SCMA网络中基于保密能量效率最大化的分层安全多播策略

SCMA网络中基于保密能量效率最大化的分层安全多播策略 Layered Security Multicast Scheme Based on Security Energy Efficiency Maximization in SCMA Networks 1 first-author 陈 雷，男，讲师，博士，主要从事无线网络安全、无线资源优化和协作通信方面的研究。 E-mail：chenleikb@bupt.edu.cn 陈 雷，男，讲师，博士，主要从事无线网络安全、无线资源优化和协作通信方面的研究。 E-mail：chenleikb@bupt.edu.cn chenleikb@bupt.edu.cn 中国刑事警察学院 公安情报学系，辽宁 沈阳 110035 中国刑事警察学院 公安情报学系，辽宁 沈阳 110035 Department of Public Security Intelligence, Criminal Investigation Police University of China, Shenyang 110035, Liaoning, China Department of Public Security Intelligence, Criminal Investigation Police University of China, Shenyang 110035, Liaoning, China 首先，研究稀疏码多址接入（sparse code multiple access,SCMA）网络中的分层安全多播策略，来克服多播组的保密容量受限于组内最差用户信道条件的问题。其次，解决了网络保密能量效率最大化的最优化问题。为了减少解决最优化问题的计算复杂度，又提出了添加人工噪声下码本分配与功率分配的次优化算法（sub-optimization algorithm,SA）。仿真结果表明，本文提出的SA算法在基于SCMA网络中的网络保密能量效率方面要优于采用传统安全多播（conventional security multicast, CSM）策略的SCMA网络和基于正交频分多址（orthogonal frequency division multiple access, OFDMA）网络，并且提出的SA算法在SCMA网络中的安全多播系统功效要优于在OFDMA网络中的安全多播系统功效。 In this paper, we first study the layered security multicast scheme in SCMA（sparse code multiple access） network,whose system security capacity is no longer limited by the worst channel quality user in the multicast group. Secondly, we formulate a network security energy efficiency(SEE) maximization problem. In order to reduce the computational complexity, we then propose a sub-optimization algorithm(SA) by separating codebook assignment and power allocation with artificial noise. Finally, the simulation results verify that the network SEE of the proposed algorithm significantly outperforms the conventional security multicast (CSM) scheme in SCMA systems and orthogonal frequency division multiple access (OFDMA) systems. Moreover, the security multicast performance of SCMA systems with SA is better than that of OFDMA systems with SA. 稀疏码多址接入 分层安全多播 码本分配 功率分配 网络保密能量效率 SCMA（sparse code multiple access） layered security multicast codebook assignment power allocation network security energy efficiency 0　 引 言 1 为了达到未来移动网络的需求，第5代移动通信技术（5G）需要满足近千倍的数据增长和近百倍的能量效率的提升 [ 1 1 ] 。稀疏码多址接入（sparse code multiple access，SCMA）技术是一种基于码本的频谱效率接近最优化的新型非正交多址接入技术，该技术可以显著提升系统性能 [ 2 2 , 3 3 ] 。在SCMA技术中，每个子载波可以使用多个码本，因此需要多用户检测技术（MUD, multi-user detection）来分离出多个用户的信号。文献[ 4 4 ]研究了一种基于上行内容的SCMA网络。文献[ 5 5 ]研究了下行多用户的SCMA网络。文献[ 6 6 ]通过仿真验证了SCMA网络在合理的计算复杂度和适当的能量消耗时，可以显著提升网络容量，但是文献[ 6 6 ]中并未考虑采用资源分配算法在SCMA网络中的应用问题。然而，SCMA作为媒体接入控制层（MAC, media access control）的一项新技术，应该从资源分配角度（包括码本分配和功率分配）考虑网络性能的提升。在5G网络中不可避免地会考虑到安全问题，近些年，无线物理层安全技术得到了长足的发展。因此，本文从资源分配策略结合物理层安全技术方面考虑提升SCMA网络的保密容量和提高网络的保密能量效率。 目前，关于SCMA网络的研究多集中于单播网络。文献[ 7 7 ]在SCMA网络中提出了高效消耗算法来最大化能量效率。文献[ 8 8 ]研究了速率和能量效率最大化问题，提出了一种基于坐标轮换算法的迭代算法。文献[ 9 9 ]在非正交多址接入网络中研究了通过动态功率分配来提高能量效率的方法。文献[ 10 10 ]将研究扩展到多播网络中，在SCMA网络中采用分层编码策略，并且研究了最大化分层多播网络容量最大化的资源分配算法。在物理层安全传输技术中其研究多集中在单播传输中，主要采用添加人工噪声技术 [ 11 11 , 12 12 ] 。如果在多播传输中使用物理层安全传输技术，则需要在每次传输时，保证发送天线的数量多于接收用户天线的数量。在文献[ 13 13 ]中基站端根据用户反馈的不同信道条件，在合法用户的信道零空间添加人工噪声技术，防止窃听用户正确接收原始数据。文献[ 14 14 ]将二维波束赋形的Alamouti编码应用于安全多播中。文献[ 15 15 ]根据信道条件将多播用户分为多个多播簇，在每个簇中形成的信道矢量正交空间中产生人工噪声，从而实现物理层安全多播传输。以上这些安全多播，是在正交频分多址（orthogonal frequency division multiple access,OFDMA）网络中进行的，其基本物理资源单元是子载波，其研究还未在SCMA网络中进行。SCMA网络中码本是基本物理资源单元，这类似于OFDMA网络中的子载波。因此，在SCMA的安全多播网络中，基本层和增强层数据所分配的不是子载波而是码本。在我们所研究的场景中，所有用户共享相同的码本集合。 本文研究在SCMA网络中采用分层编码策略下的安全多播通信技术，提出了最大化网络保密能量效率的最优化问题。该最优化问题是一个混合组合问题，对该类问题求其全局最优解的计算消耗很大并且很难做到实际应用。因此，为了促进算法的实际应用性，又提出了次优化算法（SA, suboptimal algorithm）将最优化问题分解为添加人工噪声下的码本分配和功率分配两个阶段进行。 1　 网络模型 1 图1 图1 所示为SCMA网络中基于分层安全多播策略的网络模型。其中存在一个非法窃听用户 E ，在多播组中的 U 个合法用户有不同的服务质量（QoS, quality of service）需求即 <math id="M1"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416124&type= 7.70466709 3.97933340 。基站端采用分层编码策略，将原始的多播数据首先编码为一个基本层和多个增强层。基本层数据要求多播组中的所有用户能够正确接收，多播组用户只有正确接收了基本层的数据，才能正确的解码后续的数据。增强层数据是根据多播组中不同信道质量的用户采用差异化接收的，也就是说，信道质量好的用户能够接收越多增强层的数据，用户接收越多增强层的数据，也就能恢复越高质量的原始数据 [ 10 10 ] 。本文首先需要根据合法多播用户反馈的QoS需求和信道状态信息计算所需要的物理资源;其次，基站分配足够的码本给基本层数据，如果有剩余码本再分配给增强层的数据。 本文考虑SCMA网络中存在单小区多播组。 <math id="M2"><mi>U</mi><mo>=</mo><mo>{</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>u</mi><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>U</mi><mo>}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416129&type= 33.61266708 3.55599999 表示接收相同内容的一组多播用户，所有移动终端都是单天线。基站端的发射天线数目为 <math id="M3"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416132&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416131&type= 4.14866638 3.72533321 ，多播组内窃听者e的天线数目为 <math id="M4"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416134&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416133&type= 4.48733330 3.72533321 ，并且要求每次传输时 <math id="M5"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo><</mo><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416136&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416135&type= 13.71600056 3.72533321 。假设每个多播用户反馈其QoS需求和其信道状态信息给基站。定义主信道为多播组内用户与基站端的链路；窃听信道为窃听者与基站端的链路。为了可以在每次安全多播传输中保证在主信道的零空间添加人工噪声，因此在每次传输时 <math id="M6"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416137&type= 4.14866638 3.72533321 需要大于每次接收多播数据的用户数。由于在基站端添加的人工噪声是在主信道的零空间内，因此不会对主信道用户产生影响，但对窃听信道而言，添加的人工噪声一般不会落入窃听信道的零空间内，因此会降低窃听用户的接收信噪比（SNR, signal to noise ratio），进而使窃听者不能正确解码数据，从而起到防止窃听的目的 [ 15 15 ] 。 <math id="M7"><mi>K</mi><mo>=</mo><mo>{</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>k</mi><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>K</mi><mo>}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416140&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416139&type= 32.85066605 3.55599999 表示子载波集合。