978-1-5090-3710-0/16/$31.00 ©2016 IEEE 1116
2016 9th International Congress on Image and Signal Processing, BioMedical Engineering and Informatics(CISP-BMEI 2016)
General QAM Golay Complementary Sequences
Based on Binary Signals as Their Inputs
Fanxin Zeng
1
, Zhenyu Zhang
2,1
, and Linjie Qian
1
1
Chongqing Key Lab. of Emergency Communication, Chongqing Communication Institute, Chongqing 400035, China
2
College of Communication Engineering, Chongqing University, Chongqing 400044, China
Email: fzengx@cqu.edu.cn, zhenyu.zhang@cqu.edu.cn, chargle@126.com
Abstract—A novel family of quadrature amplitude modulation
(QAM) Golay complementary sequences (GCSs), in this paper,
is presented. The resultant QAM GCSs employ binary signals
rather than quaternary signals as their inputs. New sequences are
fit for being applied to such QAM systems whose inputs merely
are binary signals. In addition, the proposed sequences have the
larger family size than the previously-known relevant sequences
with the same peak envelope power (PEP) upper bounds.
Index Terms—complementary sequences, QAM constellation,
standard Golay-Davis-Jedwab complementary sequences, gener-
alized Boolean function, peak envelope power.
I. INTRODUCTION
The quadrature amplitude modulation (QAM) Golay com-
plementary sequences (GCSs) have fairly good advantages
in reducing peak envelope power (PEP) in an orthogonal
frequency-division multiplexing (OFDM) system [1]-[3]. It has
been proved that in an OFDM system encoded by GCSs, the
PEP upper bounds of its transmission signals don’t excess 2
[3]. Hence, the constructions of QAM GCSs are explored [4]-
[22].
In the known constructions producing QAM GCSs, the
majority of those constructions is based on quaternary con-
stellation {0,1,2,3} [4]-[17]. However, there exist such QAM
systems that merely use binary signals {0,1} as their inputs,
such as in [18]. As a consequence, the constructions of
QAM GCSs based on binary constellation are necessary and
crucial. Unfortunately, only a few QAM GCSs from binary
constellation are known [19]-[22], which are far from the
requirements of the practical applications.
In this paper, we try to construct new QAM GCSs on basis
of binary constellation. The resultant sequences are educed
from B-type-2 expression of QAM constellation [23], and
include those sequences in [19] and [21] as special cases. More
importantly, the family size of the new GCSs is considerably
improved.
The rest of this paper is organized as follows. Some neces-
sary concepts, in Section II, are briefly recalled. We state the
B-type-2 description of QAM constellation in the following
section. New QAM GCSs from B-type-2 are presented in
Section IV, and their family size and PEP performance are
discussed in Section V and VI, respectively. In addition, a
numerical example appears in Section VII. Finally, we give
the conclusion remarks in Section VIII.
II. P
RELIMINARIES
Let 𝐴 =(𝐴
0
,𝐴
1
, ⋅⋅⋅ ,𝐴
𝑁−1
) and 𝐵 =
(𝐵
0
,𝐵
1
, ⋅⋅⋅ ,𝐵
𝑁−1
) be two complex sequences of length 𝑁.
The aperiodic correlation function of these two sequences is
defined by
𝐶
𝐴,𝐵
(𝜏)=
𝑁−1−𝜏
𝑖=0
𝐴
𝑖
𝐵
𝑖+𝜏
0 ≤ 𝜏 ≤ 𝑁 − 1
𝑁−1+𝜏
𝑖=0
𝐴
𝑖−𝜏
𝐵
𝑖
1 − 𝑁 ≤ 𝜏<0
0 ∣𝜏∣≥𝑁.
(1)
In particular, when 𝐴
= 𝐵, the aforementioned aperiodic
correlation function is referred to as aperiodic autocorrelation
function, denoted by 𝐶
𝐴,𝐴
(𝜏) for short.
For the sequences 𝐴
and 𝐵,ifwehave
𝐶
𝐴,𝐴
(𝜏)+𝐶
𝐵,𝐵
(𝜏)=0 (∀ 𝜏 ∕=0), (2)
these two sequences are said to be GCSs, and each of them
is referred to as a Golay sequence.
A large number of the methods producing GCSs have been
known up to now. For the more information, we recommend
the reader to refer to [24]. For saving the reader’s trouble,
however, we briefly recall standard 2
ℎ
phase shift keying
(PSK) Golay-Davis-Jedweb (GDJ) CSs [3], which are based
on standard generalized Boolean functions (GBFs) and are
considerably related to our coming discussions.
Let 𝑚 and 𝜋 denote a positive integer and a permutation of
the symbol set {1, 2, ⋅⋅⋅,𝑚}, respectively. Again set 𝑍
2
ℎ
=
{0, 1, 2, ⋅⋅⋅ , 2
ℎ
− 1} (integer ℎ ≥ 1) and 𝑐, 𝑐
𝑘
∈ 𝑍
2
ℎ
(1 ≤
𝑘 ≤ 𝑚). The standard GBFs are defined by
𝑓(𝑥
)=2
ℎ−1
𝑚−1
𝑘=1
𝑥
𝜋(𝑘)
𝑥
𝜋(𝑘+1)
+
𝑚
𝑘=1
𝑐
𝑘
𝑥
𝑘
+ 𝑐. (3)
In particular, the functions in Eq. (3) are the binary standard
GBFs when ℎ =1. Obviously, whenever we give an 𝑚-
dimensional vector 𝑥
=(𝑥
1
,𝑥
2
, ⋅⋅⋅ ,𝑥
𝑚
) ∈ 𝑍
𝑚
2
, the Boolean
function value 𝑓(𝑥
) ∈ 𝑍
2
ℎ
is produced. Hence, if we let the
𝑚-dimensional vector 𝑥
range over the range from (0, ⋅⋅⋅, 0)
to (1, ⋅⋅⋅ , 1), 2
𝑚
− 1 function values in 𝑍
2
ℎ
appear. Notice
that there exists a bijection between the 𝑚-dimensional vectors
from (0, ⋅⋅⋅ , 0) to (1, ⋅⋅⋅ , 1) and the binary representations of
the integers from 0 to 2
𝑚
−1. This means that we can obtain
a sequence 𝑓
=(𝑓
0
,𝑓
1
, ⋅⋅⋅ ,𝑓
2
𝑚
−1
), whose components