A NORVEL POWER ALLOCATION ALGORITHM under CoMP with CA

NOMA下行链路中用户匹配和功率分配算法杨路; 吴芳炜; 龙恳; 陈德建【期刊名称】《《计算机工程与设计》》【年(卷),期】2019(040)011【总页数】7页(P3061-3066,3157)【关键词】非正交多址; 加权二分图; 用户匹配; 比例公平; 功率分配【作者】杨路; 吴芳炜; 龙恳; 陈德建【作者单位】重庆邮电大学通信与信息工程学院重庆400065【正文语种】中文【中图分类】TN929.50 引言非正交多址接入技术(non-orthogonal multiple access，NOMA)通过将多个用户叠加在同一资源块上传输，可以提升系统设备接入量，提高频谱效率[1,2]，而在实际的NOMA下行链路中，合理的用户匹配和功率分配算法影响着系统的和速率和用户间的公平性[3,4]。

目前关于NOMA下行链路用户匹配的研究中，文献[5,6]介绍了一种随机用户配对方法，算法将小区中所有用户分为若干个用户数相等的集合，然后在每个用户集合中选出信道增益差异最大的用户进行配对。

由于该算法中用户集合通过随机选择得到，不能保证总体系统性能较优。

文献[7]通过设置限制条件对候选用户集进行筛选,减小用户集的范围,降低算法复杂度,但不能保证每个用户的数据速率。

文献[8]提出一种基于用户信道状态排序的用户匹配算法(channel state sorting-pairing algorithm，CSS-PA)，首先对所有用户信道状态信息按升序或者降序排列，然后按照折半配对的方式进行用户配对，算法的复杂度较低，也可以保证配对用户间具有一定的信道状态差异，但无法保证每次都可以使系统总吞吐量达到较优。

当用户完成匹配后，发射端需要在配对用户间要进行功率分配。

全搜索功率分配算法FSPA(full search power allocation)[9]通过对候选用户集遍历所有的功率分配方案，来达到最优的系统和速率，但算法的复杂度较高。

Dynamic topology:As the channel of communicationchanges, some of the neighbors who were reachable on theprevious channel might not be reachable on the currentchannel and vice versa. As a result the topology of the network changes with the change in frequency of operation resulting in route failures and packet loss.Heterogeneity:Different channels may support differenttransmission ranges, data rates and delay characteristics.Spectrum-Handoff delay:For each transition from onechannel to another channel due to the PU’s activity, thereis a delay involved in the transition called Spectrum- Handoff delay.All these factors decrease the predictability of the cause oftransit-delay and subsequent packet loss on the network. Thetime latency during channel hand-off in cognitive networksmight cause the TCP round trip timer to time out. TCP willwrongly recognize the delays and losses due to the abovefactors as network congestion and immediately take steps toreduce the congestion window size knowing not the cause ofpacket delay. This reduces the efficiency of the protocol insuch environments.动态技术：随着信道通信的变化，一些邻进信道的用户在原信道没有发生变化而在新信道发生变化，或者相反。

