不相交差集并集认知无线电网络中对称信道跳变策略

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"基于不相交差集并集的认知无线电网络中会合的对称信道跳变" 在认知无线电网络（Cognitive Radio Networks, CRN）中，相邻的认知用户（Cognitive Users, CUs）需要交换控制信息，但不能依赖固定的控制通道。为此，他们可以在多频道环境下调整各自的收发器，按照一定的跳频模式在共享的一组频谱通道中切换，以盲方式在某一共同频道上通信，这种现象被称为“盲会合”（Blind Rendezvous）。这篇研究论文关注的是如何在认知无线电网络中实现高效的盲会合策略。该论文基于组合数学中的不相交差集并集（Union of Disjoint Difference Sets, UDDS）概念，提出了两种新颖的对称信道跳变（Channel Hopping, CH）方案：UDDS-SCH（适用于时钟同步的CU）和UDDS-ACH（适用于时钟未同步的CU），以实现无需预先协调的盲会合。UDDS-SCH和UDDS-ACH这两个方案分别考虑了时钟同步和非同步的情况，旨在提高会合的成功率和效率。通过对这些方案的性能指标进行分析，作者发现，无论是UDDS-SCH还是UDDS-ACH，其性能都可以通过采用一种特殊的UDDS来优化，这种特殊UDDS可以被分割成尽可能多的互不相交的最小差集（Minimal Difference Sets, MDSs）。这一发现促使研究人员进一步发展全面的搜索算法，以寻找最佳的UDDS配置，从而最大化会合的成功概率和系统效率。不相交差集并集（UDDS）是组合数学中的一个重要工具，它在频谱资源管理和调度中起着关键作用。UDDS允许用户在一组频道中以预定义的周期性模式独立跳变，同时确保用户之间有较高的概率会在同一频道上相遇，从而实现数据交换。在时钟同步的情况下，UDDS-SCH可能提供更稳定的表现，而在时钟不同步的环境中，UDDS-ACH则能适应不确定性，保持一定的会合成功率。为了优化这些方案，论文可能还探讨了如何构造和选择合适的UDDS，以及如何设计相应的信道跳变算法。这可能包括了对各种潜在的UDDS结构的评估，以及针对不同网络环境和用户条件的性能比较。此外，论文可能还讨论了如何处理潜在的冲突，例如多个用户同时选择相同的频道，以及如何通过调整参数或算法来降低这种冲突的可能性。这篇论文在认知无线电网络的盲会合问题上提出了新的解决方案，利用不相交差集并集理论，为时钟同步和非同步的CU提供了有效的信道跳变策略，从而提高了频谱利用效率和网络的可靠性。通过深入研究UDDS的结构和性能优化，这项工作为未来认知无线电网络的设计和优化提供了重要的理论基础和实践指导。

10236 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 66, NO. 11, NOVEMBER 2017

TABLE I I

UMMARY OF SYMBOLS AND ABBREVIATIONS

Symbols & abbreviations Explanation

DoR Degree of rendezvous

MTTR Maximum time-to-rendezvous

MCTTR Maximum conditional time-to-rendezvous

ATTR Average time-to-rendezvous

CL Channel loading

DS Relaxed cyclic difference set

MDS Minimal DS

NRDS Non-relaxed DS

UDDS Union of disjoint DSs

UDMDS Union of disjoint MDSs

CQS Cyclic quorum system

The set of integers modulo n

Jump-Stay (EJS) [22], and Channel Rendezvous Sequence

(CRSEQ) [20] avoid utilizing the ID sequences of CUs for the

construction of CH systems. However, they work inefﬁciently

when neighboring CUs have heterogeneous channel resources

and hence are only suitable for a limited environment. To im-

prove the rendezvous efﬁciency for CUs with heterogeneous

spectrum resources, the Interlocking CH (ICH) [31], Heteroge-

neous Hopping (HH) [32], Conversion Based Hopping (CBH)

[33], and Single-radio Sunﬂower-Sets-based (SSS) rendezvous

[25] allow each CU to only hop among its locally available chan-

nels for blind rendezvous. These CH systems, however, suffer

from various constraints of each CU, e.g., sensing ability, lack

of other CUs’ IDs, and the agility of channel availability.

In summary, as mentioned by [15], the gap between the theo-

retical lower bound on the MCTTR of symmetric asynchronous

CH systems and the best known MCTTR of the existing CH

systems with an arbitrary DoR, i.e., CRSEQ, is still very large.

In view of this, the present paper proposes the UDDS-based

Asynchronous CH (UDDS-ACH) to concurrently afford an ar-

bitrary DoR and approach the theoretical lower bound on the

MCTTR more closely than all known symmetric asynchronous

CH systems.

III. T

HE PROPOSED TWO SYMMETRIC CH SYSTEMS

To solve (*), this section ﬁrst formulates the concept of UDDS

and then describes the construction algorithm for UDDS-SCH

and -ACH based on this concept. For the simplicity of descrip-

tion, Table II summarizes all symbols and abbreviations to be

adopted in the remaining of this paper.

