具有低命中区的跳频序列的非周期性汉明相关的下界

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本文研究的是准同步频率跳变多址接入系统中低命中区的频率跳变序列设计问题。频率跳变序列（FHS）是一种在通信系统中常用的技术，用于减少多址干扰。序列设计的质量常通过周期性汉明相关性来衡量，但非周期性汉明相关性在实际应用中也非常重要，尽管在文献中对其关注较少。文章的核心内容是建立了低命中区频率跳变序列的非周期性汉明相关性下界。这些下界与频率槽集的大小、序列长度、低命中区、序列族的大小、最大非周期性汉明自相关值和最大非周期性汉明互相关值有关。新提出的下界比以往的界限更紧，特别是新的界限包括了最大非周期性汉明自相关和互相关值的平方项，而之前的界限并未包含这些项。为了理解这些概念，我们首先需要弄清楚一些基础概念： 1. **频率跳变序列（Frequency Hopping Sequence, FHS）**：在通信系统中，FHS是一系列跳变的频率，用于在通信过程中不断改变载波频率，以此来对抗干扰和提高信号的隐蔽性。 2. **周期性汉明相关性**：该概念用于衡量两个序列在循环位移时汉明距离的变化，汉明距离是指两个等长序列之间不同位置上的数字不同的数量。在频率跳变序列设计中，周期性汉明相关性被用来评估序列的性能。 3. **非周期性汉明相关性**：与周期性相关性不同，非周期性相关性考虑的是序列在整个长度内的汉明距离，而不考虑循环位移的影响，这更贴近实际应用中的情况。 4. **低命中区（Low Hit Zone, LHZ）**：低命中区是频率跳变序列中的一种特性，指的是序列在特定区间内自相关或互相关值较低的特性。低命中区的设计用于最小化多址干扰，提高通信质量。 5. **汉明自相关与汉明互相关**：汉明自相关是指序列与其自身的汉明距离，汉明互相关是指两个不同序列之间的汉明距离。它们在评估序列的干扰性能时发挥重要作用。 6. **准同步频率跳变通信**：在该通信系统中，尽管收发双方的频率跳变序列不能完全同步，但有一定的同步程度，使得跳变序列在一定的时间内对齐。本文中提出的新下界，基于序列族大小、最大非周期性汉明自相关值和最大非周期性汉明互相关值等参数。研究者认为，这些新界限更准确地反映了实际应用中的序列性能。文章还提到了数学分类号：94A55 和 94B05，这是关于编码理论和序列设计在数学领域的分类。在研究背景中，作者指出，尽管周期性汉明相关性在评价频率跳变序列设计质量中起到了重要作用，但非周期性汉明相关性在实际应用中同样重要。因此，本研究的目标是建立更为严格的非周期性汉明相关性的下界，以期能更好地指导实际应用中的序列设计。本文的研究内容对于通信系统中频率跳变序列设计及其性能评估具有重要意义。通过对非周期性汉明相关性下界的研究，能够为实际通信系统的抗干扰能力和信号隐蔽性提供理论支持。这一研究将有助于推动通信技术的发展，特别是对于准同步频率跳变通信系统，能够在多用户同时通信时，有效减少干扰和提升信号质量。

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Des. Codes Cryptogr. (2015) 75:157–174

DOI 10.1007/s10623-013-9900-x

New lower bounds on the aperiodic Hamming

correlations of frequency hopping sequences

with low hit zone

Hongyu Han · Daiyuan Peng · Xing Liu

Received: 3 May 2013 / Revised: 22 November 2013 / Accepted: 25 November 2013 /

Published online: 5 December 2013

Abstract In quasi-synchronous frequency hopping code division multiple-access systems,

low hit zone frequency hopping sequences are commonly employed to minimize multiple-

access interferences. In order to evaluate the goodness of low hit zone frequency hopping

sequences design, the periodic Hamming correlation is used as an important measure. But

aperiodic Hamming correlation of frequency hopping sequences matters in real applications,

although it received little attraction in the literatures compared with the periodic Hamming

correlation. In this paper, the new lower bounds on aperiodic Hamming correlation of the

low hit zone frequency hopping sequences, with respect to the size of the frequency slot

set, the sequence length, low hit zone, the family size, the maximum aperiodic Hamming

autocorrelation and the maximum aperiodic Hamming crosscorrelation are established. The

new bounds are tighter than the previous bounds. Specially, the new bounds include the second

powers of the maximum aperiodic Hamming autocorrelation and the maximum aperiodic

Hamming crosscorrelation but the previous bounds do not include them.