最佳中继协作通信系统的功率分配算法李国兵1,朱世华1,惠 1,2(11西安交通大学电子与信息工程学院,陕西西安710049;21西安理工大学自动化与信息工程学院,陕西西安710048) 摘　要:　为提高基于最佳中继选择的协作通信系统的性能,提出了以最小化系统中断概率为目标的功率分配算法.首先建立了系统的优化模型并证明了待解的优化问题实质是凸优化问题,由此提出了最优功率分配算法并给出了算法步骤.其次,在此基础上提出了一种有效的次最优功率分配算法,该算法计算简单且仅需已知各个中继节点的平均信道状态信息,无需在传输中实时更新,因而不增加系统的额外开销.仿真结果表明,本文提出的最优算法和次最优算法所得到的功率分配方案与穷举搜索方法的结果非常接近;与等功率分配方案相比,这两种算法均能显著提高系统的中断概率性能.关键词:　协作通信;无线中继;功率分配;凸优化中图分类号:　T N925 文献标识码:　A 文章编号:　037222112(2008)1021944205Power Allocation in Opportunistic Cooperative Relaying SystemsLI G uo 2bing 1,ZH U Shi 2hua 1,H UI Hui 1,2(11School o f Electronics and Information Engineering ,Xi ’an Jiaotong Univer sity ,Xi ’an ,Shaanxi 710049,China ; 21School o f Automation and Information Engineering ,Xi ’an Univer sity o f Technology ,Xi ’an ,Shaanxi 710048,China )Abstract :　To enhance the performance of opportunistic cooperative relaying systems ,this paper proposes the power allocation algorithms aiming at minimizing the outage probability.The optimization model of the system is constructed and proved to be a con 2vex optimization problem.Then the optimal power allocation algorithm and its operation steps are presented.Furthermore a simple and effective near 2optimal power allocation strategy is developed ,which only depends on the average channel gains of the relays and thus incurs little overhead.Simulation and numerical results show that significant performance gains can be achieved by the two pro 2posed power allocation algorithms.K ey words :　cooperative communications ;wireless relaying ;power allocation ;Convex optimization1　引言 协作通信技术通过用户之间彼此共享天线而引入空间分集,从而有效对抗无线信道的多径衰落,成为近年来无线通信领域的研究热点[1～3].针对存在多个中继节点的无线网络,分布式空时码受到了广泛关注,精心设计的分布式空时码可以带来显著的性能增益[4].但是随着参与协作的节点数增多,同步、节点间协调等问题成为分布式空时码的设计难点.合理地取舍协作节点是解决这一难题的方法之一.文献[5～8]提出了基于最佳中继选择的协作通信策略,即每次传输只选择一个最佳的中继参与协作,使协作过程得以简化.文献[5～11]分析并证明了在中继已知信道状态信息的情况下,该策略性能甚至优于所有中继参与的协作.针对上述文献中尚未完全解决的功率分配问题[10],本文将针对解码转发型的最佳中继选择协作通信系统,提出以最小化系统中断概率为目标的最优和次最优功率分配算法.2　系统模型 考虑一个通用的包含K +2个节点的半双工两跳无线中继网络,源节点S 通过中继向目的节点D 发送信息,信息传输过程存在K 个潜在的中继节点R ={1,2,…,K }.一次传输过程包括两个阶段:第一阶段,源节点向所有中继广播信息,中继进行监听;第二阶段,从所有中继中挑选出最佳中继,仅由该中继向目的节点转发信息.由于第二个阶段中只有一个节点在发送信号,因此目的端不存在接收信号不同步的问题.假设信道服从准静态瑞利平坦衰落,并且在一次传输中保持不变,而在各次传输中相互独立.任意节点i 和节点j 之间的信道增益h ij 是均值为0,方差为Ωij 的复高斯随机变量,因此信道的功率增益|h ij |2服从均值为Ω-1ij 的负收稿日期:2007209210;修回日期:200824220基金项目:国家自然科学基金(N o.60372055);教育部高等学校博士学科点专项科研基金(N o.20030698027)第10期2008年10月电 子 学 报ACT A E LECTRONICA SINICA V ol.36　N o.10Oct.　2008指数分布.信道中的噪声是均值为0,单边功率谱密度为N 0的加性复高斯白噪声.另外,为了在传输的第二阶段确定最佳中继,假设中继已知自身瞬时信道状态信息.进一步地,我们假设一次传输过程中的总功率受限,即有P s =ζP tot ,P r =(1-ζ)P tot .其中,P tot 表示一次传输的总功率,ζ∈(0,1]是功率分配的比例因子,P s 、P r 分别表示源和中继节点的发射功率.对于最佳中继的选择,目前有两种方式[10,11],一种称为“Reactive ”方式,即传输时在所有能够正确解码的节点中选择与目的端信道条件最好的中继进行传输;另一种称为“Proactive ”方式,即在每次传输之前首先根据信道条件选择最好的中继,然后在传输的第二阶段使用该中继进行合作.现有文献的分析和仿真都表明,这两种合作策略具有完全相同的中断概率性能.其中断概率为[10,11]P outage =∏Kk =11-e-22R-1SNR1ζΩSk +1(1-ζ)ΩkD(1)其中,R 表示系统端到端的频谱效率,单位是H z ;SNR >P tot /N 0,ΩSk 和ΩkD 分别表示源节点S 到第k 个中继、第k 个中继到目的节点D 的平均信道功率增益.3　最优功率分配算法 由式(1)可知,基于最小中断概率的最优功率分配问题等效为求解ζ3=arg m in 0<ζ≤1∏K k =11-e-22R-1SNR1ζΩSk +1(1-ζ)ΩkD(2)当信噪比较高时,式(2)可近似为ζ3=arg m in 0<ζ<1∏Kk =11ζΩSk+1(1-ζ)Ω(3)由于式(2)中的变量R 和SNR 对最终结果没有影响,为简化表达式,它们在式(3)中没有出现.