Performance Analysis of Discrete-Phase-Shifter IRS-aided Amplify-and-Forward Relay Network
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摘要: 作为一种通过算法智能地控制信号反射来重构无线通信环境的新技术，智能反射面(IRS)近年来受到了广泛关注。与传统的中继系统相比，IRS辅助的中继系统可有效节约成本和能耗，并显著提高系统性能。然而，配备离散移相器的IRS会导致相位量化误差，从而降低接收机的接收性能。为了分析IRS相位量化误差导致的性能损失，该文基于弱大数定律和瑞利分布，在瑞利信道下，推导了关于移相器量化比特数的双IRS辅助放大转发中继网络的信噪比性能损失与可达速率的闭合表达式。此外，基于Taylor级数展开表达式，推导了其近似性能损失闭合表达式。仿真结果表明，系统的信噪比和可达速率性能损失随着量化比特数的增加而逐渐减小，而随着 IRS 相移元件数的增加而逐渐增大。当量化比特数为4时，系统的信噪比和可达速率性能损失分别小于0.06 dB 和0.03 bit/(s·Hz)。Abstract:
Objective Most existing research assumes that the Intelligent Reflecting Surface (IRS) is equipped with continuous phase shifters, which neglects the phase quantization error. However, in practice, IRS devices are typically equipped with discrete phase shifters due to hardware and cost constraints. Similar to the performance degradation caused by finite quantization bit shifters in directional modulation networks, discrete phase shifters in IRS systems introduce phase quantization errors, potentially affecting system performance. This paper analyzes the performance loss and approximate performance loss in a double IRS-aided amplify-and-forward relay network, focusing on Signal-to-Noise Ratio (SNR) and achievable rate under Rayleigh fading channels. The findings provide valuable guidance on selecting the appropriate number of quantization bit for IRS in practical applications. Methods Based on the weak law of large numbers, Euler’s formula, and Rayleigh distribution, closed-form expressions for the SNR performance loss and achievable rate of the discrete phase shifter IRS-aided amplify-and-forward relay network are derived. Additionally, corresponding approximate expressions for the performance loss are derived using the first-order Taylor series expansion. Results and Discussions The SNR performance loss at the destination is evaluated as a function of the number of IRS-1 elements (N), assuming that the number of IRS-2 elements (M) equals N ( Fig. 2 ). It is evident that, regardless of whether the scenario involves actual or approximate performance loss, the SNR performance loss decreases as the number of quantization bit (k) increases but increases as N grows. When k = 1, the gap between the actual performance loss and the approximate performance loss widens with increasing N. This gap becomes negligible when k is greater than or equal to 2. Notably, when k = 4, the SNR performance loss is less than 0.06 dB. Furthermore, both the SNR performance loss and approximate performance loss gradually decelerate as N increases towards a larger scale. The achievable rate at the destination is evaluated as a function of the N, where M equals N (Fig. 3 ). It can be observed that, in all scenarios—whether there is no performance loss, with performance loss, or approximate performance loss—the achievable rate increases gradually as N increases. This is because both IRS-1 and IRS-2 provide greater performance gains as N grows. When k = 1, the difference in achievable rate between the performance loss and approximate performance loss scenarios increases with N. As k increases, the achievable rate with performance loss and approximate performance loss converge towards the no-performance-loss scenario. For example, when N = 1 024, the performance loss in achievable rate is about 0.15 bit/(s·Hz) at k = 2 and only 0.03 bit/(s·Hz) at k = 3. The achievable rate is evaluated as a function of k (Fig. 4 ). The performance loss in achievable rate increases with N and M. When k = 3, the achievable rate with performance loss and approximate performance loss decrease by 0.04 bit/(s·Hz) compared to the no performance loss scenario. When k = 1, the differences in achievable rate between the no performance loss, performance loss, and approximate performance loss scenarios grow with increasing N and M. Remarkably, the achievable rate for the system with N = 1 024 and M = 128 outperforms that of N = 128 and M = 1 024. This suggests that increasing N provides a more significant improvement in rate performance than increasing M.Conclusions This paper investigates a double