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Secure 5G downlink NOMA cognitive relay network: joint the impact of imperfect spectrum sensing and outdated CSI

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Abstract

This article studies the interweave cognitive network with non-orthogonal multiple access (NOMA) principle under the appearance of one eavesdropper. The best relay selection method is used by considering two users of main channels at the same time. The general accuracy and asymptotic theory expressions of the system performance for two NOMA cognitive secondary users, e.g., outage probability and interception probability are derived, and they are also verified through Monte Carlo simulation. On this basis, we also investigate the effects of important system parameters such as the relay number, outdated channel side information (CSI), imperfect spectrum sensing, NOMA power allocation and imperfect successive interference cancellation (SIC) factor. It can be concluded that the outage diversity order for the NOMA users in the main channels can achieve as the relay number K when the perfect CSI appears. But the outage diversity order decreases to only one when the outdated CSI appears. However the intercept probability of the wiretap channel only got the diversity order as zero regardless of relay. And the imperfect spectrum sensing, imperfect SIC and NOMA power allocation can not affect the diversity order although they affect the system performance.

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Funding

This work was supported by Shandong Provincial Natural Science Foundation (ZR2020MF001, ZR2022MF273), A Project of Shandong Province Higher Educational Science and Technology Program (J18KA315), the National Defense Science and Technology Innovation Zone of China - Marine Science and Technology Collaborative Innovation Center Foundation of Qingdao (22-05-CXZX-04-02-06).

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Contributions

Xinjie Wang proposed the idea, checked the derivations, revised the paper. Min Zhang derived the equations, done the simulations, wrote the draft. Enyu Li and Guang Yang revised the paper. Xuhu Wang given some suggestions for simulations.

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Correspondence to Xinjie Wang.

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Appendices

Appendix A

\(I_{1}\) and \(I_{2}\) in (20) can be given as

$$\begin{aligned} {I_{1}}&=Pr({H=H_{0}}\vert {\tilde{H}=H_{0}}) \nonumber \\&=\frac{P_{a}(1-P_{f})}{P_{a}(1-P_{f})+(1-P_{a})(1-P_{d})} \end{aligned}$$
(31)
$$\begin{aligned} {I_{2}}&=Pr({H=H_{1}}\vert {\tilde{H}=H_{0}}) \nonumber \\&=\frac{(1-P_{a})(1-P_{d})}{P_{a}(1-P_{f})+(1-P_{a})(1-P_{d})} \end{aligned}$$
(32)

\(I_{3}^{k}=Pr\{{\left| {D}\right| =k}\vert {H=H_{0},\tilde{H}=H_{0}}\}\) represents the probability of successful relay decoding set under the condition that S detects the spectrum idle and matches the actual situation, and it means \(\alpha _{p}=0\). \(I_{4}^{k}=Pr\{{\left| {D}\right| =k}\vert {H=H_{1},\tilde{H}=H_{0}}\}\) is the probability of successful relay decoding if the sensing result does not match the actual situation, so it leads to \(\alpha _{p}=1\)\(I_{3}^{k}\) and \(I_{4}^{k}\) also mean that there are \(k(k=0,1,2,...,K)\) relay decodes successfully. Substitute (3) and (4) into \(I_{3}^{k}\) and \(I_{4}^{k}\),we have (33) and (34) as follows.

$$\begin{aligned} {I_{3}^{k}}&=Pr\{{\left| {D}\right| =k}\vert {H=H_{0},\tilde{H}=H_{0}}\} \nonumber \\&={{K}\atopwithdelims (){k}}{\prod _{\begin{array}{c} R \end{array}_{k}\in {C_{k_{c}}}}\Pr \left( {{\gamma }_{SR_{k}}}>{{\gamma }_{th}}\right) } \nonumber \\&\quad \times {\prod _{\begin{array}{c} R \end{array}_{k}\notin {C_{k_{c}}}}\left[ 1-\Pr \left( {{\gamma }_{SR_{k}}}>{{\gamma }_{th}}\right) \right] } \nonumber \\&={{K}\atopwithdelims (){k}}\prod _{\begin{array}{c} R \end{array}_{k}\in {C_{k_{c}}}} \int _{\gamma _{th}}^{\infty }\frac{1}{\overline{\gamma }_{SR}}\exp \left( -\frac{x}{\overline{\gamma }_{SR}}\right) dx \nonumber \\&\quad \times \prod _{\begin{array}{c} R \end{array}_{k}\notin {C_{k_{c}}}}\left[ 1-\int _{\gamma _{th}}^{\infty }\frac{1}{\overline{\gamma }_{SR}}\exp \left( -\frac{x}{\overline{\gamma }_{SR}}\right) dx\right] \nonumber \\&={{K}\atopwithdelims (){k}}\left[ 1-\exp \left( -\frac{\gamma _{th}}{\overline{\gamma }_{SR}}\right) \right] ^{K-k}\exp \left( -\frac{\gamma _{th}}{\overline{\gamma }_{SR}}\right) ^{k} \end{aligned}$$
(33)
$$\begin{aligned} {I_{4}^{k}}&={\Pr \{{{{{\left| {D}\right| }={k}}}\vert {{{H}=H_{1},\tilde{H}=H_{0}}}}\}} \nonumber \\&={{K}\atopwithdelims (){k}}{\prod _{\begin{array}{c} R \end{array}_{k}\in {C_{k_{c}}}}}\Pr ({{\gamma }_{1,R_{1k}}}\ge {{\gamma }_{th1}},{{\gamma }_{1,R_{2k}}}\ge {{\gamma }_{th2}}) \nonumber \\&\quad \times {\prod _{\begin{array}{c} R \end{array}_{k}\notin {C_{k_{c}}}}}\left[ 1-\Pr ({{\gamma }_{1,R_{1k}}}\ge {{\gamma }_{th1}},{{\gamma }_{1,R_{2k}}}\ge {{\gamma }_{th2}})\right] \nonumber \\&={{K}\atopwithdelims (){k}}{\prod _{\begin{array}{c} R \end{array}_{k}\in {C_{k_{c}}}}}\Pr \left[ {{\gamma }_{SR_{k}}}\ge {{\gamma }_{th}}(\gamma _{PR_{k}}+1)\right] \nonumber \\&\quad \times {\prod _{\begin{array}{c} R \end{array}_{k}\notin {C_{k_{c}}}}}\left\{ 1-\Pr \left[ {{\gamma }_{SR_{k}}}\ge {{\gamma }_{th}}(\gamma _{PR_{k}}+1)\right] \right\} \nonumber \\&={{K}\atopwithdelims (){k}}\prod _{\begin{array}{c} R \end{array}_{k}\in {C_{k_{c}}}} \int _{0}^{\infty }\frac{1}{\overline{\gamma }_{PR}}\exp \left( -\frac{y}{\overline{\gamma }_{PR}}\right) \nonumber \\&\quad \times \exp \left[ -\frac{\gamma _{th}(y+1)}{\overline{\gamma }_{SR}}\right] dy \nonumber \\&\quad \times \prod _{\begin{array}{c} R \end{array}_{k}\notin {C_{k_{c}}}}\left\{ 1-\int _{0}^{\infty }\frac{1}{\overline{\gamma }_{PR}}\exp \left( -\frac{y}{\overline{\gamma }_{PR}}\right) \nonumber \right. \\&\quad \times \left. \exp \left[ -\frac{\gamma _{th}(y+1)}{\overline{\gamma }_{SR}}\right] \right\} dy \nonumber \\&={{K}\atopwithdelims (){k}}\left[ \frac{\overline{\gamma }_{SR}}{\overline{\gamma }_{SR}+\overline{\gamma }_{PR}\gamma _{th}}\exp \left( -\frac{\gamma _{th}}{\overline{\gamma }_{SR}}\right) \right] ^{k} \nonumber \\&\quad \times \left[ 1-\frac{\overline{\gamma }_{SR}}{\overline{\gamma }_{SR}+\overline{\gamma }_{PR}\gamma _{th}}\exp \left( -\frac{\gamma _{th}}{\overline{\gamma }_{SR}}\right) \right] ^{K-k} \end{aligned}$$
(34)

Based on (17), \(I_{6}\) can be expressed as

$$\begin{aligned} I_{6}&= \int _{0}^{\infty }{F_{gi}\left[ (y+1){\tau }_{i}^{*}\right] }\frac{1}{\overline{\gamma }_{PD_{i}}}\exp \left( -\frac{y}{\overline{\gamma }_{PD_{i}}}\right) dy \nonumber \\&=\sum _{k_{i}=0}^{k-1}{\frac{\omega _{ik}\beta }{(k_{i}\lambda _{j}+\beta )}}\frac{1}{\overline{\gamma }_{PD_{i}}}\int _{0}^{\infty }\left\{ \exp \left[ -\frac{(y+1){\tau }_{i}^{*}}{\lambda _{i}}\right] \nonumber \right. \\&\left. \quad +\frac{{{k}_{i}}{{\lambda }_{j}}}{(k_{i}+1)\lambda _{i}}\exp \left[ -\frac{(y+1){\tau }_{i}^{*}}{A_{ik}}\right] \right\} \exp \left( \frac{y}{\overline{\gamma }_{PD_{i}}}\right) dy \nonumber \\&=\sum _{k_{i}=0}^{k-1}{\frac{\omega _{ik}\beta }{k_{i}\lambda _{j}+\beta }}\left[ \frac{\lambda _{i}}{\overline{\gamma }_{PD_{i}}{{\tau }_{i}^{*}}+\lambda _{i}}\exp \left( -\frac{{\tau }_{i}^{*}}{\lambda _{i}}\right) \nonumber \right. \\&\left. \quad +\frac{{{k}_{i}}{{\lambda }_{j}}}{(k_{i}+1)\lambda _{i}}\frac{A_{ik}}{(\overline{\gamma }_{PD_{i}}{{\tau }_{i}^{*}}+A_{ik})}\exp \left( -\frac{{\tau }_{i}^{*}}{A_{ik}}\right) \right] \end{aligned}$$
(35)

Substituting (31), (32), (33), (34) and (35) into (20), we get (21).

Appendix B

When considering the wiretap of the eavesdropper, the number of decoding relays should be larger than zero, so \(I_{3}^{k}=Pr\{{\left| {D}\right| =k}\vert {H=H_{0},\tilde{H}=H_{0}}\}\) can be represented as \(I_{3}^{k_{*}}=Pr\{{\left| {D}\right| =k,k>0}\vert {H=H_{0},\tilde{H}=H_{0}}\}\),where \(k>0\). So \(I_{3}^{k_{*}}\) can be expressed as

$$\begin{aligned} {I_{3}^{k_{*}}}&=Pr\left\{ {\left| {D}\right| =k}, k>0 \vert {H=H_{0},\tilde{H}=H_{0}}\right\} \nonumber \\&={{K}\atopwithdelims (){k}}\left[ 1-\exp \left( -\frac{\gamma _{th}}{\overline{\gamma }_{SR}}\right) \right] ^{K-k}\exp \left( -\frac{\gamma _{th}}{\overline{\gamma }_{SR}}\right) ^{k} \end{aligned}$$
(36)

Similarly, we can also obtain \({I_{4}^{k_{*}}}\) as

$$\begin{aligned} {I_{4}^{k^{*}}}&=\Pr \{{{{\left| {D}\right| }={k}}, k>0 \vert {{{H}=H_{1},\tilde{H}=H_{0}}}}\}\nonumber \\&={{K}\atopwithdelims (){k}}\left[ \frac{\overline{\gamma }_{SR}}{\overline{\gamma }_{SR}+\overline{\gamma }_{PR}\gamma _{th}}\exp \left( -\frac{\gamma _{th}}{\overline{\gamma }_{SR}}\right) \right] ^{k}\nonumber \\&\quad \times \left[ 1-\frac{\overline{\gamma }_{SR}}{\overline{\gamma }_{SR}+\overline{\gamma }_{PR}\gamma _{th}}\exp \left( -\frac{\gamma _{th}}{\overline{\gamma }_{SR}}\right) \right] ^{K-k} \end{aligned}$$
(37)

Substitute (13) into \(Pr\{\gamma _{0,E_{i}}>\gamma _{thE_{i}}^{x_{i}}\}\) and \(Pr\{\gamma _{1,E_{i}}>\gamma _{thE_{i}}^{x_{i}}\}\), we have

$$\begin{aligned}&Pr\{\gamma _{0,E_{i}}>\gamma _{thE_{i}}^{x_{i}}\}=\exp \left( -\frac{\gamma _{thE_{i}}^{x_{i}}}{\overline{\gamma }_{RE}}\right) \end{aligned}$$
(38)
$$\begin{aligned}&Pr\{\gamma _{1,E_{i}}\ge \gamma _{thE_{i}}^{x_{i}}\} =Pr\{\gamma _{RE}(\gamma _{PE}+1)\ge {\gamma _{thE_{i}}^{x_{i}}}\} \nonumber \\&\quad =\int _{0}^{\infty }\frac{1}{\overline{\gamma }_{PE}}\exp \left( -\frac{y}{\overline{\gamma }_{PE}}\right) \exp \left[ -\frac{(y+1)\gamma _{thE_{i}}^{x_{i}}}{\overline{\gamma }_{RE}}\right] dy \nonumber \\&\quad =\frac{\overline{\gamma }_{RE}}{\overline{\gamma }_{RE}+\overline{\gamma }_{PE}\gamma _{thE_{i}}^{x_{i}}} \exp \left( -\frac{\gamma _{thE_{i}}^{x_{i}}}{\overline{\gamma }_{RE}}\right) \end{aligned}$$
(39)

Substitute (20), (21), (36), (37), (38) and (39) into (24), we have the intercept probability of \(x_{i}\) by E as (25).

Appendix C

In the high SNR region, \(I_{3}^{k}\) can be gotten with the help of (26) as

$$\begin{aligned} I_{3}^{k*} \approx {{K}\atopwithdelims (){k}}\left( \frac{\gamma _{th}}{\overline{\gamma }_{SR}}\right) ^{K-k} \end{aligned}$$
(40)

\(I_{4}^{k}\) can be expressed as

$$\begin{aligned} I_{4}^{k*}&\approx {{K}\atopwithdelims (){k}} \left[ \int _{0}^{\infty } \left( \frac{\gamma _{th}(y+1)}{\overline{\gamma }_{SR}}\right) \right. \nonumber \\&\quad \times \left. \frac{1}{\overline{\gamma }_{PR}}\exp \left( \frac{y}{\overline{\gamma }_{PR} } \right) dy \right] ^{K-k}\nonumber \\&={{K}\atopwithdelims (){k}} \left[ \frac{\gamma _{th}\left( 1+{\overline{\gamma }_{PR}}^2 \right) }{{\overline{\gamma }_{SR}}{\overline{\gamma }_{PR}} } \right] ^{K-k} \end{aligned}$$
(41)

Based on (27), the asymptotic value of \(I_{6}\) can be expressed as \(I_{6}^{*}\). When \(\rho _{i}<1\)\(I_{6}^{*}\) is given as

$$\begin{aligned} I_{6}^{*,0}&\approx \sum _{k_{i}=0}^{k-1}\phi _{ik} \int _{0}^{\infty }(y+1){\tau }_{i}^{*}\frac{1}{\overline{\gamma }_{PD_{i}}}\exp \left( -\frac{y}{\overline{\gamma }_{PD_{i}}}\right) dy \nonumber \\&=\sum _{k_{i}=0}^{k-1}\phi _{ik}{\tau }_{i}^{*}\left( 1+\overline{\gamma }_{PD_{i}}\right) \end{aligned}$$
(42)

Substituting (40), (41) and (42) into (21), the outage probability with \(\rho _{i}<1\) can be gotten as (28).

when \(\rho _{i}=1\), \(I_{6}^{*}\) can be expressed as

$$\begin{aligned} I_{6}^{*,1}&\approx \frac{1}{{\beta }^{k}}\frac{\lambda _{j}}{\lambda _{i}+\lambda _{j}}\int _{0}^{\infty }\left[ (y+1){\tau }_{1}^{*}\right] ^{k}\frac{1}{\overline{\gamma }_{PD_{i}}}\exp (-\frac{y}{\overline{\gamma }_{PD_{i}}})dy\nonumber \\&=\frac{\lambda _{j}}{\lambda _{i}+\lambda _{j}}\left( \frac{{\overline{\gamma }_{PD_{i}}}{\tau }_{1}^{*}}{\beta }\right) ^{k} \exp \left( \frac{1}{\overline{\gamma }_{PD_{i}}}\right) \nonumber \\&\quad \times \Gamma \left( k+1,\frac{1}{\overline{\gamma }_{PD_{i}}}\right) \end{aligned}$$
(43)

Substituting (40), (41) and (43) into (21), the outage probability with \(\rho _{i}=1\) can be gotten as (29).

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Zhang, M., Wang, X., Li, E. et al. Secure 5G downlink NOMA cognitive relay network: joint the impact of imperfect spectrum sensing and outdated CSI. Wireless Netw 30, 1925–1937 (2024). https://doi.org/10.1007/s11276-023-03632-x

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