频谱感知4：CCS硬合并中m-out-of-K准则下m与K的联合优化问题

本文主要内容来自：
Narasimha Rao Banavathu, M. Z. A. K. (2018). Joint Optimization of both m and K for the m-out-of-K Rule for Cooperative Spectrum Sensing. European Wireless 2018; 24th European Wireless Conference. Catania, Italy.

1、参考文献综述

  在认知无线电（CR）中，频谱感知[1]-[3]是次用户（SU）检测主用户（PU）信号的基础。

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然而，仅依赖单一SU的频谱感知由于多径和阴影衰落的存在检测性能较差。为了缓解这一问题，提出了合作频谱感知（CSS）[4]-[9]，将来自多个SU的观测数据通过报告信道发送到融合中心（FC），在融合中心（fc）对数据进行合并，从而对PU是否活跃进行判决。

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然而，在FC处可以采用多种不同合并方案，例如软合并[10]-[12]、量化软合并[13]、加权软合并[14]、[15]、多选择CSS方案[16]和m-out-k融合准则[17]。如果 $K$个SU用户中至少 $m$个用户检测到PU信号，根据m-out-of-k规则判决为PU信号存在。[18]在相同SU和相同报告信道假设下，研究了CSS在错误报告信道中的检测性能。

[10] S. Atapattu, C. Tellambura, and H. Jiang, “Energy detection based
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  基于各种目标函数，现有文献研究了m-out-of-k规则下如何优化 $m$。这些目标函数包括，最小化无错误报告通道上的bayes风险[17]、最小化无错误报告通道上的总错误率（TER）[19]、最小化错误报告通道上的TER[20]、最小化错误报告信道上的虚检概率（FDP）[21]、最小化无报告信道时的TER[22]，最大化能量效率[23]，最小化多跳CR网络的TER[24]，在满足对PU的保护约束的同时最大化次网络吞吐量[25]，在全局虚检概率约束下最大化全局检测概率[26]。

[17] P. K. Varshney, Distributed Detection and Data Fusion, 1st ed. Secaucus, NJ, USA: Springer-Verlag New York, Inc., 1996.
[19] W. Zhang, R. Mallik, and K. Letaief, “Optimization of cooperative spectrum sensing with energy detection in cognitive radio networks,” Wireless Communications, IEEE Transactions on, vol. 8, no. 12, pp.5761 –5766, Dec.2009.
[20] N. R. Banavathu and M. Z. A. Khan, “Optimal n-out-of- k voting rule for cooperative spectrum sensing with energy detector over erroneous control channel,” in 2015 IEEE 81st Vehicular Technology Conference (VTC Spring), May 2015, pp. 1–5.

此外，对如何优化 $K$也开展了研究，包括OR规则时最小化TER[27-28]、AND与MAJORITY规则时最小化TER[29]、最大化CR网络的平均信道吞吐量[30]和最小化Bayes风险[31]。

[27] A. Singh, M. Bhatnagar, and R. Mallik, “Optimization of cooperative spectrum sensing with an improved energy detector over imperfect reporting channels,” in Vehicular Technology Conference (VTC Fall), 2011 IEEE, Sept 2011, pp. 1–5.
[28] A. Singh, M. R. Bhatnagar, and R. K. Mallik, “Cooperative spectrum sensing in multiple antenna based cognitive radio network using an improved energy detector,” IEEE Communications Letters, vol. 16, no. 1, pp. 64–67, January 2012.
[29] N. R. Banavathu and M. Z. A. Khan, “On cooperative spectrum sensing with improved energy detector over erroneous control channel,” in 2016 IEEE Wireless Communications and Networking Conference, April 2016, pp. 1–6.
[30] ——, “On the throughput maximization of cognitive radio using cooperative spectrum sensing over erroneous control channel,” in 2016 Twenty Second National Conference on Communication (NCC), March 2016, pp. 1–6.
[31] ——, “Optimal number of cognitive users in k -out-of- m rule,” IEEE Wireless Communications Letters, vol. 6, no. 5, pp. 606–609, Oct 2017.