IRS-assisted amplify-and-forward relay network and analyzes the system performance loss caused by phase quantization errors in IRS equipped with discrete phase shifters under Rayleigh fading channels. Using the weak law of large numbers, Euler’s formula, and Rayleigh distribution, closed-form expressions for SNR performance loss and achievable rate are derived. Approximate performance loss expressions are also derived based on a first-order Taylor series expansion. Simulation results show that the performance losses in SNR and achievable rate decrease with increasing quantization bit, but increase with the number of IRS elements. When the number of quantization bit is 4, the performance losses in SNR and achievable rate are less than 0.06 dB and 0.03 bit/(s·Hz), respectively, suggesting that the system performance loss is negligible when using 4-bit phase quantization shifters. -
1. 引言
随着6G时代的到来，无处不在的无线网络即将成为现实[1,2]。中继作为一种可有效提高无线通信质量的方式，已被学术界和工业界广泛研究及使用。当进行远程端到端通信时，使用中继作为传输节点辅助通信可提高系统的传输性能[3]。中继选择技术可以满足用户对分集增益的需求，并降低用户接收信号的衰减程度[4]。例如，文献[5]指出即使协作中继仅协作监听而不发送信息，也能给系统提供分集增益。为了减少无线系统中多径传输引起的衰落，基于时空编码协作分集协议，文献[6]证明了这些协议在解码中继数量和合作终端数量上均实现了完全的空间分集。文献[7]介绍了两种具有覆盖和功率约束的中继部署算法来提高吞吐量和覆盖范围。为了提高中继通信系统的和速率，文献[8]提出了两种有效的波束成形设计方案。与上述仅研究单个中继的网络不同，文献[9]研究了一种具有两个中继的无线网络，其可以恢复不同半双工中继方案的大部分损失。文献[10]研究了一种多源多中继网络，为了实现协作分集和多用户分集，提出了一种基于循环的公平调整方案。
然而，在无线通信系统中部署大量的有源设备会导致大量的能源消耗。新兴的智能反射面(Intelligent Reflecting Surface, IRS)作为一种低成本、节能及有效的解决方案，其由大量低成本的无源反射元件组成，每个元件均可独立地调整入射信号的相位和/幅度[11,12]。作为未来6G无线网络的潜在关键技术，IRS通过重构友好及可控的无线环境，为构建智能无线传输环境提供了一种全新的解决思路。IRS可以改善无线传输环境，有效解决毫米波通信技术大规模应用带来的信号衰落问题[13]。文献[14]研究了一种IRS辅助的毫米波大规模多输入多输出系统，并提出了一种快速多波束训练方案，该方案的训练开销明显小于基准方案。IRS 已被证明可显著提高系统的速率性能。与传统的有源波束成形/中继相比，当IRS的元件数趋于大规模时，IRS辅助的多输入多输出系统可实现与传统大规模多输入多输出系统相同的速率性能，但显著减少了有源天线的数量。文献[15]通过在小区边缘处安装IRS来提高小区边缘用户的通信性能。当存在窃听者时，可以利用IRS来提高系统性能。文献[16]研究了一种多用户多输入多输出网络，为了最大化加权安全和速率，提出了一种交替优化算法来设计发射和反射波束成形。
中继和IRS的相干结合已被证明是提高频谱效率、能量效率及速率性能的一种有效方式[17]。为了探究传统放大转发中继系统与IRS结合的优势，文献[18]研究了IRS辅助中继系统的性能，为了从概率上表述最优的信噪比，推导了一个紧的近似累积分布函数。当存在不信任的中继时，中继可能会在帮助转发隐私信息的同时对其进行窃听。为了在存在不信任的中继时最大化IRS辅助无线网络的安全速率，文献[19]提出了一种交替迭代方案来联合设计有源和无源波束成形。为了进一步提高系统的频谱效率，使用双向中继是一个自然的选择。为了探究功率分配系数对中继系统性能的影响，文献[20]提出了一种基于双向中继网络的最优功率分配方案，以达到降低干扰和提高无线中继网络信噪比的目的。文献[21]提出了一种基于黎曼流形梯度下降的方法来优化双IRS的相移矩阵，以最大化IRS辅助双向中继系统的可达和速率。
然而，上述工作均基于连续移相器的IRS展开研究，即无相位量化误差。在实际应用中，由于存在硬件和成本的限制，IRS通常配备离散移相器。与方向调制网络中使用有限量化位数移相器会造成系统性能损失类似[22]，配备离散移相器的IRS会产生相位量化误差，从而降低系统的性能[23]。为了分析双IRS辅助放大转发中继网络中由IRS相位量化误差引起的性能损失，并为实际中选择合适量化比特数的IRS提供参考，本文考虑在瑞利信道下，分析双IRS辅助放大转发中继网络的信噪比和可达速率的性能损失和近似性能损失。本文的主要贡献总结如下：
(1) 为了分析配备离散移相器的IRS对系统性能的影响及提高传统中继网络的速率，构建一种双IRS辅助的放大转发中继系统，其中IRS-1和IRS-2的相移元件数可能不同。假设所有的信道均为瑞利信道。基于弱大数定律、欧拉公式及瑞利分布，推导了关于移相器量化比特数的信噪比性能损失闭合表达式。此外，利用一阶Taylor展开表达式，推导了系统具有相位量化误差的近似性能损失闭合表达式。
(2) 此外，推导了在无性能损失、有性能损失和近似性能损失情况下系统可达速率的闭合表达式。从仿真结果可知，当量化比特数为4时，系统的信噪比和可达速率性能损失分别小于0.06 dB和0.03 bit/(s·Hz)。在无性能损失、有性能损失及近似性能损失情形下，系统的可达速率随着IRS阵元数的增加而增大。当量化比特数大于1时，无论IRS的阵元数取值如何，系统的信噪比和可达速率的性能损失与近似性能损失之间的差可忽略不计。
本文的其余部分组织如下：第2节给出了双IRS辅助放大转发中继网络的系统模型；第3节推导了系统信噪比和可达速率的性能损失与近似性能损失表达式；第4节和第5节分别给出了仿真结果和结论。
注意：在本文中，小写字母、黑体小写字母和黑体大写字母分别表示标量、向量及矩阵。符号E(⋅)，(⋅)T，(⋅)H，diag(⋅)和|⋅|m分别表示数学期望、转置、共轭转置、对角化及第m个元素的模。符号CN×N表示N×N的矩阵空间。符号CN(x,y)表示均值为x，方差为y的复高斯分布，本文用黑体h表示多维信道，非黑体h表示1维信道。
2. 系统模型
如图1所示，本文研究一种双IRS辅助放大转发中继的无线网络，其中放大转发中继工作于半双工模式。基站通过IRS-1、IRS-2和中继向终端传输隐私消息。假设由于距离较远，基站与终端、基站与IRS-2、IRS-1与终端之间无直接的信息传输。基站、终端及中继均配备单天线。IRS-1和IRS-2分别配备N和M个无源反射元件，且它们均在1个时隙内仅反射1次信号。hsi∈CN×1, hHir∈C1×N, hHsr∈C1×1, hri∈CM×1, hHid∈C1×M和hHrd∈C1×1分别表示基站到IRS-1、IRS-1到中继、基站到中继、中继到IRS-2、IRS-2 到终端及中继到终端的信道。
在第1个时隙中，基站在IRS-1的辅助下将隐私信息发送给中继，中继接收到的信号为
yr=√gsrPshHsrxs+√gsirPshHirΘ1hsixs+nr (1) 其中，xs和Ps分别表示基站处的发射信号和功率，gsr为基站到中继的路径损耗系数，gsir = gsigir表示基站到IRS-1和IRS-1到中继的等价路径损耗系数，Θ1=diag(ejϕ1,⋯,ejϕn,⋯,ejϕN)表示IRS-1的对角反射系数矩阵，ϕn∈(0,2π]为第n个反射元件的相位，nr～CN(0,σ2r)表示中继处的复加性高斯白噪声。
3. 性能损失分析
假设所有的信道均为瑞利信道，由于配备离散相位移相器的IRS-1和IRS-2可能会产生相位量化误差，从而降低系统的接收性能。接下来，本文将分别推导关于量化比特数的信噪比和可达速率的闭合形式表达式，并在第1时隙和第2时隙分析配备离散移相器的IRS-1和IRS-2对系统性能的影响。
3.1 在第1时隙中
假设所有信道均服从瑞利分布，相应的概率密度函数为
fα(x)={xα2e−x22α2, x∈[0,+∞)0, x∈(−∞,0) (2) 其中，α表示瑞利分布参数，其满足α>0。
接收信号(1)可重写为