在SCMA网络中，码本与子载波相匹配作为一个基本资源单元 [ 2 2 , 3 3 ] ，这与子载波在OFDMA网络中类似。假定每个码本与 L 个子载波匹配，那么网络共有 <math id="M8"><msubsup><mrow><mi>C</mi></mrow><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416142&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416141&type= 4.23333359 3.97933340 个码本。 <math id="M9"><mi>C</mi><mo>=</mo><mo>{</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>c</mi><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>C</mi><mo>}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416144&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416143&type= 32.00400162 3.55599999 表示码本集合。 图2 图2 为用户、码本与子载波之间的映射关系。定义 <math id="M10"><mi mathvariant="bold-italic">X</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>c</mi><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416146&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416145&type= 15.91733360 4.48733330 表示码本与子载波之间的映射关系，其中 <math id="M11"><msub><mrow><mi>X</mi></mrow><mrow><mi>c</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416147&type= 12.27666664 3.72533321 表示码本 c 使用子载波 k； 否则 <math id="M12"><msub><mrow><mi>X</mi></mrow><mrow><mi>c</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416150&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416149&type= 12.27666664 3.72533321 。定义码本分配的变量 <math id="M13"><mi mathvariant="bold-italic">S</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416151&type= 14.73199940 4.48733330 ，其中 <math id="M14"><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416153&type= 11.93799973 3.72533321 表示码本 c 分配给用户 u ，否则 <math id="M15"><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416156&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416155&type= 11.93799973 3.72533321 。可以看到在子载波与码本之间的映射关系是固定的，即 <math id="M16"><mi mathvariant="bold-italic">X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416158&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416157&type= 2.70933342 2.79399991 是已知的参数。 每个多播数据流被映射到一个码本 c ，得到一个稀疏码字矢量 <math id="M17"><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416163&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416162&type= 3.04799986 3.72533321 后， <math id="M18"><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416164&type= 16.00200081 5.58799982 ，所有的码本 <math id="M19"><mi>C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416167&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416166&type= 2.62466669 2.79399991 被复用到 K 个子载波上。 <math id="M20"><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416169&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416168&type= 2.79399991 4.57200003 表示每个码本 c 上分配的功率，并且满足 <math id="M21"><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></munder><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416170&type= 17.01799965 6.77333355 ，在这里 <math id="M22"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416173&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416172&type= 4.14866638 3.72533321 表示基站的总功率。 由于SCMA码字的稀疏性，可以使接收端采用复杂度较低的消息传递算法（MPA, message passing algorithm）和多用户联合迭代法，从而实现近似于多用户最大似然解码 [ 16 16 ] 的效果。SCMA是码域非正交多址接入技术，将来自一个或多个用户的多个数据流，通过码域扩频和非正交叠加在同一时频资源单元中发送；在接收端采用低复杂度、性能近似最优的迭代消息检测算法，分离出同一时频资源单元中的多个数据层。因此，来自不同数据流的码本在互不干扰时可以被解码 [ 17 17 ] 。本文重点研究SCMA网络中的多播网络能量效率问题。假设每一个用户通过无差错反馈信道反馈其瞬时信道信息给基站，不考虑非多播组用户码本的重复使用情况，并且多播组的链路之间没有相互干扰。 2　 添加人工噪声与最优化问题 1 2.1　添加人工噪声 2 定义用户 u 在码本 c 上所对应的主信道与窃听信道增益矩阵分别为 <math id="M23"><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416175&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416174&type= 5.92666674 3.72533321 和 <math id="M24"><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">e</mi><mo>,</mo><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416177&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416176&type= 5.84200001 3.72533321 ；发送方在主信道的零空间内随机产生均值为0、方差为 <math id="M25"><msubsup><mrow><mi>δ</mi></mrow><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416178&type= 3.13266659 3.97933340 的人工噪声 <math id="M26"><msub><mrow><mi>A</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416180&type= 3.81000018 3.72533321 ，且满足 <math id="M27"><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416183&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416182&type= 17.10266685 3.72533321 。为了在安全多播传输中使用人工噪声技术，首先，需要在每次传输中保证基站发送天线数量多于合法多播用户的接收天线数量（每个码本能够服务的用户数为 <math id="M28"><msub><mrow><mi>L</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416185&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416184&type= 26.50066566 4.57200003 个），这样做的目的是确保传输所用码本 c 上存在合法用户信道的零空间。其次，根据这 <math id="M29"><msub><mrow><mi>L</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416187&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416186&type= 3.38666677 3.72533321 个合法用户反馈的瞬时信道状态信息，在其信道零空间内引入人工噪声。 <math id="M30"><msub><mrow><mi>K</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416188&type= 3.72533321 3.72533321 表示使用码本 c 的子载波的集合。因此，用户 u 在码本 c 上的信噪比可以表示为 <math display="block" id="M31"><msub><mrow><mi>γ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub><mo>=</mo><mfrac><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>k</mi><mo>∈</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></munder><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi><mo>,</mo><mi>k</mi></mrow></msub><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">H</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msubsup><mrow><mi>δ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416191&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416190&type= 36.32199860 12.95399952 其中， <math id="M32"><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi><mo>,</mo><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416193&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416192&type= 4.65666676 4.57200003 表示在子载波 k 上采用码本 c 时加载的功率，本文 <math id="M33"><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416195&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416194&type= 12.44600010 4.57200003 ， <math id="M34"><msubsup><mrow><mi>δ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416196&type= 3.13266659 3.97933340 表示广播信道中在码本 c 上的噪声功率。