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大规模天线作为第五代移动通信的关键技术[1],能够有效的提高频谱利用率,同时异构网能提高系统的覆盖范围[2],并且可以提高整个网络的吞吐量。故异构网络中宏基站和微基站均配置大规模天线,以提高系统的传输速率[3]。大规模天线在部署中,受尺寸限制,一般采用矩形天线阵列,称之为全维度天线阵列[4]。全维度天线阵列除了可以调节水平维度外,还能调节垂直维度角度,从而消除微蜂窝之间信号的干扰,提高系统的容量。
在异构蜂窝网络中部署全维度天线可以动态调整用户信号垂直到达的方向(direction of arrival, DOA),增加垂直维度的空间自由度,从而有利于波束对准异构网络中的用户[5]。然而,正是增加了垂直维度的空间自由度,也为窃听用户得到合法信息提供了方便,从而影响异构网络的物理层安全性能。若微基站部署密度过大,则会带来更多被窃听的风险,而密度过小,则会导致整个异构网络的吞吐量下降,影响用户的服务质量[6]。因此研究全维度天线异构网络的基站部署问题,以提高整个网络的物理层安全性能是十分必要的。
配置多天线的无线系统中,主要通过波束成形算法[7]、发送人工噪声以及天线选择算法[8]等技术来优化物理层安全性能。配置多天线的异构网采用随机几何的方法,分析异构网络的能效问题以及吞吐量问题,通过基站休眠等手段[9-10],提高系统的能效和中断容量。文献[11]采用随机几何的分析方法,分析了无线传感器网络的物理层安全性能,通过增加接入点的个数来提高系统的平均安全速率。然而在以上的研究中,均未考虑配置全维度天线异构网络的安全性能问题,而全维度天线系统研究也主要集中于多用户场景下信道估计以及波束成形方法的研究。
针对以上的研究现状,本文分析配置全维度天线异构网络下行信道物理层安全性能,利用平均信漏噪比下界的最优波束成形算法,利用每个用户信道统计状态信息和随机几何分布信息推导了用户安全连接概率,根据用户安全连接概率,确定微基站的分布密度。仿真结果表明,微基站与宏基站的密度比在25~35之间时,微基站用户的安全连接概率最大,此结果为全维度天线异构网基站的部署提供了有益的参考。
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考虑配置全维度天线下行两层异构网络系统中,其基站服从二维随机泊松独立点过程(poisson point process, PPP),宏基站发射功率为$\rho $,变量${\mathit{\Phi }_m}$分布服从密度${\mathit{\lambda }_m}$的泊松分布。微基站发射功率为$\rho '$,变量${\mathit{\Phi }_f}$分布服从密度为${\mathit{\lambda }_f}$的泊松分布。宏基站和微基站所配置的天线都是$M \times N$的均匀矩形天线(uniform rectangular antenna, URA)阵列,每行有服从均匀线性阵列的$N$个天线,垂直方向有均匀线性阵列的$M$个天线,每个微小区服务$K$个单天线用户,并且用户位置服从密度为${\mathit{\lambda }_u}$的独立泊松点过程${\mathit{\Phi }_u}$。
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对于所考虑的多用户下行链路传输系统,用户终端$k$接收到的信号表示为:
$$y_{k}=\sum\limits_{i=1}^{K} \sqrt{P} \boldsymbol{h}_{k}^{\mathrm{T}} \boldsymbol{w}_{i} x_{i}+n_{k}$$ (1) 式中,${\boldsymbol{h}_k} \in {\mathbb{C}^{MN \times 1}}$表示基站和用户$k$之间的下行链路信道向量;${\boldsymbol{w}_k} \in {\mathbb{C}^{MN \times 1}}$是用户$k$的波束成形向量;${n_k}$~${{\rm CN}}(0, \sigma _k^2)$是归一化复杂加性高斯白噪声;${x_k}$是传输给用户$k$的信息,且${{\rm E}}[|{x_i}{|^2}] = 1$。通过式(1)可知用户$k$的信噪比可以表示为:
$${{\rm SIN}}{{{\rm R}}_k} = \frac{{P{{\left| {\boldsymbol{h}_k^{{\rm T}}{\boldsymbol{w}_k}} \right|}^2}}}{{\sigma _k^2 + \sum\limits_{i = 1, i \ne k}^K {P{{\left| {\boldsymbol{h}_i^{{\rm T}}{\boldsymbol{w}_k}} \right|}^2}} }}$$ (2) -
对于配置的URA全维度天线,其信道向量${\boldsymbol{h}_k}$可以写成:
