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SCIENCE CHINA Information Sciences, Volume 61, Issue 2: 022310(2018) https://doi.org/10.1007/s11432-016-9149-4

Physical layer security in multi-antenna cognitive heterogeneous cellular networks: a unified secrecy performance analysis

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  • ReceivedDec 25, 2016
  • AcceptedMay 24, 2017
  • PublishedSep 22, 2017

Abstract

Cognitive heterogeneous cellular networks (CHCNs) are emerging as a promising approach to next-generation wireless communications owing to their seamless coverage and high network throughput. In this paper, we describe our reliance on multi-antenna technology and a secrecy transmission protocol to ensure the reliability and security of downlink underlay CHCNs. First, we introduce a two-tier CHCN model using a stochastic geometry framework, and derive the probability distribution of the indicator function for a secrecy transmission scheme. We then investigate the connection outage probability, secrecy outage probability (SOP), and transmission SOP of both primary and cognitive users under a secrecy guard scheme and a threshold-based scheme. Furthermore, we reveal some insights into the secrecy performance by properly setting the predetermined access threshold and the radius of detection region for the secrecy transmission scheme. Finally, simulation results are provided to show the influence of the antenna system, eavesdropper density, predetermined access threshold, and radius of the detection region on the reliability and security performance of a CHCN.


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61401510, 61379006, 61601514, 61521003), National High Technology Research and Development Program of China (863) (Grant No. 2015AA01A708).


Supplement

The activation probability of the CFBS is derived by

\begin{align}P_{{\rm c}\_{\rm act}} = &\left[ {1 - {{\rm E}_{\Phi _{{\text{CU}}}^{}}}\left[ {\prod\limits_{{x_{{\text{CU}}}} \in \Phi _{{\text{CU}}}^{}} {\mathbb{P}\left( {{x_{{\text{CU}}}}{\text{ is not associated with }}{B_{\rm c}}} \right)} } \right]} \right]\mathbb{P}\left( {{\text{No eavesdropper within detection region}}} \right) \\ =& \left[ {1 - {{\rm E}_{\Phi _{{\text{CU}}}^{}}}\left[ {\prod\limits_{{x_{{\text{CU}}}} \in \Phi _{{\text{CU}}}^{}} {\mathbb{P}\left( {{P_{\rm c}}{\Delta _{\rm c}}{{\left\| {{x_{{x_{{\text{CU}}}},{B_{\rm c}}}}} \right\|}^{ - \alpha }} < \mathop {\max }\limits_{{x_{\rm c}} \in {\Phi _{\rm c}}\backslash{B_{\rm c}}} {P_{\rm c}}{\Delta _{\rm c}}{{\left\| {{x_{{\rm c},{x_{{\text{CU}}}}}}} \right\|}^{ - \alpha }}} \right)} } \right]} \right]{{\rm e}^{ - \pi {\lambda _{\text{E}}}R_\mu ^2}} \\ \mathop = \limits^{\left( {\text{b}} \right)}& \left[ {1 - {{\rm E}_{\Phi _{{\text{CU}}}^{}}}\left[ {\prod\limits_{{x_{{\text{CU}}}} \in \Phi _{{\text{CU}}}^{}} {\left( {1 - {{\rm e}^{ - \pi {{\bar \lambda }_{\rm c}}{{\left\| {{x_{{x_{{\text{CU}}}},{B_{\rm c}}}}} \right\|}^2}}}} \right)} } \right]} \right]{{\rm e}^{ - \pi {\lambda _{\text{E}}}R_\mu ^2}} \\ \mathop = \limits^{\left( {\text{c}} \right)}& \left[ {1 - \exp \left[ { - {\lambda _{{\text{CU}}}}\bar \lambda _{\rm c}^{ - 1}\left( {1 - {{\rm e}^{ - \pi {{\bar \lambda }_{\rm c}}{{\left( {{P_{\rm c}}{\Delta _{\rm c}}\gamma _\mu ^{ - 1}} \right)}^{{2 / \alpha }}}}}} \right)} \right]} \right]{{\rm e}^{ - \pi {\lambda _{\text{E}}}R_\mu ^2}}, \tag{30} \end{align}

where $\mathbb{P}\{~{{\text{No~eavesdropper~within~detection~region}}}~\}$ is given in 7, $\|~{{x_{{x_{{\text{CU}}}},{B_{\rm~c}}}}}~\|~=~\|~{{x_{{\text{CU}}}}~-~{B_{\rm~c}}}~\|$, $\|~{{x_{{\rm~c},{x_{{\text{CU}}}}}}}~\|~=~\|~{{x_{{\text{CU}}}}~-~{x_{\rm~c}}}~\|$, the derivation of step (b) is given in [20], and step (c) is obtained based on the probability generating functional lemma (PGFL) over HPPP ${\Phi~_{\text{E}}}$ [22]. The SOP of PU is derived as