yr=√gsrPshHsrxs+√gsirPshHirΘ1hsixs+nr=√gsrPs|hsr|e−jφsrxs+√gsirPsN∑n=1|hir(n)|⋅|hsi(n)|ej(−2πΨθir(n)+ϕn+2πΨθsi(n))xs+nr=(√gsir/gsrN∑n=1ej(−2πΨθir(n)+ϕn+2πΨθsi(n)+φsr)⋅|hir(n)||hsi(n)|+|hsr|)√gsrPse−jφsrxs+nr (3) 其中，φsr表示hsr的相位，2πΨθir(n)和2πΨθsi(n)分别表示hir和hsi第n个元素的相位。为推导方便，假设φsr=0且ϕn=−φsr+2πΨθir(n)−2πΨθsi(n)。此时，基于弱大数定律和瑞利分布，|hir(n)|和|hsi(n)|均服从独立同分布的瑞利分布，其参数分别为αir和αsi，式(3)可简化为
yr=√Ps(√gsr|hsr|+√gsirN⋅1NN∑n=1|hir(n)||hsi(n)|)xs+nr=√Ps(√gsrE(|hsr|)+√gsirNE(|hir(n)||hsi(n)|)xs+nr=√Ps(√gsr+∞∫0|hsr|fαsr(|hsr|)d(|hsr|)+√gsirN+∞∫0|hir(n)|fαir(|hir(n)|)d(|hir(n)|)⋅+∞∫0|hsi(n)|fαsi(|hsi(n)|)d(|hsi(n)|))xs+nr=√Ps(√π2gsrαsr+√gsirNπ2αirαsi)xs+nr (4) 其中，αsr表示基站到中继信道的瑞利分布参数。
假设配备离散相位移相器的IRS采用k比特的相位量化器，IRS每个反射元件的相位可行集为
Ω={12kπ,32kπ,⋯,2k+1−12kπ} (5) 假设IRS第n个元件的实际离散相位¯ϕn从式(5)的可行集Ω中选取，记为
¯ϕn=argmin¯ϕn∈Ω‖ (6) 其中， {\phi _n} 为期望的连续相位。一般情形下， \overline {{\phi _n}} \ne {\phi _n} ，此意味着相位不匹配，从而造成系统性能损失。定义IRS的第 n 个相位的量化误差为
\Delta {\phi _n} = \overline {{\phi _n}} - {\phi _n} (7) 假设 \Delta {\phi _n} 服从均匀分布，其概率密度函数为
f(x) = \left\{ \begin{aligned} & {\frac{1}{{2\Delta x}},}\;\;{{\text{ }}x \in [ - \Delta x,\Delta x]} \\ & {0,}\qquad\;\; { x \notin [ - \Delta x,\Delta x]} \end{aligned} \right. (8) 其中， \Delta x = {{\pi/}}{2^k} 。
当使用具有离散相位移相器的IRS时，由于相位不匹配会导致相位量化误差，从而造成性能损失。当存在相位量化误差时，接收信号(3)可转化为
\begin{split} {{\hat y}_{\text{r}}} = \;&\sqrt {{g_{{\text{sr}}}}{P_{\text{s}}}} h_{{\text{sr}}}^{\text{H}}{x_{\text{s}}} + \sqrt {{g_{{\text{sir}}}}{P_{\text{s}}}} {{\boldsymbol{h}}}_{{\text{ir}}}^{\text{H}}{{\boldsymbol{\varTheta }}_1}{{{\boldsymbol{h}}}_{{\text{si}}}}{x_{\text{s}}} + {n_{\text{r}}} \\ = \;&\sqrt {{g_{{\text{sr}}}}{P_{\text{s}}}} \left| {{h_{{\text{sr}}}}} \right|{{\text{e}}^{ - {\text{j}}{\varphi _{{\text{sr}}}}}}{x_{\text{s}}} + \sqrt {{g_{{\text{sir}}}}{P_{\text{s}}}} \sum\limits_{n = 1}^N {\left| {{h_{{\text{ir}}}}(n)} \right|} \left| {{h_{{\text{si}}}}(n)} \right|\\ & \cdot {{\text{e}}^{{\text{j}}\overbrace {( - 2{\pi }{\varPsi _{{\theta _{{\text{ir}}}}}}(n) + {\phi _n} + 2{\pi }{\varPsi _{{\theta _{{\text{si}}}}}}(n))}^{\Delta {\phi _n}}}}{x_{\text{s}}} + {n_{\text{r}}} \\ =\;& \sqrt {{P_{\text{s}}}} \left(\sqrt {{g_{{\text{sr}}}}} \left| {{h_{{\text{sr}}}}} \right| + \sqrt {{g_{{\text{sir}}}}} N \right.\\ & \cdot \left.\frac{1}{N}\sum\limits_{n = 1}^N {\left| {{h_{{\text{ir}}}}(n)} \right|} \left| {{h_{{\text{si}}}}(n)} \right|{{\text{e}}^{{\text{j}}\Delta {\phi _n}}}\right.){x_{\text{s}}} + {n_{\text{r}}} \\[-1pt] \end{split} (9) 由于 \left| {{h_{{\text{ir}}}}(n)} \right| ， \left| {{h_{{\text{si}}}}(n)} \right| 和 {{\text{e}}^{{\text{j}}\Delta {\phi _n}}} 相互独立，通过使用与式(4)相同的方法、式(8)及欧拉公式，即 {{\text{e}}^{{\text{j}}\Delta {\phi _n}}} = {\text{cos}}(\Delta {\phi _n}) + {\text{jsin}}(\Delta {\phi _n}) ，则式(9)可重写为
\begin{split} {{\hat y}_{\text{r}}} = \;&\sqrt {{P_{\text{s}}}} (\sqrt {{g_{{\text{sr}}}}} \left| {{h_{{\text{sr}}}}} \right| + \sqrt {{g_{{\text{sir}}}}} N\mathbb{E}(|{h_{{\text{ir}}}}(n)||{h_{{\text{si}}}}(n)|{{\text{e}}^{{\text{j}}\Delta {\phi _n}}}))\\ & \cdot {x_{\text{s}}} + {n_{\text{r}}} \\ =\;& \sqrt {{P_{\text{s}}}} \Biggr(\sqrt {{g_{{\text{sr}}}}} \int\limits_{\text{0}}^{ + \infty } {\left| {{h_{{\text{sr}}}}} \right|} {f_{{\alpha _{{\text{sr}}}}}}(|{h_{{\text{sr}}}}|){\text{d}}(|{h_{{\text{sr}}}}|) \\ & + \sqrt {{g_{{\text{sir}}}}} N\int\limits_{\text{0}}^{ + \infty } {\left| {{h_{{\text{ir}}}}(n)} \right|} {f_{{\alpha _{{\text{ir}}}}}}(|{h_{{\text{ir}}}}(n)|){\text{d}}(|{h_{{\text{ir}}}}(n)|) \\ \;&\cdot \int\limits_{\text{0}}^{ + \infty } {\left| {{h_{{\text{si}}}}(n)} \right|} {f_{{\alpha _{{\text{si}}}}}}(|{h_{{\text{si}}}}(n)|){\text{d}}(|{h_{{\text{si}}}}(n)|)\\ & \cdot \int\limits_{ - \Delta x}^{\Delta x} {{{\text{e}}^{{\text{j}}\Delta {\phi _n}}}} f(\Delta {\phi _n}){\mathrm{d}}(\Delta {\phi _n})\Biggr){x_{\text{s}}} + {n_{\text{r}}} \\ = \;&\sqrt {{P_{\text{s}}}} \left( \sqrt {\frac{{\pi }}{2}{g_{{\text{sr}}}}} {\alpha _{{\text{sr}}}} + \sqrt {{g_{{\text{sir}}}}} N\frac{1}{{2\Delta x}}\int\limits_{ - \Delta x}^{\Delta x} {\text{cos}}(\Delta {\phi _n})\right.\\ & \cdot{\text{d}}(\Delta {\phi _n}) \frac{{\pi }}{2}{\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}} \Biggr){x_{\text{s}}} + {n_{\text{r}}} \\ = \;&\sqrt {{P_{\text{s}}}} \left( {\sqrt {\frac{{\pi }}{2}{g_{{\text{sr}}}}} {\alpha _{{\text{sr}}}} + \sqrt {{g_{{\text{sir}}}}} N\frac{{{\text{sin}}(\Delta x)}}{{\Delta x}}\frac{{\pi }}{2}{\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}}} \right)\\ & \cdot {x_{\text{s}}} + {n_{\text{r}}} \\ =\;& \sqrt {{P_{\text{s}}}} \left( {\sqrt {\frac{{\pi }}{2}{g_{{\text{sr}}}}} {\alpha _{{\text{sr}}}} + \sqrt {{g_{{\text{sir}}}}} N{\text{sinc}}\left( {\frac{{\pi }}{{{2^{{k_1}}}}}} \right)\frac{{\pi }}{2}{\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}}} \right)\\ & \cdot {x_{\text{s}}} + {n_{\text{r}}} \\[-1pt] \end{split} (10) 其中， {k_1} 表示IRS-1的量化比特数，其为有限正整数。
