Skip to main content
Log in

Secrecy Performance of Amplify-and-Forward Relay Networks with Relay Selection under Nakagami-m Fading

  • Published:
Wireless Personal Communications Aims and scope Submit manuscript

Abstract

In this paper, we examine the secrecy outage performance of a dual-hop relay network under Nakagami-m fading. Here, we consider a three-terminal wireless communication system, where one node (source) communicates to the other node (destination) via a group of relay nodes. Due to severe fading and heavy shadowing, the direct link between two communicating nodes is not suitable to achieve the desired performance. Therefore, one relay is opportunistically selected amongst K relay nodes to establish their communications. For providing relay cooperation, the best relay is selected based on criteria that maximize the secrecy outage performance. Selected relay applies amplify-and-forward operation to facilitate relay assistance by broadcasting the received signal. Along with the destination node, an eavesdropper also receives the broadcasted signals. For this setup, we derive the closed-form expressions of secrecy outage probability with the assumption that wireless channels experience Nakagami-m fading. Various numerical results are illustrated to highlight the key performance impact of different system and channel parameters. We also verify the accuracy of our derived expressions by comparing the results with similar work done in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Wang, H.-M., & Xia, X.-G. (2015). Enhancing wireless secrecy via cooperation: Signal design and optimization. IEEE Communications Magazine, 53(12), 47–53.

    Article  Google Scholar 

  2. Pahuja, S., & Jindal, P. (2019). Cooperative communication in physical layer security: Technologies and challenges. Wireless Personal Communications,. https://doi.org/10.1007/s11277-019-06430-x.

    Article  Google Scholar 

  3. Zhang, Y., Sun, A., Liang, T., & Qiao, X. (2015). Max-ratio relay selection for secure communication in amplify-and-forward buffer-aided cooperative networks. In IEEE international conference on signal processing, communications and computing (ICSPCC) (pp. 1–4). Ningbo.

  4. Yang, N., Wang, L., Geraci, G., Elkashlan, M., Yuan, J., & Renzo, M. D. (2015). Safeguarding 5G wireless communication networks using physical layer security. IEEE Communications Magazine, 53(4), 20–27.

    Article  Google Scholar 

  5. Gurjar, D. S., & Upadhyay, P. K. (2018). Overlay device-to-device communications in asymmetric two-way cellular systems with hybrid relaying. IEEE Systems Journal, 12(4), 3713–3724.

    Article  Google Scholar 

  6. Li, J., Petropulu, A. P., & Weber, S. (2011). On cooperative relaying schemes for wireless physical layer security. IEEE Transactions on Signal Processing, 59(10), 4985–4996.

    Article  MathSciNet  Google Scholar 

  7. Gurjar, D. S., Upadhyay, P. K., da Costa, D. B., & de Sousa, R. T. (2017). Beamforming in traffic-aware two-way relay systems with channel estimation error and feedback delay. IEEE Transactions on Vehicular Technology, 66(10), 8807–8820.

    Article  Google Scholar 

  8. Wu, N. E., & Li, H. J. (2013). Effect of feedback delay on secure cooperative networks with joint relay and jammer selection. IEEE Wireless Communications Letters, 2(4), 415–418.

    Article  Google Scholar 

  9. Liu, Y., Wang, L., Duy, T. T., Elkashlan, M., & Duong, T. Q. (2015). Relay selection for security enhancement in cognitive relay networks. IEEE Wireless Communications Letters, 4(1), 46–49.

    Article  Google Scholar 

  10. Hoang, T. M., Duong, T. Q., Vo, N. S., & Kundu, C. (2017). Physical layer security in cooperative energy harvesting networks with a friendly jammer. IEEE Wireless Communications Letters, 6(2), 28–31.

    Article  Google Scholar 

  11. Wang, W., Teh, K. C., & Li, K. H. (2016). Generalized relay selection for improved security in cooperative DF relay networks. IIEEE Wireless Communications Letters, 5(1), 28–31.

    Article  Google Scholar 

  12. Bouallegue, T., & Sethom, K. (2017). Green and secure relay selection algorithm in cooperative networks. In Sixth international conference on communications and networking (ComNet), Hammamet, Tunisia.

  13. Wang, K., Yuan, L., Miyazaki, T., Zeng, D., Guo, S., & Sun, Y. (2017). Strategic antieavesdropping game for physical layer security in wireless cooperative networks. IEEE Transactions on Vehicular Technology, 66(10), 9448–9457.

