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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:
≤
√
2
M
2
√
N
+2
.
N
+2
M
2
−
1
N
−
M
2
2
This implies that the sequences in family
are approximately bal-
anced, i.e., all points from the
M
2
-QAM constellation occur approximately
equally often in sequences of long period.
I
SQ
M
2
−B
The period is clearly
N
=2(2
r
−
1) since the period of each of the component
sequences is 2
r
are computed
in the next subsection. For lack of space, we omit the rest of the proofs.
−
1. The correlation properties of family
I
2
SQ
16
−B
2.4 Correlation Properties of Family
I
2
SQ
16
−B
2
We define basic sequences in family
as sequences corresponding to
κ
0
=
κ
1
= 0 in (4). We analyze the correlation between two basic sequences
from family
I
SQ
16
−B
2
; it is straightforward to extend the results to the case
of modulated sequences.
Let
I
SQ
16
−B
{
s
(
g
1
,
0
,t
)
}
and
{
s
(
g
2
,
0
,t
)
}
be two basic sequences belonging to family
2
I
SQ
16
−B
, i.e.,
s
(
g
1
,
0
,t
)=
√
2
ı
ı
u
1
(
t
)
+2
ı
u
0
(
t
)
,t
even
2
ı
u
0
(
t
)
,t
odd
s
(
g
2
,
0
,t
)=
√
2
ı
ı
v
1
(
t
)
+2
ı
v
0
(
t
)
,t
even
√
2
ıı
ı
u
1
(
t
)
−
√
2
ıı
ı
v
1
(
t
)
2
ı
v
0
(
t
)
,t
odd
−
where
u
0
(
t
)=
T
([1 + 2
g
1
]
ξ
t
)
,
u
1
(
t
)=
T
([1 + 2 (
g
1
+
δ
1
)]
ξ
t
+
τ
1
)
,
v
0
(
t
)=
T
([1 + 2
g
2
]
ξ
t
)and
v
1
(
t
)=
T
([1 + 2 (
g
2
+
δ
1
)]
ξ
t
+
τ
1
)
.
The expression for the correlation between the two QAM sequences would
take one of the two forms depending on whether
τ
is even or odd.
Assume that
τ
≡
0 (mod 2). The correlation between the two sequences can
be written as:
