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Table 1.
Families of sequences over
M
2
-QAM and 2
M
-ary Q-PAM,
M
=2
m
,m≥
2
Family
Constellation
Period
N
Data rate
θ
max
1
.
80
√
N
to 3
.
00
√
N
16-QAM to
M
2
-QAM
2
r
CQ
[1]
−
1
2
m
1
.
41
√
N
to 2
.
12
√
N
2
16-QAM to
M
2
-QAM
2(2
r
I
CQ
[5]
−
1)
2
m
m
+1 1
.
61
√
N
to 2
.
76
√
N
16-QAM to
M
2
-QAM
2
r
SQ
[1]
−
1
m
+1 1
.
41
√
N
to 1
.
99
√
N
2
16-QAM to
M
2
-QAM
2(2
r
I
SQ−A
[1,5]
−
1)
m
+1 1
.
17
√
N
to 1
.
99
√
N
2
16-QAM to
M
2
-QAM
2(2
r
I
SQ−B
(new)
−
1)
m
+1 1
.
60
√
N
to 1
.
64
√
N
4
64-QAM to
M
2
-QAM
4(2
r
I
SQ
(new)
−
1)
m
+1 1
.
34
√
N
to 2
.
24
√
N
8-ary Q-PAM to 2
M
-ary Q-PAM 2
r
P
[1]
−
1
√
N
to 1
.
72
√
N
2
8-ary Q-PAM to 2
M
-ary Q-PAM 2(2
r
I
P
[1,5]
−
1)
m
+1
In 1996, Boztas [2] proposed a 16-QAM CDMA sequence family built from
quaternary sequences drawn from family
[4].Thefirstconcertedlookatlow
correlation sequence families over QAM constellations was presented by Anand
and Kumar (AK) [1] recently. Extending the concept of interleaving proposed
by AK [1], Garg, Kumar, and Madhavan (GKM) [5] proposed three interleaved
sequence families with improved correlation properties.
The main properties of the five sequence families proposed in [1], the three
sequence families proposed in [5], and the two sequence families proposed in this
paper are summarized in Table 1. In the present paper, we are interested in the
middle block of rows of the table, i.e., the families of selected sequences viz.
A
SQ
,
I
2
SQ−A
I
2
SQ−B
I
4
SQ
,
and
.
was proposed by AK in [1]. Over
M
2
-QAM, it permits (
m
+1)-bit
data modulation. As can be seen from the table, the upper bound on
θ
max
for
Family
SQ
sequences in this family ranges from 1
.
61
√
N
in the case of 16-QAM to 2
.
76
√
N
for large
M
.
In the same paper, AK propose the interleaved selected family
2
I
SQ
16
−A
1
.
41
√
N
. Motivated by this and building
upon the techniques introduced in [1], GKM [5] extend the interleaved con-
struction of family
over 16-QAM. This family has
θ
max
2
I
SQ
16
−A
to 64-QAM and beyond. The upper bound on
ranges from 1
.
89
√
N
for 64-QAM to 1
.
99
√
N
for
large
M
. This implies that family
2
θ
max
for family
I
SQ
M
2
−A
2
I
SQ−A
has better correlation properties
as compared to family
. This can also be seen from the table.
In the present paper, we propose two new interleaved families. Both these
families have better correlation properties than family
SQ
2
I
SQ−A
.
2
,with
M
=2
m
,m
2. This family
is obtained by interleaving 2 selected sequences. Over 16-QAM, the
θ
max
for
We first construct family
I
SQ
M
2
−B
≥
sequences in this family is upper bounded by 1
.
17
√
N
,whichisthelowestover
all known 16-QAM sequence families.
We then construct family
4
SQ
M
2
,with
M
=2
m
,m
3. This family is
obtained by interleaving 4 selected sequences. The upper bound on
θ
max
for
I
≥
sequences in this family ranges from 1
.
60
√
N
for 64-QAM to 1
.
64
√
N
for large
M
. From the table, we can see that this value of
θ
max
for large
M
is substantially
lower than the previous upper bound of 1
.
99
√
N
.