    Article  Google Scholar 

  14. Lee, J. H. (2015). Cooperative relaying protocol for improving physical layer security in wireless decode-and-forward relaying networks. Wireless Personal Communications, 83(4), 3033–3044.

    Article  Google Scholar 

  15. Guo, H., Yang, Z., Zhang, L., Zhu, J., & Zou, Y. (2017). Joint cooperative beamforming and jamming for physical-layer security of decode-and-forward relay networks. IEEE Access, 5, 19620–19630.

    Article  Google Scholar 

  16. Zhang, C., Ge, J., Xia, Z., & Du, H. (2017). Graph theory based cooperative transmission for physical-layer security in 5G large-scale wireless relay networks. IEEE Access, 5, 21640–21649.

    Article  Google Scholar 

  17. Saeidi-Khabisi, F. S., Vakili, V. T., & Abbasi-Moghadam, D. (2017). Improving the physical layer security in cooperative networks with multiple eavesdroppers. Wireless Personal Communications, 95(3), 3295–3320.

    Article  Google Scholar 

  18. Rahmanpour, A., Vakili, V. T., & Razavizadeh, S. M. (2017). Enhancement of physical layer security using destination artificial noise based on outage probability. Wireless Personal Communications, 95(2), 1553–1565.

    Article  Google Scholar 

  19. Qing, L., Guangyao, H., & Xiaomei, F. (2018). Physical layer security in multi-hop AF relay network based on compressed sensing. IEEE Communications Letters, 22(9), 1882–1885.

    Article  Google Scholar 

  20. Nguyen, B. V., & Kim, K. (2015). Secrecy outage probability of optimal relay selection for secure AnF cooperative networks. IEEE Communications Letters, 19(12), 2086–2089.

    Article  Google Scholar 

  21. Zou, Y., Champagne, B., Zhu, W. P., & Hanzo, L. (2015). Relay-selection improves the security-reliability trade-off in cognitive radio systems. IEEE Transactions on Communications, 63(1), 215–228.

    Article  Google Scholar 

  22. Amarasuriya, G., Ardakani, M., & Tellambura, C. (2010). Output-threshold multiple relay selection scheme for cooperative wireless networks. IEEE Transactions on Vehicular Technology, 59(6), 3091–3097.

    Article  Google Scholar 

  23. Ikki, S., & Ahmed, M. H. (2007). Performance analysis of cooperative diversity wireless networks over Nakagami-\(m\) fading channel. IEEE Communications Letters, 11(4), 334–336.

    Article  Google Scholar 

  24. Gradshteyn, I., & Ryzhik, I. (2007). Table of integrals, series, and products (7th ed.). San Diego: Academic Press.

    MATH  Google Scholar 

  25. Simon, M. K., & Alouini, M.-S. (2005). Digital communication over fading channels: A unified approach to performance analysis (2nd ed.). New York: Wiley.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Devendra S. Gurjar.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A

Here, we provide the detailed step-by-step solution for (12). Based on (12), the expression of \(I_{11}\) can be given as

$$\begin{aligned} I_{11}&=\int _0^\infty F_X(x)f_Y(x)dx. \end{aligned}$$
(32)

After incorporating the respective CDF and PDF in (32) and solving further, we get

$$\begin{aligned} I_{11}=\int _0^\infty \Bigg (1-\sum _{k=0}^{m_x-1}\dfrac{(x\beta _x)^k}{k!}e^{-\beta _xx}\Bigg )\dfrac{\beta _y^{m_y}}{\varGamma [m_y]}x^{m_y-1}e^{-\beta _yx}dx. \end{aligned}$$
(33)

On applying [24, eqs. 3.381.1, 3.471.9] and doing some adjustments, one can obtain

$$\begin{aligned} I_{11}=1-\dfrac{1}{\varGamma [m_y]}\left( \dfrac{\beta _y}{\beta _x+\beta _y}\right) ^{m_y}\sum _{k=0}^{m_x-1}\dfrac{1}{k!}\left( \dfrac{\beta _x}{\beta _x+\beta _y}\right) ^k\varGamma [k+m_y]. \end{aligned}$$
(34)

Similarly, we derive the expression of \(I_{12}\) as

$$\begin{aligned} I_{12} = 1-\dfrac{1}{\varGamma [m_z]}\left( \dfrac{\beta _z}{\beta _x+\beta _z}\right) ^{m_z} \sum _{k=0}^{m_x-1}\dfrac{1}{k!}\left( \dfrac{\beta _x}{\beta _x+\beta _z}\right) ^k\varGamma [k+m_z]. \end{aligned}$$
(35)

