在认知协作中继网络(CCRN，cognitive cooperative relay network)中，次网络(SN，secondary network)信源节点既能通过选择的中继节点将数据转发给目的节点，从而提高SN传输的可靠性，又能通过与主网络(PN，primary network)共享频谱的方式，提高网络的频谱利用率，因此CCRN引起了研究者们的极大兴趣^[1].由于无线通信的开放性，认知协作中继网络正面临着严峻的安全性问题^[2].近几年来，以香农信息论为基础的CCRN物理层安全(PLS，physical layer security)性能分析逐渐成为了研究热点之一.

近来，对CCRN的PLS性能研究主要集中在分析网络的安全中断概率(SOP，secrecy outage probability)和窃听中断概率(IOP，intercept outage probability)这两方面. Nasr等^[3]采用基于最小安全速率的中继选择方式来降低SN的SOP. Liu等^[4]将友好人为干扰机引入CCRN，并推导了在四种中继选择方式下SN的SOP闭合式. Chakraborty等^[5]推导了CCRN中SN源节点与目的节点和窃听节点间具有直接链路情况下SN的SOP闭合式. Jia等^[6]研究了窃听链路信道状态信息(CSI，channel state information)无法实时获取以及具有人为噪声信号的CCRN，并求出了在2种机会中继选择方式下SN的SOP闭合式. Lei等^[7]求导了CCRN在Nakagami-m信道下SN采用最优中继选择(ORS, optimal relay selection)、次优中继选择(SRS，suboptimal relay selection)和多中继合并(MRC，multiple relays combining)等3种中继选择方式的SOP闭合式. Ho-Van等^[8]则分析了CCRN在SN源节点和窃听节点间具有直接链路情况下SN的近似和渐近IOP.

为了延长能量受限认知中继网络(CRN，cognitive relay network)的生存期，射频能量采集(RF-EH，radio frequency energy harvesting)技术被引入CRN，构建起了新的RF-EH-CRN.在RF-EH-CRN中，SN中继节点主要从PN发送节点^[9]、SN源节点^[10]和专用功率信标(PB，power beacon)^[11]的射频信号中采集能量.目前，对RF-EH-CCRN的PLS性能研究方兴未艾. Li等^[12]构建了一个覆盖式(overlay)RF-EH-CCRN，并得出了基于两个EH协议的最优中继选择方式下PN的精确和渐近SOP闭合式. Maji等^[13]则研究了一个衬底式(underlay)RF-EH-CCRN，并在SN非完美CSI条件下对采用被动式中继选择方式的SOP进行了数值计算.现有的研究存在一定的局限性：①文献[12, 13]构建的RF-EH-CCRN中SN中继从PN节点或SN源节点的RF信号中采集能量.由于PN节点或SN源节点的发射功率远低于PB信号功率，导致SN中继从PN节点或SN源节点采集的能量远低于从PB信号中采集的^[11]. ② Maji等^[13]并未推导出SOP的闭合式.当前，对衬底式RF-EH-CCRN进行PLS性能研究的文献还很少.

在前人研究的基础上，笔者建立一个SN节点从PB信号采集能量的衬底式RF-EH-CCRN，推导出了SN采用ORS和SRS方式的SOP闭合式，并对2种不同中继选择方式下SN的安全中断性能进行了分析.笔者的主要贡献包括：①次网络中各中继节点分散分布，网络信道功率增益为独立非同分布；②推导出了更加科学的SOP闭合式.