\begin{eqnarray}{P_{{\rm so},{\rm p}}}\left( {{{\hat R}_{{\rm s},{\rm p}}}} \right) = 1 - {F_{{\text{SIN}}{{\text{R}}_{{\text{E}},{\rm p}}}}}\left( {{2^{{{\hat R}_{\rm p}} - {{\hat R}_{{\rm s},{\rm p}}}}} - 1} \right) = 1 - \exp \left( {\frac{{ - {\lambda _{\text{E}}}\alpha {{\left( {{2^{{{\hat R}_{\rm p}} - {{\hat R}_{{\rm s},{\rm p}}}}}} \right)}^{ - ({\Psi _{\rm p}} - 1)}}}}{{{A_{\rm p}}{{\left( {{2^{{{\hat R}_{\rm p}} - {{\hat R}_{{\rm s},{\rm p}}}}} + 1} \right)}^{2/\alpha }}}}} \right). \tag{31} \end{eqnarray}

Then, the CDF of ${\text{SIN}}{{\text{R}}_{{\text{E}},{\rm~p}}}$ is derived as

\begin{align}{F_{{\text{SINR}}_{{\text{E}},{\rm p}}^{}}}\left( \gamma \right)\mathop = \limits^{\left( {\text{d}} \right)}& {{\rm E}_{{\Phi _{\text{E}}}}}\left[ {\prod\limits_{e \in {\Phi _{\text{E}}}} {\left[ {1 - {\rm E}\left[ {\exp \left( {\frac{{\gamma {I_{{\text{E,}}p}}}}{{{P_{\rm p}}{{\left\| {{x_{e,{\rm p}}} - e} \right\|}^{ - \alpha }}}}} \right)} \right]} \right]} } \right] \\ \mathop = \limits^{\left( {\text{e}} \right)}& \exp \left( { - 2\pi {\lambda _{\text{E}}}\int_0^\infty {{L_{I_{{\rm p},{\rm p}}^{{\text{E,intra}}}}}\left( s \right){L_{I_{{\rm p},{\rm p}}^{{\text{E,inter}}} + I_{{\rm c},{\rm p}}^{\text{E}}}}\left( s \right)x{\rm d}x} } \right), \tag{32} \end{align}

where step (d) is derived by the PDF of ${h_{e,{\rm~p}}}$ with ${h_{e,{\rm~p}}}~\sim~\exp~(1)$, and step (e) is achieved based on the PGFL over PPP ${\Phi~_{\text{E}}}$ [22]. Additionally, the Laplace transform of $I_{{\rm~p},{\rm~p}}^{{\text{E,intra}}}$ and $I_{{\rm~p},{\rm~p}}^{{\text{E,inter}}}~+~I_{{\rm~c},{\rm~p}}^{\text{E}}$ can be given by

\begin{eqnarray}{L_{I_{{\rm p},{\rm p}}^{{\text{E,intra}}}}}\left( s \right) = {\rm E}\left[ {\exp \left( { - \frac{{\gamma I_{{\rm p},{\rm p}}^{{\text{E,intra}}}{x^\alpha }}}{{{P_{\rm p}}}}} \right)} \right] = {\left( {\gamma + 1} \right)^{ - ({\Psi _{\rm p}} - 1)}}, \tag{33} \end{eqnarray}

and

\begin{eqnarray}{L_{I_{{\rm p},{\rm p}}^{{\text{E,inter}}} + I_{{\rm c},{\rm p}}^{\text{E}}}}\left( s \right) = {\rm E}\left[ {\exp \left( { - \frac{{\gamma \left( {I_{{\rm p},{\rm p}}^{{\text{E,inter}}} + I_{{\rm c},{\rm p}}^{\text{E}}} \right)x_{}^\alpha }}{{{P_{\rm p}}}}} \right)} \right]{\text{ = }}\exp \left( { - \frac{{\pi {x^2}{A_{\rm p}}}}{{\alpha {\gamma ^{{{ - 2} \mathord{\left/ {\vphantom {{ - 2} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }}}}}} \right). \tag{34} \end{eqnarray}

Eq. 34 is achieved from the PGFL over PPP ${\Phi~_{\rm~c}}$ and the result of (3.241)

Gradshteyn I S, Ryzhik I M. Table of Integrals, Series and Products. 7th ed. San Diego: Academic Press, 2007. 322–323.

