Resource Allocation Algorithm for Intelligent Reflecting Surface-assisted Secure Integrated Sensing And Communications System
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摘要: 为了解决6G通感一体化系统(ISAC)中信息传输安全以及频谱紧张的问题,该文提出一种智能反射面(IRS)辅助ISAC系统安全资源分配算法。首先,在IRS-ISAC系统中,用户受到窃听者的恶意攻击时,通过干扰机发射的干扰信号和IRS智能地调节反射相移,重新配置传输环境,以提高系统的物理层安全。其次,考虑在基站和干扰机的最大发射功率约束,IRS反射相移约束以及雷达的信干噪比约束下,建立一个联合优化基站发射波束成形、干扰机预编码和IRS相移的系统保密率最大化优化问题。然后,利用交替优化和半正定松弛(SDR)算法等方法对原非凸优化问题进行转换,求出一个能够得到确定解的凸优化问题。最后提出一种基于交替迭代的安全资源分配算法。仿真结果验证了所提算法的安全性和有效性以及IRS-ISAC系统的优越性。Abstract:
Objective In the 6G era, the rapid increase in wireless devices coupled with a scarcity of spectrum resources necessitates the enhancement of system capacity, data rates, and latency. To meet these demands, Integrated Sensing And Communications (ISAC) technology has been proposed. Unlike traditional methods where communication and radar functionalities operate separately, ISAC merges wireless communication with radar sensing, utilizing a shared infrastructure and spectrum. This innovative approach maximizes the efficiency of compact wireless hardware and improves spectral efficiency. However, the integration of communication and radar signals into transmitted beams introduces vulnerabilities, as these signals can be intercepted by potential eavesdroppers, increasing the risk of data leakage. As a result, Physical Layer Security (PLS) becomes essential for ISAC systems. PLS capitalizes on the randomness and diversity inherent in wireless channels to create transmission schemes that mitigate eavesdropping risks and bolster system security. Nevertheless, PLS’s effectiveness is contingent on the quality of wireless channels, and the inherently fluctuating nature of these channels leads to inconsistent security performance, posing significant challenges for system adaptability and optimization. Moreover, Intelligent Reflecting Surfaces (IRS) emerge as a pivotal technology in 6G networks, offering the capability to control wireless propagation and the environment by adjusting reflection phase shifts. This advancement facilitates the establishment of reliable communication and sensing links, thereby enhancing the ISAC system’s sensing coverage, accuracy, wireless communication performance, and overall security. Consequently, IRS presents a vital solution for addressing PLS challenges in ISAC systems. In light of this, the paper proposes a design study focused on IRS-assisted ISAC systems incorporating cooperative jamming to effectively tackle security concerns. Methods This paper examines the impact of eavesdroppers on the security performance of ISAC systems and proposes the secure IRS-ISAC system model. The proposed model features a dual-functional base station equipped with antennas, an IRS with reflective elements, single-antenna legitimate users, and an eavesdropping device. To enhance system security, a jammer equipped with antennas is integrated into the system, transmitting interference signals to mitigate the effects of eavesdroppers. Given the constraints on maximum transmit power for both the base station and the jammer, as well as the IRS reflection phase shifts and radar Signal-to-Interference-plus-Noise Ratio (SINR), a joint optimization problem is formulated to maximize the system’s secrecy rate. This optimization involves adjusting base station transmission beamforming, jammer