1 网络模型和安全容量 1.1 网络模型

网络模型如图 1所示. M个PN接收节点Q_m，m∈{1, 2, …, M}构成PN.源节点S、中继节点集合Φ={R_i|i=1, 2, …, N}和目的节点B则构成SN.单窃听节点F被随机设置在中继节点R_i附近，节点F会窃听R_i发送的信号，以图截获信息.假设PN对SN的干扰忽略不计，同时假设S与B和F之间的信道存在严重衰落，因此S与B和F之间不存在直接链路.网络中所有节点均使用单天线和半双工方式工作. SN采用underlay模式与PN共享频谱，即S采用主网络的授权频谱，通过选定的使用DF模式的中继节点R_i^*向B转发信息.安装了RF-EH装置的Φ中所有中继和S同时从Z收集能量.网络中的所有链路均为独立Rayleigh块衰落信道，即信道系数在同一个时隙周期T内保持不变，在不同时隙间则独立变化. S到R_i以及R_i到B的数据链路信道系数分别表示为h_{S, Ri}和h_{Ri, B}，Z到R_i和S的EH链路信道系数分别表示为f_{Z, Ri}和f_{Z, S}，S和R_i到Q_m的干扰链路信道系数分别用g_{S, Qm}和g_{Ri, Qm}表示，e_{Ri, F}则表示R_i到F的窃听链路信道系数.信道功率增益|h_{S, Ri}|²、|h_{Ri, B}|²、|f_{Z, Ri}|²、|f_{Z, S}|²、|g_{S, Qm}|²、|g_Ri, Q_m|²和|e_{Ri, F}|²分别服从数学期望为1/λ_{S, Ri}、1/λ_{Ri, B}、1/β_Ri、1/β_S、1/ω_{S, Qm}、1/ω_{Ri, Qm}和1/φ_{Ri, E}的指数分布；例如，λ_{S, Ri}=d_{S, Ri}^ξ，其中d_{S, Ri}为节点S到R_i的距离，ξ为路径损耗指数.为了简单起见，假设所有的Q紧绕同一个中心点设置^[9].因此，S和R_i到Q_m链路的信道增益分别是独立同分布的，即ω_{S, Qm}= ω_S和ω_{Ri, Qm}=ω_Ri.

网络系统模型采用的传输协议时隙结构如图 2所示，协议分为EH和数据传输(DT，date transmission)2个阶段.时隙的前αT时段(α为EH比率，0 < α < 1)为EH阶段，节点q，q∈{S, R_i}收集到的能量为

$ {E_q} = \eta {P_{\rm{t}}}|{f_{Z,q}}{|^2}\alpha T $

(1)

其中：0 < η < 1为能量转换效率，P_t为发射功率.式(1)忽略了q收集的噪声能量.假设q的蓄电池容量无穷大，能将q收集到的能量全部存储，并且蓄电池存在漏电，在时隙结束时因其存储的能量被全部泄露，而不会用于下一时隙中.

在underlay模式下，SN节点发射功率严格受限，以保证Q接收到的SN节点信号功率不大于PN干扰约束P_I，因此q的发射功率为

$ {P_q} = {\rm{min}}\{ \rho {P_{\rm{t}}}{Z_q},{P_{\rm{I}}}/{Y_q}\} $

(2)

其中：$Z_{q}=|f_{Z, q}|^{2}，Y_{q}=\max \limits_{m=1, 2, …, M} |g_{q, Q_m}|^{2}，ρ=2αη/(1－α)$.注意到，式(2)忽略了节点q的电路功率消耗.

在DT阶段的前(1－α)T/2时段，Φ中所有中继均收到S发送的信号，R_i的瞬时接收端信噪比(SNR，signal-to-noise ratio)以及S到R_i的信道容量分别为

$ {{\gamma _{S,{R_i}}} = {P_S}{X_{S,{R_i}}}/{\sigma ^2}} $

(3)

$ {{C_{S,{R_i}}} = 0.5(1 - \alpha ) {\rm{lb}} (1 + {\gamma _{S,{R_i}}})} $

(4)

其中：X_{S, Ri}=|h_{S, Ri}|²，σ²为接收端加性复高斯白噪声方差.在前(1－α)T/2时段最后，Φ中所有中继对接收自S的数据进行译码.根据文献[7, 11]，当C_{S, Ri}大于数据传输速率阈值R_d时，R_i能成功译码.将成功译码的中继组成为集合D，由于Φ的基数为N，故集合D有2^N种组合方式，即成功译码的结果集Ω={∅, D₁, D₂, …, D_n, …, D_2^N－1}，其中∅表示空集，D_n则表示非空集合.若Ω=∅，则表明所有中继均未能成功解码S发送的信息，因此在DT阶段的后(1－α)T/2时段，所有中继将保持静默；若Ω=D_n，则在DT阶段的后(1－α)T/2时段，R_i(R_i∈D_n)将译码的S数据转发给B，B的瞬时接收端SNR为

$ {\gamma _{{R_i},B}} = {P_{{R_i}}}{X_{{R_i},B}}/{\sigma ^2} $

(5)

其中X_{Ri, B}=|h_{Ri, B}|².与此同时，F也接到R_i发送的信号，其瞬时接收端SNR为

$ {\gamma _{{R_i},F}} = {P_{{R_i}}}{X_{{R_i},F}}/{\sigma ^2} $

(6)

其中X_{Ri, F}=|h_{Ri, F}|². R_i到B和F链路信道容量分别为

$ {{C_{{R_i},B}} = 0.5(1 - \alpha ){\rm{lb}}(1 + {\gamma _{{R_i},B}})} $

(7)

$ {{C_{{R_i},F}} = 0.5(1 - \alpha ){\rm{lb}}(1 + {\gamma _{{R_i},F}})} $

(8)

为了降低系统的复杂度和减小SN的SOP，当Ω=D_n时，将采用以下2种中继选择方式.

