优化周期N=1 mod 4的平衡二进制序列对的自相关特性研究

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本文主要探讨了具有最佳互相关性质的平衡二进制序列对（Balanced Binary Sequence Pair, BBSP）在周期N满足N ≡ 1 (mod 4)的情况下，其自相关幅度的相关理论。首先，作者针对这类序列对，确定了它们交叉相关优化性的标准，并提出了一个关于具有最优交叉相关性的BBSP的自相关幅度下界。这种优化性对于序列设计在通信系统中的应用至关重要，特别是在异步码分多址（Asynchronous Code Division Multiple Access, CDMA）系统中，低自相关性和良好的互相关性能有助于提高系统的信号传输效率和抗干扰能力。具体来说，研究集中在以下几点： 1. **定义与基本概念**：文章定义了周期为N的二进制序列，以及特性集U，其中包含所有使得ui = -1的序列位置i。平衡二进制序列对指的是两个具有相同长度且元素交替出现+1和-1的序列。 2. **交叉相关分析**：当序列对的周期满足N ≡ 1 (mod 4)时，作者着重研究了这些序列对的交叉相关性，这是衡量序列间关联程度的重要指标。优化的交叉相关性意味着序列对之间的干扰最小，这对于通信系统的同步和误码率控制具有积极影响。 3. **自相关幅度下界**：文中建立了对于具有最优交叉相关性的BBSP，其最大出相位自相关幅度的下界。这个下界对于设计者来说提供了性能目标，帮助他们构建更高效的序列。 4. **构造新序列**：基于循环四次幂的思想，作者提出了新的BBSP构造方法，这些序列不仅可以达到最优的交叉相关性，而且它们的总平方相关值也符合Ganapathy-Pados-Karystinos界限。这表明它们在满足互相关性能的同时，还具备其他重要的序列特性，如良好的功率谱密度和良好的码元间干扰抑制。 5. **实际应用**：由于这些序列满足异步CDMA系统的性能需求，因此它们被视作一种有潜力的选择，可以在无线通信系统中作为基带信号的编码工具，尤其是在需要抵抗多径衰落和符号间干扰的环境中。这篇研究论文对具有周期N ≡ 1 (mod 4)的平衡二进制序列对的自相关特性进行了深入分析，提供了新的设计策略和理论支持，为异步CDMA系统的信号处理和性能优化提供了有价值的研究成果。

IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 4, APRIL 2015 585

The Autocorrelation Magnitude of Balanced Binary Sequence Pairs of Prime

Period

N ≡ 1(mod 4) With Optimal Cross-Correlation

Yang Yang, Xiaohu Tang, Member, IEEE, and Zhengchun Zhou

Abstract—In this paper, for balanced binary sequence pairs

(BBSPs for short) of period N ≡ 1(mod 4), the optimality of

their cross-correlation is determined. Next, a lower bound on the

maximal out-of-phase autocorrelation magnitude is established for

BBSPs of period N ≡ 1(mod 4) with optimal cross-correlation.

Furthermore, BBSPs with optimal cross-correlation whose max-

imal out-of-phase autocorrelation magnitude achieves the new

lower bound are constructed based on the cyclotomy of order 4.

These proposed sequences have a periodic total squared correla-

tion value meeting the Ganapathy-Pados-Karystinos bound, and

are therefore an option for use in asynchronous CDMA systems.

Index Terms—Sequence pairs, balanced, binary sequences,

cyclotomy.

I. INTRODUCTION

SEQUENCE u = {u

}

N−1

i=0

, where u

∈{±1}, is called a

binary sequence of period N.ThesetU = {0 ≤ i<N :

= −1} is called the characteristic set of u.If|U| = N/2 for

even N or |U| =(N ± 1)/2 for odd N, where |U| denotes the

cardinality of U , then such a sequence u is called a balanced

sequence.