precoding, and IRS phase shifts. The problem, characterized by multiple coupled variables, exhibits non-convexity, complicating direct solutions. To address this non-convex challenge, Alternating Optimization (AO) methods are first employed to decompose the original problem into two sub-problems. Semi-Definite Relaxation (SDR) algorithms, along with auxiliary variable introductions, are then applied to transform the non-convex optimization issue into a convex form, enabling a definitive solution. Finally, a resource allocation algorithm based on alternating iterations is proposed to ensure secure operational efficiency. Results and Discussions The simulation results substantiate the security and efficacy of the proposed algorithm, as well as the superiority of the IRS-ISAC system. Specifically, the system secrecy rate in relation to the number of iterations is illustrated, demonstrating the convergence of the proposed algorithm across varying numbers of base station transmit antennas. The findings indicate that the algorithm reaches the maximum system secrecy rate and stabilizes at the fifth iteration, which shows its excellent convergence characteristics. Furthermore, an increase in the number of transmit antennas correlates with a notable enhancement in the system secrecy rate. This improvement can be attributed to the additional spatial degrees of freedom afforded by the base station’s antennas, which enable the projection of legitimate information into the null space of all eavesdropper channels—effectively reducing the information received by eavesdroppers and boosting the overall system secrecy rate. The system secrecy rate is presented as a function of the transmit power of the base station. The results indicate that an increase in the base station’s maximum transmit power corresponds with an increase in the system secrecy rate. This enhancement occurs because higher transmit power effectively mitigates path loss, thereby improving the quality of the signal. The IRS-assisted ISAC system significantly outperforms scenarios without IRS, thanks to the introduction of additional non-line-of-sight links. Additionally, the proposed scheme demonstrates superior performance compared to the random scheme in the joint design of transmit beamforming and reflection coefficients, validating the effectiveness of the algorithm. The system secrecy rate is illustrated in relation to the number of IRS reflection elements. The results reveal that the system secrecy rates for both the proposed and random methods increase as the number of IRS elements rises. This can be attributed to the incorporation of additional reflective elements, which facilitate enhanced passive beamforming gain and expand the spatial freedom available for optimizing the propagation environment, thereby strengthening anti-eavesdropping capabilities. In contrast, the system secrecy rate for the scheme without IRS remains constant. Notably, as the number of IRS elements increases, the gap in secrecy rates between the proposed scheme and the random scheme expands, highlighting the significant advantage of optimizing the IRS phase shift in improving system performance. The radar SINR is depicted concerning the transmit power of the base station. The results indicate that as the maximum transmit power of the base station increases, the SINR of the radar likewise improves. The