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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
.
3.2 Properties
4
Let
m
SQ
M
2
be the family of sequences over
M
2
-QAM,
M
=2
m
, defined in the previous subsection. Then,
≥
3 be a positive integer and let
I
4
SQ
M
2
have period
N
=4(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
I
−
2
3
(
M
2
E≈
−
1)
N.
3. For large values of
m
and
N
, the maximum normalized correlation
θ
max
of
family
4
I
SQ
M
2
is bounded as
√
689
16
√
N
1
.
64
√
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.
6. The normalized minimum squared Euclidean distance between all sequences
assigned to a user is given by
12
M
2
d
2
min
≈
1
N.
−
7. The number
N
of times an element from the
M
2
-QAM constellation occurs
in sequences of large period can be bounded as:
≤
√
N
+4
.
M
2
N
+4
2
M
2
−
1
N
−
M
2
4
This implies that the sequences in family
SQ
M
2
are approximately bal-
anced, i.e., all points from the
M
2
-QAM constellation occur approximately
equally often in sequences of long period.
I
4 Summary and Further Work
Various properties of the four sequence families discussed in this paper (
SQ
,
2
2
4
I
SQ−A
,
I
SQ−B
and
I
SQ
) have been summarized in Table 2. From this
2
table, we can see that family
I
SQ−B
has the lowest correlation value over