Let u = {u

}

N−1

i=0

and v = {v

}

N−1

i=0

be two binary se-

quences of period N.Theperiodic cross-correlation function

of u and v is deﬁned as

u,v

(τ)=

N−1



i=0

i+τ

, 0 ≤ τ<N.

where the subscript i + τ is performed modulo N.Ifu =

v, then R

u,v

is called the periodic autocorrelation function,

denoted by R

for short.

Binary sequences with low correlation have very signiﬁcant

applications in communication systems, radar and cryptogra-

phy [5], [6]. For example, in an asynchronous direct-sequence

code division multiple access (CDMA), balanced or almost

balanced sequences with good autocorrelation property are

considered to be good sequences from the point of view of

randomness [11]. Sequences should have low autocorrelation

to eliminate the effect of multipath, and low cross-correlation

to extract the desired user’s signal from the rest of the users.

Manuscript received January 2, 2015; revised February 5, 2015; accepted

February 6, 2015. Date of publication February 10, 2015; date of current

version April 8, 2015. This work was supported in part by the NSFC under

Grants 61401376 and 61201243, and the Sichuan Provincial Youth Science and

Technology Fund under Grant 2015JQO004. The associate editor coordinating

the review of this paper and approving it for publication was M. F. Flanagan.

Y. Yang and Z. Zhou are with the School of Mathematics, Southwest Jiao-

tong University, Chengdu 610031, China (e-mail: yang_data@swjtu.edu.cn;

zczhou@swjtu.edu.cn).

X. Tang is with the Information Security and National Computing Grid

Laboratory, Southwest Jiaotong University, Chengdu 610031, China (e-mail:

xhutang@swjtu.edu.cn).

Digital Object Identiﬁer 10.1109/LCOMM.2015.2402278

TABLE I

OMPARISON OF KNOWN OPTIMAL BBSPs OF

ODD PRIME PERIOD N ≡ 1(mod 4)

In the past decades, a number of constructions of balanced

binary sequences with optimal autocorrelation have been pro-

posed (see [1], [2], [6], [8], [12] and the references therein).

In [3], Ding and Tang established lower bounds on the cross-

correlation magnitude of two balanced binary sequences with

optimal autocorrelation, which are better than the Sarwate

bound [9] on the cross-correlation of binary sequence pairs

with optimal autocorrelation. As a result, optimal balanced

binary sequence pairs (BBSPs for short) of period N were

obtained such that two balanced binary sequences have the op-

timal out-of-phase autocorrelation {1, −3} and also possess the

minimal maximum cross-correlation magnitude 

√

N, where

N ≡ 1(mod 4) is an odd prime.

In this paper, we will study the BBSPs which are optimal in

another way, namely that two balanced binary sequences with

optimal cross-correlation have the minimal maximum autocor-

relation magnitude as well. We ﬁrstly determine that the opti-

mal cross-correlation of BBSPs of period N ≡ 1(mod 4) takes

the value 1 or −3. Secondly, we discuss the lower bound on

the maximal out-of-phase autocorrelation magnitude of BBSPs

with optimal cross-correlation. Finally, based on the cyclotomic

numbers of order 4 [10], we present a class of pairs of bal-

anced binary sequences with cross-correlation {1, −3} and the

maximal out-of-phase autocorrelation magnitude 

√

N, which

achieves the new lower bound. For comparison, the parameters

of the two types of optimal BBSPs are listed in Table I. Those

sequence pairs have a periodic total squared correlation ( PTSC)

value meeting the Ganapathy-Pados-Karystinos bound [7], and

are therefore suited for asynchronous CDMA systems.

II. C

YCLOTOMIC NUMBERS

In this section, we shall give a short introduction to cyclo-

tomic numbers which will be used in the sequel to construct

new BBSPs.

In this paper, assume that N =4f +1, where f is a positive

integer. In particular, let N =4f +1 be an odd prime in this

section and in Section V.

Let α be a primitive element of the integer residue ring Z

{0, 1, ···,N−1} such that for any j ∈ Z

\{0}, there exists

an integer k satisfying j = α

. Denote by D

a multiplicative

IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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