proposed scheme outperforms the two benchmark schemes in this respect, attributable to the optimization of the IRS phase shift matrix, which not only enhances system security but also effectively conserves energy resources within the communication system. This enables a more efficient allocation of resources to improve sensing performance. By incorporating IRS into the ISAC system, performance in the sensing direction is markedly enhanced while simultaneously bolstering system security. Conclusions This paper addresses the potential for eavesdropping by proposing a secure resource allocation algorithm for ISAC systems with the support of IRS. A secrecy rate maximization problem is formulated, subject to constraints on the transmit power of the base station and jammer, the IRS reflection phase shifts, and the radar SINR. This formulation involves the joint design of transmit beamforming, jammer precoding, and IRS reflection beamforming. The interdependencies among these variables create significant challenges for direct solution methods. To overcome these complexities, the AO algorithm is employed to decompose the non-convex problem into two sub-problems. SDR techniques are then applied to transform these sub-problems into convex forms, enabling their resolution with convex optimization tools. Our simulation results indicate that the proposed method considerably outperforms two benchmark schemes, confirming the algorithm’s effectiveness. These findings highlight the considerable potential of IRS in bolstering the security performance of ISAC systems. -
1. 引言
在6G时代,需要更大的系统容量、更高的数据速率以及更低的延迟来应对频谱资源短缺问题[1–3]。面对这一挑战,最近提出了通感一体化技术(Integrated Sensing And Communication, ISAC)作为解决方案,其将无线通信与雷达感知相结合,共享相同的基础设施和频谱,目前这一技术已成为研究的热点[4–6]。文献[4]将雷达和通信集成到6G大规模物联网系统中,实现了目标检测和海量连接。文献[5]提出了一种基于多域非正交多址的ISAC方案,即所提出的ISAC系统在感知潜在目标的同时,能够在时域、频域和时延多普勒域中并行和非正交地传输数据流。文献[6]为最大化ISAC系统中的波束赋形增益,设计了一个全息波束成形优化算法来优化系统中的波束,实现了全息波束与ISAC的结合。
除了频谱资源短缺,安全通信在6G无线网络中也至关重要[7–9]。与其他通信系统一样,ISAC系统容易受到恶意窃听者的攻击。因此,物理层安全在ISAC系统中也备受关注和亟待解决。文献[10]考虑了ISAC系统中针对窃听者的多用户安全下行通信方案,在保密性能和功率预算约束的概率中断约束下,利用半正定松弛(Semi-Definite Relaxation, SDR)技术和伯恩施坦型不等式方法,最小化设计波束与期望波束之间的平方误差。文献[11]研究了双功能雷达通信(Dual-Functional Radar Communication, DFRC)系统的安全解决方案,其中雷达目标被视为潜在的窃听者,在满足安全和功率预算约束条件下,最大限度地提高雷达的信噪比(Signal-to-Noise Ratio, SNR)。此外,通过利用人工噪声(Artificial Noise, AN)或者外部干扰机发射干扰信号能够有效地对抗窃听,也是提高系统安全性的主要手段。文献[12]提出了ISAC系统中用于安全通信的鲁棒资源分配设计方案,其中基站利用AN进行联合传感和物理层安全配置,仿真结果表明可显著提高物理层的性能。文献[13]针对多输入多输出(Multi-Input Multi-Output, MIMO)-DFRC系统的物理层安全问题,在基站处引入了AN,最小化雷达目标处接收到的SNR,同时满足合法用户处信干噪比(Signal-to-Interference-plus Noise Ratio, SINR)的要求。文献[14]研究了利用强雷达信号的作为固有干扰来提高通信和雷达共存系统的物理层安全性。
智能反射面(Intelligent Reflecting Surface, IRS)作为6G潜在关键技术,在各种通信系统中得到广泛应用[15–19]。IRS是由一组无源元件组成,这些无源元件通过相移来控制入射无线电波的反射,可以实时动态调整无线环境,以支持无线通信和雷达传感,在ISAC系统中的应用具有巨大的潜在优势。IRS的引入可以提供额外的视距(Line-of-Sight, LoS)链路,使得ISAC系统具有更大的传感覆盖范围、更高的传感精度、更好的无线通信性能和更强的系统安全性。文献[20]提出了一种在毫米波波段工作的IRS辅助ISAC系统,通过联合设计雷达信号协方差矩阵、通信系统的波束成形矢量和IRS相移,利用2次变换技术和交替优化算法进行求解,最大化了ISAC系统的传输速率。文献[21]利用IRS来增强ISAC系统的物理层安全性,其中目标作为可疑的窃听者,在满足窃听者/目标的最大信息泄漏阈值和用户通信要求的最小SINR条件下,最大限度地提高目标的感知波束图增益。文献[22]研究了IRS辅助安全雷达通信系统。其中DFRC基站与多个合法用户进行通信,并感知恶意雷达目标,IRS可智能调控每个反射元件的相移来提高整个系统的安全性。但目前,IRS辅助具有协同干扰能力的ISAC系统少有研究。
针对以上频谱资源短缺和物理层安全问题,本文提出IRS辅助具有协同干扰的ISAC系统安全问题的设计研究。本文的主要贡献可概括如下:
(1) 本文考虑了窃听者对ISAC系统安全性的影响,提出了IRS-ISAC安全系统模型。在满足基站和干扰机发射最大发射功率、雷达的SINR和IRS反射相移的约束条件下,通过联合优化基站发射波束成形、干扰机预编码和IRS相移,构建了系统保密率最大化问题。该问题是含有多耦合变量的非凸问题,很难直接求解。
