based on the comparing of the main and
side peaks magnitudes, the detection may have a high
probability of false alarm when the signal-to-noise
ratio (SNR) is low. Not only that, but this technique
needs time to detect and recover from false lock, so it
is inapplicable in some critical applications.
Side-peaks cancellation techniques are another
way to solve the ambiguity problem for sine-BOC
modulated signals. The first side-peaks cancellation
technique is proposed in [9]. This approach removes
the ambiguities of the correlation function, but one
drawback is that this method destroys the sharp peak
of the correlation function. For accurate tracking,
preserving a sharp peak of the correlation function is
a prerequisite. An innovative unambiguous tracking
technique, which is referred to as ASPeCT, is
described in [10]. This technique uses ten correlation
channels, completely removing the side peaks from
the correlation function and keeping the sharp main
peak. However, this technique has some limitations,
for it is only applicable to sine-BOC(n,n) signals.
Some other side-peaks cancellation methods have
been proposed recently [11—16]. All of the side-peaks
cancellation techniques mentioned above use new
local replica signals whose modulated symbols
are different from that of the received signal and
noncoherently combine outputs of the correlation
channels to completely remove the side peaks from
the correlation function.
The key of the side-peaks cancellation technique is
the design of local signals’ modulated symbols, which
need to be analyzed by the correlation function of the
signals with various step-shape modulated symbols. In
[20], the analytical expression of the autocorrelation
function for BOC modulated signals was given.
However, to the authors’ knowledge, there is no
available theoretical formula of the cross-correlation
function of two signals with step-shape modulated
symbols. In the next section, an analytical expression
of that correlation function is given out, which is
the basis of the investigation of a newly developed
side-peaks cancellation technique in Section IV.
III. DERIVATIONS OF CORRELATION FUNCTION
Consider two spreading signals with step-shape
modulated symbols s(t)ands
0
(t) which have the
same f
c
and M, while the spreading sequence and the
shapes of the modulated symbols may be different.
Using (1) and (2), these two signals are specified by
s(t)=
1
X
i=¡1
M¡1
X
k=0
(¡1)
c
i
d
k
Ã
k
(t ¡iMT
s
)
s
0
(t)=
1
X
j=¡1
M¡1
X
k=0
(¡1)
c
0
j
d
0
k
Ã
k
(t ¡jMT
s
):
(5)
The cross-correlation function of these two signals is
R
ss
0
(¿)=
1
T
Z
T
0
s(t) s
0
(t + ¿)dt
=
1
T
X
i
X
j
M¡1
X
k=0
M¡1
X
q=0
(¡1)
c
i
+c
0
j
d
k
d
0
q
£
Z
T
0
Ã
k
(t ¡iMT
s
)Ã
q
(t ¡jMT
s
+ ¿)dt (6)
where T = NT
c
is the period of the spreading signals.
1784 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 46, NO. 4 OCTOBER 2010