接下来，为了简化式(10)，考虑到当量化比特数 {k_1} 很大时， \Delta {\phi _n} \to 0 ，通过使用一阶Taylor级数展开表达式，可得
{\text{cos}}(\Delta {\phi _n}) \approx 1 - \Delta \phi _n^2{\text{/2}} (11) 将式(11)代入式(10)，则具有近似相位量化误差的接收信号(3)可转化为
\begin{split} {{\tilde y}_{\text{r}}} =\;& \sqrt {{P_{\text{s}}}} \left( \sqrt {\frac{{\pi }}{2}{g_{{\text{sr}}}}} {\alpha _{{\text{sr}}}} + \sqrt {{g_{{\text{sir}}}}} N\frac{1}{{2\Delta x}}\right.\\ & \left.\cdot \int\limits_{ - \Delta x}^{\Delta x} {(1 - \frac{{\Delta \phi _n^2}}{2}){\text{d}}(\Delta {\phi _n})} \frac{{\pi }}{2}{\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}} \right){x_{\text{s}}} + {n_{\text{r}}} \\ = \;&\sqrt {{P_{\text{s}}}} \left( \sqrt {\frac{{\pi }}{2}{g_{{\text{sr}}}}} {\alpha _{{\text{sr}}}} + \sqrt {{g_{{\text{sir}}}}} N\left( {1 - \frac{1}{6}{{\left( {\frac{{\pi }}{{{2^{{k_1}}}}}} \right)}^2}} \right)\right.\\ & \cdot \frac{{\pi }}{2}{\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}} \Biggr){x_{\text{s}}} + {n_{\text{r}}}\\[-1pt] \end{split} (12) 3.2 在第2时隙中
基于式(1)，假设中继接收信号后成功地将其放大及转发，终端处接收到的信号为
{y_{\text{d}}} = \sqrt {{g_{{\text{rd}}}}{P_{\text{r}}}} h_{{\text{rd}}}^{\text{H}}{x_{\text{r}}} + \sqrt {{g_{{\text{rid}}}}{P_{\text{r}}}} {{\boldsymbol{h}}}_{{\text{id}}}^{\text{H}}{{\boldsymbol{\varTheta }}_2}{{{\boldsymbol{h}}}_{{\text{ri}}}}{x_{\text{r}}} + {n_{\text{d}}} (13) 其中， {x_{\text{r}}} 和 {P_{\text{r}}} 分别表示中继处的发射信号和功率， {g_{{\text{rd}}}} 表示中继到终端信道的路径损耗系数， {g_{{\text{rid}}}} = {g_{{\text{ri}}}}{g_{{\text{id}}}} 表示中继到IRS-2信道与IRS-2到终端信道的等价路径损耗系数， {{\boldsymbol{\varTheta }}_2} = {\text{diag}}({{\text{e}}^{{\text{j}}{\phi _1}}}, \cdots ,{{\text{e}}^{{\text{j}}{\phi _m}}}, \cdots ,{{\text{e}}^{{\text{j}}{\phi _M}}}) 表示IRS-2的反射系数矩阵，其中 {\phi _m} \in (0,2{\pi }] 表示第 m 个反射元件的相移， {n_{\text{d}}}{\text{～}}\mathcal{C}\mathcal{N}(0,\sigma _{\text{d}}^2) 表示终端处的复加性高斯白噪声。 {x_{\text{r}}} 的表达式为
\begin{split} {x_{\text{r}}} =\,& \beta {y_{\text{r}}} = \beta (\sqrt {{g_{{\text{sr}}}}{P_{\text{s}}}} h_{{\text{sr}}}^{\text{H}}{x_{\text{s}}} + \sqrt {{g_{{\text{sir}}}}{P_{\text{s}}}} {{\boldsymbol{h}}}_{{\text{ir}}}^{\text{H}}{{\boldsymbol{\varTheta }}_1}{{{\boldsymbol{h}}}_{{\text{si}}}}{x_{\text{s}}} + {n_{\text{r}}}) \\ =\,& \beta \sqrt {{P_{\text{s}}}} \left( {\sqrt {{g_{{\text{sr}}}}} h_{{\text{sr}}}^{\text{H}} + \sqrt {{g_{{\text{sir}}}}} {{\boldsymbol{h}}}_{{\text{ir}}}^{\text{H}}{{\boldsymbol{\varTheta }}_1}{{{\boldsymbol{h}}}_{{\text{si}}}}} \right){x_{\text{s}}} + \beta {n_{\text{r}}}\\[-1pt] \end{split} (14) 其中， \beta 为放大转发中继的放大因子，其表达式为
\beta = \frac{{\sqrt {{P_{\text{r}}}} }}{{\sqrt {{P_{\text{s}}}{{\left| {\sqrt {{g_{{\text{sr}}}}} h_{{\text{sr}}}^{\text{H}} + \sqrt {{g_{{\text{sir}}}}} {{\boldsymbol{h}}}_{{\text{ir}}}^{\text{H}}{{\boldsymbol{\varTheta }}_1}{{{\boldsymbol{h}}}_{{\text{si}}}}} \right|}^2} + \sigma _{\text{r}}^2} }} (15) 在无相位量化误差的情形下，基于式(4)，放大转发中继的放大因子(15)可重写为
\beta = \frac{{\sqrt {{P_{\text{r}}}} }}{{\sqrt {{P_{\text{s}}}{{\left( {\sqrt {\dfrac{{\pi }}{2}{g_{{\text{sr}}}}} {\alpha _{{\text{sr}}}} + \sqrt {{g_{{\text{sir}}}}} N\dfrac{{\pi }}{2}{\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}}} \right)}^2} + \sigma _{\text{r}}^2} }} (16) 同理，基于式(10)和式(12)，具有相位量化误差和近似量化误差的放大转发中继功率分配因子分别为
\hat \beta = \frac{{\sqrt {{P_{\text{r}}}} }}{{\sqrt A }}, \tilde \beta = \frac{{\sqrt {{P_{\text{r}}}} }}{{\sqrt B }} (17) 其中，