1.2 ORS方式

假设节点F采用主动窃听方式，此时Dn中的元素Ri可以实时获得其到B和F链路的瞬时CSI，此时采用ORS方式从Dn中选出中继Ri*ORS对B进行转发，其选择标准为

$ {i^*} = {\rm{arg}}\mathop {{\rm{max}}}\limits_{{R_i} \in {D_n}} {[{C_{{R_i},B}} - {C_{{R_i},F}}]^ + } $

(9)

其中[x]⁺=max(x, 0).

采用ORS方式的SN安全容量为^[5-7]

$ \begin{array}{l} C_S^{{\rm{ORS}}} = {[{C_{R_{{i^ * }}^{{\rm{ORS}}},B}} - {C_{R_{{i^ * }}^{{\rm{ORS}}},F}}]^ + } = \\ {\left[ {5(1 - \alpha ){\rm{lb}}\left( {\frac{{1 + {\gamma _{R_{{i^ * }}^{{\rm{ORS}}},B}}}}{{1 + {\gamma _{R_{{i^ * }}^{{\rm{ORS}}},F}}}}} \right)} \right]^ + } \end{array} $

(10)

其中：CRi*ORS, B和γRi*ORS, B分别表示Ri*ORS到B链路的信道容量以及B的接收端SNR，CRi*ORS, F和γRi*ORS, F则分别表示Ri*ORS到F链路的信道容量以及F的接收端SNR.

1.3 SRS方式

假设节点F采用被动窃听方式，即F在窃听过程中始终保持静默，因此Dn中的元素Ri只能实时获得其到B链路的瞬时CSI，而无法实时获得其到E链路的瞬时CSI.此时，选择Dn中到B链路信道功率增益最大的中继Ri*SRS对B进行转发，其选择标准可以写为

$ {i^*} = {\rm{arg}}\mathop {{\rm{max}}}\limits_{{R_i} \in {D_n}} {X_{{R_i},B}} $

(11)

采用SRS方式的SN安全容量为

$ \begin{array}{l} C_S^{{\rm{SRS}}} = {[{C_{R_{{i^ * }}^{{\rm{SRS}}},B}} - {C_{R_{{i^ * }}^{{\rm{SRS}}},F}}]^ + } = \\ {\left[ {5(1 - \alpha ){\rm{lb}}\left( {\frac{{1 + {\gamma _{R_{{i^ * }}^{{\rm{SRS}}},B}}}}{{1 + {\gamma _{R_{{i^ * }}^{{\rm{SRS}}},F}}}}} \right)} \right]^ + } \end{array} $

(12)

其中：CRi*SRS, B和γRi*SRS, B分别表示Ri*SRS到B链路的信道容量以及B的接收端SNR，CRi*SRS, F和γRi*SRS, F则分别表示Ri*SRS到F链路的信道容量以及F的接收端SNR.

2 SN的SOP分析

SOP是评价无线网络系统安全性能的重要指标之一，将瞬时安全容量低于预设目标速率阈值R_s的概率定义为SOP.根据全概率公式，SOP可以表示为

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {P_{{\rm{ out }}}} = {\rm{Pr}}\{ {C_S} < {R_s}\} = \\ {\rm{Pr}} \{ {C_S} < {R_s},\varOmega = \varnothing \} + \sum\limits_{n = 1}^{{2^N} - 1} { {\rm{Pr}} } \{ {C_S} < {R_s},\varOmega = {D_n}\} \end{array} $

(13)

其中：Pr { ·}表示求概率运算.当Ω=∅时，Φ中所有中继均因未成功解码而无法向B转发数据，导致C_S=0.因此，式(13)可以简化为

$ {P_{{\rm{ out }}}} = {\rm{Pr}} \{ \varOmega = \varnothing \} + \sum\limits_{n = 1}^{{2^N} - 1} { {\rm{Pr}} } \{ {C_S} < {R_s},\varOmega = {D_n}\} $