(2) 针对上述非凸问题,首先使用交替优化的方法将原有非凸问题转化为两个子问题;然后利用SDR算法以及引入辅助变量将两个子问题转化为凸形式;最后再利用现有的凸优化工具对子问题进行求解,得到该问题的确定解。
(3) 仿真结果验证了所提出算法的有效性和收敛性,表明了所提出的IRS辅助具有协同干扰的ISAC系统能够有效提高系统的保密率。与基准方案相比,IRS应用在ISAC系统中在提高系统物理层安全性方面具有很大优势。
本文的结构如下:第2节首先介绍IRS辅助的ISAC安全系统模型,并提出优化问题;第3节、第4节给出了基站和干扰机发射波束成形设计方案和IRS发射波束成形设计,并提出了一种基于SDR的交替优化算法进行求解;第5节给出了IRS辅助ISAC安全设计的仿真结果分析。
2. 系统模型
2.1 通信模型
图1给出了IRS辅助具有协同干扰的ISAC安全系统模型。在图1中,可以看到一个配备M根天线的基站将保密信息通过直接链路以及经过具有N个反射元件的IRS的反射链路发送给单天线的合法用户,同时感知目标信息。此外,K个单天线窃听者试图窃听用户的信息。为了提高系统的安全性,在系统中部署了配有M根天线的干扰机,干扰机发射干扰信号对抗系统中的窃听者。为了有效地设计基站发射和IRS反射波束成形和干扰信号,假设所有信道增益都采用理想的信道状态信息模式,并采用准静态平坦衰落信道模型[23]。
根据图1的安全系统模型,从基站发送的信号可表示为
x1=ws (1) 来自干扰机的干扰信号可表示为
x2=vc (2) 其中,s∼CN(0,1)和c∼CN(0,1)分别为独立信息和干扰信号。w∈CM×1和v∈CM×1分别表示基站的波束成形向量和干扰机的预编码矢量。设PB和PJ分别为基站和干扰机处的最大发射功率,可得|wHw|≤PB和|vHv|≤PJ。合法用户接收到的信号表示为
yU=(hHI,UΘHB,I+hHB,U)ws+(gHI,UΘGJ,I+gHJ,U)⋅vc+nU (3) 其中,HB,I∈CN×M, hB,U∈CM×1分别表示从基站到IRS和用户的信道增益,hI,U∈CN×1表示从基站经过IRS再到用户的反射信道增益。Θ=diag[ejθ1,ejθ2,⋯,ejθN]表示IRS的对角相移矩阵。其中,在其主对角线上,θn=(0,2π)表示IRS第n个反射单元对应的相移,n=1,2,⋯,N。GJ,I∈CN×M, gJ,U∈CM×1分别表示从干扰机到IRS和用户的信道增益。gI,U∈CN×1表示从干扰机经过IRS再到用户的反射信道增益。nU~CN(0,σ20)是加性高斯白噪声(Additive White Gaussian Noise, AWGN)。同样,第k个窃听者处的接收信号表示为
yE,k=(hHI,E,kΘHB,I+hHB,E,k)ws+(gHI,E,kΘGJ,I+gHJ,E,k)vc+nE,k (4) 其中,hB,E,k∈CM×1表示从基站到第k个窃听者的信道增益,hI,E,k∈CN×1表示从基站经过IRS再到第k个窃听者的反射信道增益。gJ,E,k∈CM×1表示从干扰机到第k个窃听者的信道增益。gI,E,k∈CN×1表示从干扰机经过IRS再到第k个窃听者的反射信道增益。nE,k~CN(0,σ20)是AWGN。令θH=[θ1,θ2,⋯,θn],则有hHI,mΘHB,I=θHHI,m, 其中HI,m=diag(hHI,m)HB,I,m∈{U,(E,k)}; gHI,mΘGJ,I=θHGI,m, 其中GI,m=diag(gHI,m)GJ,I。因此,用户和第k个窃听者的SINR的表达式分别为
SINRs=|(θHHI,U+hHB,U)w|2|(θHGI,U+gHJ,U)v|2 + σ20 (5) SINRe,k=|(θHHI,E,k+hHB,E,k)w|2|(θHGI,E,k+gHJ,E,k)v|2 + σ20 (6) 可实现的保密率可定义为
R=Rs−maxk∈KRe,k=log2(1+SINRs)−maxk∈Klog2(1+SINRe,k) (7) 2.2 感知模型
基站接收到的回波信号,包括目标的回波信号、干扰机的干扰信号和噪声,可表示为
yr=αa(β)aH(β)ws+HHJ,Bvc+nr (8) 其中,HJ,B∈CM×M表示从干扰机到基站的信道增益,α是信道衰落系数和目标的反射系数的乘积。nr~CN(0,σ2rI)是AWGN。a(β)=[1,ej2πdλsinβ,⋯,ej2π(M−1)dλsinβ]T为转向矢量,其中d是相邻天线元件之间的间距,λ是波长,β是目标所在的角度。则用于目标探测的雷达输出SINR的计算公式为
SINRr=wHAwvHQv+σ2r (9) 其中,A=(αa(β)aH(β))H(αa(β)aH(β)), Q=HJ,BHHJ,B。在雷达系统中,雷达SINR是衡量雷达传感性能的一个重要指标。因此,为了获得更好的目标探测性能,必须确保雷达SINR不小于预设阈值。
2.3 问题建模
如下节所述通过联合优化基站的发射波束成形向量w,干扰机的干扰预编码矢量v和IRS的相移Θ,研究了可实现保密率最大化问题。因此,可实现系统保密率最大化问题为
maxw,v,Θ R=Rs−maxk∈KRe,k (10a) s.t. SINRr≥Γt (10b) |wHw|≤PB (10c) |vHv|≤PJ (10d) 0≤θn≤2π,∀n∈N (10e) 其中,式(10b)为雷达SINR约束,其中Γt表示雷达SINR的预设阈值;式(10c)为基站的最大发射功率约束;式(10d)为干扰机的最大发射功率约束;式(10e)为IRS的反射相移约束;在问题式(10)中,由于目标函数和约束条件的非凸性,以及耦合优化变量的存在,问题式(10)很难求解。但是,当固定(w,v)和Θ其中的一个变量时,问题式(10)可以通过交替优化算法进行有效求解。如下节所述。
3. 交替优化算法
3.1 发射波束优化设计
首先,固定IRS的相移矩阵Θ,优化基站发射波束成形向量w和干扰机的预编码矢量v。定义Hm=(HHI,mθ+hB,m)(θHHI,m+hHB,m), Gm=(GHI,mθ+gJ,m)(θHGI,m+gHJ,m),优化问题可重述为问题式(11)
maxW,F{log2(1+a0Tr(HUW)a0Tr(GUF) + 1)−maxk∈Klog2(1+a0Tr(HE,kW)a0Tr(GE,kF) + 1)} (11a) s.t. Tr(HUW)Tr(QF)+σ2r≥Γt (11b) (W,F)∈F (11c) 其中,a0=1/σ20, W=wwH, F=vvH以及F={(W,F)∣Tr(W)≤PB,Tr(F)≤PJ,W≽。但是,由于秩–1约束,问题式(11)仍然是非凸的。在忽略秩–1约束的情况下,可以使用SDR方法。然而,由于耦合变量 {\boldsymbol{W}} 和 {\boldsymbol{F}} 的存在,问题式(11)仍就不可直接求解。为解决上述问题,引用下面的定理1[24]来解决以上所求的优化问题。
定理1 对于任意 x >0 ,存在一个函数 \varphi(t)=tx+\ln t+1 。那么,有
- \ln x = \mathop {\max }\limits_{t \gt 0} \varphi (t) (12) 问题式(12)的最优解可在 t=1/x 时给出。之后,让 x = {a_0}{\text{Tr}}({{\boldsymbol{G}}_{\text{U}}}{\boldsymbol{F}}){\text{ + }}1 和 t = {t_{\text{s}}} ,可得
