Information Technology Reference
In-Depth Information
of sequences. The
κ
-th sequence
s
(
g, κ, t
)for
m
=2,
m
=3and
m
≥
4isgiven
by (4), (5) and (6) respectively,
s
(
g, κ, t
)=
√
2
ı
ı
u
1
(
t
)
(
1)
κ
1
+2
ı
u
0
(
t
)
ı
κ
0
−
,t
even
√
2
ıı
ı
u
1
(
t
)
(
2
ı
u
0
(
t
)
ı
κ
0
,t
odd
(4)
1)
κ
1
−
−
s
(
g, κ, t
)=
√
2
ı
ı
u
2
(
t
)
(
1)
κ
1
+4
ı
u
0
(
t
)
ı
κ
0
1)
κ
2
+2
ı
u
1
(
t
)
(
−
−
,t
even
√
2
ıı
ı
u
2
(
t
)
(
−
1)
κ
2
−
4
ı
u
0
(
t
)
ı
κ
0
,t
odd
(5)
+2
ı
u
1
(
t
)
(
−
1)
κ
1
⎧
⎨
⎩
√
2
ı
m−
1
+2
m−
1
ı
u
0
(
t
)
ı
κ
0
k
=1
2
m−k−
1
ı
u
k
(
t
)
(
1)
κ
k
−
,t
even
√
2
ıı
m−
1
s
(
g, κ, t
)=
k
=3
2
m−k−
1
ı
u
k
(
t
)
(
−
1)
κ
k
−
2
m−
3
ı
u
2
(
t
)
(
−
1)
κ
2
+
(6)
2
m−
1
ı
u
0
(
t
)
ı
κ
0
2
m−
2
ı
u
1
(
t
)
(
1)
κ
1
−
−
,t
odd
where
u
0
(
t
)=
T
([1 + 2
g
]
ξ
t
)
u
k
(
t
)=
T
([1 + 2(
g
+
δ
k
)]
ξ
t
+
τ
k
)
,
k
=1
,
2
,...,m
−
1
.
We refer to the element
g
as the
ground coecient
. Note that given the ground
coecient and the set
{
δ
1
,
···
,δ
m−
1
}
, the set of coecients used by a user are
uniquely determined. The elements
provide a selection of the component
sequences that leads to low correlation values.
{
δ
k
}
2.3 Properties
2
Let
m
be the family of sequences
over
M
2
-QAM,
M
=2
m
, defined in the previous subsection. Then,
≥
2 be a positive integer and let
I
SQ
M
2
−B
I
2
SQ
M
2
−B
have period
N
=2(2
r
−
1. All sequences in the family
1).
2. For large values of
N
, the energy of the sequences in the family is given by
2
3
(
M
2
E≈
−
1)
N.
3. For large values of
m
and
N
, the maximum normalized correlation
θ
max
of
family
2
I
SQ
M
2
−B
is bounded as
45
√
2
32
√
N
1
.
99
√
N.
θ
max
≈
4. The family size is given by (3). Note from (1) that this can potentially be
improved by a different construction of the set
G
.
5. Each user in the family can transmit
m
+ 1 bits of information per sequence
period.