\begin{split} A = \;& {P_{\mathrm{s}}}{\left( {\sqrt {{g_{{\text{sr}}}}{\pi }/2} {\alpha _{{\text{sr}}}} + \sqrt {{g_{{\text{sir}}}}} N{\text{sinc}}\left( {{\pi }/{2^{{k_1}}}} \right){\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}}{\pi }/2} \right)^2} \\ & + \sigma _{\text{r}}^2 \\ B =\;& {P_{\mathrm{s}}}{\left( {\sqrt {{g_{{\text{sr}}}}{\pi }/2} {\alpha _{{\text{sr}}}}{\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}}{\pi }/2} \right)^2} \\ & + \sqrt {{g_{{\text{sir}}}}} N\left( {1 - {{\left( {{\pi }/{2^{{k_1}}}} \right)}^2}/6} \right) + \sigma _{\text{r}}^2 \\[-1pt] \end{split} (18) 在无相位量化误差的情形下，同理于式(4)，式(13)可重写为
\begin{split} {y_{\text{d}}} = \;&\beta \sqrt {{P_{\text{r}}}{P_{\text{s}}}} \Biggr(\sqrt {{g_{{\text{sr}}}}{g_{{\text{rd}}}}} \frac{{\pi }}{2}{\alpha _{{\text{rd}}}}{\alpha _{{\text{sr}}}} + \sqrt {{g_{{\text{sir}}}}{g_{{\text{rd}}}}} N{\left( {\frac{{\pi }}{2}} \right)^{{\text{3/2}}}}\\ & \cdot {\alpha _{{\text{rd}}}}{\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}} + \sqrt {{g_{{\text{sr}}}}{g_{{\text{rid}}}}} M{\left( {\frac{{\pi }}{2}} \right)^{{\text{3/2}}}}{\alpha _{{\text{id}}}}{\alpha _{{\text{ri}}}}{\alpha _{{\text{sr}}}} \\ &+ \sqrt {{g_{{\text{sir}}}}{g_{{\text{rid}}}}} MN\frac{{{{\pi }^2}}}{4}{\alpha _{{\text{id}}}}{\alpha _{{\text{ri}}}}{\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}}\Biggr){x_{\text{s}}} \\ & + \beta \sqrt {{P_{\text{r}}}} \left( {\sqrt {\frac{{\pi }}{2}{g_{{\text{rd}}}}} {\alpha _{{\text{rd}}}} + \sqrt {{g_{{\text{rid}}}}} M\frac{{\pi }}{2}{\alpha _{{\text{id}}}}{\alpha _{{\text{ri}}}}} \right){n_{\text{r}}} + {n_{\text{d}}} \end{split} (19) 在存在相位量化误差与近似量化误差情形下，终端处的接收信号可分别写为
\begin{split} {{\hat y}_{\text{d}}} =\;& \hat \beta \sqrt {{P_{\text{r}}}{P_{\text{s}}}} \left(\sqrt {{g_{{\text{sr}}}}{g_{{\text{rd}}}}} \frac{{\pi }}{2}{\alpha _{{\text{rd}}}}{\alpha _{{\text{sr}}}} + \sqrt {{g_{{\text{sir}}}}{g_{{\text{rd}}}}} N{\left( {\frac{{\pi }}{2}} \right)^{{\text{3/2}}}}\right.\\ & \cdot{\alpha _{{\text{rd}}}}{\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}}{\text{sinc}}\left( {\frac{{\pi }}{{{2^{{k_1}}}}}} \right) + \sqrt {{g_{{\text{sr}}}}{g_{{\text{rid}}}}} M \\ & \cdot {\left( {\frac{{\pi }}{2}} \right)^{{\text{3/2}}}}{\alpha _{{\text{id}}}}{\alpha _{{\text{ri}}}}{\alpha _{{\text{sr}}}}{\text{sinc}}\left( {\frac{{\pi }}{{{2^{{k_2}}}}}} \right) \\ &+ \sqrt {{g_{{\text{sir}}}}{g_{{\text{rid}}}}} MN\frac{{{{\pi }^2}}}{4}{\alpha _{{\text{id}}}}{\alpha _{{\text{ri}}}}{\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}}\\ & \left.\cdot{\text{sinc}}\left( {\frac{{\pi }}{{{2^{{k_1}}}}}} \right){\text{sinc}}\left( {\frac{{\pi }}{{{2^{{k_2}}}}}} \right)\right){x_s} \\ & + \hat \beta \sqrt {{P_{\text{r}}}} \Biggr(\sqrt {\frac{{\pi }}{2}{g_{{\text{rd}}}}} {\alpha _{{\text{rd}}}} + \sqrt {{g_{{\text{rid}}}}} M\frac{{\pi }}{2}\\ & \cdot {\alpha _{{\text{id}}}}{\alpha _{{\text{ri}}}}{\text{sinc}}\left( {\frac{{\pi }}{{{2^{{k_2}}}}}} \right) \Biggr){n_{\text{r}}} + {n_{\text{d}}} \end{split} (20) \begin{split} {{\tilde y}_{\text{d}}} = \;&\tilde \beta \sqrt {{P_{\text{r}}}{P_{\text{s}}}} \Biggr(\sqrt {{g_{{\text{sr}}}}{g_{{\text{rd}}}}} \frac{{\pi }}{2}{\alpha _{{\text{rd}}}}{\alpha _{{\text{sr}}}} + \sqrt {{g_{{\text{sir}}}}{g_{{\text{rd}}}}} N{\left( {\frac{{\pi }}{2}} \right)^{{\text{3/2}}}}\\ & \cdot {\alpha _{{\text{rd}}}}{\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}}\left( {1 - \frac{1}{6}{{\left( {\frac{{\pi }}{{{2^{{k_1}}}}}} \right)}^2}} \right) + \sqrt {{g_{{\text{sr}}}}{g_{{\text{rid}}}}} M{\left( {\frac{{\pi }}{2}} \right)^{{\text{3/2}}}} \\ \;&\cdot {\alpha _{{\text{id}}}}{\alpha _{{\text{ri}}}}{\alpha _{{\text{sr}}}}\left( {1 - \frac{1}{6}{{\left( {\frac{{\pi }}{{{2^{{k_2}}}}}} \right)}^2}} \right) + \sqrt {{g_{{\text{sir}}}}{g_{{\text{rid}}}}} MN\frac{{{{\pi }^2}}}{4}\\ & \cdot{\alpha _{{\text{id}}}}{\alpha _{{\text{ri}}}}{\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}}\left( {1 - \frac{1}{6}{{\left( {\frac{{\pi }}{{{2^{{k_1}}}}}} \right)}^2}} \right)\left( {1 - \frac{1}{6}{{\left( {\frac{{\pi }}{{{2^{{k_2}}}}}} \right)}^2}} \right)\Biggr) \\ \;& \cdot{x_{\text{s}}}+ \tilde \beta \sqrt {{P_{\text{r}}}} \Biggr( \sqrt {\frac{{\pi }}{2}{g_{{\text{rd}}}}} {\alpha _{{\text{rd}}}} + \sqrt {{g_{{\text{rid}}}}} M\frac{{\pi }}{2}{\alpha _{{\text{id}}}}{\alpha _{{\text{ri}}}}\\ & \cdot\left( {1 - \frac{1}{6}{{\left( {\frac{{\pi }}{{{2^{{k_2}}}}}} \right)}^2}} \right) \Biggr){n_{\text{r}}} + {n_{\text{d}}} \end{split} (21) 其中， {k_2} 表示IRS-2的量化比特数。