假设每个码本上的噪声功率都是相等的，例如： <math id="M35"><msubsup><mrow><mi>δ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>δ</mi></mrow><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><msup><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416198&type= 19.30399895 3.97933340 , <math id="M36"><mo>∀</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416201&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416200&type= 12.61533356 3.72533321 。当使用码本 c 时，定义多播组中的主信道与窃听信道的信噪比表示为 <math display="block" id="M37"><msub><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi>u</mi><mo>∈</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></munder><mfenced open="{" close="}" separators="|"><mrow><mrow><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>k</mi><mo>∈</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></munder><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi><mo>,</mo><mi>k</mi></mrow></msub><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">H</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>/</mo><mrow><msubsup><mrow><mi>δ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mrow></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416203&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416202&type= 50.03800201 12.10733318 <math display="block" id="M38"><msub><mrow><mi>β</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><mrow><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>k</mi><mo>∈</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></munder><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi><mo>,</mo><mi>k</mi></mrow></msub><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">H</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>/</mo><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi>δ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>δ</mi></mrow><mrow><mi>v</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416205&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416204&type= 51.30799866 8.88999939 其中， <math id="M39"><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">H</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416207&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416206&type= 14.47799969 5.58799982 表示合法用户 u 与基站间在码本 c 上的主信道增益， <math id="M40"><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">H</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>e</mi><mo>,</mo><mi>c</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416209&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416208&type= 14.30866623 5.58799982 表示窃听用户 e 与基站间在码本 c 上的窃听信道增益， <math id="M41"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416211&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416210&type= 3.55599999 3.72533321 为码本 c 的波束赋形矢量， <math id="M42"><msub><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi><mo>,</mo><mn mathvariant="normal">1</mn><mo> </mo></mrow></msub><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><msub><mrow><mi>h</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi><mo>,</mo><mn mathvariant="normal">2</mn></mrow></msub><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>⋯</mo><mo> </mo><mo> </mo><mo> </mo><msub><mrow><mi>h</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>j</mi></mrow></msub><mo> </mo><mo> </mo><mo> </mo><mo>⋯</mo><mo> </mo><msub><mrow><mi>h</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub><msup><mrow><mo> </mo><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416213&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416212&type= 60.70600128 5.33399963 表示多播组中的第 u 个用户在第 c 个码本上的信道矢量，其中 <math id="M43"><msub><mrow><mi>h</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><mroot><mrow><mi>η</mi></mrow><mrow/></mroot><msubsup><mrow><mi>d</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>j</mi></mrow><mrow><mfrac><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msubsup><msub><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416215&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416214&type= 28.78666687 6.26533318 表示用户 u 与基站端天线 j 之间在码本 c 上的瞬时信道增益， <math id="M44"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416217&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416216&type= 1.86266661 3.64066648 表示路损常数， <math id="M45"><msub><mrow><mi>d</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416218&type= 4.82599974 3.72533321 表示从用户 u 到基站端天线 j 的距离， <math id="M46"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416220&type= 1.94733334 2.79399991 表示路损系数， <math id="M47"><msub><mrow><mi>ρ</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416223&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416222&type= 6.26533318 4.57200003 表示用户 u 与基站端天线 j 之间在码本 c 上的瑞利衰落。 