\begin{split} {R_{\text{s}}}\ln 2 = \,&\ln \left( {{a_0}{\text{Tr(}}{{\boldsymbol{H}}_{\text{U}}}{\boldsymbol{W}}{\text{)}} + {a_0}{\text{Tr(}}{{\boldsymbol{G}}_{\text{U}}}{\boldsymbol{F}}{\text{) + }}1} \right) \\ & - \ln \left( {{a_0}{\text{Tr(}}{{\boldsymbol{G}}_{\text{U}}}{\boldsymbol{F}}{\text{) + }}1} \right) \\ =\,& \mathop {\max }\limits_{{t_{\text{s}}} \gt 0} {\varphi _{\text{s}}}({\boldsymbol{W}},{\boldsymbol{F}},{t_{\text{s}}}) \end{split} (13) 其中
\begin{split} {\varphi _{\text{s}}}({\boldsymbol{W}},{\boldsymbol{F}},{t_{\text{s}}}) =\,& \ln \left( {{a_0}{\text{Tr(}}{{\boldsymbol{H}}_{\text{U}}}{\boldsymbol{W}}{\text{)}} + {a_0}{\text{Tr(}}{{\boldsymbol{G}}_{\text{U}}}{\boldsymbol{F}}{\text{) + }}1} \right) \\ & - {t_{\text{s}}}\left( {{a_0}{\text{Tr(}}{{\boldsymbol{G}}_{\text{U}}}{\boldsymbol{F}}{\text{) + }}1} \right) + \ln {t_{\text{s}}}{\text{ + }}1 \end{split} (14) 同样,定义 x = {a_0}{\text{Tr}}({{\boldsymbol{H}}_{{\text{E,}}k}}{\boldsymbol{W}}) + {a_0}{\text{Tr}}({{\boldsymbol{G}}_{{\text{E,}}k}}{\boldsymbol{F}}) {{ + }}1 和 t = {t_{{\text{e,}}k}} ,可得式(15)
\begin{split} {R_{\text{e}}}\ln 2 = \,& \ln \left( {{a_0}{\text{Tr(}}{{\boldsymbol{H}}_{{\text{E,}}k}}{\boldsymbol{W}}{\text{)}} + {a_0}{\text{Tr(}}{{\boldsymbol{G}}_{{\text{E,}}k}}{\boldsymbol{F}}{\text{) + }}1} \right) \\ & - \ln \left( {{a_0}{\text{Tr(}}{{\boldsymbol{G}}_{{\text{E,}}k}}{\boldsymbol{F}}{\text{) + }}1} \right) \\ = & \mathop {\min }\limits_{{t_{{\text{e,}}k}} \gt 0} {\varphi _{\text{e}}}({\boldsymbol{W}},{\boldsymbol{F}},{t_{{\text{e,}}k}})\\[-1pt] \end{split} (15) 其中
\begin{split} {\varphi _{{\text{e,}}k}}({\boldsymbol{W}},{\boldsymbol{F}},{t_{{\text{e,}}k}}) =\,& {t_{{\text{e,}}k}}\left( {a_0}{\text{Tr(}}{{\boldsymbol{H}}_{{\mathrm{E}},k}}{\boldsymbol{W}}{\text{)}} \right.\\ & \left.+ {a_0}{\text{Tr(}}{{\boldsymbol{G}}_{{\text{E,}}k}}{\boldsymbol{F}}{\text{) + }}1 \right) \\ & - \ln \left( {{a_0}{\text{Tr(}}{{\boldsymbol{G}}_{{\text{E,}}k}}{\boldsymbol{F}}{\text{) + }}1} \right) \\ &- \ln {t_{{\text{e,}}k}} - 1 \end{split} (16) 因此,本文可以利用Sion极小极大定理[25],将问题式(11)重新整理为式(17)
\mathop {\max }\limits_{{\boldsymbol{W}},{\boldsymbol{F}},{t_{\text{s}}},{t_{{\text{e,}}k}}} \left\{ {{\varphi _{\text{s}}}({\boldsymbol{W}},{\boldsymbol{F}},{t_{\text{s}}}) - } \right.\mathop {\max }\limits_{k \in K} \left. {{\varphi _{{\text{e,}}k}}({\boldsymbol{W}},{\boldsymbol{F}},{t_{{\text{e,}}k}})} \right\} (17a) {\text{s}}.{\text{t}}.{\text{ }}{t_{\text{s}}},{t_{{\text{e,}}k}} \gt 0,式(11{\mathrm{b}})、式(11{\mathrm{c}}) (17b) 这里,在不损失最优性的情况下,在目标函数中省略了常数“ln 2”。很明显,当 \left( {{\boldsymbol{W}},{\boldsymbol{F}}} \right) 或 \left( {{t_{\text{s}}},{t_{{\text{e,}}k}}} \right) 其中之一固定时,交替优化算法可以有效地对问题式(17)进行求解。
在定理1的基础上,当给定 \left( {{\boldsymbol{W}},{\boldsymbol{F}}} \right) 时,可以得到 \left( {{t_{\text{s}}},{t_{{\text{e,}}k}}} \right) 的最优解为
t_{\text{s}}^* = {\left( {{a_0}{\text{Tr(}}{{\boldsymbol{G}}_{\text{U}}}{\boldsymbol{F}}{\text{) + }}1} \right)^{ - 1}} (18) t_{{\text{e,}}k}^* = {\left( {{a_0}{\text{Tr(}}{{\boldsymbol{H}}_{{\text{E,}}k}}{\boldsymbol{W}}{\text{)}} + {a_0}{\text{Tr(}}{{\boldsymbol{G}}_{{\text{E,}}k}}{\boldsymbol{F}}{\text{) + }}1} \right)^{ - 1}} (19) 在得到 \left( {t_{\text{s}}^*,t_{{\text{e,}}k}^{{*}}} \right) 以后,就可以求出 \left( {{\boldsymbol{W}},{\boldsymbol{F}}} \right) 的最优解。同时,引入 l 作为松弛变量。原问题可转化为优化问题式(20)