On invoking the PDFs and CDF in the respective part of (12), we represent \(I_{13}\) as

$$\begin{aligned} I_{13}&=\int _0^\infty \Big [1- \sum _{k=0}^{m_x-1}\dfrac{(x\beta _x)^k}{k!}e^{-\beta _x x}\Big ]\Big [1- \sum _{l=0}^{m_y-1}\dfrac{(x\beta _y)^l}{l!}e^{-x\beta _y}\Big ]\nonumber \\&\quad \times \dfrac{\beta _z^{m_z}}{\varGamma [m_z]} x^{m_z-1}e^{-\beta _zx}dx, \end{aligned}$$
(36)

which can be further expressed as

$$\begin{aligned} I_{13}&=1-\dfrac{\beta _z^{m_z}}{\varGamma [m_z]}\sum _{k=0}^{m_x-1}\dfrac{\beta _x^k}{k!}\dfrac{\varGamma [k+m_z]}{(\beta _x+\beta _z)^{k+m_z}}-\dfrac{\beta _z^{m_z}}{\varGamma [m_z]}\nonumber \\&\quad \times \sum _{l=0}^{m_y-1}\dfrac{\beta _y^l}{l!}\dfrac{\varGamma [l+m_z]}{(\beta _y+\beta _z)^{l+m_z}}+\dfrac{\beta _z^{m_z}}{\varGamma [m_z]}\sum _{k=0}^{m_x-1}\nonumber \\&\quad \times \sum _{l=0}^{m_y-1}\dfrac{\beta _x^k}{k!}\dfrac{\beta _y^l}{l!}\dfrac{\varGamma [k+l+m_z]}{(\beta _x+\beta _y+\beta _z)^{k+l+m_z}} \end{aligned}$$
(37)

In the same way

$$\begin{aligned} I_{14}&=\int _0^\infty \Big [1-\sum _{k=0}^{m_x-1}\dfrac{(x\beta _x)^k}{k!}e^{-\beta _xx}\Big ]\dfrac{\beta _y^{m_y}}{\varGamma [m_y]}x^{m_y-1}e^{-\beta _yx}\nonumber \\&\quad \times \Big [1-\sum _{l=0}^{m_z-1}\dfrac{(x\beta _z)^l}{l!}e^{-\beta _z x}\Big ]dx, \end{aligned}$$
(38)
$$\begin{aligned} I_{14}&=1-\dfrac{\beta _y^{m_y}}{\varGamma [m_y]}\sum _{k=0}^{m_x-1}\dfrac{\beta _x^k}{k!}\dfrac{\varGamma [k+m_y]}{(\beta _x+\beta _y)^{k+m_y}}-\dfrac{\beta _y^{m_y}}{\varGamma [m_y]}\nonumber \\&\quad \times \sum _{l=0}^{m_z-1}\dfrac{\beta _z^l}{l!}\dfrac{\varGamma [l+m_y]}{(\beta _y+\beta _z)^{l+m_y}}+\dfrac{\beta _y^{m_y}}{\varGamma [m_y]}\sum _{k=0}^{m_x-1}\nonumber \\&\quad \times \sum _{l=0}^{m_z-1}\dfrac{\beta _x^k}{k!}\dfrac{\beta _z^l}{l!}\dfrac{\varGamma [k+l+m_y]}{(\beta _x+\beta _y+\beta _z)^{k+l+m_y}}. \end{aligned}$$
(39)

Appendix B

On invoking the expressions of respective CDF in the Integral of (17) and solving further using the similar step as followed for \(I_{1}\), the expression of \(I_{2}\) can be given as

$$\begin{aligned} I_2&=\int _0^\infty f_Z(z)\int _z^{a+bz}\dfrac{\beta _x^{m_x}}{\varGamma [m_x]}x^{m_x-1}e^{-\beta _x x}e^{-\beta _y x}\nonumber \\&\quad \times \sum _{k=0}^{m_y-1}\dfrac{(\beta _yx)^k}{k!}dxdz. \end{aligned}$$
(40)

Now, simplifying \(I_{2}\) using equations [24, eqs. 3.381.1, 8.352], we have

$$\begin{aligned} I_2&=\int _0^\infty \beta _z^{m_z}\dfrac{z^{m_z-1}}{\varGamma [m_z]}e^{-\beta _zz}\dfrac{\beta _x^{m_x}}{\varGamma [m_x]}\sum _{k=0}^{m_y-1}\dfrac{\beta _y^k}{k!}(\beta _x+\beta _y)^{-(k+m_x)}\nonumber \\&\quad \times \Big [\varUpsilon (k+m_x,(\beta _x+\beta _y)(a+bz))-\varUpsilon (k+m_x),(\beta _x+\beta _y)z\Big ]dz. \end{aligned}$$
(41)

On applying the binomial expansion in (41), the required solution can be obtained as given in (18).

