388 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 1, NO. 4, AUGUST 2012
Perfect 8-QAM+ Sequences
Fanxin Zeng, Xiaoping Zeng, Xiangyong Zeng, Zhenyu Zhang, and Guixin Xuan
Abstract—This letter investigates the construction methods
of perfect 8-QAM+ sequences, which relate to one of three
open problems proposed by Boztas¸ and Parampalli in 2010,
and two constructions that transform the known perfect ternary
sequences (PTSs) into the required sequences are presented. The
resultant sequences and the employed sequences have the same
period, or the former’s length is twice as long as the latter’s,
depending on the differently proposed methods. In addition, the
simulation results by a computer show that energy efficiency of
the proposed sequences in Construction 2 is higher than that of
the employed PTSs. Incidentally, one of our constructions can
also produce a family of almost perfect 8-QAM+ sequences with
only two nonzero nontrivial autocorrelation values.
Index Terms—Boztas¸ and Parampalli’s open problems, energy
efficiency, perfect sequences, QAM+ constellation.
I. INTRODUCTION
R
ECENTLY, more and more attention was placed in the
QAM signals due to applications of the 64-QAM and
16-QAM to 3GPP’s communication standard [1]. Apart from
the aforementioned advantages, QAM transmission possesses
high bandwidth efficiency and transmission data rate [2]. Up
to now, lots of QAM sequences have been found, including
the QAM sequences with low correlation [2], the ZCZ QAM
sequences [3] [4], and the QAM Golay complementary se-
quences [5]. To the best of the authors’ knowledge, however,
the QAM perfect sequences, which are mainly applied to
synchronization of communication systems, are absent.
In order to satisfy the requirement of communication sys-
tems, Boztas¸ and Parampalli firstly introduced a new concept:
QAM+ constellation, and presented a class of almost perfect
QAM+ sequences in 2010 [6], where the QAM+ means
enlarged QAM alphabets, that is, the union of the QAM
alphabets and the number “0”. In the conclusion of their paper,
three open problems were given. One of them did be the
existence of perfect QAM+ sequences.
Manuscript received April 30, 2012. The associate editor coordinating the
review of this letter and approving it for publication was B. S. Rajan.
F. X. Zeng is with the College of Communication Engineering, Chongqing
University, Chongqing 400044, China, and also with the Chongqing Key Lab-
oratory of Emergency Communication, Chongqing Communication Institute,
Chongqing 400035, China (e-mail: fzengx@yahoo.com.cn).
X. P. Zeng is with the College of Communication Engineering, Chongqing
University, Chongqing 400044, China (e-mail: zxp@cqu.edu.cn).
X. Y. Zeng is with the Faculty of Mathematics and Computer Science,
Hubei University, Wuhan 430062, China (e-mail: xzeng@hubu.edu.cn).
Z. Y. Zhang and G. X. Xuan are with the Chongqing Key Laboratory of
Emergency Communication, Chongqing Communication Institute, Chongqing
400035, China (e-mail: {cqzhangzy, guixinxuan}@yahoo.com.cn).
This work was supported by the National Natural Science Foundation of
China (NSFC) under Grants 60872164, 61002034, and 61171089, the Ministry
of Industry and Information Technology of China (No.Equipment[2010]307),
the Natural Science Project of CQ (CSTC, 2009BA2063, 2009DA0001,
2009AB2147, and 2010BB2203), and the Open Research Foundation of
Chongqing Key Laboratory of Signal and Information Processing under Grant
CQSIP-2010-01.
Digital Object Identifier 10.1109/WCL.2012.053112.120319
In this letter, the existence of perfect 8-QAM+ sequences
is positively answered, and a family of such sequences is pro-
posed. More clearly, the resultant perfect 8-QAM+ sequences
are derived from the known perfect ternary sequences (PTSs),
and the proposed sequences and the PTSs employed have
the same period, or the former’s length is twice as long as
the latter’s, depending on the proposed construction methods.
In addition, the simulation results by a computer show that
energy efficiency of the proposed sequences in Construction 2
is higher than that of the employed PTSs. Incidentally, a family
of almost perfect 8-QAM+ sequences with only two nonzero
nontrivial autocorrelation values can be given by one of the
proposed methods. However, the existence of other perfect
QAM+ sequences is still unclear.
II. P
RELIMINARIES
For the convenience of presentation, some relevant defini-
tions in this letter will be given below.
A. QAM+ Constellation
In 2010, Boztas¸ and Parampalli [6] defined the following
expanded constellation
{0, 1, 1+j, j, −1+j, −1, −1 − j, −j, 1 − j}, (1)
and presented a family of almost perfect QAM+ sequences
over it. In accordance with Boztas¸ and Parampalli’s convention
in [6], the set in Eq. (1) is referred to as the 8-QAM+
constellation and denoted by Ω
+
8
.
This letter is to follow their discussions, in other words, we
will design a class of perfect 8-QAM+ sequences over this
constellation.
B. A Perfect Sequence
Let u
= {u(t)} =(u(0),u(1),u(2), ··· ,u(N − 1)) and
v
= {v(t)} =(v(0),v(1),v(2), ··· ,v(N − 1)) be two com-
plex sequences with the same length N . We define periodic
correlation function R
u,v
(τ) between the sequences u and v
as follows.
R
u,v
(τ)=
N−1
t=0
u(t)v(t + τ ), (2)
where
x denotes the complex conjugate of x, and the addition
t + τ is counted modulo N.Ifu
= v, we refer to R
u
(τ) as an
autocorrelation function, or else a crosscorrelation function.
A sequence u
is said to be perfect if its autocorrelation
satisfies
R
u
(τ)=
E
u
τ ≡ 0(modN)
0 τ ≡ 0(modN),
(3)
2162-2337/12$31.00
c
2012 IEEE
下载后可阅读完整内容，剩余3页未读，立即下载
weixin_38736529
- 粉丝: 2
- 资源: 875
我的内容管理
展开
最新资源
- C++标准程序库：权威指南
- Java解惑：奇数判断误区与改进方法
- C++编程必读：20种设计模式详解与实战
- LM3S8962微控制器数据手册
- 51单片机C语言实战教程：从入门到精通
- Spring3.0权威指南：JavaEE6实战
- Win32多线程程序设计详解
- Lucene2.9.1开发全攻略：从环境配置到索引创建
- 内存虚拟硬盘技术：提升电脑速度的秘密武器
- Java操作数据库：保存与显示图片到数据库及页面
- ISO14001：2004环境管理体系要求详解
- ShopExV4.8二次开发详解
- 企业形象与产品推广一站式网站建设技术方案揭秘
- Shopex二次开发：触发器与控制器重定向技术详解
- FPGA开发实战指南：创新设计与进阶技巧
- ShopExV4.8二次开发入门：解决升级问题与功能扩展
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