依据香农公式计算码本 c 上传输速率为 <math display="block" id="M48"><msub><mrow><mi>R</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416224&type= 27.17800140 4.65666676 当系统中存在窃听者时，在码本 c 上的保密速率 <math id="M49"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416226&type= 3.64066648 3.97933340 表示为主信道传输速率与窃听信道传输速率之差,即 <math display="block" id="M50"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">[</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>c</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416229&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416228&type= 55.79533386 5.07999992 其中 <math id="M51"><msup><mrow><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416230&type= 28.78666687 3.81000018 ，并且需要满足 <math id="M52"><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><msub><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>></mo><msub><mrow><mi>β</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416232&type= 12.44600010 4.74133348 。（5）式表示，只有在主信道的信道增益高于窃听信道时，才存在保密速率。 用户 u 在码本 c 上的保密速率可以表示为 <math display="block" id="M53"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">[</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>k</mi><mo>∈</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></munder><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi><mo>,</mo><mi>k</mi></mrow></msub><mrow><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">H</mi></mrow></msubsup><msub><mrow><mi>H</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>/</mo><mrow><msubsup><mrow><mi>δ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mrow><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>c</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416235&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416234&type= 63.66933441 15.07066631 <math id="M54"><msub><mrow><mi>U</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416237&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416236&type= 4.06400013 3.72533321 表示使用码本 c 的用户集合，因此用户 u 的保密速率可以表示为： <math display="block" id="M55"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>=</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>u</mi><mo>∈</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></munder><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416239&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416238&type= 20.57399940 7.45066643 2.2　最优化问题 2 <math id="M56"><mi>B</mi><mo>=</mo><mo>{</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>b</mi><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>B</mi><mo>}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416241&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416240&type= 32.00400162 3.55599999 表示分配给基本层数据的码本集合， <math id="M57"><mi>ε</mi><mo>=</mo><mo>{</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>e</mi><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>E</mi><mo>}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416243&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416242&type= 30.90333366 3.55599999 表示分配给增强层数据的码本集合。 <math id="M58"><mi>M</mi><mo>=</mo><mo>{</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>m</mi><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>M</mi><mo>}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416245&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416244&type= 35.39066696 3.55599999 表示满足增强层最小保密速率需求的用户集合即增强层用户集合。本文中 <math id="M59"><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416247&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416246&type= 7.95866680 3.97933340 表示传输基本层数据需要的最小保密速率， <math id="M60"><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416248&type= 7.95866680 3.97933340 表示传输增强层数据需要的最小保密速率。 基本层多播数据的保密速率可以表示为基本层码本保密速率之和。因此，基本层保密速率可以表示为 <math display="block" id="M61"><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>=</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>c</mi><mo>∈</mo><mi>B</mi></mrow></munder><msubsup><mrow><mi>R</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>≥</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416251&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416250&type= 31.49600029 6.77333355 增强层的最小保密速率可以表示为 <math display="block" id="M62"><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>=</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>c</mi><mo>∈</mo><mi>ε</mi></mrow></munder><msubsup><mrow><mi>R</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>≥</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416252&type= 31.49600029 6.77333355 由于能够获得增强层数据的用户的信道质量不同，为了最大化增强层用户的保密容量（保密容量是用户传输保密速率之和），需要充分利用每个码本，分配原则是码本分配给部分增强层用户，要求能够在该码本上得到的增强层数据的用户的保密容量最大。因此，增强层在码本 c 上的保密容量可以表示为 <math display="block" id="M63"><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>=</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>u</mi><mo>∈</mo><mi>M</mi></mrow></munder><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mn mathvariant="normal">1</mn><mo stretchy="false">(</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>≥</mo><msubsup><mrow><mi>R</mi></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416254&type= 48.59866714 6.77333355 这里1(A)用来表示，当A满足条件时，1(A)的值取1，否则取0。 <math id="M64"><msubsup><mrow><mi>R</mi></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416256&type= 6.34999990 4.14866638 表示在使用码本 c 的增强层用户中，用户 <math id="M65"><msup><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416258&type= 2.96333337 3.04799986 的速率，该速率作为码本 c 的速率时，可以使 <math id="M66"><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416261&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416260&type= 4.06400013 3.97933340 最大化。 