\mathop {\max }\limits_{{\boldsymbol{W}},{\boldsymbol{F}}} \left\{ {{\varphi _{\text{s}}}({\boldsymbol{W}},{\boldsymbol{F}},{t_{\text{s}}}) - } \right.l\} (20a) \begin{split} & {\text{s}}.{\text{t}}.{\text{ }}{\varphi _{{\text{e,}}k}}({\boldsymbol{W}},{\boldsymbol{F}},{t_{{\text{e,}}k}}) \le l,k = 1,2,\cdots,K\\ & \quad\;\; 式(11{\mathrm{b}})、式(11{\mathrm{c}}) \end{split} (20b) 显然,优化问题式(20)是凸问题,可以直接用CVX工具求解。这里,如果 {\boldsymbol{W}} 和 {\boldsymbol{F}} 的秩都为1,则对 {\boldsymbol{W}} = {{\boldsymbol{w}}}{{{\boldsymbol{w}}}^{{\mathrm{H}}} } 和 {\boldsymbol{F}} = {{\boldsymbol{v}}}{{{\boldsymbol{v}}}^{{\mathrm{H}}} } 进行特征值分解,得到最优的 {\boldsymbol{w}} 和 {{\boldsymbol{v}}} 。否则,当 {\boldsymbol{W}} 和 {\boldsymbol{F}} 的秩不为1时,可以应用标准的高斯随机化方法来得到问题式(20)的可行解。在本小节中,通过交替优化优化更新 \left( {{\boldsymbol{W}},{\boldsymbol{F}}} \right) 和 \left( {{t_{\text{s}}},{t_{{\text{e,}}k}}} \right) 来获得问题式(11)的可行解。
3.2 反射波束优化设计
当已知 {\boldsymbol{W}} 和 {\boldsymbol{F}} 时,令 {\overline {\boldsymbol{H}} _m} = \left[ {{{\boldsymbol{H}}_{{\text{I,}}m}};{{\boldsymbol{h}}}_{{\text{B,}}m}^{{\mathrm{H}}} } \right] , {\overline {\boldsymbol{G}} _m} = \left[ {{{\boldsymbol{G}}_{{\text{I,}}m}};{{\boldsymbol{g}}}_{{\text{J,}}m}^{{\mathrm{H}}} } \right] ,为简便计算,设 {{\boldsymbol{\tilde H}}_m} = {\overline {\boldsymbol{H}} _m}{{\boldsymbol{w}}} , {{\boldsymbol{\tilde G}}_m} = {\overline {\boldsymbol{G}} _m}{{\boldsymbol{v}}} ,那么,问题式(10)转化为式(21)
\begin{split} & \mathop {\max }\limits_{\boldsymbol{\varPhi }} \left\{ {{\log }_2}\left( {1 + \frac{{{a_0}{\text{Tr(}}{{\widehat {\boldsymbol{H}}}_{\text{U}}}{\boldsymbol{\varPhi }}{\text{)}}}}{{{a_0}{\text{Tr(}}{{\widehat {\boldsymbol{G}}}_{\text{U}}}{\boldsymbol{\varPhi }}{\text{) + }}1}}} \right) \right.\\ & \quad \left.- \mathop {\max }\limits_{k \in K} {{\log }_2}\left( {1 + \frac{{{a_0}{\text{Tr(}}{{\widehat {\boldsymbol{H}}}_{{\text{E,}}k}}{\boldsymbol{\varPhi }}{\text{)}}}}{{{a_0}{\text{Tr(}}{{\widehat {\boldsymbol{G}}}_{{\text{E,}}k}}{\boldsymbol{\varPhi }}{\text{) + }}1}}} \right) \right\} \end{split} (21a) {\text{s}}.{\text{t}}.{\text{ }}{\boldsymbol{\varPhi }} \succcurlyeq {{{{0}}}},{{\boldsymbol{\varPhi }}_{n,n}} = 1,n = 1,2,\cdots,N + 1 (21b) 其中, {\overline {{\boldsymbol{\theta}} } ^{{\mathrm{H}}} } = \left[ {\begin{array}{*{20}{c}} {{{{\boldsymbol{\theta}} }^{\rm H} }}&1 \end{array}} \right] , {\boldsymbol{\varPhi }} = \overline {{\boldsymbol{\theta}} }\, {\overline {{\boldsymbol{\theta}} } ^{{\mathrm{H}}} } 及 {\widehat {\boldsymbol{H}}_m} = {{\boldsymbol{\tilde H}}_m}{\boldsymbol{\tilde H}}_m^{{\mathrm{H}}} , {\widehat {\boldsymbol{G}}_m} = {{\boldsymbol{\tilde G}}_m}{\boldsymbol{\tilde G}}_m^{{\mathrm{H}}} 。与问题式(11)的解法类似,应用定理1和SDR算法,并引入 {l_{\boldsymbol{\varPhi }}} 作为辅助变量。问题式(21)可重新表述为
\mathop {\max }\limits_{{\boldsymbol{\varPhi }},{z_{\text{s}}},{z_{{\text{e,}}k}}} \left\{ {{\varphi _{\text{s}}}({\boldsymbol{\varPhi }},{z_{\text{s}}}) - } \right.\left. {{l_{\boldsymbol{\varPhi }}}} \right\} (22a) {\text{s}}.{\text{t}}.{\text{ }}{\varphi _{{\text{e,}}k}}({\boldsymbol{\varPhi }},{z_{{\text{e,}}k}}) \le {l_{\boldsymbol{\varPhi }}},k = 1,2,\cdots,K (22b) {z_{\text{s}}},{z_{{\text{e,}}k}} \gt 0 (22c) 其中