. Substituting 33 and 34 into 32, we obtain step (g). Moreover, we can derive the PDF of ${\text{SIN}}{{\text{R}}_{{\text{E}},{\rm~p}}}$ as follows:

\begin{eqnarray}{f_{{\text{SIN}}{{\text{R}}_{{\text{E}},{\rm p}}}}}\left( \gamma \right){\text{ = }}\exp \left( {\frac{{ - {\lambda _{\text{E}}}\alpha {{\left( {\gamma + 1} \right)}^{ - ({\Psi _{\rm p}} - 1)}}}}{{{A_{\rm p}}{\gamma ^{2/\alpha }}}}} \right)\left( {({\Psi _{\rm p}} - 1)\frac{{{\lambda _{\text{E}}}\alpha {{\left( {\gamma + 1} \right)}^{ - {\Psi _{\rm p}}}}}}{{{A_{\rm p}}{\gamma ^{2/\alpha }}}} + \frac{{2{\lambda _{\text{E}}}{{\left( {\gamma + 1} \right)}^{ - ({\Psi _{\rm p}} - 1)}}}}{{{A_{\rm p}}}}{\gamma ^{ - 2/\alpha - 1}}} \right). \tag{35} \end{eqnarray}

The SOP in a CUN is derived as

\begin{align}P_{{\rm so},{\rm c}}^{}\left( {{{\hat R}_{{\rm s},{\rm c}}}} \right) = &\mathbb{P}\left( {{{\log }_2}\left( {1 + {\text{SINR}}_{{\text{E}},{\rm c}}^{}} \right) > {{\hat R}_{\rm c}} - {{\hat R}_{{\rm s},{\rm c}}}\left| {\mu = 1} \right.} \right) \\ =& 1 - {{\rm E}_{{\Phi _{\text{E}}}}}\left[ {\prod\limits_{e \in {\Phi _{\text{E}}}} { {\mathbb{P}\left( {\frac{{{P_{\rm c}}{h_{e,{\rm c}}}{{\left\| {{x_{e,{\rm c}}}} \right\|}^{ - \alpha }}}}{{I_{{\rm c},{\rm c}}^{{\text{E,intra}}} + I_{{\rm c},{\rm c}}^{{\text{E,inter}}} + I_{{\rm p},{\rm c}}^{\text{E}}}} \leqslant {2^{{{\hat R}_{\rm c}} - {{\hat R}_{{\rm s},{\rm c}}}}} - 1\left| {\left\| {{x_{e,{\rm c}}}} \right\| > {R_\mu }} \right.} \right)} } } \right] \\ =& 1 - {F_{{\text{SIN}}{{\text{R}}_{{\text{E}},{\rm c}}}}}\left( {{2^{{{\hat R}_{\rm c}} - {{\hat R}_{{\rm s},{\rm c}}}}} - 1\left| {\left\| {{x_{e,{\rm c}}}} \right\| > {R_\mu }} \right.} \right), \tag{36} \end{align}

where ${F_{{\text{SIN}}{{\text{R}}_{{\text{E}},{\rm~c}}}}}\left(~{\gamma~\left|~{\left\|~{{x_{e,{\rm~c}}}}~\right\|~>~{R_\mu~}}~\right.}~\right)~=~{F_{{\text{SIN}}{{\text{R}}_{{\text{E}},{\rm~c}}}}}\left(~{\gamma~\left|~{\mu~=~1}~\right.}~\right)$ is the CDF of ${\text{SIN}}{{\text{R}}_{{\text{E}},{\rm~c}}}$ under the condition of $\mu~=~1$, and its detailed derivation process is given by

\begin{align}{F_{{\text{SIN}}{{\text{R}}_{{\text{E}},{\rm c}}}}}\left( {\gamma \left| {\mu = 1} \right.} \right) =& \exp \left( { - 2\pi {\lambda _{\text{E}}}\int_{{R_\mu }}^\infty {\left( {1 - \mathbb{P}\left( {{h_{e,{\rm c}}} \leqslant \frac{{I_{{\rm c},{\rm c}}^{{\text{E,intra}}} + I_{{\rm c},{\rm c}}^{{\text{E,inter}}} + I_{{\rm p},{\rm c}}^{\text{E}}}}{{{P_{\rm c}}{\gamma ^{ - 1}}{x^{ - \alpha }}}}} \right)} \right)x{\rm d}x} } \right) \\ =& \exp \left( { - 2\pi {\lambda _{\text{E}}}\int_{{R_\mu }}^\infty {{L_{I_{{\rm c},{\rm c}}^{{\text{E,intra}}}}}\left( s \right){L_{I_{{\rm c},{\rm c}}^{{\text{E,inter}}} + I_{{\rm p},{\rm c}}^{\text{E}}}}\left( s \right)x{\rm d}x} } \right) \\ =& \exp \left( {\frac{{ - {\lambda _{\text{E}}}\alpha {{\left( {\gamma + 1} \right)}^{ - \left( {{\Psi _{\rm c}} - 1} \right)}}}}{{{A_{\rm c}}{\gamma ^{2/\alpha }}}}\exp \left( { - \frac{{\pi R_\mu ^2{A_{\rm c}}}}{{\alpha {\gamma ^{{{ - 2} /\alpha }}}}}} \right)} \right), \tag{37} \end{align}