在SCMA网络中基于分层编码策略的安全多播网络的保密容量可以定义为在多播组中所有用户的保密速率之和，基于（5）和（10）式，网络的保密容量表示为 <math display="block" id="M67"><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>=</mo><mi>U</mi><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>+</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>c</mi><mo>∈</mo><mi>ε</mi></mrow></munder><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">f</mi><mo> </mo><mo> </mo><mtext> </mtext><mi>M</mi><mo>≠</mo><mi mathvariant="normal">∅</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416262&type= 49.69933319 6.77333355 网络消耗的总功率可以表示为： <math display="block" id="M68"><msub><mrow><mi>P</mi></mrow><mrow><mi>c</mi><mo>_</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi></mrow></msub><mo>=</mo><mi>θ</mi><mo stretchy="false">(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi><mi>'</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416265&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416264&type= 38.18466568 4.48733330 其中, <math id="M69"><mi>θ</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416267&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416266&type= 8.55133343 3.30199981 表示功率放大系数， <math id="M70"><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi><mi>'</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416269&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416268&type= 11.17599964 4.14866638 表示基站端电压消耗系数, <math id="M71"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mo>=</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>c</mi><mo>∈</mo><mi>B</mi></mrow></munder><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416270&type= 17.10266685 6.77333355 和 <math id="M72"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub><mo>=</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>c</mi><mo>∈</mo><mi>ε</mi></mrow></munder><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416273&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416272&type= 17.10266685 6.77333355 分别表示传输基本层和增强层的总功率。从（11）和（12）式可以计算出网络保密能量效率(security energy efficiency,SEE)的表达式 <math display="block" id="M73"><msubsup><mrow><mi>η</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>c</mi><mo>_</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>U</mi><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>+</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>c</mi><mo>∈</mo><mi>ε</mi></mrow></munder><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></mrow><mrow><mi>θ</mi><mo stretchy="false">(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi><mi>'</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416275&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416274&type= 51.98533249 11.68400002 信道质量很差的情况，即 <math id="M74"><mi>M</mi><mo>=</mo><mi mathvariant="normal">∅</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416277&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416276&type= 12.27666664 3.64066648 且 <math id="M75"><mi>B</mi><mo>=</mo><mi>C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416278&type= 9.82133293 2.79399991 ，此时没有用户能正确接收增强层的数据。这是由于没有更多的资源用来传输增强层数据。因此，分配所有的码本给基本层的数据，这时分层安全多播策略退化成传统安全多播策略，即 <math id="M76"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416280&type= 10.83733368 3.72533321 ，此时 <math id="M77"><msubsup><mrow><mi>η</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416282&type= 5.84200001 4.82599974 可以表示为 <math display="block" id="M78"><msubsup><mrow><mi>η</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>=</mo><mo>=</mo><mfrac><mrow><mi>U</mi><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></mrow><mrow><mi>θ</mi><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi><mi>'</mi></mrow></msub></mrow></mfrac><mo>,</mo><mo> </mo><mo> </mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">f</mi><mo> </mo><mo> </mo><mtext> </mtext><mi>M</mi><mo>=</mo><mi mathvariant="normal">∅</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416285&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416284&type= 48.25999832 8.72066689 分层安全多播策略退化成传统安全多播策略,其退化概率可以表示为 <math display="block" id="M79"><msubsup><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo><</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup><mfenced open="|" close="" separators="|"><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>≥</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup></mrow></mfenced><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo><</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416287&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416286&type= 79.07866669 4.48733330 本文研究在功率受限的情况下，考虑最优化的分配码本和分配功率使得网络保密能量效率最大化，因此，最优化问题可以表示为 <math display="block" id="M80"><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>ε</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub></mrow></munder><msubsup><mrow><mi>η</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416289&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416288&type= 19.13466644 5.58799982 满足 <math display="block" id="M81"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">C</mi><mn mathvariant="normal">1</mn><mo>:</mo><mtext> </mtext><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>≥</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>≥</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup></mtd></mtr><mtr><mtd><mi mathvariant="normal">C</mi><mn mathvariant="normal">2</mn><mo>:</mo><mtext> </mtext><mi>B</mi><mo>⋂</mo><mi>ε</mi><mo>=</mo><mi mathvariant="normal">∅</mi><mo>,</mo><mi>B</mi><mo>⋃</mo><mi>ε</mi><mo>⊆</mo><mi>C</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">C</mi><mn mathvariant="normal">3</mn><mo>:</mo><mtext> </mtext><mover><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>c</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>C</mi></mrow></mover><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">C</mi><mn