\begin{split} {\varphi _{\text{s}}}({\boldsymbol{\varPhi }},{z_{\text{s}}}) = \,& \ln \left( {{a_0}{\text{Tr(}}{{\widehat {\boldsymbol{H}}}_{\text{U}}} + {{\widehat {\boldsymbol{G}}}_{\text{U}}}{\text{)}}{\boldsymbol{\varPhi }}{\text{ + }}1} \right) \\ & - {z_s}\left( {{a_0}{\text{Tr(}}{{\widehat {\boldsymbol{G}}}_{\text{U}}}{\boldsymbol{\varPhi }}{\text{) + }}1} \right) + \ln {z_{\text{s}}}{\text{ + }}1 \end{split} (23) \begin{split} {\varphi _{{\text{e,}}k}}({\boldsymbol{\varPhi }},{z_{{\text{e,}}k}}) = \,& {z_{{\text{e,}}k}}\left( {{a_0}{\text{Tr(}}{{\widehat {\boldsymbol{G}}}_{{\text{E,}}k}} + {{\widehat {\boldsymbol{H}}}_{{\text{E,}}k}}{\text{)}}{\boldsymbol{\varPhi }}{\text{ + }}1} \right) \\ & - \ln \left( {{a_0}{\text{Tr(}}{{\widehat {\boldsymbol{G}}}_{{\text{E,}}k}}{\boldsymbol{\varPhi }}{\text{) + }}1} \right) - \ln {z_{{\text{e,}}k}} - 1 \end{split} (24) 同样的,也可以通过交替优化方法优化 {\boldsymbol{\varPhi }} 和 \left( {{z_{\text{s}}},{z_{{\text{e,}}k}}} \right) 来近似求解。首先,当 {\boldsymbol{\varPhi }} 固定时,可以得到最优的 \left( {{z_{\text{s}}},{z_{{\text{e,}}k}}} \right)
z_{\text{s}}^* = {\left( {{a_0}{\text{Tr(}}{{\widehat {\boldsymbol{G}}}_{\text{U}}}{\boldsymbol{\varPhi }}{\text{) + }}1} \right)^{ - 1}} (25) z_{{\text{e,}}k}^* = {\left( {{a_0}{\text{Tr(}}{{\widehat {\boldsymbol{G}}}_{{\text{E,}}k}} + {{\widehat {\boldsymbol{H}}}_{{\text{E,}}k}}{\text{)}}{\boldsymbol{\varPhi }}{\text{ + }}1} \right)^{ - 1}} (26) 对于给定的 \left( {z_{\text{s}}^*,z_{{\text{e,}}k}^*} \right) ,优化 {\boldsymbol{\varPhi }} 可通过优化问题式(27)得到
\begin{split} & \mathop {\max }\limits_{\boldsymbol{\varPhi }} \left\{ {{\varphi _{\text{s}}}({\boldsymbol{\varPhi }},{z_{\text{s}}}) - } \right.\left. {{l_{\boldsymbol{\varPhi }}}} \right\}, \\ & {\text{s}}.{\text{t}}.{\text{ }}式(21{\mathrm{b}})、式(22{\mathrm{b}}) \end{split} (27) 与问题式(20)类似,该问题是凸形式,可以用现有的凸优化工具来解决。再通过具有高斯随机化的特征值分解从 {\boldsymbol{\varPhi }} 中提取 {\theta } ,IRS反射系数为
{\theta _n} = {{\text{e}}^{{\text{j}}\angle \left( {\frac{{{{\overline \theta }_n}}}{{{{\overline \theta }_{N + 1}}}}} \right)}},n = 1,2, \cdots ,N (28) 其中, \angle \left( O \right) 表示 O 的相位,并满足约束{\text{0}} \le {{\theta}_n} \le 2\pi , \forall n。本文整体算法如算法1所示。
表 1 求解式(10)的交替优化算法输入: {P_{\text{B}}} , {P_{\text{J}}} , {\varGamma _{\text{t}}}, {{\boldsymbol{H}}_{{\text{I, }}m}} , {{\boldsymbol{G}}_{{\text{I, }}m}} , {{\boldsymbol{h}}}_{{\text{B, }}m}^{H} , {{\boldsymbol{g}}}_{{\text{J, }}m}^{H} , \varepsilon , L 输出: {{\boldsymbol{w}}} , {{\boldsymbol{v}}} , {{\boldsymbol{\theta}} } (1) 初始化 {{{\boldsymbol{w}}}^{(0)}} , {{{\boldsymbol{v}}}^{(0)}} 和 {{{\boldsymbol{\theta}} }^{(0)}} ; (2) 设置迭代次数 r = 1 , {{\boldsymbol{W}}^{(0)}} = {{\boldsymbol{w}}}{{{\boldsymbol{w}}}^{{\mathrm{H}}} } , {{\boldsymbol{F}}^{(0)}} = {{\boldsymbol{v}}}{{{\boldsymbol{v}}}^{{\mathrm{H}}} } ; (3) 重复 (4) 在给定 {{{\boldsymbol{\theta}} }^{(r - 1)}} , {{\boldsymbol{W}}^{(r - 1)}} 和 {{\boldsymbol{F}}^{(r - 1)}} 时,求解式(11);根据
式(18)和式(19)分别找到最优的 {t}_{\text{s}}^{(r)} 和 t_{{\text{e, }}k}^{(r)} ;(5) 在给定 {t}_{\text{s}}^{(r)} 和 t_{{\text{e, }}k}^{(r)} 时,通过求解式(20),找到最优的 {{\boldsymbol{W}}}^{(r)}
和 {{\boldsymbol{F}}^{(r)}} ,通过特征值分解得出 {{{\boldsymbol{w}}}^{(r)}} 和 {{{\boldsymbol{v}}}^{(r)}} ;(6) 在给定 {{{\boldsymbol{w}}}^{(r)}} 和 {{{\boldsymbol{v}}}^{(r)}} 时,方法同上,通过求解式(21),找到
最优的 {{{\boldsymbol{\theta }}}^{(r)}} ;(7) 更新r{\text{ = }}r{\text{ + 1}} (8) 直到问题式(10)的目标中的目标值下降 \le \varepsilon 或者r = L。 4. 复杂度分析
本节给出了所提算法的复杂度分析。在算法1中给出了求解问题式(10)的整体迭代算法,其中\varepsilon 表示一个最小阈值,L是最大迭代次数。本文中所提算法的复杂度主要来源于求解问题式(11)和式(21)。具体来说,设{L_1}为求解问题式(11)的迭代次数,问题式(11)的复杂度为\mathcal{O}\left( {{L_1}\max {{\{ M,K\} }^4}{M^{1/2}}} \right)。类似地,设{L_2}为求解问题式(21)的迭代次数,问题式(21)的复杂度为\mathcal{O}\left( {{L_2}\max {{\{ N,K\} }^4}{N^{1/2}}} \right)。因此,设{L_3}是算法1收敛所需的迭代次数,求解问题式(10)的总体复杂度为\mathcal{O}\left( {L_3}\left( {L_1}\max {{\{ M,K\} }^4}{M^{1/2}} + {L_2}\max {{\{ N,K\} }^4}{N^{1/2}} \right) \right)。
5. 仿真分析