因此最优化问题可以表示为min∏Kk =11ζΩSk+1(1-ζ)ΩkDs.t.　0<ζ<1(4)这是一个有约束非线性规划问题.为求最优功率分配方案,令f (ζ)>∏Kk =11ζΩSk+1(1-ζ)ΩkD ,其驻点满足f ′(ζ)=∑Kk =1-1ζ2ΩSk+1(1-ζ)2ΩkD・∏Kj =1,j ≠k1ζΩSj +1(1-ζ)ΩjD =0(5)由式(5)可以看出,尽管目标函数的导数形式不难得到,但是试图通过解式(5)这样的方程直接得到驻点的解析表达式却非常困难.为此,我们考虑使用数值计算方法得到最优解.定理1　最优化问题(4)是一个凸优化问题,且目标函数f (ζ)>∏Kk =11ζΩSk+1(1-ζ)ΩkD 是严格下凸函数.证明:由于约束条件0<ζ<1显然是一个凸集,因此只需证明目标函数f (ζ)>∏Kk =11ζΩSk+1(1-ζ)ΩkD 是定义在凸集0<ζ<1上的严格下凸函数.这一问题可以转化为证明如下两个结论:(1)ln f (ζ)在0<ζ<1上是严格下凸函数(见附录A );(2)ln f (ζ)在0<ζ<1上严格下凸是f (ζ)在0<ζ<1上严格下凸的充分条件(见附录B ).证毕.根据定理1可知,凸优化问题具有唯一的全局最优解,且该最优解是目标函数f (ζ)在ζ∈(0,1)上的极小点[12].由此我们提出计算最优功率分配因子ζ3以及最小中断概率的迭代算法:步骤1　定义初始值ζ(n )=ζ0=015,ε=10-4.若f ′(ζ0)<ε,则迭代过程结束,输出ζ3=ζ0,否则转步骤2;步骤2　ζ(n +1)=ζ(n )+Δζ(n ),Δζ(n )=-f ′(ζ(n ))・λ(n ),其中-f ′(ζ(n ))为下降方向,λ(n )=2-m 表示步长,m 取使0<ζ(n +1)<1且f (ζ(n +1))-f (ζ(n ))<0的最小非负整数;步骤3　若|f (ζ(n +1))-f (ζ(n ))|<ε或|Δζ(n )|<ε或|f ′(ζ(n +1))|<ε,则迭代过程结束,输出ζ3=ζ(n +1),否则令n =n +1,转步骤2.因此,我们得到采用最优功率分配时的系统中断概率:P minout=∏Kk =11-e-22R-1SNR1ζ3ΩSk +1(1-ζ3)ΩkD(6)尽管不能得到最佳功率分配因子ζ3的闭式表达,我们从式(5)仍然可以看出:最终的功率分配方案只与平均信道增益有关,而与信息速率、发射功率等因素无关.因此在平均信道增益变化较慢的场合,完全可以在传输开始前事先计算功率分配因子并通知所有协作节点,并不需要过多的系统开销.4　次最优功率分配算法 根据定理1可知,f (ζ)在ζ∈(0,1)上存在极小值点,那么f ′(ζ)=0在ζ∈(0,1)上必有解,因此考虑从式(5)出发寻找进一步简化功率分配的算法.考虑到在一般情况下,式(5)中∏Kj =1,j ≠k1ζΩSj+1(1-ζ)ΩjD的值对所有节点k 相差不大且不为0,因此我们将式(5)近似为∑Kk =1-1ζ2ΩSk+1(1-ζ)2ΩkD>0(7)由此,我们得到功率分配的一种次最优方案:5491第　10　期李国兵:最佳中继协作通信系统的功率分配算法ζ3=∑Kk =1Ω-1Sk∑Kk =1Ω-1Sk+∑Kk =1Ω-1kD (8)可见,次最优功率分配方案只与∑Kk =1Ω-1Sk和∑Kk =1Ω-1kD的比值有关,在二者相差不大时,等功率分配接近或等于最优功率分配;而当二者相差较大时,次最优功率分配方案将比等功率分配带来更大的性能增益.5　仿真实验结果 本文在准静态瑞利平坦衰落信道下对功率分配算法的效果进行了仿真.在下面的仿真中,取R =1bps/H z ,节点i 和j 节点之间信道的平均功率增益与距离d ij的α次方成反比,即Ωij=cd -αij ,其中c 是与传播环境有关的常数,α是路径损耗指数,取值一般在2～5之间.不失一般性,在仿真中我们取α=3,c =1.图1和图2给出了系统中存在6个中继节点且源与中继、中继与目的之间的距离都相等时的仿真结果,此时取信道的平均信道增益{ΩSk }K k =1={ΩkD }K k =1=1.图1对不同功率分配因子时的系统中断概率进行了数值计算和蒙特卡罗仿真.通过图1我们可以看到,在这种情况下,等功率分配是最佳的功率分配方案.而观察式(8)可以发现,由于此时∑Kk =1Ω-1Sk和∑Kk =1Ω-1kD相等,因此根据次最优算法得到的功率分配方案就是等功率分配.进一步地,从图2中的数值仿真结果也可以看出,本文的最优功率分配、次最优功率分配算法得到的分配方案与等功率分配的中断概率性能相吻合. 采用文献[10]中使用的非对称网络拓扑,取信道的平均信道功率增益{ΩSk }K k =1和{ΩkD }K k =1为{415,015,014,013,012,011}.也可以看到,尽管此时6个中继节点的信道增益各不相同,但由于∑Kk =1Ω-1Sk和∑Kk =1Ω-1kD相等,因此根据式(8)可知,与{ΩSk }Kk =1={ΩkD }K k =1=1类似,等功率分配是最佳的功率分配方案.如图3、图4所示: 上述仿真表明,本文提出最优功率分配和次最优功率分配算法能够准确获得最优的功率分配方案.为了进一步说明本文算法的性能,图5和图6给出了一种源与中继、中继与目的之间的距离不相等时的仿真结果.为使结果具有一般意义,我们将源与目的之间的距离归一化,并选取了6个与源节点相对接近的点作为中继.在本次仿真中,取{d Sk }K k =1为{010962,011556,012279,012460,012747,013213},取{d kD }K k =1为{019055,018508,018004,018004,017379,016791}.相应地,{ΩSk }Kk =1为{112316,26516,8414,6712,4813,3012},6491 电 子 学 报2008年{ΩkD}K k=1为{113469,116238,119505,119505,214890, 311927}.从图5的仿真结果可以看到,最优的功率分配因子的值在0115左右,这可由穷举搜索所有的功率分配可能性而得到准确结果.仿真中由最优和次最优功率分配算法得到的结果见表1:表1　几种方法获得的功率分配因子所用算法最优功率次最优功率穷举分配算法分配算法搜索功率分配因子0.14730.14230.1501 如表1所示,本文所提出的两种功率分配算法都与穷举搜索方法得到的功率分配因子非常接近,其误差小于10-2.事实上,通过调整精度ε,最优功率分配算法还可以进一步逼近穷举搜索.另外,从图6可以进一步看到,本文提出的这两种方法具有几乎相同的中断概率性能,同时二者与等功率分配方案相比,都可获得2dB左右的性能增益.此外,由于本文的两种算法是在中断概率的近似式(3)提出的,因此精度略低于穷举搜索方法.但是从计算复杂度来看,本文的两种算法远低于穷举搜索方法.首先,为了获得不同信噪比下的功率分配因子,穷举搜索对每一个给定的信噪比都需要重新计算.而本文的两种算法都与信噪比无关,计算一次即可获得;其次,在信噪比给定的情况下,本文的算法复杂度也低于穷举搜索.以本文仿真为例,为了获得如表1所示的功率分配方案,使用穷举搜索方法需要进行10000次搜索和比较,而本文提出的最优功率分配算法只需4次迭代即可完成.