(14)

为了简化SOP的计算，本节首先给出Y_q和U_q=P_q/σ²，q∈{S, R_i}的累积分布函数(CDF，cumulative distribution function)和概率密度函数(PDF，probability density function).由于Y_q是M个独立同分布的|g_{q, PRm}|²，m∈{1, 2, …, M}最大值，且|g_q, Q_m|²均服从参数为1/ω_q的指数分布，因此CDF和PDF分别为

$ {F_{{Y_q}}}({y_q}) = {(1 - {{\rm{e}}^{ - {\omega _q}{y_q}}})^M} = \sum\limits_{m = 0}^M {\left( {\begin{array}{*{20}{l}} M\\ m \end{array}} \right)} {( - 1)^m}{{\rm{e}}^{ - m{\omega _q}{y_q}}} $

(15)

$ {f_{{Y_q}}}({y_q}) = M{\omega _q}\sum\limits_{m = 0}^{M - 1} {\left( {\begin{array}{*{20}{c}} {M - 1}\\ m \end{array}} \right)} {( - 1)^m}{{\rm{e}}^{ - (m + 1){\omega _q}{y_q}}} $

(16)

U_q的CDF和PDF分别为

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {F_{{U_q}}}({u_q}) = \\ 1 - {\rm{Pr }}\{ {Z_q} > {u_q}{\sigma ^2}/(\rho {P_{\rm{t}}})\} {\rm{Pr}} \{ {Y_q} < {P_{\rm{I}}}/({\sigma ^2}{u_q})\} = \\ 1 - \sum\limits_{m = 0}^M {\left( {\begin{array}{*{20}{l}} M\\ m \end{array}} \right)} {( - 1)^m}{{\rm{e}}^{ - {\beta _q}{u_q}{\sigma ^2}/(\rho {P_{\rm{t}}}) - m{\omega _q}{P_{\rm{I}}}/({\sigma ^2}{u_q})}} \end{array} $

(17)

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {f_{{U_q}}}({u_q}) = \frac{{\partial {F_{{U_q}}}({u_q})}}{{\partial {u_q}}} = \\ \sum\limits_{m = 0}^M {\left( {\begin{array}{*{20}{l}} M\\ m \end{array}} \right)} {( - 1)^m}\left( {\frac{{{\beta _q}{\sigma ^2}}}{{\rho {P_{\rm{t}}}}} - \frac{{m{\omega _q}{P_{\rm{I}}}}}{{{\sigma ^2}u_q^2}}} \right){{\rm{e}}^{ - \frac{{{\beta _q}{u_q}{\sigma ^2}}}{{\rho {P_{\rm{t}}}}} - \frac{{m{\omega _q}{P_{\rm{I}}}}}{{{\sigma ^2}{u_q}}}}} \end{array} $

(18)

本节将分别推导出采用ORS和SRS方式下SN的SOP闭式解，为后续的SN安全中断性能分析提供依据.

2.1 ORS方式下SN的SOP

根据式(14)，并使用全概率公式，可将采用ORS方式的SN精确安全中断概率改写为

$ \begin{array}{l} P_{{\rm{ out }}}^{{\rm{ORS}}}({\gamma _d},{\gamma _s}) = \underbrace {{\rm{Pr}} \{ \varOmega = \varnothing \} }_{{J_1}({\gamma _d})} + \\ \underbrace {\sum\limits_{n = 1}^{{2^N} - 1} {{\rm{Pr}}} \{ C_S^{{\rm{ORS}}} < {R_s},\varOmega = {D_n}\} }_{{J_2}({\gamma _d},{\gamma _s})} \end{array} $

(19)

其中：J₁(γ_d)可以计算为^[5]

$ \begin{array}{l} {J_1}({\gamma _d}) = \int_0^\infty {\prod\limits_{i = 1}^N {{\rm{ Pr }}} } \{ {\gamma _{S,{R_i}}} < {\gamma _d}|{U_S} = u\} {f_{{U_S}}}(u){\rm{d}}u = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1 + \sum\limits_{k = 1}^{{2^N} - 1} {\sum\limits_{m = 0}^M {\left( {\begin{array}{*{20}{l}} M\\ m \end{array}} \right)} } {( - 1)^{m + \left| {{A_k}} \right|}} \times \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \int_0^\infty {\left( {\frac{{{\beta _S}{\sigma ^2}}}{{\rho {P_{\rm{t}}}}} - \frac{{m{\omega _S}{P_{\rm{I}}}}}{{{\sigma ^2}{u^2}}}} \right)} {{\rm{e}}^{ - \frac{{{\beta _{{S^{u\sigma }}^2}}}}{{\rho {P_{\rm{t}}}}} - \frac{{m{\omega _S}{P_{\rm{I}}}}}{{{\sigma ^2}u}} - \frac{{\sum\limits_{{R_i} \in {A_k}} {{\lambda _{S,{R_i}}}} {\gamma _d}}}{u}}}{\rm{d}}u \end{array} $