$${\boldsymbol{h}_k} = {\boldsymbol{a}_t}^{(v)}({\theta _k}) \otimes {\boldsymbol{a}_t}^{(h)}({\theta _k}, {\varphi _k})$$ (3) $${\boldsymbol{a}_t}^{(v)}({\theta _k}) = {\left[ {1, {{\rm e} ^{ - {{\rm j}}2\pi \frac{{{d_t}^{(v)}}}{\lambda }\sin {\theta _k}}}, \cdots , {{\rm e} ^{ - {{\rm j}}2\pi (M - 1)\frac{{{d_t}^{(v)}}}{\lambda }\sin {\theta _k}}}} \right]^{{\rm T}}}$$ (4) $${\boldsymbol{a}_t}^{(h)}({\theta _k}, {\varphi _k}) = {\left[ {1, {{\rm e} ^{ - {{\rm j}}2\pi \frac{{{d_t}^{(h)}}}{\lambda }\sin {\phi _k}}}, \cdots , {{\rm e} ^{ - {{\rm j}}2\pi (N - 1)\frac{{{d_t}^{(h)}}}{\lambda }\sin {\phi _k}}}} \right]^{{\rm T}}}$$ (5) 式中,${\phi _k} = \cos {\theta _k}\sin {\varphi _k}$;${\theta _k} \in \left( { - \frac{\pi }{2}, \frac{\pi }{2}} \right)$和${\varphi _k} \in \left( { - \frac{\pi }{2}, \frac{\pi }{2}} \right)$分别代表用户$k$在垂直和水平方向的发射角;${d_t}^{(v)}$和${d_t}^{(h)}$是每行和每列两个相邻天线间的距离;$\lambda $是载波波长。假定每行和每列两个相邻天线间的距离都是。因此可以把式(4)和式(5)写成:
$${\boldsymbol{a}_t}^{(v)}({\theta _k}) = {[1, {{\rm e} ^{ - {{\rm j \mathit{ π} }}\sin {\theta _k}}}, \cdots , {{\rm e} ^{ - {{\rm j}}(M - 1)\sin {\theta _k}}}]^{{\rm T}}}$$ (6) $${\boldsymbol{a}_t}^{(h)}({\theta _k}, {\varphi _k}) = {[1, {{\rm e} ^{ - {{\rm j}}\pi {\phi _k}}}, \cdots , {{\rm e} ^{ - {{\rm j}}\pi (N - 1){\phi _k}}}]^{{\rm T}}}$$ (7) -
$${\boldsymbol{H}_k} = \boldsymbol{R}_{V, k}^{1/2}{\boldsymbol{H}_{w, k}}\boldsymbol{R}_{H, k}^{1/2}$$ (8) 式中,${\boldsymbol{R}}_{V, k}^{} \in {\mathbb{C}^{M \times M}}$和$\boldsymbol{R}_{H, k}^{} \in {\mathbb{C}^{N \times N}}$是垂直信道和水平信道相关矩阵,可以表示为:
$$\boldsymbol{R}_{V, k}^{} = {\boldsymbol{U}_{V, k}}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{V, k}}\boldsymbol{U}_{V, k}^{{\rm H}}$$ (9) $${\boldsymbol{R}}_{H, k}^{} = {{\boldsymbol{U}}_{H, k}}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{H, k}}{\boldsymbol{U}}_{H, k}^{{\rm H}}$$ (10) 式中,对角矩阵${\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{V, k}} = {{\rm diag}}\left\{ {\mathit{\lambda }_{V, k}^{(1)}, \mathit{\lambda }_{V, k}^{(2)}, \cdots , \mathit{\lambda }_{V, k}^{(M)}} \right\}$和${{\boldsymbol{\Lambda }}_{H, k}} = {{\rm diag}}\left\{ {\mathit{\lambda }_{H{{\rm , }}k}^{(1)}, \mathit{\lambda }_{H{{\rm , }}k}^{(2)}, \cdots , \mathit{\lambda }_{H{{\rm , }}k}^{(N)}} \right\}$的元素由奇异值组成;${{\boldsymbol{U}}_{H, k}}$和${\boldsymbol{U}_{V, k}}$是酉矩阵;${\boldsymbol{H}_{w, k}}$是一个$M \times N$的零均值和单位方差独立同分布的随机复杂高斯矩阵。当$k1 \ne k2$时${{\boldsymbol{H}}_{w, k1}}$和${\boldsymbol{H}_{w, k2}}$是独立的。
假设天线阵列很大,$M \gg 1$,$N \gg 1$。大规模天线阵列时,相关矩阵的特征向量可以用统一的离散傅里叶变换(discrete Fourier transform, DFT)矩阵近似表示。所以$\boldsymbol{R}_{V, k}^{}$和$\boldsymbol{R}_{H, k}^{}$可以近似表示为:
$${\boldsymbol{R}}_{V, k}^{}\mathop = \limits^{M \to \infty } {{\boldsymbol{F}}_M}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{V, k}}{\boldsymbol{F}}_M^{{\rm H}}$$ (11) $${\boldsymbol{R}}_{H, k}^{}\mathop = \limits^{N \to \infty } {{\boldsymbol{F}}_N}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{V, k}}{\boldsymbol{F}}_N^{{\rm H}}$$ (12) 式中,${[{{\boldsymbol{F}}_M}]_{m, n}} = \frac{1}{{\sqrt M }}{{{\rm e}}^{{{\rm j}}2\pi (m - 1)\left( {n - \frac{M}{2}} \right)/M}}$;${[{{\boldsymbol{F}}_N}]_{m, n}} = \frac{1}{{\sqrt N }} \times $ ${{{\rm e}}^{{{\rm j}}2\pi (m - 1)\left( {n - \frac{N}{2}} \right)/N}}$,有:
$${\boldsymbol{F}} \triangleq {{\boldsymbol{F}}_N} \otimes {\boldsymbol{F}}_M^c$$ (13) 根据定义,推导出用户$k$的信漏噪比下限,从而推导出最大化此下限的波束成形矢量${{\boldsymbol{w}}_k}$:
$${{\rm {\rm E}}}[{{\rm SLN}}{{{\rm R}}_k}] = {{\rm {\rm E}}}\left[ {\frac{{{{\left| {{\boldsymbol{h}}_k^{{\rm T}}{{\boldsymbol{w}}_k}} \right|}^2}}}{{\sum\limits_{i = 1, i \ne k}^K {P{{\left| {{\boldsymbol{h}}_i^{{\rm T}}{{\boldsymbol{w}}_k}} \right|}^2}} + 1}}} \right] \geqslant $$ $$\frac{{{{\rm {\rm E}}}\left[ {{{\left| {{\boldsymbol{h}}_k^{{\rm T}}{{\boldsymbol{w}}_k}} \right|}^2}} \right]}}{{{{\rm {\rm E}}}\left[ {\sum\limits_{i = 1, i \ne k}^K {P{{\left| {{\boldsymbol{h}}_i^{{\rm T}}{{\boldsymbol{w}}_k}} \right|}^2}} + 1} \right]}}$$ (14) $${{\rm {\rm E}}}[{{\rm SLN}}{{{\rm R}}_k}] \geqslant \frac{{\boldsymbol{w}_k^{{\rm H}}{\boldsymbol{F}}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_k}{{\boldsymbol{F}}^{{\rm H}}}{\boldsymbol{w}_k}}}{{\boldsymbol{w}_k^{{\rm H}}{\boldsymbol{F}}\left( {\sum\limits_{i = 1, i \ne k}^K {{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}} } \right){{\boldsymbol{F}}^{{\rm H}}}{\boldsymbol{w}_k} + 1}}\mathop = \limits^\Delta {L_k}$$ (15) 式中,${\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i} \triangleq {\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{H, i}} \otimes {\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{V, i}}$,$i = 1, 2, \cdots , K$。
根据文献[11]可以得出:在这种情况下,最优波束成形向量${{\boldsymbol{w}}_k} = {({{\boldsymbol{F}}_N})_{{l_k}}} \otimes {({\boldsymbol{F}}_M^c)_{{j_k}}}$,${{\boldsymbol{h}}_k}{{\boldsymbol{w}}_k}$~${{\rm CN}}(0, 1)$。
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考虑窃听用户的分布也是服从密度为${\mathit{\lambda }_e}$的独立泊松点分布,合法用户接收到的信号为:
$$\begin{gathered} {y_m} = \sqrt {\rho '{l^{ - \alpha }}} {{\boldsymbol{h}}_1}{{\boldsymbol{w}}_1}{x_1} + \sum\limits_{i = 1}^K {\sqrt {\rho '{d^{ - \alpha }}} } {{\boldsymbol{h}}_i}{{\boldsymbol{w}}_i}{x_i} + \\ \sum\limits_{k \in {\mathit{\Phi }_m}} {\sqrt {\rho {D^{ - \alpha }}} {{\boldsymbol{h}}_m}} {{\boldsymbol{w}}_m}{x_m} + {n_m} \\ \end{gathered} $$ (16) 式中,$\rho '$为微基站发射功率;$l$为微基站到用户1的距离;$d$为微基站到同小区其他用户的距离;$D$为宏基站到用户的距离;$\rho $为宏基站的发射功率。
同时,窃听用户接收到的信号为:
$$\begin{gathered} {y_e} = \sqrt {\rho '{G^{ - \alpha }}} {{\boldsymbol{h}}_e}{{\boldsymbol{w}}_1}{x_1} + \sum\limits_{j \in {\mathit{\Phi }_m}} {\sqrt {\rho {D^{ - \alpha }}} {{\boldsymbol{h}}_m}{{\boldsymbol{w}}_m}{x_m}} + \\ \sum\limits_{i = 2}^K {\sqrt {\rho '{d^{ - \alpha }}} } {{\boldsymbol{h}}_i}{{\boldsymbol{w}}_i}{x_i} + {n_f} \\ \end{gathered} $$ (17) 忽略合法信道与窃听信道噪声功率,则可达安全速率为:
$$\begin{gathered} {{R}_s} = {\log _2}\left( {1 + \frac{{\rho '{l^{ - \alpha }}{{\left| {{{\boldsymbol{h}}_1}{{\boldsymbol{w}}_1}} \right|}^2}}}{{\sum\limits_{i = 2}^K {\rho '{d^{ - \alpha }}{{\left| {{{\boldsymbol{h}}_i}{{\boldsymbol{w}}_i}} \right|}^2}} + \sum\limits_{k \in {\mathit{\Phi }_m}} {\rho {D^{ - \alpha }}{{\left| {{{\boldsymbol{h}}_m}{{\boldsymbol{w}}_m}} \right|}^2}} }}} \right) - \\ {\log _2}\left( {1 + \frac{{\rho '{G^{ - \alpha }}{{\left| {{{\boldsymbol{h}}_e}{{\boldsymbol{w}}_1}} \right|}^2}}}{{\sum\limits_{k \in {\mathit{\Phi }_m}} {\rho {D^{ - \alpha }}{{\left| {{{\boldsymbol{h}}_m}{{\boldsymbol{w}}_m}} \right|}^2}} + \sum\limits_{i = 2}^K {\rho '{d^{ - \alpha }}{{\left| {{{\boldsymbol{h}}_i}{{\boldsymbol{w}}_i}} \right|}^2}} }}} \right) \\ \end{gathered} $$ (18) 设安全速率限制指标常数为$\delta (\delta \geqslant 0)$,如果满足$R > \delta $,则认为基站与合法用户间可实现安全连接,否则,标记为安全中断事件。因此,定义安全连接概率为:
$$\begin{gathered} \Pr ({{R}_s} > \delta ) = {{\rm {\rm Pr}}}\{ {{R}_m} - {{R}_e} > \delta \} \mathop \approx \limits^{(a)} \\ {{\rm {\rm Pr}}}\{ {{\rm SIN}}{{{\rm R}}_m} > {2^{\delta + { E}({{R}_e})}} - 1\} \\ \end{gathered} $$ (19) 式中,窃听者的位置都是相互独立的,${{R}_e}$只与部署基站的位置和窃听者有关。所有${{R}_m}$和${{R}_e}$是相互独立的。为了获得计算结果,本文使用$E({{R}_e})$来取代${{R}_e}$。根据文献[12],此近似符合一致收敛,可以推导出:
$$\begin{gathered} E({{R}_e}) = 2\pi {\mathit{\lambda }_e}\int_{{{\rm 0}}}^{{{\rm }}\infty } {\int_{{{\rm 0}}}^{{{\rm }}\infty } {x{{\rm d}}t{{\rm d}}x} } \times \\ {{\rm exp}}\left\{ { - \frac{{({2^t} - 1)}}{{\rho '}}{x^\alpha } - \pi {x^2}y({2^t} - 1, \alpha , 1)} \right\} \\ \end{gathered} $$ (20) 式中,