Keywords Frequency hopping sequences · Aperiodic Hamming correlation ·

Low hit zone · Quasi-synchronous frequency hopping communication · Multiple-access

Mathematics Subject Classiﬁcation 94A55 · 94B05

Communicated by C. J. Colbourn.

H. Han (

) · D. Peng · X. Liu

Provincial Key Lab of Information Coding and Transmission, Institute of Mobile

Communications, Southwest Jiaotong University, Chengdu, China

e-mail: hyhan@my.swjtu.edu.cn

D. Peng

e-mail: dypeng@swjtu.edu.cn

X. Liu

e-mail: liuxing4@126.com

123

158 H. Han et al.

1 Introduction

For a positive integer q,letF beasetofq available frequency slots. A sequence x ={x

}

L−1

k=0

called a frequency hopping sequence (FHS)oflengthL over F if x

∈ F for all 0 ≤ k ≤ L−1.

For any two FHSs x ={x

}

L−1

k=0

, y ={y

}

L−1

k=0

of length L over F,theirperiodic Hamming

correlation (PHC) function H(x, y; τ) and aperiodic Hamming correlation (AHC) function

H(x, y; τ) at time delay τ are deﬁned as follows, respectively:

H(x, y; τ) =

L−1



k=0

h(x

, y

k+τ

)(τ= 0, 1,...,L − 1),

H(x, y; τ) =

L−1−τ



k=0

h(x

, y

k+τ

)(τ= 0, 1,...,L − 1)

where h(m, n) = 1ifm = n, and 0 otherwise, and all operations among the position indices

are performed modulo L.

Let S beasetofM FHSs of length L over frequency slot set F of size q.Themaximum

periodic Hamming autocorrelation H

(S) and the maximum periodic Hamming crosscor-

relation H

(S) of S are deﬁned as follows, respectively:

(S) = max{H(x, x; τ)|x ∈ S,τ = 1, 2,...,L − 1},

(S) = max{H(x, y; τ)|x, y ∈ S, x = y,τ = 0, 1,...,L − 1}.

Similarly, the maximum aperiodic Hamming autocorrelation

(S) and the maximum

aperiodic Hamming crosscorrelation

(S) of S are deﬁned as follows, respectively:

(S) = max{H(x, x; τ)|x ∈ S,τ = 1, 2,...,L − 1},

(S) = max{H(x, y; τ)|x, y ∈ S, x = y,τ = 0, 1,...,L − 1}.

For simplicity and convenience, let H

= H

(S), H

= H

(S), H

= H

(S),

= H

(S),andletH

max

= max{H

, H

} and H

max

= max{H

, H

} be the

maximum PHC and maximum AHC respectively.

For the FHS set S, let integers H

≥ 0, H

≥ 0. Then the autocorrelation low hit zone

(LHZ) L

PAHZ

, the crosscorrelation LHZ L

PCHZ

and the LHZ L

PHZ

of S with respect to

PHC are deﬁned as follows, respectively:

PAHZ

= max{T |H(x, x; τ) ≤ H

for any x ∈ S and for any positive τ ≤ T },

PCHZ

= max{T |H(x, y; τ)≤ H

for any x, y ∈ S, x = y, and for any nonnegative τ ≤ T },

PHZ

= min{L

PAHZ

, L

PCHZ

When H

= H

= 0, we use the term no hit zone (NHZ) instead of LHZ. In this case, the

parameter L

PHZ

is written as N

PHZ

.AnFHSsetS with L

PHZ

≥ 0orN

PHZ

≥ 0 is called

an LHZ FHS set or NHZ FHS set with respect to PHC, respectively.