(20)

其中：γ_d=2^{2R_d/(1－α)}－1，A_k为Φ的子集，|A_k|表示集合A_k的基数.根据文献[15]中式(3.324.1)和式(3.471.9)，J₁(γ_d)可以写为

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {J_1}({\gamma _d}) = 1 + \sum\limits_{k = 1}^{{2^N} - 1} {\sum\limits_{m = 0}^M {\left( {\begin{array}{*{20}{l}} M\\ m \end{array}} \right)} } {( - 1)^{m + |{A_k}|}} \times \\ [\begin{array}{*{20}{c}} 2&{\sqrt {{a_1}} ({\gamma _d}) - 2}&{\sqrt {{b_1}({\gamma _d})} } \end{array}]{{\rm{K}}_1}\left[ {\begin{array}{*{20}{l}} 2&{\sqrt {{a_1}} ({\gamma _d})} \end{array}} \right] \end{array} $

(21)

其中：$ a_{1}(γ_{d})= \frac{{(\sum\limits_{ R_{i}∈A_{k} }λ_{S, R_i}γ_{d}σ^{2}+mω_{S}P_{\rm I}) β_{S}}}{{ ρP_{\rm t}}} ，b_{1}(γ_{d})= \frac{{ β_{S}m^{2}ω^{2}_{S}P^{2}_{\rm I}}}{{ρP_{\rm t}(\sum\limits_{ R_{i}∈A_{k} }λ_{S, R_i}γ_{d}σ^{2}+mω_{S}P_{\rm I}}}) $，K₁(·)为一阶第二类修正Bessel函数.

J₂(γ_d, γ_s)则可以推导为

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {J_2}({\gamma _d},{\gamma _s}) = \\ \sum\limits_{n = 1}^{{2^N} - 1} {\underbrace { {\rm{Pr}} \{ C_S^{{\rm{ORS}}} < {R_S}|\varOmega = {D_n}\} }_{{\vartheta _1}({\gamma _s})}} \underbrace {{\rm{Pr}} \{ \varOmega = {D_n}\} }_{{\vartheta _2}({\gamma _d})} \end{array} $

(22)

其中：ϑ₁(γ_s)可以依据式(10)改写为

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\vartheta _1}({\gamma _s}) = {\rm{ Pr }}\left\{ {\mathop {{\rm{max}}}\limits_{{R_i} \in {D_n}} \left( {\frac{{1 + {\gamma _{{R_i},B}}}}{{1 + {\gamma _{{R_i},F}}}}} \right) < {\gamma _s}} \right\} = \\ \prod\limits_{{R_i} \in {D_n}} {\left\{ {1 - \sum\limits_{m = 0}^M {\left( {\begin{array}{*{20}{c}} M\\ m \end{array}} \right)} {{( - 1)}^m}\frac{{{\varphi _{{R_i},F}}}}{{{\varphi _{{R_i},F}} + {\lambda _{{R_i},B}}{\gamma _s}}} \times } \right.} \\ \left. {[\begin{array}{*{20}{c}} 2&{\sqrt {{a_2}} ({\gamma _s}) - 2}&{\sqrt {{b_2}({\gamma _s})} } \end{array}]{{\rm{K}}_1}[\begin{array}{*{20}{c}} 2&{\sqrt {{a_2}({\gamma _s})} } \end{array}]} \right\} \end{array} $

(23)