次最优算法则可根据式直接计算出结果,其计算复杂度更低.6　结论 本文研究了在基于最佳中继选择的协作通信系统中的功率分配策略,给出了以最小化系统中断概率为目标的非线性规划模型并证明待解的优化问题是一个凸优化问题,提出了针对凸优化问题求解的功率分配算法并给出了算法步骤.在此基础上提出了一种次最优的功率分配算法.该算法简化了计算过程,同时其性能与最优功率分配算法以及穷举搜索得到的结果非常接近.本文提出的算法复杂度低,且只需已知平均信道增益信息,因而对系统负担小,有助于协作通信技术的实用.附录A证明ln f(ζ)在ζ∈(0,1)是下凸函数:ln f(ζ)=∑Kk=1ln1ζΩSk+1(1-ζ)ΩkD是多个对数函数之和,对ln1ζΩSk+1(1-ζ)ΩkD的二阶导数进行整理,可得ln1ζΩSk+1(1-ζ)ΩkD″=1ζΩSk+1(1-ζ)ΩkD-1-1ζ2ΩSk+1(1-ζ)2ΩkD′=-1ζΩSk+1(1-ζ)ΩkD-2-1ζ2ΩSk+1(1-ζ)2ΩkD2 +1ζΩSk+1(1-ζ)ΩkD-12ζ3ΩSk+2(1-ζ)3ΩkD=1ζΩSk+1(1-ζ)ΩkD-2・1ζ2ΩSk+1(1-ζ)2ΩkD2　+2ζ3(1-ζ)ΩSkΩkD+2ζ(1-ζ)3ΩSkΩkD由上式不难看出,任取ζ∈(0,1),对所有k=1,…,K,有ln1ζΩSk+1(1-ζ)ΩkD″>0,显然有[ln f(ζ)]″=∑Kk=1ln1ζΩSk+1(1-ζ)ΩkD″>0,即ln f(ζ)的二阶导数严格大于0,因此ln f(ζ)是严格下凸函数.证毕.附录B证明f(ζ)是严格下凸函数:7491第　10　期李国兵:最佳中继协作通信系统的功率分配算法因为ln f (ζ)是严格下凸函数,因此对于任意ζ1,ζ2∈(0,1),λ1,λ2∈(0,1)且λ1+λ2=1,有λ1ln f (ζ1)+λ2ln f (ζ2)>ln f (λ1ζ1+λ2ζ2)(9)另一方面,将f (ζ)视为ln f (ζ)的自变量,那么由于自然对数函数在实数范围内都是严格上凸函数,所以ln f (ζ)是关于f (ζ)的下凸函数.因此有λ1ln f (ζ1)+λ2ln f (ζ2)<ln [λ1f (ζ1)+λ2f (ζ2)](10)综合式(9)、(10),并由对数函数的单调性可得,对于任意ζ1,ζ2∈(0,1),λ1λ2∈(0,1)且λ1+λ2=1,有λ1f (ζ1)+λ2f (ζ2)>f (λ1ζ1+λ2ζ2)(11)亦即f (ζ)是严格下凸函数.证毕.参考文献:[1]殷勤业,张莹,丁乐等.协作分集:一种新的空域分集技术[J ].西安交通大学学报,2005,39(6):551-557.Y in Qinye ,Zhang Y ing ,Ding Le ,et al.Cooperation diversity :a new spatial diversity technique [J ].J ournal of Xi ′an Jiaotong University ,2005,39(6):551-557.(in Chinese )[2]A Sendonaris ,E Erkip ,B er cooperation diversity 2Part 1:system description [J ].IEEE Transactions on Communi 2cation ,2003,51(11):1927-1938.[3]A Nosratinia ,T E Hunter ,A Hedayat.Cooperative communica 2tion in wireless networks [J ].IEEE Communications Maga 2zine ,2004,42(10):74-80.[4]J N 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一种低复杂度非正交多址接入功率分配算法谭歆;肖杰;高翔;吴广富【期刊名称】《电子技术应用》【年(卷),期】2017(43)4【摘要】Power allocation is an important research issue of resource allocation in non-orthogonal multiple access system.Optimal water-filling power allocation algorithm can improve the system performance,but the complexity of the algorithm is relatively high.This paper presents a low complexity power allocation algorithm.Firstly,the algorithm uses water-filling principle to subcarriers in order to get the total multiplexed power.Then the fractional transmission power allocation algorithm is used to reallocate the total multiplexed power for multiplexed users on single subcarrier.The simulation result shows that the proposed algorithm can obviously reduce the computational complexity under the circumstance of the performance loss of less than 3％ compared with the optimal water-filling transmission power allocation.%功率分配是非正交多址系统(NOMA)资源分配中的一个重要研究问题.最优迭代注水功率分配算法能提高系统性能,但是算法复杂度较高.提出一种低复杂度的功率分配算法,首先对予载波采用注水原理得到总的复用功率,然后在单个子载波上叠加用户间采用分数阶功率分配方法进行功率再分配.通过仿真分析,与最优迭代注水功率分配算法相比,该算法在性能损失不超过3％的情况下,大幅减低了计算复杂度.【总页数】4页(P126-128,132)【作者】谭歆;肖杰;高翔;吴广富【作者单位】重庆邮电大学移动通信技术重庆市重点实验室,重庆400065;重庆邮电大学移动通信技术重庆市重点实验室,重庆400065;重庆邮电大学移动通信技术重庆市重点实验室,重庆400065;重庆邮电大学移动通信技术重庆市重点实验室,重庆400065【正文语种】中文【中图分类】TN929.5【相关文献】1.OFDM系统中一种改进的低复杂度自适应比特功率分配算法 [J], 朱继华;王竟鑫;申茜;邱飘玉;王永;袁建国2.一种低复杂度OFDMA功率分配算法 [J], 邵朝;张成程;朱婷鸽3.MIMO系统一种新的低复杂度功率分配算法 [J], 黄鹏;姜永权;林元模;谌开元4.一种虚拟MIMO中的低复杂度功率分配算法 [J], 王大鸣;吕璐;窦冬冬;崔维嘉5.一种低复杂度OFDMA功率分配算法 [J], 邵朝;张成程;朱婷鸽;因版权原因，仅展示原文概要，查看原文内容请购买。

