TELKOMNIKA ISSN: 1693-6930
Unambiguous Sine-Phased BOC kn,n Signal Acquisition Based on .... Deng Zhongliang 503
In this paper, we will focus on the acquisition of the sine-phased BOCkn,n signals, where k is the ratio of the sub-carrier frequency
s
f
to the spreading code rate
c
f
. At first, the signal model will be given. Then, two kinds of correlation functions are obtained which are
composed of sub-correlations. Side peaks of auto correlation function will be shown. Through combining of different sub-correlations, we will have new correlation without side-peaks. Finally,
theoretical results will be given.
2. Signal Model
The generic Binary Coded Symbols BCS [
1 2
, ,...
n
s s s
],
c
f
baseband signal can be expressed as [5]
1
1
i c
n s
c BCS
T n i
iT s
t p
t n
1 where
{ 1,1}
i
s Î -
is the
i
th chip of the binary sequence [
1 2
, ,...
n
s s s
],
c
T
is the PRN code chip period,
c
T n
p t
is a unit rectangular sub-carrier pulse waveform over [0,
c
T n
. Sin-phased BOC baseband signal is a special case of BCS signal with a
representation vector formed by +1’s and -1’s alternating in a particular defined way. The sine- phased BOCkn,n baseband signal can be expressed as
2 1
1
s
k u
T c
s u
s t p
t iT
uT
- =
= -
- -
å
2 Where
s
T is the sub-carrier pulse duration of
2 =1 2 1.023
c
T k
kn MHz
´ ,
s
T
p t
is the unit rectangular sub-carrier pulse waveform over [0,
s
T . The full expression of sine-phased
BOCkn,n signal will contain the spreading code and the navigation data, which is
sin
BOC c
c i
s t
P c t
iT d t iT s t
¥ = - ¥
= -
-
å
3 Where
c t is the spreading code and
d t is the navigation data. For the purpose of
focusing on the ambiguous acquisition of
sin
BOC
s t
signal, we assume that the navigation data is always 1, which means that we choose a pilot channel for acquisition and furthermore we
also don’t consider the effect of secondary code. During the process of acquisition, the spreading code or the sub-carrier will be
wiped off. Therefore, there are two kinds of autocorrelation function which depends on the local generated signal. One is the correlation of the sine-phased BOCkn,n with the
spreading code only. The other is the correlation of the sine-phased BOCkn,n with the spreading code and sub-carrier both. Without considering the front-end filtering, the
normalized BOC correlation function of the sine-phased BOCkn,n with the spreading code and sub-carrier both can be expressed as:
1 1
1
1 1
= {[
1 ] [
1 ]}
sin sin
c s
s
T SC
BOC BOC
T T N
N i
j c
T c
s c
T c
s m
i j
R s
t s t
dt PT
Pc mT p
mT iT
Pc mT p
mT jT
PT t
t t
t
- -
- =
= =
= +
- -
´ +
- +
-
ò å
å å
4
ISSN: 1693-6930
TELKOMNIKA Vol. 13, No. 2, June 2015 : 502 – 509
504
1 1
1 1
1 1
1 1
1 {
[ 1
1 ]}
1 [
1 1
] 1
1 1
c s
s c
s s
s
T T N
N i
j c
c T
c s
T c
s m
i j
T T N
N i
j T
c s
T c
s m
i j
N N
i j
s T
s s
i j
c i
j j
c mT c mT p
mT iT
p mT
jT T
p mT
iT p
mT jT
T T
jT iT
T N
t t
t t
- -
- =
= =
- -
- =
= =
- -
+ =
= +
=
= +
- -
´ -
+ -
= -
- ´
- +
- =
- L
- +
= -
å å
å å å
å å å
1 1
1
s
N N
T s
s i
N i
SCsub i
jT iT
R t
t
- -
= -
=
L -
+ =
å å
å
Where
1 ,
0,
s
s T
s s
T T
T t
t t
t ìï
ï - £
ïï L
= í ïï
ï ³
ïî 5
is a triangular function, which is the correlation function of two rectangular pulse waveform, and
1
1 1
s
N i
i j
SCsub T
s s
j
R jT
iT N
t t
- +
=
= -
L -
+
å
6 is the sub-correlation function of the sine-phased BOCkn,n with the spreading code
and sub-carrier both. We can see that this sub-correlation function
i SCsub
R t is a combination
of triangular functions with different phases. And
SC
R t
is the combination of different
i SCsub
R t , which is the reason of the ambiguity problem.
The second kind of correlation function is the correlation function of the sine-phased BOCkn,n
with the spreading code only, which can be expressed as 7 in case that the front-end filtering is not considered.
1 1
1 1
1 1
1
1 1
= {[
1 ] [
]} 1
[ 1
] 1
[ 1
sin c
s c
s s
c s
T C
BOC T T
N i
c T
c s
c m
i T T
N N
i c
c T
c s
T c
s m
i j
T T N
i T
c m
i
R s
t c t dt
PT Pc mT
p mT
iT Pc mT
PT c mT c mT
p mT
iT p
mT jT
T p
mT T
t t
t t
t
- -
= =
- -
- =
= =
- -
= =
= +
- -
´ +
= +
- -
´ +
- =
- -
ò å
å å
å å
å å
1 1
1 1
1 1
] 1
1 1
s s
s
N s
T c
s j
N N
i s
T s
s i
j c
N N
i T
s s
i j
N i
Csub i
iT p
mT jT
T jT
iT T
jT iT
N R
t t
t t
- =
- -
= =
- -
= =
- =
´ +
- =
- L
- +
= -
L -
+ =
å å å
å å
å
7
Where
1
1 1
s
N i
i Csub
T s
s j
R jT
iT N
t t
- =
= -
L -
+
å
8
TELKOMNIKA ISSN: 1693-6930
Unambiguous Sine-Phased BOC kn,n Signal Acquisition Based on .... Deng Zhongliang 505
is the second kind of sub-correlation function, which is also the combination of triangular functions with different phases. However, the factor of triangular functions in two kinds of
sub-correlation function is different. Two kinds of sub-correlation functions are shown in Figure 1.
Figure 1. Correlation functions and sub-correlation functions
3. Proposed Method
