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In-Depth Information
3D gnof2
T
-Periodic
M
-Ary LCZ Sequence Sets
The interleaved technique [2] may be applied to construct
nT
-periodic sequences
with low correlation from a
T
-periodic sequence with good autocorrelation,
where both
n
and
T
are positive integers. The technique may also be used to
construct 2
T
-periodic
M
-ary LCZ sequences from a
T
-periodic
M
-ary sequence
with good autocorrelation.
Let
{
s
(
t
)
}
be a
T
-periodic
M
-ary sequence satisfying
|
C
s
(
τ
)
|≤
≤
T−
2
.
for some nonnegative constant
. Assume that
M
is even and 1
≤
d
Let
⎧
⎨
2
T−
1
4
d
if
d
T−
1
2
f
=
(1)
2
T−
3
4
d
if
d
⎩
T−
1
2
|
.
The set
IS
of 2
T
-periodic sequences is defined as
IS
=
{{
s
(
i,m
)
(
t
)
}|
0
≤
i
≤
f
−
1
,m
=0
,
1
}
(2)
where
s
(
i,
0)
(2
t
)=
s
(
t
−
id
)
,s
(
i,
0)
(2
t
+1)=
s
(
t
+1+(
i
+1)
d
)
,
id
)+
M
2
s
(
i,
1)
(2
t
)=
s
(
t
−
,s
(
i,
1)
(2
t
+1)=
s
(
t
+1+(
i
+1)
d
)
T−
1
for
d
,and
2
s
(
i,
0)
(2
t
)=
s
(
t
−
id
)
,s
(
i,
0)
(2
t
+1)=
s
(
t
+2+(
i
+1)
d
)
,
id
)+
M
2
s
(
i,
1)
(2
t
)=
s
(
t
−
,s
(
i,
1)
(2
t
+1)=
s
(
t
+2+(
i
+1)
d
)
T−
1
2
for
d
contains 2
fM
-ary sequences of period 2
T
.The
crosscorrelation between any two sequences in
|
. Note that the set
IS
IS
is derived in the following
lemma.
Lemma 1.
Let
τ
=2
τ
1
+
τ
0
,
τ
0
∈{
0
,
1
}
,and
0
≤
τ
1
≤
T
−
1
.For
0
≤
i, j
≤
f
−
1
,
m, n
∈{
0
,
1
}
,let
a
=
j
−
i
and
b
=
j
+
i
+1
. Then the crosscorrelation
C
(
i,m
)
,
(
j,n
)
(
τ
)
between
{
s
(
i,m
)
(
t
)
}
and
{
s
(
j,n
)
(
t
)
}
is given as follows:
1
2
:
C
(
i,
0)
,
(
j,
0)
(
τ
)=
C
s
(
τ
1
+
ad
)+
C
s
(
τ
1
−
T
−
Case i)
d
ad
)
,
if
τ
0
=0;
C
s
(
τ
1
+
bd
+1)+
C
s
(
τ
1
−
bd
)
,
if
τ
0
=1;
C
(
i,
1)
,
(
j,
1)
(
τ
)=
C
s
(
τ
1
+
ad
)+
C
s
(
τ
1
−
ad
)
,
if
τ
0
=0;
bd
)
,
if
τ
0
=1;
C
(
i,
0)
,
(
j,
1)
(
τ
)=
−C
s
(
τ
1
+
ad
)+
C
s
(
τ
1
− ad
)
,
if
τ
0
=0;
−
C
s
(
τ
1
+
bd
+1)
−
C
s
(
τ
1
−
C
s
(
τ
1
+
bd
+1)
− C
s
(
τ
1
− bd
)
,
if
τ
0
=1;
C
(
i,
1)
,
(
j,
0)
(
τ
)=
−
C
s
(
τ
1
+
ad
)+
C
s
(
τ
1
−
ad
)
,
if
τ
0
=0;
−
C
s
(
τ
1
+
bd
+1)+
C
s
(
τ
1
−
bd
)
,
if
τ
0
=1;