定义
\begin{split} & {v_1} = \sqrt {{g_{{\text{sir}}}}{g_{{\text{rd}}}}} N{\left( {{{\pi/2}}} \right)^{{\text{3/2}}}}{\alpha _{{\text{rd}}}}{\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}}, \\ & {v_2} = \sqrt {{g_{{\text{sr}}}}{g_{{\text{rid}}}}} M{\left( {{{\pi /2}}} \right)^{{\text{3/2}}}}{\alpha _{{\text{id}}}}{\alpha _{{\text{ri}}}}{\alpha _{{\text{sr}}}} \end{split} (22) \begin{split} & {v_3} = \sqrt {{g_{{\text{sir}}}}{g_{{\text{rid}}}}} MN{\alpha _{{\text{id}}}}{\alpha _{{\text{ri}}}}{\alpha _{{\text{ir}}}}{\alpha _{{\text{si}}}}{{\pi }^2}/4, \\ & {v_4} = {\beta ^2}{P_{\text{r}}}{\left( {\sqrt {{g_{{\text{rd}}}}{\pi }/2} {\alpha _{{\text{rd}}}} + \sqrt {{g_{{\text{rid}}}}} M{\alpha _{{\text{id}}}}{\alpha _{{\text{ri}}}}{\pi }/2} \right)^2}\sigma _{\text{r}}^2 + \sigma _{\text{d}}^2 \end{split} (23) 基于式(19)、式(22)和式(23)，终端处无性能损失时的信噪比表达式为
{\text{SN}}{{\text{R}}_{\text{d}}} = {\beta ^2}{P_{\text{r}}}{P_{\text{s}}}{\left( {\sqrt {{g_{{\text{sr}}}}{g_{{\text{rd}}}}} {\alpha _{{\text{rd}}}}{\alpha _{{\text{sr}}}}{{\pi /2}} + {v_1} + {v_2} + {v_3}} \right)^2}{\text{/}}{v_4} (24) 定义
\begin{split} & {u_1} = {v_1}{\text{sinc}}\left( {{\pi }/{2^{{k_1}}}} \right), {u_2} = {v_2}{\text{sinc}}\left( {{\pi }/{2^{{k_2}}}} \right),\\ & {u_3} = {v_3}{\text{sinc}}\left( {{\pi }/{2^{{k_1}}}} \right){\text{sinc}}\left( {{\pi }/{2^{{k_2}}}} \right) \end{split} (25) \begin{split} {u_4} =\;& {\hat \beta ^2}{P_{\text{r}}}\left( \sqrt {{g_{{\text{rd}}}}{\pi }/2} {\alpha _{{\text{rd}}}} + \sqrt {{g_{{\text{rid}}}}} M{\alpha _{{\text{id}}}}{\alpha _{{\text{ri}}}}\right.\;\;\\ & \left.\cdot {\text{sinc}}\left( {{\pi }/{2^{{k_2}}}} \right){\pi }/2 \right)^2\sigma _{\text{r}}^2 + \sigma _{\text{d}}^2 \end{split} (26) 则终端处有性能损失时的信噪比表达式为
{{{\mathrm{S}}\hat {\mathrm{N}}}}{{\text{R}}_{\text{d}}} = {\hat \beta ^2}{P_{\text{r}}}{P_{\text{s}}}{\left( {\sqrt {{g_{{\text{sr}}}}{g_{{\text{rd}}}}} {\alpha _{{\text{rd}}}}{\alpha _{{\text{sr}}}}{{\pi /2}} + {u_1} + {u_2} + {u_3}} \right)^2}{\text{/}}{u_4} (27) 定义
\begin{split} & {q_1} = {v_1}\left( {1 - \frac{1}{6}{{\left( {\frac{{\pi }}{{{2^{{k_1}}}}}} \right)}^2}} \right), \\ & {q_2} = {v_2}\left( {1 - \frac{1}{6}{{\left( {\frac{{\pi }}{{{2^{{k_2}}}}}} \right)}^2}} \right), \\ & {q_3} = {v_3}\left( {1 - \frac{1}{6}{{\left( {\frac{{\pi }}{{{2^{{k_1}}}}}} \right)}^2}} \right)\left( {1 - \frac{1}{6}{{\left( {\frac{{\pi }}{{{2^{{k_2}}}}}} \right)}^2}} \right) \end{split} (28) \begin{split} {q_4} =\;& {\tilde \beta ^2}{P_{\text{r}}}\left( \sqrt {{g_{{\text{rd}}}}{\pi }/2} {\alpha _{{\text{rd}}}} + \sqrt {{g_{{\text{rid}}}}} M{\alpha _{{\text{id}}}}{\alpha _{{\text{ri}}}}\right.\;\;\\ & \left.\cdot \left( {1 - {{\left( {{\pi }/{2^{{k_2}}}} \right)}^2}/6} \right){\pi }/2 \right)^2\sigma _{\text{r}}^2 + \sigma _{\text{d}}^2 \end{split} (29) 基于式(21)、式(28)和式(29)，可得终端处近似性能损失的信噪比表达式为
{{{\mathrm{S}}\tilde {\mathrm{N}}}}{{\text{R}}_{\text{d}}} = {\tilde \beta ^2}{P_{\text{r}}}{P_{\text{s}}}{(\sqrt {{g_{{\text{sr}}}}{g_{{\text{rd}}}}} {\alpha _{{\text{rd}}}}{\alpha _{{\text{sr}}}}{{\pi /2}} + {q_1} + {q_2} + {q_3})^2}{\text{/}}{q_4} (30) 因此，终端处的系统性能损失和近似性能损失的表达式分别为
{\hat L_{\text{d}}} = \frac{{{\text{SN}}{{\text{R}}_{\text{d}}}}}{{{{{\mathrm{S}}\hat {\mathrm{N}}}}{{\text{R}}_{\text{d}}}}} = \frac{{{u_4}{\beta ^2}{{\left( {\sqrt {{g_{{\text{sr}}}}{g_{{\text{rd}}}}} {\alpha _{{\text{rd}}}}{\alpha _{{\text{sr}}}}{{\pi /2}} + {v_1} + {v_2} + {v_3}} \right)}^2}}}{{{v_4}{{\hat \beta }^2}{{\left( {\sqrt {{g_{{\text{sr}}}}{g_{{\text{rd}}}}} {\alpha _{{\text{rd}}}}{\alpha _{{\text{sr}}}}{{\pi /2}} + {u_1} + {u_2} + {u_3}} \right)}^2}}} (31) {\tilde L_{\text{d}}} = \frac{{{\text{SN}}{{\text{R}}_{\text{d}}}}}{{{{{\mathrm{S}}\tilde {\mathrm{N}}}}{{\text{R}}_{\text{d}}}}} = \frac{{{q_4}{\beta ^2}{{\left( {\sqrt {{g_{{\text{sr}}}}{g_{{\text{rd}}}}} {\alpha _{{\text{rd}}}}{\alpha _{{\text{sr}}}}{{\pi /2}} + {v_1} + {v_2} + {v_3}} \right)}^2}}}{{{v_4}{{\tilde \beta }^2}{{\left( {\sqrt {{g_{{\text{sr}}}}{g_{{\text{rd}}}}} {\alpha _{{\text{rd}}}}{\alpha _{{\text{sr}}}}{{\pi /2}} + {q_1} + {q_2} + {q_3}} \right)}^2}}} (32) 相应地，基于式(19)—式(21)，可得在无性能损失、有性能损失及近似性能损失时的系统可达速率分别为