A. Union of Disjoint Difference Sets

Deﬁnition 1: A subset {a

,...,a

k−1

} of Z

is called a

relaxed cyclic (n, k)-difference set or an (n, k)-DS in short if,

for each d ∈{1, 2,...,n−1}, there exists at least one ordered

pair (a

) subject to a

− a

= dmodn.

Deﬁnition 2: Let A be a k-element set {a

,...,a

k−1

}⊆

. Then the rotation of A by the distance i ∈ [0,n− 1],

denoted by ROT (A, i), means the set {a

+ imodn,a

imodn,...,a

k−1

+ imodn}. In particular, A = ROT (A, 0).

Corollary 1: If a k-element set A ⊂ Z

is an (n, k)-DS, so

is the set ROT (A, i) ∀i ∈ [1,n−1].

For example, the set A = {0, 1, 3}⊂Z

is a (6, 3)-DS

because 1 ≡ 1 − 0 mod 6, 2 ≡ 3 − 1 mod 6, 3 ≡ 3 − 0 ≡ 0 −

3 mod 6, 4 ≡ 1 − 3 mod 6, and 5 ≡ 0 − 1 mod 6. So are the

5 rotations of A, i.e., ROT (A, 1)={1, 2, 4}, ROT (A, 2)=

{2, 3, 5}, ROT (A, 3)={3, 4, 0}, ROT (A, 4)={4, 5, 1}, and

ROT (A, 5)={5, 0,

2}.

As shown by [16], when the parameter k of an (n, k)-DS

is ﬁxed, the parameter n will be upper bounded by k

− k + 1.

This fact, together with k ≥ 2, implies t hat k>

√

n for all (n, k)-

DSs. In particular, when k approaches

√

n as much as possible,

an (n, k)-DS is further called an (n, k)-minimal difference set

or an (n, k)-MDS in short.

Meanwhile, the following concept is closely related to DS:

Deﬁnition 3: A subset {a

,...,a

β −1

} of Z

is called

an (n, β, λ)-non-relaxed difference set or an (n, β, λ)-NRDS in

short if, for each d ∈{1, 2,...,α− 1}, there exist exactly λ

ordered pairs (a

) subject to a

− a

= dmodn.

Thus an (n, β, 1)-NRDS is an (n, β)-MDS, but not vice

versa. While (n, k)-DSs do exist for all n ≥ 2, it is known

that (n, β, λ)-NRDSs only exist for some special n ≥ 2, e.g.,

n = β

− β + 1 with β − 1 being a prime and λ = 1 [34]. As an

(n, k)-MDS contains the minimum number of elements among

all DSs and NRDSs over Z

,wehave

Corollary 2: The size β of an (n, β, λ)-NRDS will always

be not smaller than the size k of an (n, k)-MDS.

Deﬁnition 4: A cyclic quorum system,oraCQS in short,

over Z

is a group of the n subsets U

, U

, ..., U

n−1

of Z

which satisﬁes the following two properties:

(1) U

∩ U

= ∅∀i, j ∈ [0,n−1];

(2) U

= ROT (U

,i) ∀i ∈ [0,n− 1].

Thus a CQS {U

,...,U

n−1

} should satisfy the rotation

closure property, i.e., ∀i, j, h ∈ [0,n− 1], ROT (U

,h) ∩ U

=

∅. The following theorem quotes from [16]:

Theorem 1: The group of nk-element sets A, ROT (A, 1),

..., ROT (A, n − 1) form a CQS over Z

if and only if A is an

(n, k)-DS.

Proof: First, if A = {a

,...,a

k−1

} is an (n, k)-DS, we

shall prove that ROT (A, x)



ROT (A, y) = ∅∀x, y ∈ [0,n−

1]. Denote by b

x,l

and b

y,m

, respectively, the l

element

of ROT (A, x) and the m

element of ROT (A, y), where

l, m ∈ [0,k− 1]. Thus b

x,l

= a

+ xmodnand b

y,m

= a

ymodn. By Deﬁnition 1, whenever A is an (n, k)-DS, there

should exist some l and m such that a

− a

= y − xmodn

or, equivalently, b

x,l

− b

y,m

= 0 mod n. This, together with

x,l

∈ Z

and b

y,m

∈ Z

, implies that b

x,l

= b

y,m

and hence

ROT (A, x)



ROT (A, y) = ∅.

Next, to show that the inverse is true, we shall prove it

by contradiction. Assume that ROT (A, x)



ROT (A, y) = ∅

and A is not an (n, k)-DS. Thus there should exist at least

one d ∈{1, 2,...,n−1} such that a

− a

= dmodn∀i, j.

Note that, because b

d,l

= a

+ dmodn and b

0,m

= a

,we

have b

d,l

− b

0,m

= a

− a

+ dmodn. However, the fact

ROT (A, 0)



ROT (A, d) = ∅implies the equality b

d,l

= b

0,m

for some l and m. This leads to a

− a

= dmodnfor some l

and m, which contradicts the initial assumption. 

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