当前,计算机技术与网络技术得到了较快发展,计算机软件工程进入到社会各个领域当中,使很多操作实现了自动化,得到了人们的普遍欢迎,解放了大量的人力.为了适应时代的发展,社会各个领域大力引进计算机软件工程.下面是软件工程英文参考文献105个，供大家参考阅读。

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EPC知识考试(试卷编号161)1.[单选题]SGSN与CG对接采用单地址对接起始端口号（）。

When the SGSN uses a single address to interconnect with the CG, the starting port number is ( ).A)15001B)25001C)35001D)45001答案:C解析:2.[单选题]在xGW上，20槽位MPU的standby口的IP地址是（）。

In xGW,the IP address of the standby port on MPU in slot 20 is ( ).A)168.0.31.1B)168.0.31.9C)128.0.31.1D)128.0.31.9答案:A解析:3.[单选题]中兴ZXUN SSS的网元逻辑功能不包含哪一个A)MMTELB)SCC ASC)MRFCD)MRFP答案:D解析:4.[单选题]在Gb口上, PVC是由（ ） 来标识的。

A)DLCIB)NSVCIC)BVCID)NSEI答案:A解析:5.[单选题]如何在SGSN告警管理中上报电源和温度等环境的告警？To report environment alarms, such as power and temperature alarms, in alarm management of the SGSN, which of the following steps should we perform? ( )B)检查SGSN机架前面的PDU单板的拨码设置，然后在配置数据中进行PDU的配置。

We need to check the DIP switch setting of the PDU board in front of the SGSN rack, and then make PDU configuration in data configuration.C)默认电源和环境的告警会自动上报，不需要做任何的配置。