其中γ_s=2^{2R_s/(1－α)}，

$ \begin{array}{l} {a_2}({\gamma _s}) = \frac{{[m{\omega _{{R_i}}}{P_{\rm{I}}} + {\lambda _{{R_i},B}}({\gamma _s} - 1){\sigma ^2}]{\beta _{{R_i}}}}}{{\rho {P_{\rm{t}}}}}\\ {b_2}({\gamma _s}) = \frac{{{\beta _{{R_i}}}{m^2}\omega _{{R_i}}^2P_{\rm{I}}^2}}{{\rho {P_{\rm{t}}}[m{\omega _{{R_i}}}{P_{\rm I}} + {\lambda _{{R_i},B}}({\gamma _s} - 1){\sigma ^2}]}}\\ {\vartheta _2}({\gamma _d}) = \int_0^\infty {\prod\limits_{{R_i} \in {D_n}} { \rm Pr } } \{ {U_S}{X_{S,{R_i}}} > {\gamma _d}|{U_S} = u\} \times \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \prod\limits_{{R_i} \in {{\bar D}_n}} { {\rm{Pr}} } \{ {U_S}{X_{S,{R_i}}} < {\gamma _d}|{U_S} = u\} {f_{{U_S}}}(u){\rm{d}}u = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{m = 0}^M {\left( {\begin{array}{*{20}{l}} M\\ m \end{array}} \right)} {( - 1)^m}2[\sqrt {{a_3}({\gamma _d})} - \sqrt {{b_3}({\gamma _d})} ] \times \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {K_1}[2{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sqrt {{a_3}({\gamma _d})} ] + \\ \sum\limits_{k = 1}^{2\left| {{{\bar D}_n}} \right| - 1} {\sum\limits_{m = 0}^M {\left( {\begin{array}{*{20}{l}} M\\ m \end{array}} \right)} } {( - 1)^{|{C_k}| + m}}2[\sqrt {{a_4}({\gamma _d})} - \sqrt {{b_4}({\gamma _d})} ] \times \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {K_1}[2{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sqrt {{a_4}({\gamma _d})} )] \end{array} $

(24)

其中：D_n是D_n的补集，C_k是D_n的子集，|D_n|和|C_k|分别表示D_n和C_k的基数，

$ {a_3}({\gamma _d}) = \frac{{{\beta _S}\left( {m{\omega _S}{P_{\rm I}} + \sum\limits_{{R_i} \in {D_n}} {{\lambda _{S,{R_i}}}} {\gamma _d}{\sigma ^2}} \right)}}{{\rho {P_{\rm{t}}}}} $

$ {b_3}({\gamma _d}) = \frac{{{\beta _s}{m^2}\omega _S^2P_{\rm{I}}^2}}{{\rho {P_{\rm{t}}}\left( {m{\omega _S}{P_{\rm{I}}} + \sum\limits_{{R_i} \in {D_n}} {{\lambda _{S,{R_i}}}} {\gamma _d}{\sigma ^2}} \right)}} $

$ {a_4}({\gamma _d}) = \frac{{{\beta _s}\left( {m{\omega _s}{P_{\rm{I}}} + \sum\limits_{{R_i} \in {D_n}} {{\lambda _{s,{R_i}}}} {\gamma _d}{\sigma ^2} + \sum\limits_{{R_i} \in {C_k}} {{\lambda _{s,{R_i}}}} {\gamma _d}{\sigma ^2}} \right)}}{{\rho {P_{\rm{t}}}}} $

$ {b_4}({\gamma _d}) = \frac{{{\beta _S}{m^2}\omega _S^2P_{\rm{I}}^2}}{{\rho {P_{\rm{t}}}\left( {m{\omega _S}{P_{\rm{I}}} + \sum\limits_{{R_i} \in {D_n}} {{\lambda _{S,{R_i}}}} {\gamma _d}{\sigma ^2} + \sum\limits_{{R_i} \in {C_k}} {{\lambda _{S,{R_i}}}} {\gamma _d}{\sigma ^2}} \right)}} $

将式(21)和式(22)代入式(19)，可得采用ORS方式的SN安全中断概率P_out^ORS.

2.2 SRS方式下SN的SOP

根据式(13)，采用SRS方式的SN精确SOP可以表示为

$ \begin{array}{l} P_{{\rm{ out }}}^{{\rm{SRS}}}({\gamma _d},{\gamma _s}) = \underbrace {{\rm{ Pr }}\{ \varOmega = \varnothing \} }_{{J_1}({\gamma _d})} + \\ \underbrace {\sum\limits_{n = 1}^{{2^N} - 1} { {\rm{Pr }}} \{ C_S^{{\rm{SRS}}} < {R_s},\varOmega = {D_n}\} }_{{J_3}({\gamma _d},{\gamma _s})} \end{array} $

(25)