where

\begin{eqnarray}{L_{I_{{\rm c},{\rm c}}^{{\text{E,intra}}}}}\left( s \right) = {\rm E}\left[ {\exp \left( { - \frac{{\gamma I_{{\rm c},{\rm c}}^{{\text{E,intra}}}{x^\alpha }}}{{{P_{\rm c}}}}} \right)} \right] = {\left( {\gamma + 1} \right)^{ - ({\Psi _{\rm c}} - 1)}} \tag{38} \end{eqnarray}

and

\begin{eqnarray}{L_{{I_{{\rm c},{\rm c}}^{{\text{E,inter}}}}+I_{{\rm p},{\rm c}}^{\text{E}}}}\left( s \right) = {\rm E}\left[ {\exp \left( { - \frac{{\gamma \left( {I_{{\rm c},{\rm c}}^{{\text{E,inter}}}}+I_{{\rm p},{\rm c}}^{\text{E}}\right) x_{}^\alpha }}{{{P_{\rm c}}}}} \right)} \right] = \exp \left( {\frac{{ - {\lambda _{\text{E}}}\alpha }}{{{A_{\rm c}}{\gamma ^{2/\alpha }}}}\exp \left( { - \frac{{\pi R_\mu ^2{A_{\rm c}}}}{{\alpha {\gamma ^{{{ - 2}/\alpha }}}}}} \right)} \right). \tag{39} \end{eqnarray}

The expression of TSOP in a CUN is derived as follows:

\begin{align}{P_{{\rm tso},{\rm c}}} = &1 - \mathbb{P}\left( {\begin{array}{*{20}{c}} {{{\log }_2}\left( {1 + {\text{SINR}}_{{\text{U}},{\rm c}}^{}} \right) > {{\hat R}_{\rm c}}}&\& &{{{\log }_2}\left( {1 + {\text{SINR}}_{{\text{E}},{\rm c}}^{}} \right) < {{\hat R}_{\rm c}} - {{\hat R}_{{\rm s},{\rm c}}}} \end{array}\left| {\mu = 1} \right.} \right) \\ =& 1 - \mathbb{P}\left( {{\text{SINR}}_{{\text{U}},{\rm c}}^{} > {2^{{{\hat R}_{\rm c}}}} - 1,\left\| {{x_{\rm c}}} \right\| < {{\left( {{P_{\rm c}}{\Delta _{\rm c}}\gamma _\mu ^{ - 1}} \right)}^{{1/ \alpha }}},{\text{SINR}}_{{\text{E}},{\rm c}}^{} < {2^{{{\hat R}_{\rm c}} - {{\hat R}_{{\rm s},{\rm c}}}}} - 1,\left\| {{x_{e,{\rm c}}}} \right\| > {R_\mu }} \right)/{P_{\mu = 1}} \\ =& 1 - \left( {1 - {F_{{\text{SIN}}{{\text{R}}_{{\text{U}},{\rm c}}}}}\left( {{2^{{{\hat R}_{\rm c}}}} - 1\left| {{P_{\rm c}}{\Delta _{\rm c}}{{\left\| {{x_{\rm c}}} \right\|}^{ - \alpha }} > {\gamma _\mu }} \right.} \right)} \right){F_{{\text{SIN}}{{\text{R}}_{{\text{E}},{\rm c}}}}}\left( {{2^{{{\hat R}_{\rm c}} - {{\hat R}_{{\rm s},{\rm c}}}}} - 1\left| {\left\| {{x_{e,{\rm c}}}} \right\| > {R_\mu }} \right.} \right) \\ =& 1 - \left( {1 - {P_{{\rm co},{\rm c}}}\left( {{{\hat R}_{\rm c}}} \right)} \right)\left( {1 - {P_{{\rm so},{\rm c}}}\left( {{{\hat R}_{{\rm s},{\rm c}}}} \right)} \right), \end{align}

where ${F_{{\text{SIN}}{{\text{R}}_{{\text{U}},{\rm~c}}}}}(~{{2^{{{\hat~R}_{\rm~c}}}}~-~1|~{{P_{\rm~c}}{\Delta~_{\rm~c}}{{\|~{{x_{\rm~c}}}~\|}^{~-~\alpha~}}~>~{\gamma~_\mu~}}~}~)$ and ${F_{{\text{SIN}}{{\text{R}}_{{\text{E}},{\rm~c}}}}}(~{{2^{{{\hat~R}_{\rm~c}}~-~{{\hat~R}_{{\rm~s},{\rm~c}}}}}~-~1|~{\|~{{x_{e,{\rm~c}}}}~\|~>~{R_\mu~}}~}~)$ are derived in sect. 4.1 and 37, respectively.


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