LetterJoint Slot Scheduling and Power Allocation in ClusteredUnderwater Acoustic Sensor NetworksZhi-Xin Liu, Xiao-Cao Jin, Yuan-Ai Xie, and Yi YangDear Editor,This letter deals with the joint slot scheduling and power alloca-tion in clustered underwater acoustic sensor networks (UASNs),based on the known clustering and routing information, to maximize the network’s energy efficiency (EE). Based on the block coordi-nated decent (BCD) method, the formulated mixed-integer non-con-vex problem is alternatively optimized by leveraging the Kuhn-Munkres algorithm, the Dinkelbach’s method and the successive con-vex approximation (SCA) technique. Numerical results show that the proposed scheme has a better performance in maximizing EE com-pared to the separate optimization methods.Recently, the interest in the research and development of underwa-ter medium access control (MAC) protocol is growing due to its potentially large impact on the network throughput. However, the focus of many previous works is at the MAC layer only, which may lead to inefficiency in utilizing the network resources [1]. To obtain a better network performance, the approach of cross-layer design has been considered. In [1], Shi and Fapojuwo proposed a cross-layer optimization scheme to the scheduling problem in clustered UASNs.However, power allocation and slot scheduling were separately designed in [1], which cannot guarantee a global optimum solution.In [2], a power control strategy was introduced to achieve the mini-mum-frame-length slot scheduling. However, EE, as a non-negligi-ble aspect of network performance, is not being considered in [2].In this letter, we formulate a joint slot scheduling and power allo-cation optimization problem to maximize the network’s EE in clus-tered UASNs. The formulated problem with coupled variables is non-convex and mixed-integer, which is challenging to be solved.We propose an efficient iterative algorithm to solve it. Numerical results demonstrate the effectiveness of our proposed algorithm.N ≜{1,2,...,N }K ≜{1,2,...,K }M ≜{1,2,...,M }Problem statement: A clustered UASN with N sensor nodes grouped into K clusters is considered in this article, with the sets and . Sensor nodes’ operation time in a frame consists of M equal and length-fixed time slots with the index set . The sensor nodes send carriers at the same frequency. The half-duplex (HD) mode and the decode-and-for-ward (DF) mode are adopted for data relaying. The data packet length is assumed equal to the length of the time slot. Since packet collisions occur at the receiver but not the sender, we optimize the slot scheduling from the perspective of signal arrival time. As shown in Fig. 1, packets are scheduled to reach the destination at specific time slots. To avoid collisions, the arriving packets cannot overlap with each other as shown in the example of the packets at the sink from CH1, CH2 and CH3 in Fig. 1.z n =(M ,[t ,1])∈R M ×1M −1We use the sparse vector (which means the t -th element is 1 and the rest elements are 0) to represent the scheduling indicator, i.e., the t -th time slot is assigned to node n to Z ≜{z 1,z 2,...,z N }p n P ≜{p 1,p 2,...,p N }B log 2(1+γn )γn deliver data. Then the slot scheduling of the overall network can beexpressed by the set . The allocated transmission power of node n is denoted as , with the corresponding set of the overall network . By Shannon’s law, the achiev-able link rate of node n to its receiver can be given by R n =, where B is the bandwidth, and is the signal-to-interference-and-noise ratio (SINR) at the receiver of node n , which can be written asg nn n N 0(f )where is the link’s channel gain from node to the receiver of node n , is the power spectral density (p.s.d.) ofthe ambient noises at the receiver (refer to [3]), and binary variablen n n ∈N tionships between node n and node , where .We regard the links connecting to the sink (i.e., sea surface buoy node) directly as the bottleneck links, then the EE maximization˙KK Q k ≜{1,2,...,Q k }C (z k )k ∈˙Kγth NR th T p C 1C 2C 3C 4M min M max C 5C 6z k i ,j where is the set of the links connecting to the sinkdirectly, is the remaining part of after removing the cluster con-taining the sink, is the set of cluster members (CMs) in k -th cluster, represents the set of time slots occupiedby the cluster head (CH) to transmit data, is the required SINR threshold for each link, is the required threshold of net-work rate for the entire network, and is a constant integer. is the transmission power constraint. is the SINR constraint to ensure that the signals can be correctly demoduled as shown in the example of CH3 in Fig. 1. indicates that the output link rate of CH k is restrained by the rate of its subnetwork, which ensures that the links connecting to the sink directly are the bottleneck links. is the integer constraint of M ranging from to . is the required minimum network rate constraint. denotes is a binary variable, which is set as 1 when the j -th time slot is occupied by theCorresponding author: Zhi-Xin Liu.Citation: Z.-X. Liu, X.-C. Jin, Y.-A. Xie, and Y. Yang, “Joint slot scheduling and power allocation in clustered underwater acoustic sensor networks,” IEEE/CAA J. Autom. Sinica , vol. 10, no. 6, pp. 1501–1503, Jun.2023.The authors are with the School of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China (e-mail:Color versions of one or more of the figures in this paper are available online at .Digital Object Identifier 10.1109/JAS.2022.106031CH1CH2CH2CH1CH3CH3Slot1Signal packetInterference packetSlot2Slot3Slot4Slot5SinkFig. 