其中：J₃(γ_d, γ_s)可以计算为

$ {J_3}({\gamma _d},{\gamma _s}) = \sum\limits_{n = 1}^{{2^N} - 1} {\underbrace { {\rm{Pr}} \{ C_S^{{\rm{SRS}}} < {R_s}|\varOmega = {D_n}\} }_{{\vartheta _3}({\gamma _s})}} \underbrace {{\rm{Pr}} \{ \varOmega = {D_n}\} }_{{\vartheta _2}({\gamma _d})} $

(26)

其中：ϑ₃(γ_s)可以依据式(12)表示为

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\vartheta _3}({\gamma _s}) = \\ \sum\limits_{{R_i} \in {D_n}} { {\rm{Pr}} } \{ {X_{{R_i},B}} > \mathop {{\rm{max}}}\limits_{{R_l} \in \{ {D_n} - {R_i}\} } {X_{{R_l},B}}\} {\rm{Pr }}\left\{ {\frac{{1 + {\gamma _{{R_i},B}}}}{{1 + {\gamma _{{R_i},F}}}} < {\gamma _s}} \right\} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{{R_i} \in {D_n}} {\left\{ {1 - \sum\limits_{m = 0}^M {\left( {\begin{array}{*{20}{l}} M\\ m \end{array}} \right)} \frac{{{{( - 1)}^m}{\varphi _{{R_i},F}}}}{{{\varphi _{{R_i},F}} + {\gamma _s}{\lambda _{{R_i},B}}}} \times } \right.} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2[\sqrt {{a_2}({\gamma _s})} - \sqrt {{b_2}({\gamma _s})} ] \times \\ {{\rm{K}}_1}[2{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sqrt {{a_2}({\gamma _s})} ] + \sum\limits_{k = 1}^{{2^{|{D_n}| - 1}} - 1} {\frac{{{{( - 1)}^{\left| {{G_k}} \right|}}{\lambda _{{R_i},B}}}}{{{\lambda _{{R_i},B}} + \sum\limits_{{R_l} \in {G_k}} {{\lambda _{{R_l},B}}} }}} \times \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} [1 - \sum\limits_{m = 0}^M {\left( {\begin{array}{*{20}{l}} M\\ m \end{array}} \right)} {( - 1)^m}2[\sqrt {{a_5}({\gamma _s})} - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sqrt {{b_5}({\gamma _s})} ]{{\rm{K}}_1}[2{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sqrt {{a_5}({\gamma _s})} ]] + \\ \sum\limits_{k = 1}^{{2^{\left| {{D_n}} \right| - 1}} - 1} {\sum\limits_{m = 0}^M {\left( {\begin{array}{*{20}{l}} M\\ m \end{array}} \right)} } \frac{{{{( - 1)}^{|{G_k}| + m}}{\gamma _s}{\lambda _{{R_i},B}}}}{{{\varphi _{{R_i},F}} + {\gamma _s}{\lambda _{{R_i},B}} + {\gamma _s}\sum\limits_{{R_l} \in {G_k}} {{\lambda _{{R_l},B}}} }} \times \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2[\sqrt {{a_5}({\gamma _s})} - \sqrt {{b_5}({\gamma _s})} ]{{\rm{K}}_1}[2{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sqrt {{a_5}({\gamma _s})} ]\} \end{array} $

(27)

其中：{D_n－R_i}表示D_n去除元素R_i后的中继集合，G_k为{D_n－R_i}的子集，|G_k|为G_k的基数，

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {a_5}({\gamma _s}) = \\ \frac{{{\beta _{{R_i}}}\left[ {m{\omega _{{R_i}}}{P_{\rm{I}}} + \left( {{\lambda _{{R_i},B}} + \sum\limits_{{R_l} \in {G_k}} {{\lambda _{{R_l},B}}} } \right)({\gamma _s} - 1){\sigma ^2}} \right]}}{{\rho {P_{\rm{t}}}}} \end{array} $

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {b_5}({\gamma _s}) = \\ \frac{{{\beta _{{R_i}}}{m^2}\omega _{{R_i}}^2P_{\rm{I}}^2}}{{\rho {P_{\rm{t}}}\left[ {m{\omega _{{R_i}}}{P_{\rm{I}}} + \left( {{\lambda _{{R_i},B}} + \sum\limits_{{R_l} \in {G_k}} {{\lambda _{{R_l},B}}} } \right)({\gamma _s} - 1){\sigma ^2}} \right]}} \end{array} $

将式(24)和式(27)代入式(26)可得J₃(γ_d, γ_s)；最终将式(21)和式(26)代入式(25)可得SRS方式下的SN安全中断概率P_out^SRS.

