Global Positioning System Reference
In-Depth Information
order
M
=
. Note that in an odd order SCS signal, the chip waveform is time-varying. For
example, for sin-BOC signal with
2
M
= , the shape vector of the spreading chip is (1,-1,1)
T
3
)
)
)
(
(
)
in the time interval
t
∈⎡
2, 21
c
TnT
+
, while it is (-1,1,-1)
T
in
t
∈⎡
21, 2
c
n
−
TnT
. We
⎣
⎣
c
c
assume that
M
is even hereafter.
4.3 CCF of SCS signals
All the SC techniques are based on the shapes of CCF between the received signal and the
local signal. Here we consider the CCF of two SCS signals which have the same chip rate
c
f
,
spreading sequence
{ }
c
, and the order
M
, while the chip waveform are difference.
i
By using (2) and (9), a SCS baseband signal can be expressed as
+∞
M
−
1
( )
∑∑
( )
c
(
)
g t
;
d
=
−
1
d
ψ
t
−
nMT
(13)
n
kk
s
n
=−∞ =
k
0
The CCF of two SCS signals is
1
T
(
)
() ( )
∫
R
τ
;,
dd
′
=
g t g
′
t
+
τ
dt
gg
′
T
0
(14)
1
MM
−−
11
T
∑∑∑∑
()
cc
+
(
)(
)
′
∫
=
−
1
nm
dd
ψ
t
−
nMT
ψ
t
−
mMT
+
τ
dt
kq
k
s
q
s
T
0
nmk
==
00
q
where
TNT
=
is the period of the spreading sequence. The integral in (14) is nonzero only
c
when
(
)
and
(
)
− + have the overlapping parts. The delay τ can be
expressed as the summation of three parts
ψ −
tnMT
ψ
tmMT
τ
k
s
q
s
τ
=
aT
++, where
a
is an integer,
bT
ε
c
s
[
)
b
=
0,1,
,
M
−
1
，
and
ε∈
0,
T
. And after some algebraic simplifications, (14) can be
rewritten as (Yao & Lu, 2011)
( )
(
)
R
τ
=
R
aT
+
bT
+
ε
gg
′
gg
′
c
s
⎡
⎤
⎡
⎤
(15)
⎛
⎞
⎛ ⎞
⎛
⎞
⎛ ⎞
ε
ε
ε
ε
()
(
)
=
Rar
1
−
+
r
+
Ra
+
1
r
1
−
+
r
⎢
⎜
⎟
⎜ ⎟
⎥
⎢
⎜
⎟
⎜ ⎟
⎥
c
b
b
+
1
c
bM
−
bM
− +
1
T
T
T
T
⎢
⎥
⎢
⎥
⎝
⎠
⎝ ⎠
⎝
⎠
⎝ ⎠
⎣
⎦
⎣
⎦
s
s
s
s
where
1
N
−
1
()
∑
()
cc
+
Ra
−
1
(16)
nna
+
c
N
n
=
0
and the aperiodic cross-correlation function (ACCF) of
d
and
d
is
⎧
1
Mb
−−
1
∑
dd
′
,
0
≤ ≤−
b
M
1
⎪
⎪
ibi
+
M
i
Mb
=
−−
0
1
1
(
)
∑
r
dd
,
′
=
d
d
′
,
1
−≤<
M
b
0
(17)
⎪
⎪
b
i
−
b
i
M
i
=
0
0,
bM
≥
⎩
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