\begin{split} {R_{\text{d}}} =\;& {\log _2}\Bigr( 1 + {\beta ^2}{P_{\text{r}}}{P_{\text{s}}}\left( \sqrt {{g_{{\text{sr}}}}{g_{{\text{rd}}}}} {\alpha _{{\text{rd}}}}{\alpha _{{\text{sr}}}}{{\pi /2}} \right.\\ & \left. + {v_1} + {v_2} + {v_3} \right)^2/{v_4} \Bigr) \end{split} (33) \begin{split} {\hat R_{\text{d}}} =\;& {\log _2}\Bigr( 1 + {{\hat \beta }^2}{P_{\text{r}}}{P_{\text{s}}}\left( \sqrt {{g_{{\text{sr}}}}{g_{{\text{rd}}}}} {\alpha _{{\text{rd}}}}{\alpha _{{\text{sr}}}}{{\pi /2}}\right. \\ & \left.+ {u_1} + {u_2} + {u_3} \right)^2/{u_4} \Bigr) \end{split} (34) \begin{split} {\tilde R_{\text{d}}} = \;& {\log _2}\Bigr( 1 + {{\tilde \beta }^2}{P_{\text{r}}}{P_{\text{s}}}\left( \sqrt {{g_{{\text{sr}}}}{g_{{\text{rd}}}}} {\alpha _{{\text{rd}}}}{\alpha _{{\text{sr}}}}{{\pi /2}} \right.\\ & \left.+ {q_1} + {q_2} + {q_3} \right)^2/{q_4} \Bigr) \end{split} (35) 4. 仿真结果
接下来，本文分析配备离散移相器的IRS由于相位不匹配对系统信噪比和可达速率的影响。假设距离为 {\bar d_0} 的路径损耗模型为 g({\bar d_0}) = P{L_0} - 10\gamma {\log _{10}} {\bar d_0}/{d_0} ，其中， P{L_0} = - 30 dBm表示在参考距离 {d_0} = 1 时的路径损耗， \gamma 为路径损耗指数。基站到IRS-1、IRS-1到中继、中继到IRS-2、IRS-2到终端及中继到终端信道的路径损耗指数分别为2.6, 2.6, 3.5, 2.6, 2.6和3.5。基站、中继、终端、IRS-1和IRS-2分别位于(0 m, 0 m), (150 m, 0 m), (300 m, 0 m), ( 25\sqrt 2 m, 25\sqrt 2 m)和( (150 + 25\sqrt 2 ) m, 25\sqrt 2 m)。基站功率 {P_{\text{s}}} = 30 \;{\text{dBm}} ，中继功率 {P_{\text{r}}} = 35 \;{\text{dBm}} 。所有信道的瑞利分布参数均设置为0.5。
图2为终端处信噪比性能损失随IRS-1阵元个数 N 变化的关系曲线图。假设IRS-2的阵元个数 M = N 。从图中可知，无论在有性能损失还是在近似性能损失的情形下，系统的信噪比性能损失均随着量化比特数 k 的增大而逐渐减小，而随着 N 的增大而逐渐增大。当 k = 1 时，系统的性能损失与近似性能损失的差随着 N 的增大而逐渐增大。而当 k \ge 2 时，两者的差可忽略不计。当 k = 4 时，系统的信噪比损失小于0.06 dB。此外，随着 N 趋于大规模，系统的信噪比性能损失和近似性能损失速度逐渐减慢。
图3为终端处可达速率随IRS-1阵元个数 N 变化的关系曲线图，其中，IRS-2的阵元个数 M 与IRS-1的阵元个数相同，即 M = N 。从图中可知，无论在无性能损失、有性能损失还是近似性能损失情形下，系统的可达速率均随着 N 的增大而逐渐增大。这是由于随着 N 的增大，IRS-1和IRS-2能给系统提供更多的性能增益。当 k = 1 时，系统在有性能损失和近似性能损失时的可达速率的差随着 N 的增大而逐渐增大。随着 k 逐渐增大，有性能损失和近似损失的可达速率逐渐逼近无性能损失时的可达速率。例如，当 N = 1\;024 时，系统的可达速率性能损失在 k = 2 时约为0.15 bit/(s·Hz)，而在 k = 3 时仅为0.03 bit/(s·Hz)。
图4为系统的可达速率随量化比特数 k 变化的曲线图。从图中可发现，系统的可达速率性能损失随着IRS-1的阵元个数 N 和IRS-2的阵元个数 M 的增大而逐渐增大。当 k = 3 时，相比于无性能损失情形，系统的可达速率在有性能损失和近似性能损失情形下均降低了0.04 bit/(s·Hz)。当 k = 1 时，随着 N 和 M 的增大，系统在无性能损失、有性能损失和近似性能损失情形下的可达速率之间的差也逐渐增大。无论在何种情形下，系统在 N = 1\;024 ， M = 128 情形下的可达速率均优于 N = 128 ， M = 1024 时的可达速率。此揭示了与增加 M 相比，增加 N 对可达速率性能的提升更显著。
5. 结束语
本文研究了一种双 IRS 辅助的放大转发中继网络，并分析了在瑞利信道中，由于配备离散移相器的IRS相位量化误差而导致的系统性能损失。基于弱大数定律、欧拉公式及瑞利分布，推导了信噪比性能损失和可达速率的闭合表达式。此外，根据一阶Taylor级数展开表达式，推导了相应的近似性能损失闭合表达式。仿真结果表明：系统的信噪比和可达速率性能损失及近似性能损失均随着量化比特数的增加而逐渐减小，而随着 IRS元件数的增加而逐渐增大。当量化比特数为4时，系统的信噪比和可达速率性能损失分别小于0.06 dB和0.03 bit/(s·Hz)。因此，使用4比特的相位量化移相器即可实现可忽略的系统性能损失。
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[1] LIN Ruiquan, QIU Hangding, WANG Jun, et al. Physical-layer security enhancement in energy-harvesting-based cognitive internet of things: A GAN-powered deep reinforcement learning approach[J]. IEEE Internet of Things Journal, 2024, 11(3): 4899–4913. doi: 10.1109/JIOT.2023.3300770. [2] LIN Ruiquan, LI Fushuai, WANG Jun, et al. A blockchain-based method to defend against massive SSDF attacks in cognitive internet of vehicles[J]. IEEE Transactions on Vehicular Technology, 2024, 73(5): 6954–6967. doi: 10.1109/TVT.2023.3347430. [3] COVER T and GAMAL A E. Capacity theorems for the relay channel[J]. IEEE Transactions on Information Theory, 1979, 25(5): 572–584. doi: 10.1109/TIT.1979.1056084. [4] DING Haiyang, GE Jianhua, DA COSTA D B, et al. Diversity and coding gains of fixed-gain amplify-and-forward with partial relay selection in Nakagami-m fading[J]. IEEE Communications Letters, 2010, 14(8): 734–736. doi: 10.1109/LCOMM.2010.08.100530. [5] BLETSAS A, SHIN H, and WIN M Z. Cooperative communications with outage-optimal opportunistic relaying[J]. IEEE Transactions on Wireless Communications, 2007, 6(9): 3450–3460. doi: 10.1109/TWC.2007.06020050. [6] LANEMAN J N and WORNELL G W. Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks[J]. IEEE Transactions on Information Theory, 2023, 49(10): 2415–2425. doi: 10.1109/TIT.2003.817829. [7] ARTHI M, JOY J J, ARULMOZHIVARMAN P, et al. An efficient relay station deployment scheme based on the coverage and budget constraints in multi-hop relay networks[C]. 2015 International Conference on Communications and Signal Processing (ICCSP), Melmaruvathur, India, 2015: 124–128. doi: 10.1109/ICCSP.2015.7322702. [8] YILMAZ E, ZAKHOUR R, GESBERT D, et al. Multi-pair two-way relay channel with multiple antenna relay station[C]. 2010 IEEE International Conference on Communications, Cape Town, South Africa, 2010: 1–5. doi: 10.1109/ICC.2010.5502396. [9] RANKOV B and WITTNEBEN A. Spectral efficient protocols for half-duplex fading relay channels[J]. IEEE Journal on Selected Areas in Communications, 2007, 25(2): 379–389. doi: 10.1109/JSAC.2007.070213. [10] ZHANG Zhang, LÜ Tiejun, and SU Xin. Combining cooperative diversity and multiuser diversity: a fair scheduling scheme for multi-source multi-relay networks[J]. IEEE Communications Letters, 2011, 15(12): 1353–1355. doi: 10.1109/LCOMM.2011.102611.111715. [11] 张在琛, 江浩. 智能超表面使能无人机高能效通信信道建模与传输机理分析[J]. 电子学报, 2023, 51(10): 2623–2634. doi: 10.12263/DZXB.20221352.ZHANG Zaichen and JIANG Hao. Channel modeling and characteristics analysis for high energy-efficient RIS-assisted UAV communications[J]. Acta Electronica Sinica, 2023, 51(10): 2623–2634. doi: 10.12263/DZXB.20221352. [12] CHEN Kangjian, QI Chenhao, DOBRE O A, et al. Simultaneous beam training and target sensing in ISAC systems with RIS[J]. IEEE Transactions on Wireless Communications, 2024, 23(4): 2696–2710. doi: 10.1109/TWC.2023.3302319. [13] JIANG Hao, RUAN Chengyao, ZHANG Zaichen, et al. A general wideband non-stationary stochastic channel model for intelligent reflecting surface-assisted MIMO communications[J]. IEEE Transactions on Wireless Communications, 2021, 20(8): 5314–5328. doi: 10.1109/TWC.2021.3066806. [14] ZHANG Chencheng, QI Chenhao, and NALLANATHAN A. Fast multibeam training for RIS-assisted millimeter wave massive MIMO[J]. IEEE Communications Letters, 2024, 28(1): 168–172. doi: 10.1109/LCOMM.2023.3333683. [15] PAN Cunhua, REN Hong, WANG Kezhi, et al. Multicell MIMO communications relying on intelligent reflecting surfaces[J]. IEEE Transactions on Wireless Communications, 2020, 19(8): 5218–5233. doi: 10.1109/TWC.2020.2990766. [16] NIU Hehao, CHU Zheng, ZHOU Fuhui, et al. Weighted sum secrecy rate maximization using intelligent reflecting surface[J]. IEEE Transactions on Communications, 2021, 69(9): 6170–6184. doi: 10.1109/TCOMM.2021.3085780. [17] YILDIRIM I, KILINC F, BASAR E, et al. Hybrid RIS-empowered reflection and decode-and-forward relaying for coverage extension[J]. IEEE Communications Letters, 2021, 25(5): 1692–1696. doi: 10.1109/LCOMM.2021.3054819. [18] GALAPPATHTHIGE D L, DEVKOTA A, and AMARASURIYA G. On the performance of IRS-assisted relay systems[C]. 2021 IEEE Global Communications Conference (GLOBECOM), Madrid, Spain, 2021: 1–6. doi: 10.1109/GLOBECOM46510.2021.9685500. [19] LIU Chang, ZHOU Jiayu, GAO Ying, et al. IRS-aided secure communications over an untrusted AF relay system[J]. IEEE Transactions on Wireless Communications, 2023, 22(12): 8620–8633. doi: 10.1109/TWC.2023.3264626. [20] TALWAR S, JING Yindi, and SHAHBAZPANAHI S. Joint relay selection and power allocation for two-way relay networks[J]. IEEE Signal Processing Letters, 2011, 18(2): 91–94. doi: 10.1109/LSP.2010.2096466. [21] TAO Ye, LI Qiang, and GE Xiaohu. Sum rate optimization for IRS-aided two-way AF relay systems[C]. 2021 IEEE/CIC International Conference on Communications in China (ICCC), Xiamen, China, 2021: 823–828. doi: 10.1109/ICCC52777.2021.9580369. [22] DONG Rongen, SHI Baihua, ZHAN Xichao, et al. Performance analysis of massive hybrid directional modulation with mixed phase shifters[J]. IEEE Transactions on Vehicular Technology, 2022, 71(5): 5604–5608. doi: 10.1109/TVT.2022.3152807. [23] DONG Rongen, TENG Yin, SUN Zhongwen, et al. Performance analysis of wireless network aided by discrete-phase-shifter IRS[J]. Journal of Communications and Networks, 2022, 24(5): 603–612. doi: 10.23919/JCN.2022.000029. -
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