3 仿真结果与分析

本节给出SOP的仿真和数值计算结果与分析. Monte Carlo仿真次数设置为10⁵.除非另有说明，网络模型中的各参数设置如表 1所示. P_t和P_I均被σ²归一化.

从图 3~7可以看出：① Monte Carlo仿真曲线与数值计算曲线高度吻合，证明了理论推导的正确性. ②在相同参数设置情况下，P_out^ORS低于P_out^SRS.如P_t=15 dB，P_I=10 dB，M=N=3时，P_out^ORS和P_out^SRS分别为1.51%和2.89%，ORS方式下的SN安全中断性能较SRS方式下的提升了91.39%.

图 3所示为η=0.4, 0.8，P_I=10, 20 dB，N=3时，SOP与Q发射功率P_t之间的关系.可以看出：①当P_I和η给定时，P_out^ORS和P_out^SRS均随P_t的增大而单调下降，并趋于饱和.随着P_t的增大，SN节点因采集到的能量增加而导致其发射功率变大，SOP降低，但SN节点的发射功率因受P_I限制而不会无限制增加，最终其发射功率和SN的SOP均将趋于饱和；②当P_I和P_t给定时，η的增大有助于降低SOP；③当η和P_t给定时，P_I的提高将使SOP降低.

图 4和图 5展示了M=2, 3, 4，N=3，P_t=20 dB，以及R_d和R_s取不同值时，SOP与P_I之间的关系. 可以发现：①与图 3类似，当P_t、M、R_d和R_s给定时，P_out^ORS和P_out^SRS也随P_I的增大而单调下降，并趋于饱和.由于CCRN是干扰受限网络，P_t给定，致使SN节点采集的能量一定，随着P_I的提高，SN节点能将更多能量用于提高发射功率直至耗尽所有存储能量，此时SN节点发射功率P_q、P_out^ORS和P_out^SRS也将饱和. ②当P_t、P_I、R_d和R_s给定时，随着M的增大，Y_q获得更大值的概率变大，导致P_q减小，从而使得P_out^ORS和P_out^SRS增大. ③当P_t、P_I、M给定时，P_out^ORS和P_out^SRS随R_d和R_s的增大而提高，且ORS方式下R_d对SOP的影响大于R_s对SOP的影响，在SRS方式下则相反.

图 6体现了中继R₁、R₂和R₃处于2种不同位置设置方式时，SOP与d_F之间的关系，其中：P_t=20 dB，P_I=15 dB，d_F表示F沿X正半轴移动时其与原点之间的距离.可以看出：①当中继位置确定时，随着d_F的增大，F逐步远离EH-CCRN，其接收端SNR将逐步降低，并趋近于0，导致P_out^ORS和P_out^SRS随之减小，并趋于饱和. ②当d_F给定时，不同的中继位置设置方式对P_out^ORS和P_out^SRS会产生较大影响.

图 7所示为N=2, 3时，P_out^ORS和P_out^SRS与α之间的关系，其中：P_t=20 dB，P_I=15 dB.可以发现：①当N给定时，P_out^ORS和P_out^SRS随α的增大逐步从1减小到最小值，再增大到1. ②当N取不同值时，P_out^ORS和P_out^SRS最小值所对应的α各不相同.如N=2，α为0.52和0.54时，P_out^ORS和P_out^SRS可分别取得最小值0.77%和1.14%；N=3，则α为0.66和0.6时，P_out^ORS和P_out^SRS可分别取得最小值0.23%和0.58%. ③随着N值增大，P_out^ORS和P_out^SRS减小.

4 结束语

构建了一个新的单向两跳EH-CCRN.在时隙的EH阶段，S和所有的中继节点均从PB的射频信号中采集能量.在随后的DT阶段，考虑单窃听节点采用主动和被动窃听方式，分别采用ORS和SRS方式选择中继，推导出了2种中继选择方式下SN的精确SOP闭合式.理论和仿真结果表明，ORS方式下SN安全中断性能较SRS方式下提升明显；SN的SOP都会随着P_t或P_I的增大而单调下降，最终出现饱和现象；随着M的增大，SN的SOP也增大；F逐步远离EH-CCRN，SN的SOP随之减小，并趋于饱和；实时调整α和N值有助于提高SN中断性能.