mathvariant="normal">4</mn><mo>:</mo><mtext> </mtext><msub><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>></mo><msub><mrow><mi>β</mi></mrow><mrow><mi>c</mi></mrow></msub><mi>c</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">C</mi><mn mathvariant="normal">5</mn><mo>:</mo><mtext> </mtext><mn mathvariant="normal">1</mn><mo>≤</mo><mover><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>u</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>U</mi></mrow></mover><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub><mo>≤</mo><mi>K</mi><mo>,</mo><mo>∀</mo><mi>c</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">C</mi><mn mathvariant="normal">6</mn><mo>:</mo><mtext> </mtext><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub><mo>=</mo><mo>{</mo><mn mathvariant="normal">0,1</mn><mo>}</mo><mo>,</mo><mo>∀</mo><mi>u</mi><mo>,</mo><mo>∀</mo><mi>c</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416291&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416290&type= 43.26466751 41.40200043 在（16）式中，约束项C1表示确保基本层和增强层速率的QoS，约束项C2表示确保在码本集合中的码本不能同时安排给基本层和增强层数据，约束项C3表示码本的功率之和不会超过基站端的总功率，约束项C4表示主信道的信道条件优于窃听信道的信道条件，约束项C5表示码本数量的限制，约束项C6表示码本分配给哪些用户。 3　 次优化算法（SA） 1 上文的最优化问题是一个复杂的混合问题。其最优解可以通过遍历的方法解决，但是其计算复杂度高，很难应用于实际系统。因此，本文提出了两阶段快速的次优化算法，第一阶段为在添加人工噪声下的码本分配，在该阶段根据采用的分层编码的特点，又进一步分解为保密基本层算法和保密增强层算法。第二阶段为在添加人工噪声下的功率分配。 3.1　添加人工噪声的码本分配 2 添加人工噪声时的码本分配是次优化算法的第一阶段。该阶段首先假设每个码本上有相等的功率。我们通过考虑满足添加人工噪声下的安全多播条件下来分配码本，使得网络保密能量效率最大化。因此最优化问题可以重新表示为： <math display="block" id="M82"><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi>B</mi><mo>,</mo><mi>ε</mi><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub></mrow></munder><msubsup><mrow><mi>η</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416292&type= 13.54666710 5.58799982 满足： <math display="block" id="M83"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">C</mi><mn mathvariant="normal">1</mn><mo>:</mo><mtext> </mtext><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>≥</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>≥</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup></mtd></mtr><mtr><mtd><mi mathvariant="normal">C</mi><mn mathvariant="normal">2</mn><mo>:</mo><mtext> </mtext><mi>B</mi><mo>⋂</mo><mi>ε</mi><mo>=</mo><mi mathvariant="normal">∅</mi><mo>,</mo><mi>B</mi><mo>⋃</mo><mi>ε</mi><mo>⊆</mo><mi>C</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">C</mi><mn mathvariant="normal">3</mn><mo>:</mo><mtext> </mtext><mover><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>c</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>C</mi></mrow></mover><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">C</mi><mn mathvariant="normal">4</mn><mo>:</mo><mtext> </mtext><msub><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>></mo><msub><mrow><mi>β</mi></mrow><mrow><mi>c</mi></mrow></msub></mtd></mtr><mtr><mtd><mi mathvariant="normal">C</mi><mn mathvariant="normal">5</mn><mo>:</mo><mtext> </mtext><mn mathvariant="normal">1</mn><mo>≤</mo><mover><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>u</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>U</mi></mrow></mover><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub><mo>≤</mo><mi>K</mi><mo>,</mo><mo>∀</mo><mi>c</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">C</mi><mn mathvariant="normal">6</mn><mo>:</mo><mtext> </mtext><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub><mo>=</mo><mo>{</mo><mn mathvariant="normal">0,1</mn><mo>}</mo><mo>,</mo><mo>∀</mo><mi>u</mi><mo>,</mo><mo>∀</mo><mi>c</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416295&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416294&type= 43.26466751 41.40200043 由于（16）式是最优化算法，需要码本功率与码本分配同时进行联合分配，计算复杂度高，因此我们采用次优化算法，分为码本分配和功率分配。在码本分配阶段，假设每个码本是等功率分配，在码本分配后，再对分配的码本进行重新的功率分配，以降低计算复杂度。 1）　添加人工噪声的保密基本层算法 3 本文中需要使用尽量少的码本来传输基本层数据，剩余尽量多的码本来传输增强层数据，这是由于增强层保密吞吐量提升，更利于提升网络的保密能量效率。添加人工噪声的保密基本层算法如算法1所示。 算法 1 保密基本层码本分配算法 1：计算任意用户 u 在码本 c 上的保密速率 <math id="M84"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> 2： <math id="M85"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>=</mo><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi></mrow><mrow><mi>u</mi><mo>∈</mo><mi>U</mi></mrow></munder><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> ，要求 <math id="M86"><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup><mo>≤</mo><mstyle displaystyle="true"><mover><mrow><munder><mrow><mo largeop="true" movablelimits="false">∑</mo></mrow><mrow><mi>c</mi><mo>=</mo><mn>1</mn></mrow></munder></mrow><mrow><mi>C</mi></mrow></mover></mstyle><msubsup><mrow><mi>R</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> 3：for c = 1 to C 4：找一个码本 <math id="M87"><msup><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></munder><msubsup><mrow><mi>R</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> 5: 分配 <math id="M88"><msup><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> 给 <math id="M89"><msub><mrow><mi>L</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><mn>1</mn><mo stretchy="false">)</mo></math> 个用户 6：更新 <math id="M90"><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>R</mi></mrow><mrow><msup><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> ，更新码本集合 <math id="M91"><mi>C</mi></math> 7：直到所有的用户都满足 <math id="M92"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>≥</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup></math> 8: 计算剩余码本集合，并提供给增强层 2）　添加人工噪声的保密增强层算法 3 在基本层码本分配完毕后，将剩余的码本根据每个用户不同的QoS需求，分配给增强层数据使用，增强层码本分配原则是以能够最大化增强层保密容量为原则。添加人工噪声的保密增强层算法如算法2所示。 算法 2 保密增强层码本分配算法 1: 计算用户在增强层码本上的保密速率 <math id="M93"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> 2: <math id="M94"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> 按递减次序排列。 