多天线中继系统鲁棒协作波束成型与人工噪声优化设计

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本文主要探讨了在多天线中继系统中增强物理层安全的问题，特别是在存在多个窃听者的情况下。研究针对的是放大和转发(Amplify-and-Forward, AF)架构，这是无线通信中的一种常见中继策略，通过接力传输来扩大信号覆盖范围和提高通信质量。文章的核心内容聚焦于鲁棒联合波束形成(Robust Joint Beamforming, RJB)和人工噪声(Artificial Noise, AN)的设计。在实际应用中，由于目的地和窃听者之间的信道状态信息(Channel State Information, CSI)可能存在不确定性，因此设计必须具备一定的鲁棒性，能够应对这些不确定性。研究目标是最大化最差情况下的秘密速率，同时考虑中继节点的总功率和每个天线的功率限制。这是一个非凸优化问题，具有一定的复杂性。作者采用了半定义放松(Semi-Definite Relaxation, SDR)技术，结合S-procedure方法和顺序凸逼近(Sequential Convex Approximation, SCA)算法，将原问题转化为一系列可以利用内点法有效求解的半定义程序(Semidefinite Programs, SDPs)。通过仿真结果，作者验证了所提设计的有效性和优越性，它能够在保证通信质量和安全性的前提下，有效地对抗窃听威胁，提高了系统的抗干扰能力和保密性能。这对于现代无线网络，特别是物联网(IoT)、5G和未来6G网络中的安全通信具有重要意义，因为多天线技术以及安全措施的发展对于保障数据隐私至关重要。总结来说，这篇研究论文提供了一种创新的解决方案，通过联合优化波束形成和人工噪声生成，使得多天线中继系统在面对多路窃听者时依然能够维持良好的安全性能，对提升无线通信的可靠性和安全性具有实际价值。

ROBUST JOINT BEAMFORMING AND ARTIFICIAL NOISE DESIGN FOR

AMPLIFY-AND-FORWARD MULTI-ANTENNA RELAY SYSTEMS

Lijian Zhang, Liang Jin, Wenyu Luo, Yanqun Tang and Dingjiu Yu

Zhengzhou Information Science and Technology Institute, Zhengzhou 450002, China

ABSTRACT

In this paper, we address physical layer security for amplify-and-

forward (AF) multi-antenna relay systems in the presence of multi-

ple eavesdroppers. A robust joint design of cooperative beamform-

ing (CB) and artiﬁcial noise (AN) is proposed with imperfect chan-

nel state information (CSI) of both the destination and the eavesdrop-

pers. We aim to maximize the worst-case secrecy rate subject to the

sum power and the per-antenna power constraints at the relay. Such

joint design problem is non-convex. By utilizing the semideﬁnite

relaxation (SDR) technique, S-procedure and the successive convex

approximation (SCA) algorithm, the original non-convex optimiza-

tion problem is recast into a series of semideﬁnite programs (SDPs)

which can be efﬁciently solved using interior-methods. Simulation

results are presented to verify the effectiveness of the proposed de-

sign.

Index Terms— Physical layer security, amplify-and-forward re-

laying, cooperative beamforming, artiﬁcial noise, secrecy rate

1. INTRODUCTION

As a complement to cryptographic methods on upper layers, physi-

cal layer security has been regarded as a promising technique to pro-

vide secure data communication. Recently, substantial research has

been dedicated toward improving the secrecy rate of various wire-

less communication systems [1], [2], among which the cooperative

relay system has attracted considerable attention [3]. The secrecy

rate maximization problem for the single-antenna relay systems has

been explored in [4–9]. And the secure transmission approaches for

multi-antenna relay systems have been investigated in [10–14].

In this paper, we focus on the robust joint design of coopera-

tive beamforming (CB) and artiﬁcial noise (AN) for amplify-and-

forward (AF) multi-antenna relay systems. The imperfect channel

state information (CSI) of both the destination and the eavesdrop-

pers is assumed to be available at the relay. The worst-case secrecy

rate is maximized by jointly optimizing the beamforming and AN

covariance matrices at the relay subject to the sum power and the

per-antenna power constraints. Even if the worst-case secrecy rate

maximization (WCSRM) problem is non-convex, we can ﬁnd the

suboptimal solution using the semideﬁnite relaxation (SDR) tech-

nique [15], S-procedure [16], and the successive convex approxima-

tion (SCA) algorithm [17], [18].

This paper addresses the WCSRM problem for AF multi-

antenna relay systems. It is worthwhile to mention some related

works. In [8] and [9], the robust design for secure single-antenna

relay systems was investigated, while we consider the case with a

Corresponding author: Liang Jin, Email: liangjin ndsc@163.com. This

work was supported in part by the National Natural Science Foundation of

China under Grants 61171108, 61401510 and 61379006.

multi-antenna relay. The works [10] and [11] considered a multi-

antenna relay scenario, but the method is only applicable for the

single eavesdropper case. Besides, it is assumed that only the CSI

of the eavesdropper(s) is imperfect in [8–11, 19]. Actually, the CSI

of the destination may be also imperfect, which is considered in

this paper. To the best of our knowledge, robust approaches to the

WCSRM problem for AF multi-antenna relay systems overheard by

multiple eavesdroppers are not available in the literature.

Notations: We use ⊗, , I

and 1

N×1

to denote the Hadamard

product, Kronecker product, identity matrix of dimension N and the

all-one column vector of dimension N, respectively. D(q) repre-

sents a diagonal matrix with q on the main diagonal. Re(·) extracts

the real part of a complex variable. q=vec(Q) denotes a column

vector by stacking all the elements of Q and vec

−1

(q) is the inverse

operation of vec(Q) for recovering Q.

2. SYSTEM MODEL AND PROBLEM FORMULATION

We consider a two-hop AF relay system, which consists of one

source (Alice), one relay, one legitimate destination (Bob) and mul-

tiple eavesdroppers (Eves). All the nodes are equipped with a single

antenna, except that the relay is equipped with N (N ≥ 2) anten-

nas. We assumed that the direct links between Alice and Bob as

well as Alice and Eves can be ignored due to the weak quality of the

channels. Alice intends to transmit conﬁdential information to Bob

aided by the trusted relay, while keeping it secret from the Eves.

The whole information transmission includes two phases. In the

ﬁrst phase, Alice transmits a symbol s with the average power P

the relay. The received signal vector at the relay is

= f s + n

, (1)

where f ∈ C

represents the channel vector from Alice to the re-

lay; n

∼CN(0, I) is the additive white Gaussian noise (AWGN)

vector received at the relay. In the second phase, the relay forwards

the signal multiplied by a beamforming matrix A ∈ C

N×N

. Con-

currently, the AN is transmitted for confusing the Eves. Hence, the

signal vector to be transmitted by the relay is x

= Af s+An

+v.

Here v ∼CN(0, Σ) is the artiﬁcial noise vector with Σ  0 being

the AN covariance matrix.

According to the expression of x

, the sum power of all anten-

nas and the power of the nth antenna can be computed, respectively,

= Tr(P

Af f

)+Tr(AA

)+Tr(Σ), (2a)

= e

Af f

+ AA

+ Σ)e

, ∀n ∈N, (2b)

where e

is a unit vector with the nth entry being one; and

N  {1, ··· ,N}. Let h ∈ C

and g

∈ C

, ∀k ∈K,

K  {1, ··· ,K}, denote the channel vector from the relay to

Bob and the channel vector from the relay to the kth Eve, respec-

tively. Then the signals received at Bob and the kth Eve is expressed,

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