Appendix C

The Integral expression of \(I_{42}\) can written from (22) as

$$\begin{aligned} I_{42}=\int _a^\infty f_X(x)\int _0^{\dfrac{x-a}{b}}f_Z(z)\int _0^{a+bz}f_Y(y)dydzdx. \end{aligned}$$
(42)

After incorporating the respective PDFs, we get

$$\begin{aligned} I_{42}&= \int _a^\infty f_X(x)\int _0^{\dfrac{x-a}{b}} f_Z(z)\dfrac{\varUpsilon (m_y,\beta _y(a+bz))}{\varGamma [m_y]}dzdx. \end{aligned}$$
(43)

After solving further

$$\begin{aligned} I_{42}&=\int _a^\infty f_X(x)\Bigg (\dfrac{\varUpsilon \left( m_z,\beta _z\left( \dfrac{x-a}{b}\right) \right) }{\varGamma [m_z]}-\dfrac{\beta _z^{m_z}}{\varGamma [m_z]}e^{-a\beta _y}\sum _{k=0}^{m_y-1}\nonumber \\&\quad \times \sum _{l=0}^k\dfrac{\beta _y^k}{k!} C_l^ka^{k-l}b^l\dfrac{\varUpsilon \left( l+m_z,(b\beta _y+\beta _z)\left( \dfrac{x-a}{b}\right) \right) }{(b\beta _y+\beta _z)^{l+m_z}}\Bigg )dx. \end{aligned}$$
(44)

On incorporating the PDF \(f_{X}(x)\), we get

$$\begin{aligned} I_{42}&=\dfrac{\beta _x^{m_x}}{\varGamma [m_x]}\int _a^\infty x^{m_x-1}e^{-\beta _xx}dx -\dfrac{\beta _x^{m_x}}{\varGamma [m_x]}e^{a\beta _z/b}\sum _{g=0}^{m_z-1}\sum _{h=0}^g\nonumber \\&\qquad \times \dfrac{(\beta _z/b)^g}{g!}(-a)^{g-h}C^g_h \int _a^\infty x^{h+m_x-1}e^{-(\beta _x+\beta _z/b)x}dx\nonumber \\&\quad -\dfrac{\beta _x^{m_x}\beta _z^{m_z}}{\varGamma [m_x]\varGamma [m_z]}e^{-a\beta _y}\sum _{k=0}^{m_y-1}\sum _{l=0}^k\dfrac{a^{k-l}b^l\beta _y^k}{k!}C^k_l\nonumber \\&\quad \times (b\beta _y+\beta _z)^{-(l+m_z)}\varGamma [l+m_z]\int _a^\infty x^{m_x-1}e^{-\beta _xx}dx\nonumber \\&\quad +\dfrac{\beta _x^{m_x}\beta _z^{m_z}}{\varGamma [m_x]\varGamma [m_z]}e^{-a\beta _y}\sum _{k=0}^{m_y-1}\sum _{l=0}^k\sum _{p=0}^{l+m_z-1}\sum _{q=0}^p\nonumber \\&\quad \times \dfrac{a^{k-l}b^l\beta _y^k(-a)^{p-q}}{k!p!}C^k_lC^p_q\dfrac{((b\beta _y+\beta _z)/b)^p}{(b\beta _y+\beta _z)^{l+m_z}}e^{a\beta _y+a\beta _z/b}\nonumber \\&\quad \times \varGamma [l+m_z]\int _a^\infty x^{p+m_x-1}e^{-(\beta _x+\beta _y+\beta _z/b)x}dx. \end{aligned}$$
(45)

On solving the simple integration, one can get the final expression of \(I_{42}\) as in (28).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Datta, R., Gurjar, D.S., Manohar Reddy, T.K. et al. Secrecy Performance of Amplify-and-Forward Relay Networks with Relay Selection under Nakagami-m Fading. Wireless Pers Commun 112, 2233–2251 (2020). https://doi.org/10.1007/s11277-020-07147-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11277-020-07147-y

Keywords

Navigation