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Table 2.
Properties of families
SQ
,
I
2
SQ−A
,
I
2
SQ−B
and
I
4
SQ
Family
Constellation Period
N
Euclidean distance
θ
max
1
.
61
√
N
2
r
−
1
SQ
16
[1]
16-QAM
0
.
8
N
1
.
41
√
N
I
2
SQ
16
−A
[1]
2(2
r
−
1)
16-QAM
2
.
0
N
1
.
17
√
N
I
2
SQ
16
−B
(new)
2(2
r
−
1)
16-QAM
0
.
8
N
11
√
N
2
r
−
SQ
64
[1]
64-QAM
1
0
.
19
N
2
.
1
.
89
√
N
I
2
SQ
64
−A
[5]
2(2
r
−
1)
64-QAM
0
.
19
N
1
.
62
√
N
I
2
SQ
64
−B
(new)
2(2
r
−
1)
64-QAM
0
.
19
N
1
.
60
√
N
I
4
SQ
64
4(2
r
−
1)
(new)
64-QAM
0
.
19
N
2
.
41
√
N
2
r
−
1
SQ
256
0
.
047
N
[1]
256-QAM
1
.
95
√
N
I
2
SQ
256
−A
[5]
2(2
r
−
1)
0
.
12
N
256-QAM
1
.
77
√
N
I
2
SQ
256
−B
(new)
2(2
r
−
1)
256-QAM
0
.
047
N
1
.
58
√
N
I
4
SQ
256
4(2
r
−
1)
(new)
256-QAM
0
.
047
N
2
.
76
√
N
M
2
-QAM
2
r
−
1
12
N/
(
M
2
−
1)
SQ
M
2
[1]
1
.
99
√
N
I
2
SQ
M
2
−A
[5]
M
2
-QAM
2(2
r
−
1)
30
N/
(
M
2
−
1)
1
.
99
√
N
I
2
SQ
M
2
−B
(new)
M
2
-QAM
2(2
r
−
1)
12
N/
(
M
2
−
1)
1
.
64
√
N
I
4
SQ
M
2
(new)
M
2
-QAM
4(2
r
−
1)
12
N/
(
M
2
−
1)
16-QAM. Over 64-QAM, 256-QAM and large
M
2
-QAM constellations, family
I
4
SQ
has the lowest correlation value.
However, family
2
has a higher value of normalized minimum squared
Euclidean distance as compared to the new families
I
SQ−A
2
4
.
From Table 1 (of subsection 1.1), we note that although the selected families
have a lower value of correlation, they also have a lower data rate as compared
to the canonical families
I
SQ−B
and
I
SQ
2
. It would be interesting to reduce the
correlation bounds of these canonical families without changing their data rate.
In all the interleaved sequence families proposed in [1] and [5], and in the
interleaved sequence families proposed in this paper, we note that interleaving
lowers the correlation parameter without effecting the data rate. It also increases
the normalized minimum squared-Euclidean distance in most cases.
From these tables, we can also see that the Welch lower bound on sequence
correlation [13] has not been achieved for any sequence family over
M
2
-QAM,
M
=2
m
,m
CQ
and
I
CQ
≥
2.
References
1. Anand, M., Kumar, P.V.: Low Correlation Sequences over QAM and AM-PSK
Constellations. IEEE Trans. Inform. Theory 54(2), 791-810 (2008)
2. Boztas, S.: CDMA over QAM and other Arbitrary Energy Constellations. Proc.
IEEE Int. Conf. on Comm. Systems 2, 21.7.1-21.7.5 (1996)
3. Boztas, S.: New Lower Bounds on the Periodic Crosscorrelation of QAM Codes
with Arbitrary Energy. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds.)
AAECC 1999. LNCS, vol. 1719, pp. 410-419. Springer, Heidelberg (1999)