本节通过仿真分析验证了所提算法的性能。假设基站,干扰机,IRS和用户的位置分别为(5 m, 0 m, 20 m),(5 m, 0 m, 15 m),(0 m, 100 m, 0 m)和(3 m, 100 m, 0 m)。窃听者的位置分别为(2 m, 105 m, 0 m)和(2 m, 102.5 m, 0 m)。目标所处的方位角\beta = {135^ \circ }。各个信道为{{{\boldsymbol{h}}}_{i,j}} = \sqrt {{G_0}d_{i,j}^{ - {c_{i,j}}}} {b_{i,j}},其中{G_0} = - 30{\text{ dB}}为参考距离1 m处的路径损耗。{d_{i,j}}, {c_{i,j}}和{b_{i,j}}分别表示i和j之间的距离、路径损耗指数和衰落,其中i \in \{ {\text{B,J,I}}\} {\text{ }},j \in \{ {\text{U,(E,}}k{\text{)}}\} 。路径损耗指数分别设置为: {c_{{\text{B,U}}}} = {c_{{\text{B,E,}}k}} = {c_{{\text{I,E,}}k}} = {c_{{\text{J,E,}}k}} = 5 , {c_{{\text{I,U}}}} = 2 , {c_{{\text{B,I}}}} = {c_{{\text{J,B}}}} = 3.5 。{b_{i,j}}服从瑞利分布。本文其他参数设置为: {P_{\text{B}}} = 30{\text{ dBm}} , {P_{\text{J}}} = 30{\text{ dBm}} , {\varGamma _{\text{t}}} = 10{\text{ dB}}, N = 20, \sigma _0^2 = - 10{\text{5 dB}}, \sigma _{\text{r}}^2 = - 80{\text{ dB}}, \alpha = {10^{ - 6}}。另外,本文所提出的方案记为“IRS-ISAC”。同时,考虑了另外两种方案作为基准方案,与所提出的方法进行比较:(1)使用IRS,但不优化其相移,记为“Random IRS-ISAC”;(2)不使用IRS,记为“NO IRS-ISAC”。
图2给出了系统保密率与迭代次数的关系。从图2可以看到,所提算法在不同的基站发射天线数目 M ( M = 4 , M = 8 , M = 16 )下的收敛性。算法在第5次迭代时就能够获得最大系统保密率并逐渐平稳,说明所提算法具有良好的收敛性。另外,发射天线越大,系统保密率越大。这是因为基站的天线能够提高足够的空间自由度来将合法信息发送到系统所有窃听者信道的零空间中,降低窃听者的信息接收,从而使系统保密率更大。
图3给出了系统保密率与基站发射功率的关系。从图3可以观察到,随着基站最大发射功率的增大,系统保密率也随之增大。这是因为基站发射功率的增加能够缓解大尺度衰落造成的路径衰落,到达用户的信号更加准确。同时,存在IRS辅助的具有协同干扰的ISAC系统与无IRS的情况相比,实现了显著的性能改进,因为IRS引入了额外的NLoS链路,可以增强下行链路通信。此外,所提方案提供的性能改进优于随机方案,这说明了所提算法在联合设计发射波束成形和反射系数方面的有效性。
图4给出了系统的保密率与IRS元素数量 N 的关系。从图4可以看到,随着IRS反射元素数目 N 增加,所提方法与随机方法的系统保密率都随之增加,很明显,更多的反射元件提供更大的无源波束形成增益,因为它们利用更多的空间自由度来操纵传播环境,从而使得系统保密率增大。无IRS方案的系统保密率则保持不变。此外,随着IRS反射元素数量 N 的增加,所提方案与随机方案之间的保密率差值逐渐增大,这意味着优化IRS相移在提高系统性能方面的优越性。
图5给出了雷达SINR与基站发射功率的关系。从图5可以看到,雷达的SINR随之增加。基站的发射功率可以增强感知的回波信号,使得对目标的探测更加精准。所提出方案优于其他两种基准方案。这是因为优化IRS相移矩阵提高了系统的安全性能,同时也节约了通信系统中的能量资源,将资源分配于感知系统中,增强信号感知。将IRS部署在ISAC系统中,在提高系统安全性的同时,感知方向的性能也有所提升。
6. 结论
针对存在窃听者的ISAC系统安全性低的问题,本文提出一种基于IRS辅助的ISAC安全资源分配算法。在满足基站和干扰机最大发射功率,IRS反射相移以及雷达的SINR的约束条件下,通过联合优化发射波束成形、干扰机预编码和IRS相移,提出一个多变量耦合的最大化系统保密率的优化问题。采用交替优化方法将原有的非凸问题转化为两个可解的子问题,然后使用SDR方法将子问题变化为凸形式,最后利用凸优化工具进行求解。数据结果表明所提方法显著优于其他两种基准方案,验证了本文所提算法的有效性,揭示了IRS在提高ISAC系统安全性能方面具有巨大潜力。
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1 求解式(10)的交替优化算法
输入: {P_{\text{B}}} , {P_{\text{J}}} , {\varGamma _{\text{t}}}, {{\boldsymbol{H}}_{{\text{I, }}m}} , {{\boldsymbol{G}}_{{\text{I, }}m}} , {{\boldsymbol{h}}}_{{\text{B, }}m}^{H} , {{\boldsymbol{g}}}_{{\text{J, }}m}^{H} , \varepsilon , L 输出: {{\boldsymbol{w}}} , {{\boldsymbol{v}}} , {{\boldsymbol{\theta}} } (1) 初始化 {{{\boldsymbol{w}}}^{(0)}} , {{{\boldsymbol{v}}}^{(0)}} 和 {{{\boldsymbol{\theta}} }^{(0)}} ; (2) 设置迭代次数 r = 1 , {{\boldsymbol{W}}^{(0)}} = {{\boldsymbol{w}}}{{{\boldsymbol{w}}}^{{\mathrm{H}}} } , {{\boldsymbol{F}}^{(0)}} = {{\boldsymbol{v}}}{{{\boldsymbol{v}}}^{{\mathrm{H}}} } ; (3) 重复 (4) 在给定 {{{\boldsymbol{\theta}} }^{(r - 1)}} , {{\boldsymbol{W}}^{(r - 1)}} 和 {{\boldsymbol{F}}^{(r - 1)}} 时,求解式(11);根据
式(18)和式(19)分别找到最优的 {t}_{\text{s}}^{(r)} 和 t_{{\text{e, }}k}^{(r)} ;(5) 在给定 {t}_{\text{s}}^{(r)} 和 t_{{\text{e, }}k}^{(r)} 时,通过求解式(20),找到最优的 {{\boldsymbol{W}}}^{(r)}
和 {{\boldsymbol{F}}^{(r)}} ,通过特征值分解得出 {{{\boldsymbol{w}}}^{(r)}} 和 {{{\boldsymbol{v}}}^{(r)}} ;(6) 在给定 {{{\boldsymbol{w}}}^{(r)}} 和 {{{\boldsymbol{v}}}^{(r)}} 时,方法同上,通过求解式(21),找到
最优的 {{{\boldsymbol{\theta }}}^{(r)}} ;(7) 更新r{\text{ = }}r{\text{ + 1}} (8) 直到问题式(10)的目标中的目标值下降 \le \varepsilon 或者r = L。 -
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