3：在增强层的码本集合中for c = 1 to C 4: 计算 <math id="M95"><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>=</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>u</mi><mo>∈</mo><mi>M</mi></mrow></munder><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mn>1</mn><mo stretchy="false">(</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi>c</mi></mrow></msubsup><mo>≥</mo><msubsup><mrow><mi>R</mi></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><mi>c</mi></mrow><mrow><mi>c</mi></mrow></msubsup><mo stretchy="false">)</mo></math> 5: 找用户满足 <math id="M96"><msup><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">*</mi><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><mi>u</mi><mo>∈</mo><mi>M</mi></mrow></munder><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> 6: 设置 <math id="M97"><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mn>1</mn><mo>,</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>≥</mo><msubsup><mrow><mi>R</mi></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo><</mo><msubsup><mrow><mi>R</mi></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> ， <math id="M98"><mo>∀</mo><mi>u</mi></math> 7: 判断 <math id="M99"><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub></math> 的数量是否满足 <math id="M100"><msub><mrow><mi>L</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>-</mo><mn>1</mn><mo stretchy="false">)</mo></math> ，如果满足，则更新 <math id="M101"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi>R</mi></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">*</mi><mi mathvariant="normal">*</mi></mrow></msup><mo>,</mo><mi>c</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> ，否则重新选择。 8: 码本分配完毕或所有用户的 <math id="M102"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup></math> 被满足 3.2　添加人工噪声的功率分配 2 在第一阶段码本分配完成后， <math id="M103"><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi><mo>,</mo><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416301&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416300&type= 5.07999992 3.72533321 可以确定，并且基本层码本集合 <math id="M104"><mi>B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416302&type= 2.53999996 2.79399991 和增强层码本集合 <math id="M105"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416304&type= 1.43933344 2.79399991 也可以确定，由于每个码本采用等功率分配，不能有效地利用有限的功率，因此，在第二阶段的功率分配中，通过调整各个码本上分配的功率来达到最大化网络保密能量效率 <math id="M106"><msubsup><mrow><mi>η</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416307&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416306&type= 5.84200001 4.82599974 ，因此第二阶段的最优化问题可以表示为 <math display="block" id="M107"><munder><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow></msub></mrow></munder><msubsup><mrow><mi>η</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416308&type= 13.20799923 5.58799982 满足 <math display="block" id="M108"><mover><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>c</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>C</mi></mrow></mover><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416310&type= 28.36333275 8.38199997 功率分配问题可采用拉格朗日方法求解，构造（18）式的拉格朗日算子函数如下 <math display="block" id="M109"><mi>L</mi><mo stretchy="false">(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mi>η</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msubsup><mo>-</mo><mi>λ</mi><mo stretchy="false">(</mo><mover><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>c</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>C</mi></mrow></mover><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>-</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416312&type= 49.53000259 8.38199997 其中 <math id="M110"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416315&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416314&type= 1.77800000 2.79399991 为拉格朗日算子。对（19）式中的 L 求偏导，即 <math id="M111"><mrow><mrow><mo>∂</mo><mi>L</mi></mrow><mo>/</mo><mrow><mo>∂</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mrow><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416317&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416316&type= 17.94933319 4.14866638 ，得 <math display="block" id="M112"><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><msub><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><msub><mrow><mi>β</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfrac><mo>-</mo><mi>λ</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416319&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416318&type= 56.64200211 8.63599968 因此，得到每个码本上可分配的功率表示为 <math display="block" id="M113"><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><mfrac><mrow><mroot><mrow><msup><mrow><mfenced separators="|"><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>-</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></msub><msub><mrow><mi>β</mi></mrow><mrow><mi>c</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>-</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow></mfenced></mrow><mrow/></mroot><mo>-</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>-</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></msub><msub><mrow><mi>β</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416321&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416320&type= 71.03533173 15.15533447 当求解 <math id="M114"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416322&type= 1.77800000 2.79399991 时可通过将 <math id="M115"><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416325&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416324&type= 2.79399991 4.57200003 代入 <math id="M116"><mover><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>c</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder></mrow><mrow><mi>C</mi></mrow></mover><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416326&type= 17.01799965 8.38199997 进而求得。