1. Receiver-synchronized slot scheduling table.C 7C 8C 9C 10i -th CM of the k -th cluster. denotes that each time slot accommo-dates at most one node in a cluster, which is given to avoid packetcollisions. denotes that each node is assigned one and only onetime slot to deliver data in a frame. denotes that the HD mode is adopted, thus CHs could not transmit and receive data simultane-ously as shown in the example of CH1 in Fig. 1. is the con-straint for CHs to ensure that frames will not affect each other.P Z C 2C 3C 5P Z Problem solution: The optimization problem (3) is a non-convex and mixed-integer optimization problem, which cannot be solved directly due to the challenge that the variables M , and are always coupled with each other in , , and the objective function. To tackle the coupled variables, firstly, the exhaustive search method is adopted to solve the variable M , then a BCD-based alternating optimization method is utilized to decouple and .P Z∗Given the , and M , sensors’ optimal slot scheduling solution Q k M Q k M each node is assigned one and only one slot to deliver data and each slot accommodates at most one node in a cluster, the slot scheduling problem in a cluster can be modeled as a weighted matching prob-lem for a bipartite graph, in which the CMs in the k -th cluster and the M time slots can be partitioned into two independent and disjoint sets and such that every edge connects a node in to a time slot in . The weight of the edge is defined as the network rate. Then problem (4) can be solved by the Kuhn-Munkres algorithm proposed in [4]. The process, that optimizing the slot scheduling of a cluster while keeping the other clusters unchangeable, continues until all the clusters are optimized. After a round of optimization, if the network rate is improved, another optimization round will be performed until the network rate no longer increases.C 2Although any two nodes in the same cluster have no mutual inter-ference, it still should be noted that a node in other clusters may be interfered by multi nodes in the optimization cluster. That means the optimal matching obtained by the Kuhn-Munkres algorithm may unsatisfy . For solving this problem, Criterion 1 is proposed to search for the eligible slot scheduling scheme.B B B Criterion 1: Supposing node A is the node unsatisfying the SINR constraint, firstly, we find its interference nodes (called set ) who belong to the optimization cluster. Then, we sort set in descending order in terms of the interference intensity to node A to find the node having the largest interference to node A (called node C). If nodeC has more than one available time slot, the previous assigned slot is forbidden to be assigned to node C. Otherwise, other nodes’ time slot will be checked and forbidden in same fashion unless there are no more time slot that can be forbidden in .Z P∗Given the and , sensors’ optimal transmission power solution can be optimized by solving the following problem:C 3C 5P C 3R k ∀k ∈˙K R k =B log 2A k −B log 2H k A k =p k g kk +∑∀k k ∈N δk (z k )p k g kk +N 0(f )B H k =∑∀k k ∈N δk (z k )p k g kk +N 0(f )B ∑i v i ≥∏i (v i /θi )θiv i ≥0θi >0∑i θi =1∑v objective function and the constraints and with respect to .To obtain a convex upper bound of the left-hand side (LHS) of ,we note that , , can be rewritten as , where , and. Making use of the deformation of arithmetic-geometric mean inequality, which states thatwith , and (the equality happensRk =B log 2A k −B f (P )R k ≤R k ,∀k ∈K ˜pn =ln p n ∀p n ∈P Letting , we have . And the equality happens when (7) holds. Letting , , it isC 3To obtain a concave lower bound of the right-hand side (RHS) of , the logarithmic approximation method used in [5] is adopted.ˇRl ,∀l ∈L ,R l ,∀l ∈L ,C 5˜pn =ln p n ∀p n ∈P of in the objective function and . Substitut-ing the undesired terms in (5) with the upper or lower boundsobtained above, and letting , , problem (5) can be12N ˜Ptional function with a concave numerator and a convex denominatorin terms of the transmission power , and the constraints are all con-vex. Therefore, we can exploit the Dinkelbach’s method [6] to trans-form it into the equivalent convex problemP The optimal solution of problem (5) can be obtained by solving the equivalent convex problem (13) iteratively, which can be tackled with existing optimization tools like CVX. The pseudocode of the optimization process in terms of sensors’ transmission power is shown in Algorithm 1.The pseudocode of the BCD-based alternating optimization algo-rithm is shown in Algorithm 2, in which two variable blocks are opti-mized alteratively corresponding to the two optimization subprob-lems (i.e., the slot scheduling subproblem and the power allocation subproblem) in each iteration of the alternating optimization process.f =10B =2P max =Simulation results: We consider a 10 km × 10 km × 200 m area,where N = 30 underwater sensor nodes deployed randomly at differ-ent sea depths are divided K clusters. We assume that the sensors are stationary, and the data in each sensor’s buffer is always sufficient.We take the carrier frequency kHz, kHz and 2 W.P 0Z 