Information Technology Reference
In-Depth Information
Comparing (7) and (8), we note that the maximum correlation given by (7)
is higher. Dividing (7) by the energy, we bound the normalized maximum cor-
relation of family
2
I
SQ
16
−B
as
1
.
166
√
N.
θ
s
(
g
1
)
,s
(
g
2
)
(
τ
)
I
4
SQ
3 Family
4
Family
is constructed by interleaving 4 selected QAM sequences. It is
defined over the
M
2
-QAM constellation, with
M
=2
m
,m
I
SQ
≥
3. Compared to
2
2
families
, this family has a lower value of normal-
ized correlation parameter
θ
max
for 64-QAM and beyond. The data rate for all
these families is the same - (
m
+ 1) bits of data per sequence period.
SQ
,
I
SQ−A
and
I
SQ−B
3.1 Sequence Definition
Let
m
≥
3. Let
{
δ
0
=0
,δ
1
,δ
2
,...,δ
m−
1
}
be elements from
F
q
such that
tr
(
δ
k
)=
1
,
∀
k
≥
1. Set
H
=
{
δ
0
,δ
1
···
,δ
m−
1
}
.Let
G
=
{
g
k
}
be the largest subset of
F
q
having the property that
g
k
+
δ
p
H,
unless
g
k
=
g
l
and
δ
p
=
δ
q
. A subspace-based construction for
G
and
H
has
been described in subsection 2.1; we refer the reader to [1] for details.
Let
=
g
l
+
δ
q
,g
k
,g
l
∈
G, δ
p
,δ
q
∈
{τ
1
,τ
2
,...,τ
m−
1
}
be a set of non-zero, distinct time-shifts with
{
1
,α
τ
1
,α
τ
2
,...,α
τ
m−
1
}
being a linearly independent set. Let
κ
=(
κ
0
,κ
1
,...,κ
m−
1
)
m−
1
2
∈
Z
4
×
F
.
4
Family
I
SQ
M
2
is then defined as
I
=
s
(
g, κ, t
)
κ
g
G
4
m
−
1
SQ
∈
Z
4
×
F
∈
2
so that each user is identified by an element of
G
.
Each user is assigned the collection
s
(
g, κ, t
)
m−
1
2
|
κ
∈
Z
4
×
F
of sequences. The
κ
-th sequence
s
(
g, κ, t
)isgivenby
s
(
g, κ, t
)=
⎧
⎨
√
2
ı
m−
1
k
=1
1)
κ
k
+2
m−
1
ı
u
0
(
t
)
ı
κ
0
2
m−k−
1
ı
u
k
(
t
)
(
−
,t
≡
0(mod4)
√
2
ı
−
m−
1
k
=3
2
m−k−
1
ı
u
k
(
t
)
(
1)
κ
k
2
m−
3
ı
u
2
(
t
)
(
1)
κ
2
+
−
−
−
1)
κ
1
+2
m−
1
ı
u
0
(
t
)
ı
κ
0
2
m−
2
ı
u
1
(
t
)
(
−
,t
≡
1(mod4)
√
2
ı
−
m−
1
=
k
=3
2
m−k−
1
ı
u
k
(
t
)
(
−
1)
κ
k
+2
m−
3
ı
u
2
(
t
)
(
−
1)
κ
2
−
⎩
1)
κ
1
+2
m−
1
ı
u
0
(
t
)
ı
κ
0
2
m−
2
ı
u
1
(
t
)
(
−
,t
≡
2(mod4)
√
2
ı
−
m−
1
k
=3
2
m−k−
1
ı
u
k
(
t
)
(
1)
κ
k
+2
m−
3
ı
u
2
(
t
)
(
1)
κ
2
+
−
−
2
m−
1
ı
u
0
(
t
)
ı
κ
0
2
m−
2
ı
u
1
(
t
)
(
1)
κ
1
−
−
,t
≡
3(mod 4)