For the FHS set S, let integers

≥ 0, H

≥ 0. Then the autocorrelation LHZ L

AAHZ

the crosscorrelation LHZ L

AC H Z

and the LHZ L

AHZ

of S with respect to AHC are deﬁned

as follows, respectively:

AAHZ

= max{T |H(x, x; τ) ≤ H

for any x ∈ S and for any positive τ ≤ T },

AC H Z

= max{T |H(x, y; τ)≤ H

for any x, y ∈ S, x = y, and for any nonnegative τ ≤ T },

AHZ

= min{L

AAHZ

, L

AC H Z

123

Aperiodic Hamming correlations 159

As in the periodic case, we use the term NHZ instead of LHZ when H

=0. In this

case, the parameter L

AHZ

is written as N

AHZ

.AnFHSsetS with L

AHZ

≥ 0orN

AHZ

≥ 0

is called an LHZ FHS set or NHZ FHS set with respect to AHC, respectively.

Frequency-hopping code-division multiple-access (FH-CDMA) is widely used in modern

communication systems such as Bluetooth, ultra-wideband (UWB), military and commer-

cial radar systems [8,9]. In such systems, each carrier frequency is selected from a set of

frequencies, which are spaced approximately the width of the data modulation bandwidth

apart. The frequencies used are chosen pseudo-randomly by FHS. As is often the case, in

a MA environment, mutual interference occurs when two or more transmitters transmit on

the same frequency at the same time. It is desirable to keep the mutual interference between

transmitters at a level as low as possible. The degree of the mutual interference is clearly

related to the Hamming correlation properties of the FHS [11]. Moreover, it is also preferred

to have more FHSs accommodating more distinct users. However, the parameters of the FHSs

are bounded by certain theoretical limits. In order to evaluate the performance of given FHSs,

it is important to understand the trade-offs between these parameters.

The AHC property of spreading sequences is more important for system performance than

the PHC property since AHC matters in real applications [7,10], while PHC is considered in

most papers [1–3,5–7,12,16,21]. In 2004, Peng and Fan [14] obtained the ﬁrst nontrivial

bound on the AHCs of an FHS set that takes into account its size.

Conventional FHS design try to minimize the Hamming correlations no matter how large

or small a shift might be. However, we may be able to achieve lower correlations for given

basic parameters such as frequency slot set size, sequence length, sequence set size if we

only have to care about a certain limited range of shifts, rather than every possible magnitude

[15]. In a quasi-synchronous communication system, it makes more sense to use optimal

sequences tailored to such limited range of shifts scenarios [8]. The FHS design with NHZ or

LHZ aims at making Hamming correlation values equal to zero or a very low value within a

correlation zone [17]. The signiﬁcance of an NHZ/LHZ sequence set is that the number of hits

between difference sequences will always be zero or very small as long as the relative delay

does not exceed certain limit (zone), thus eliminating or reducing the mutual interference.

There exist several constructions of optimal NHZ FHSs [4,18–20] with respect to the Ye

and Fan bound [18]. In 2006, Peng et al. [15] derived some bounds on the PHCs of an FHS

set with LHZ. In 2010, Niu et al. [13] derived some bounds on the AHCs of an FHS set with

LHZ. However, the bounds on the AHCs of an FHS set with LHZ derived by Niu et al. [13]

are not tight. In this paper, we will pay particular attention to the bounds on the AHCs of an

FHS set with LHZ.

The outline of this paper is as follows. In Sect. 2, some preliminaries are introduced. In Sect.

3, basic correlation properties of FHSs with LHZ are established. In Sect. 4, lower bounds

on the maximum aperiodic Hamming autocorrelation and crosscorrelation of FHSs with

LHZ are derived and compared with the previous results. Finally, we give some concluding

remarks in Sect. 5.

2 Preliminaries

Let S be a set of FHSs of family size M and length L over a given frequency slot set F of size

q.PengandFan[14] obtained the following bounds on the maximum aperiodic Hamming

autocorrelation and crosscorrelation of an FHS set:

2(L − 1)(M + q

+ (M − 1)(2L − 1)(M + q)H

≥ (3LM − Lq − M − q)L.

(1)

123

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