添加人工噪声的功率分配算法如算法3所示。 算法 3 功率分配算法 1： <math id="M117"><msub><mrow><mi>α</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>></mo><msub><mrow><mi>β</mi></mrow><mrow><mi>c</mi></mrow></msub></math> 时 2：根据（21）式计算 <math id="M118"><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub></math> 3：判断 <math id="M119"><mstyle displaystyle="true"><mover><mrow><munder><mrow><mo largeop="true" movablelimits="false">∑</mo></mrow><mrow><mi>c</mi><mo>=</mo><mn>1</mn></mrow></munder></mrow><mrow><mi>C</mi></mrow></mover></mstyle><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub></math> 是否成立，若成立，则完成功率分配，否则,下一步 4：重选 <math id="M120"><mi>λ</mi></math> ，并计算 <math id="M121"><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub></math> 5: 判断 <math id="M122"><mo>∀</mo><mi>c</mi><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>≥</mo><mn>0</mn><mi mathvariant="normal">是</mi><mi mathvariant="normal">否</mi><mi mathvariant="normal">成</mi><mi mathvariant="normal">立</mi><mo>,</mo></math> 6：直到 <math id="M123"><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>≥</mo><mn>0</mn><mo>,</mo><mo>∀</mo><mi>c</mi></math> 4　 仿真实验 1 本文分别仿真了分层编码策略在SCMA网络和OFDMA网络中采用安全多播传输时的保密能量效率、网络退化概率和系统保密速率。在OFDMA网络中，使用子载波来代替码本分配，并且假设基本层和增强的总功率是均等地分配给OFDMA网络中的子载波。同时仿真分析了SA算法分别在SCMA网络（SA-SCMA）和OFDMA网络（SA-OFDMA）中的保密能量效率、网络退化概率和系统保密速率，并与等功率分配和码本随机分配（EPCRA, equal power allocation and codebook random allocation algorithm）算法在SCMA网络（EPCRA-SCMA）与等功率分配和子载波随机分配（EPSRA, equal power allocation and subcarriers randomly allocation algorithm）算法在OFDMA网络（EPSRA-OFDMA）进行比较。同时还仿真比较了传统安全多播策略(CSM)算法在SCMA网络（CSM-SCMA）和OFDMA网络（CSM-OFDMA）中的网络保密能量效率。仿真参数为：子载波数 K =4，子载波带宽为180 kHz，噪声功率谱密度为 <math id="M124"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416331&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416330&type= 3.72533321 2.79399991 174 dBm/Hz，码本数 <math id="M125"><mi>C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416332&type= 2.62466669 2.79399991 =6，路损系数 <math id="M126"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416334&type= 1.94733334 2.79399991 =4，路损常数 <math id="M127"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416336&type= 1.86266661 3.64066648 = <math id="M128"><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416338&type= 6.85800028 3.04799986 ，用户到基站的最大距离为1 000 m， <math id="M129"><msub><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416341&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416340&type= 4.14866638 3.72533321 =30 dBm， <math id="M130"><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416343&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416342&type= 7.95866680 3.97933340 =0.2 Mbit/s， <math id="M131"><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mo>-</mo><mi mathvariant="normal">s</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416344&type= 7.95866680 3.97933340 =1 Mbit/s， <math id="M132"><msub><mrow><mi>P</mi></mrow><mrow><mi>d</mi><mi>'</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416347&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416346&type= 4.23333359 3.72533321 =412.4 W， <math id="M133"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416349&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416348&type= 1.77800000 2.79399991 =22.6，基站天线数为8。 从 图3 图3 可以看到，分层安全多播的网络保密能量效率随着用户数的增多，首先提升随后降低，最后归为平坦。这是由于起初随着多播用户数量的增加，能够收到增强层数据的用户数开始增多，但是随着用户数的增加，信道质量差的用户出现的概率增大，因此能够获得增强层数据的用户数降低。随着用户数的进一步增加，网络保密容量不会再随着用户数量的增加而提升，因此网络保密能量效率归为平坦。本文提出的SA⁃SCMA算法与EPCRA算法和在SCMA网络中的传统安全多播策略（CSM）相比，能够大幅度提升网络保密能量效率。从仿真结果还可以看出，在SCMA网络中应用本文算法所获得的网络保密能量效率，要远高于在OFDMA网络中应用类似的算法。 图4 图4 仿真结果表明，网络保密能量效率随着基站发射功率的增加，首先提升，随后降低。这是由于随着功率的提升，获得增强层数据的用户增多，网络保密吞吐量提升，但是随着功率的不断提升，网络的吞吐量会出现天花板效应，即网络保密吞吐量不会随着功率的进一步提升而提升，而是趋于平稳，因此在出现天花板效应后，如果再进一步提升发射功率，其网络保密能量效率将会随着功率的消耗而降低。 从 图5 图5 仿真结果可以看到，随着用户数量的增加，平均退化概率也在提升。这是由于随着用户数量的增多，差用户出现的概率也在增加，需要更多物理资源分配给基本层来传输，因此用来传输增强层数据的物理资源减少，平均退化概率提升。 图6 图6 仿真结果表明，随着基站总功率的增加，平均退化概率也在降低。这是因为，当基站总功率增加时，提供较少的物理资源就能满足基本层速率需求，因此提供给增强层的物理资源也就越多。从仿真实验中还可以看到，在采用相同算法时，SCMA网络的平均退化概率要低于OFDMA网络的平均退化概率。 图7 图7 仿真结果表明，当 <math id="M134"><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416363&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3416362&type= 5.41866684 3.97933340 （本节开始时该参数定义为512 Kbit/s）增加时，网络需要分配更多的物理资源给传输基本层数据，因此减少了分配给增强层的物理资源。而增强层保密容量的减少，对整个系统保密容量的提升，影响很大，所以网络保密能量效率将降低。当所有的物理资源都分配给基本层数据时，分层安全多播传输将会降解为传统安全多播传输，即没有剩余资源分配给增强层使用，所有用户只能得到基本层数据，因此网络保密能量效率趋于平坦。 当窃听者的平均信噪比为25 dB时， 图8 图8 显示了SA算法在SCMA网络下的系统保密速率与码本数目从1到64的变化关系。 C =1时，由于可用码本过少，导致可用的子载波也过少，此时不存在合法用户的信道零空间，因此系统的保密速率为零，系统无法进行物理层安全传输。当码本数量增加时，由于存在合法用户的信道零空间，因此可以采用人工噪声技术来实现系统的保密传输，并且可以看到随着码本数量的增加，系统的保密速率也在提升，这是由于系统可以提供更多的码本用来传输增强层数据，进而提升了系统的保密速率。随着合法用户的平均信噪比的提升，系统保密速率也在增加，这是因为信道条件好的用户数量在增加，会有更多的用户得到增强层的数据，因此系统的保密速率也在增加。 5　 结 语 1 本文在SCMA网络中，研究通过分层安全多播策略来最大化网络保密能量效率的最优化问题。为了减少求解最优化问题的计算复杂度，又提出了码本分配阶段和功率分配阶段的两阶段次优化算法。在码本分配阶段，根据分层编码的特点，将其又分解为添加人工噪声的基本层码本分配与添加人工噪声的增强层码本分配。仿真结果显示，本文提出的算法在SCMA网络中的网络保密能量效率要高于在OFDMA网络中的，并且本文提出的算法的系统性能要优于传统安全多播策略下得到的系统性能。 参考文献 Imt- 2020 5 G) promotion group-white paper on 5 G vision and requirements [EB/OL].[ 2014-05-02 ]. http://www.doc88.com/p-4307074793643.html http://www.doc88.com/p-4307074793643.html . 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DOI:10.14188/j.1671-8836.2019.0511. 辽宁省博士科研启动基金项目（201601083）；辽宁省自然科学基金（20180550046）；中国刑事警察学院2019年重大项目培育计划（D2019001） 2019-09-02 2020-04-24 2020-09-30 武汉大学学报（理学版） 2 66