0For assessing the performance of the proposed alternating-opti-mization-based joint slot scheduling and power allocation algorithm (denoted as AO), we present three other schemes as contrasts, which include two kinds of separate optimization methods and the power allocation scheme and the slot scheduling scheme obtained by the proposed CMS-MAC algorithm in [2]. The two separate opti-mization methods are summarized as follows:Algorithm 1 Power Control Algorithm Based on the SCA Technique andthe Dinkelbach’s Methodτεt ←−0˜P{t }←−{ln p 1,ln p 2,...,ln p N }P 1: Set the maximum number of iterations and the maximum tolerance .Initialize iteration index and , where is the input powers;2: repeath ←−0˜P {h }temp←−˜P {t }3: Initialize iteration index and ;η{t }EE ˜P{t }4:　Compute with given ;5:　repeat αl =γl (˜P {h }temp )1+γl (˜P {h }temp )βl =ln1+γl (˜P {h }temp )γαl l(˜P {h }temp ) ∀l ∈L αq =γq (˜P {h }temp )1+γq (˜P {h }temp )βq =ln 1+γq (˜P {h }temp )γαq q (˜P {h }temp) ∀q ∈Q k ∀k ∈˙K θk ∀k k ∈N θN ∀k ∈˙K˜P {h }temp 6: Compute , , , , , , , and compute ,, , by (7) with given ;η{t }EE Z ˜P {h +1}temp7: Solve (13) with the given and , and obtain the optimal ;h ←−h +18: Update ;˜P h temp ˜P ∗temp 9: until converge to the optimal solution ;˜P{t +1}←−˜P ∗temp 10: ;t ←−t +111:　Update ; f (η{t −1}EE )<ε,or t ≥τ12: until ;P ∗={e ˜p{t }1,e ˜p {t }2,...,e ˜p {t }N }13: Obtain the optimal solution ;Algorithm 2 Alternating-Optimization-Based Joint Time Slot Scheduling and Power Allocation AlgorithmM min M max P 0Z 0M min 1: Obtain the low bound and the upper bound of M , and the power allocation solution and the slot scheduling scheme under by the algorithm proposed in [2];ε2: Set the maximum tolerance ;M =M min ;M ≤M max ;M ++3: for do l ←04:　Initialize iteration index ;Z {l }M ←Z 0P {l }M ←P 05:　Initialize , and ;6:　repeatP {l }M Z {l }M Z {l +1}M 7: Solve (4) with the given and by the Kuhn-Munkres algo-rithm, and obtain the optimal slot scheduling ;P {l }M Z {l +1}M P {l +1}M 8: Solve (5) with the given and by Algorithm (1), and obtainthe optimal power allocation ;l ←−l +19: Update ;10: until the increment of ηEE is smaller than ε;η∗M Z {l }M P {l }M 11: Obtain the optimal network EE , and the optimal solution and ;12: endη∗M ∗=Max {η∗M min ,η∗M min +1,...,η∗M max }13: Let ;M ∗Z {l }M ∗P {l }M ∗14: Return the optimal solution , and ;Z 01) Optimal power allocation with fixed slot scheduling (denoted as OPA_FSS): With the fixed slot scheduling scheme , the transmis-sion powers are optimized by Algorithm 1.P 02) Optimal slot scheduling with fixed power allocation (denoted as OSS_FPA): With the fixed power allocation scheme , the slotscheduling of all of sensors are optimized by the slot schedulingalgorithm proposed above.The corresponding comparison results are shown in Fig. 2. It can be observed that the proposed AO shows the best performance. The reason is that slot scheduling and power allocation may be influ-enced by each other, thus it is unreasonable to fix one of them and then solve another. For the proposed AO, slot scheduling and power allocation could be solved in an alternating way, which leads to bet-ter solutions. Furthermore, it can be found that AO achieves signifi-cant EE gains compared to CMS-MAC algorithm.(a)(b)K SINR (dB)2500E E (b i t s /H z /J )E E (b i t s /H z /J )200015001000500γth Fig. 2. Comparisons of EE. (a) for different clustering numbers with = 10dB; (b) for different SINR constraints with K = 7.Conclusion: In this letter, an EE maximization problem withcross-layer design is considered in clustered UASNs. To tackle the non-convex and mixed-integer optimization problem, a BCD-based iterative algorithm is proposed. Numerical results show that the pro-posed joint optimization scheme achieves significant EE gains com-pared to the separate optimization methods.Acknowledgment: This work was supported by the National Natu-ral Science Foundation of China (62273298, 61873223), the Natural Science Foundation of Hebei Province (F2019203095), and Provin-cial Key Laboratory Performance Subsidy Project (22567612H).ReferencesL. Shi and A. O. Fapojuwo, “TDMA scheduling with optimized energy efficiency and minimum delay in clustered wireless sensor networks,”IEEE. Trans. Mob. Comput., vol. 9, no. 7, pp. 927–940, 2010.[1]W. Bai, H. Wang, X. Shen, and R. Zhao, “Link scheduling method for underwater acoustic sensor networks based on correlation matrix,” IEEE Sens. J., vol. 16, no. 11, pp. 4015–4022, 2016.[2]M. Stojanovic, “On the relationship between capacity and distance in an underwater acoustic communication channel,” SIGMOBILE put. Commun. Rev., vol. 11, no. 4, pp. 34–43, 2007.[3]F. Xing, H. Yin, Z. Shen, and V. C. M. Leung, “Joint relay assignment and power allocation for multiuser multirelay networks over underwater wireless optical channels,” IEEE Internet Things J., vol. 7, no. 10, pp. 9688–9701, 2020.[4]Z. Liu, Y. Xie, Y. Yuan, K. Ma, K. Y. Chan, and X. Guan, “Robust power control for clustering-based vehicle-to-vehicle communication,”IEEE Syst. J., vol. 14, no. 2, pp. 2557–2568, 2020.[5]J.-P. Crouzeix and J. A. Ferland, “Algorithms for generalized fractional programming,” Mathematical Programming , vol. 52, no. 1, pp. 191–207,1991.[6]LIU et al .: JOINT SLOT SCHEDULING AND POWER ALLOCATION IN CLUSTERED UASNS 1503。

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