本文则提出了一个联合优化问题（JOP），该问题通过最小化m-out-k规则在错误报告通道上的Bayes风险来联合优化 $m$和 $K$。就我们所知，到目前为止，还没有考虑m和k的联合优化。本文的主要贡献如下:

• 本文给出了在有错误报告通道的情况下，采用m-out-of-K规则时求解联合优化值的解析表达式。通过采用 $m$和 $K$联合优化值，显著提升了CSS性能。
• 给定 $K$情况下，JOP专门寻找最小化Bayes风险目标下，在错误的报告通道上使m-out-of-k规则的 $m$最优值。然后我们证明现有问题[17]、[19]、[20]是所提出问题的特例。
• 错误报告通道的影响：对于给定的 $K$，融合规则的最优性受到报告通道错误概率的限制。结果表明，对于某个报告信道而言，对于一定的错误概率值，每一个规则都是最优的，超过这个值，它们就永远不是最优的。此外，由于分配给全局误报和漏检概率的有效权重不相等，这些融合规则对误报信道的鲁棒性存在显著差异。

2、问题描述

  现有 $K$个次用户，每个用户的检测性能都相同，即检测概率为 $\rm P_d=1-P_m$， $\rm P_m$为漏检概率；虚检概率为 $\rm P_f$。如果融合节点(FC)采用 $m$-out-of- $K$投票准则，不考虑报告信道错误，则有合并后的检测概率与虚检概率为
$\tag{1} {\rm Q_d}=\sum_{k=m}^{K}\binom{K}{k}{\rm P}^k_{{\rm d}}(1-{\rm P}_{{\rm d}})^{K-k}$以及
$\tag{1.4} {\rm Q_f}=\sum_{k=m}^{K}\binom{K}{k}{\rm P}^k_{{\rm f}}(1-{\rm P}_{{\rm f}})^{K-k}.$根据[1]，定义Bayes风险函数
$\begin{aligned} {\mathcal R}(m,K)&=\sum_{i=0}^{1}\sum_{j=0}^{1}C_{ij}P_jP(d_{\rm FC}=i|H_j)\\ &=C_{00}P_0Q_d+C_{01}P_1Q_m+C_{10}P_0Q_f+C_{11}P_1Q_d\\ &=(C_{00}P_0+C_{11}P_1)Q_d+C_{01}P_1Q_m+C_{10}P_0Q_f\\ &=C_FQ_f+\beta_MQ_m+C_C, \end{aligned}$这里， $C_F=C_{10}P_0$， $C_M=C_{01}P_1$， $C_D=C_{00}P_0+C_{11}P_1$； $C_{ij}$为 $H_j$时判定为 $d_{\rm FC}=i$的代价， $P_j$为假设 $H_1$的概率。因此， $\mathbb{JOP}$问题为
$\begin{aligned} \mathbb{JOP}:\quad &\min_{m,K}{\mathcal R}(m,k),\\ &{\rm s.t.}{\mathcal C}:1\le m\le K. \end{aligned}$

3、问题求解

• 引理1

给定 $K$， $\mathbb{JOP}$的解为
$m^*_{\mathcal R}=\left\{ \begin{aligned} \max(1,m^*),\ &C_F<C_M,\\ \min(K,m^*),\ &C_F>C_M,\\ m^*,\quad &C_F=C_M, \end{aligned} \right.$其中，
$m^*=\lceil\frac{a+Kb}{b+c}\rceil,a=\ln\frac{C_F}{C_M},b=\ln\frac{1-P_f}{P_m},c=\ln\frac{1-P_m}{P_f}.$

• 定理1
  $\mathbb{JOP}$的联合解，即 $m$和 $K$的联合优化为
  $\left\{ \begin{aligned} m^*_{\mathcal R}=1, K^*_{\mathcal R}=\lceil \frac{c-a}{b}\rceil;\ &C_F<C_M \ {\rm and}\ m^*<1,\\ m^*_{\mathcal R}=1, K^*_{\mathcal R}=\lceil \frac{c-a}{b}\rceil;\ &C_F>C_M \ {\rm and}\ m^*>k, \end{aligned} \right.$如果 $1\le m^*\le K$，与 $C_F$和 $C_M$无关，对于给定的 $m$存在给定的 $K^*_{\mathcal R}$；对于给定的 $K$，存在给定的 $m^*_{\mathcal R}$。