$$y({2^t} - 1, \alpha , 1) = \frac{{2({2^t} - 1)}}{{\alpha - 2}}{}_2{F_1}\left( {1, 1 - \frac{2}{\alpha }, 2 - \frac{2}{\alpha }, - ({2^t} - 1)} \right)$$ (21) 用$\beta $代替${2^{\delta + {\rm E}({{R}_e})}} - 1$,这时式(19)可以写成:
$${{\rm {\rm Pr}}}\left\{ {\frac{{\rho '{l^{ - \alpha }}{{\left| {{{h}_1}{{w}_1}} \right|}^2}}}{{\sum\limits_{i = 2}^K {\rho '{d^{ - \alpha }}{{\left| {{{h}_i}{{w}_i}} \right|}^2}} + \sum\limits_{k \in {\mathit{\Phi }_m}} {\rho {D^{ - \alpha }}{{\left| {{{h}_m}{{w}_m}} \right|}^2}} }} > \beta } \right\}$$ (22) 式中,
$${I_{\Phi f}} = \sum\limits_{i = 2}^K {\rho '{d^{ - \alpha }}{{\left| {{{\boldsymbol{h}}_i}{{\boldsymbol{w}}_i}} \right|}^2}} , {I_{\Phi m}} = \sum\limits_{k \in {\mathit{\Phi }_m}} {\rho {D^{ - \alpha }}{{\left| {{{\boldsymbol{h}}_m}{{\boldsymbol{w}}_m}} \right|}^2}} $$ (23) 因此安全连接概率为:
$${\rm P}\left\{ {|{{\boldsymbol{h}}_1}{{\boldsymbol{w}}_1}{|^2} > \frac{{\beta {l^\alpha }}}{{\rho '}}({I_{\Phi f}} + {I_{\Phi m}})} \right\}$$ (24) 式中,$|{{\boldsymbol{h}}_1}{{\boldsymbol{w}}_1}{|^2}$服从$\frac{1}{2}\chi _{(2)}^2$,令$s = \frac{{\beta {l^\alpha }}}{{2\rho '}}$,因此有:
$${{\rm {\rm Pr}}}\{ {{\rm SIN}}{{{\rm R}}_m} > \beta \} = \frac{1}{4}{{\rm E}}\left[ {{{{\rm e}}^{ - \frac{{\beta {l^\alpha }}}{{2\rho '}}({I_{\Phi f}} + {I_{\Phi m}})}}} \right] = \frac{1}{4}{L_{{I_{\Phi f}}}}(s){L_{{I_{\Phi m}}}}(s)$$ (25) $${L_{{\mathit{\Phi }_f}}} = {{{\rm E}}_{{\mathit{\Phi }_f}}}\left[ {\prod\limits_{k \in {\mathit{\Phi }_f}\backslash \{ 0\} } {(\rho '|{\boldsymbol{h}_i}{\boldsymbol{w}_i}{|^2}{d^{ - \alpha }})} } \right]$$ (26) 由独立的泊松点分布,可知:
$$L_{{\mathit{\Phi }_f}}^{}(s) = \exp ( - 2\pi {\mathit{\lambda }_f}\int_{{{\rm }}r}^{{{\rm }}\infty } {(1 - {L_{|{\boldsymbol{h}_i}{\boldsymbol{w}_i}{|^2}}}(s\rho '{r^{ - \alpha }}))} rdr)$$ (27) 式中,$|{{\boldsymbol{h}}_1}{{\boldsymbol{w}}_1}{|^2}$服从$\frac{1}{2}\chi _{(2)}^2$,有:
$$\begin{gathered} L_{{\mathit{\Phi }_f}}^{}(s) = \exp ( - 2\pi {\mathit{\lambda }_f}\int_{{{\rm }}r}^{{{\rm }}\infty } {(1 - {L_{|{{h}_i}{{w}_i}{|^2}}}(s\rho '{r^{ - \alpha }}))} rdr) = \\ \exp \left( { - 2\pi {\mathit{\lambda }_f}\int_{{{\rm }}l}^{{{\rm }}\infty } {\left( {\frac{{r{{\rm d}}r}}{{1 + \frac{2}{\beta }{{\left( {\frac{r}{l}} \right)}^\alpha }}}} \right)} } \right) \\ \end{gathered} $$ (28) 根据超几何函数定义数学推导:
$$\int_{{{\rm }}l}^{{{\rm }}\infty } {\left( {\frac{{r{{\rm d}}r}}{{1 + \frac{2}{\beta }{{\left( {\frac{r}{l}} \right)}^\alpha }}}} \right)} = \frac{{\frac{\beta }{2}{l^2}}}{{\alpha - 2}}{}_1{F_2}\left( {1, 1 - \frac{2}{\alpha };2 - \frac{2}{\alpha };\frac{2}{\beta }} \right)$$ (29) 将式(29)带入式(28)可得:
$$L_{{\mathit{\Phi }_f}}^{}(s) = \exp \left( { - \frac{{\pi {\mathit{\lambda }_f}\beta {l^2}}}{{\alpha - 2}}{}_1{F_2}\left( {1, 1 - \frac{2}{\alpha };2 - \frac{2}{\alpha };\frac{2}{\beta }} \right)} \right)$$ (30) 可以推导出:
$$L_{{\mathit{\Phi }_m}}^{}(s) = \exp \left( { - \frac{{\pi {\mathit{\lambda }_m}\beta {l^2}\rho }}{{(\alpha - 2)\rho '}}{}_1{F_2}\left( {1, 1 - \frac{2}{\alpha };2 - \frac{2}{\alpha };\frac{2}{\beta }} \right)} \right)$$ (31) 将式(30)和式(31)带入式(25),得到安全连接概率为:
$$\begin{array}{l} \Pr ({R_s} > \delta ) = \\ \frac{1}{4}{{\rm{e}}^{ - {{\frac{{\pi {\lambda _f}\beta {l^2}}}{{\alpha - 2}}}_1}{F_2}\left( {1,1 - \frac{2}{\alpha };2 - \frac{2}{\alpha };\frac{2}{\beta }} \right)}}{{\rm{e}}^{ - {{\frac{{{\rm{\pi }}{\lambda _m}\beta {l^2}}}{{\alpha - 2}}}_1}{F_2}\left( {1,1 - \frac{2}{\alpha };2 - \frac{2}{\alpha };\frac{2}{\beta }} \right)}} \end{array} $$ (32) 由式(32)可得,安全连接概率与宏基站、微基站的部署密度、大尺度衰落的损耗系数$\alpha $、设定的安全给定速率限制值$\delta $等多个因素有关。
Research on the Deployment of Small Cell Base Stations for Physical Layer Security in Full-Dimensional MIMO Heterogeneous Network
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摘要: 异构网络配置全维度多天线利用垂直维度能够有效提高系统的传输速率,然而也会造成信号的泄露,影响系统的物理层安全性能。基于此,该文在异构网络下行链路中,全维度天线采用波束成形方法消除干扰,应用随机几何的分析方法,得到合法用户的安全连接概率闭合表达式,其安全连接概率与异构网络基站密度有关。仿真结构验证了安全连接概率结论的正确性,同时在给定的场景中,确定了异构网络基站的密度值。Abstract: Heterogeneous network with full-dimensional (FD) multiple-input multiple-output (MIMO) can effectively improve the transmission rate by developing vertical dimensions; however, it can cause the signal leakage and decrease the system's physical layer security performances. In this paper, using the beamforming to eliminate the interference in the downlink of the heterogeneous network, we propose an analysis method with stochastic geometric theory to obtain the closed-form expression of the secure connection probability on the legitimate users. The secure connection probability is related to the density of heterogeneous network small cell base stations (BSs). Simulation results verify the correctness of the safety connection probability, and determine the density value of heterogeneous network BS in a given scenario.
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