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Outage Performance for Relaying Aided Non-orthogonal Multiple Access

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Recent Trends in Intelligent Computing, Communication and Devices

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1006))

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Abstract

Non-orthogonal multiple access is the key technique of the fifth-generation wireless communication. In this paper, we investigate the outage performance of NOMA-based downlink amplify-and-forward half-duplex relaying networks. The closed-form representations of exact outage performance are attained. Simulation results indicate that the concluded result is in excellent agreement with the Monte Carlo simulation.

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Acknowledgements

This work was supported by Scientific Research Common Program of Beijing Municipal Commission of Education under Grant No. KM201510009008 and Scientific Research Foundation of North China University of Technology.

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Authors and Affiliations

  1. School of Information Science and Technology, North China University of Technology, Beijing, 100144, China

    Jinhong Fan & Li He

Authors
  1. Jinhong Fan
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  2. Li He
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Corresponding author

Correspondence to Jinhong Fan .

Editor information

Editors and Affiliations

  1. Victoria Business School, Victoria University of Wellington, Wellington, New Zealand

    Vipul Jain

  2. Department of Computer Science and Engineering, Faculty of Engineering and Technology, SOA University, Bhubaneswar, Odisha, India

    Srikanta Patnaik

  3. University "Politehnica" of Bucharest, Bucharest, Romania

    Florin Popențiu Vlădicescu

  4. Department of Computer Science and Engineering, Oakland University, Rochester, MI, USA

    Ishwar K. Sethi

Appendices

Appendix 1

Proof of Theorem 1

The outage probability of D1 is:

$$P_{\text{out}}^{1} = \left[ {1 - P\left( {\gamma_{s1 \to 2} \ge \gamma_{\text{th2}} ,\gamma_{s1} \ge \gamma_{\text{th1}} } \right)} \right] \times \left[ {1 - P\left( {\gamma_{R1 \to 2} \ge \gamma_{{{\text{th}}2}} ,\gamma_{R1} \ge \gamma_{\text{th1}} } \right)} \right]$$

and assumed that \(J_{1} = 1 - P\left( {\gamma_{s1 \to 2} \ge \gamma_{\text{th2}} ,\gamma_{s1} \ge \gamma_{\text{th1}} } \right)\) and \(J_{2} = \left[ {1 - P\left( {\gamma_{R1 \to 2} \ge \gamma_{\text{th2}} ,\gamma_{R1} \ge \gamma_{\text{th1}} } \right)} \right]\) where J1 can be solved as

$$\begin{aligned} J_{1} & = 1 - P\left( {\gamma_{s1 \to 2} \ge \gamma_{\text{th2}} } \right)P\left( {\gamma_{s1} \ge \gamma_{\text{th1}} } \right) \\ & = 1 - P\left( {\frac{{\left| {g_{1} } \right|^{2} a_{2} \gamma }}{{\left| {g_{1} } \right|^{2} a_{1} \gamma + 1}} \ge \gamma_{\text{th2}} } \right)P\left( {\left| {g_{1} } \right|^{2} a_{1} \gamma \ge \gamma_{\text{th1}} } \right) \\ & = 1 - P\left( {\left| {g_{1} } \right|^{2} \ge \frac{{\gamma_{\text{th2}} }}{{\gamma \left( {a_{2} - a_{1} \gamma_{\text{th2}} } \right)}} = \varepsilon } \right)P\left( {\left| {g_{1} } \right|^{2} \ge \frac{{\gamma_{\text{th1}} }}{{a_{1} \gamma }} = \beta } \right) \\ & = 1 - P\left[ {\left| {g_{1} } \right|^{2} \ge \hbox{max} \left( {\varepsilon ,\beta } \right)\underline{\underline{\Delta }} \theta } \right] \\ & = P\left( {\left| {g_{1} } \right|^{2} \le \theta } \right) = 1 - \exp \left( { - \frac{2\theta }{{\Omega_{1} }}} \right) \\ \end{aligned}$$
(15)

Now, J2 can be solved as

$$\begin{aligned} J_{2} & = 1 - P\left( {\gamma_{R1 \to 2} \ge \gamma_{\text{th2}} } \right)P\left( {\gamma_{R1} \ge \gamma_{\text{th1}} } \right) \\ & = 1 - P\left( {\frac{{a_{2} \gamma^{2} \left| {g_{r,1} } \right|^{2} \left| {g_{r} } \right|^{2} }}{{a_{1} \gamma^{2} \left| {g_{r,1} } \right|^{2} \left| {g_{r} } \right|^{2} + \gamma \left( {\left| {g_{r,1} } \right|^{2} + \left| {g_{r} } \right|^{2} } \right) + 1}} \ge \gamma_{\text{th2}} } \right)\\ & \quad \,\,P\left( {\frac{{a_{1} \gamma^{2} \left| {g_{r,1} } \right|^{2} \left| {g_{r} } \right|^{2} }}{{\gamma \left( {\left| {g_{r,1} } \right|^{2} + \left| {g_{r} } \right|^{2} } \right) + 1}} \ge \gamma_{\text{th1}} } \right) \\ & = 1 - P\left( {\left| {g_{r} } \right|^{2} \ge \frac{{\theta \left( {1 + \gamma \left| {g_{r,1} } \right|^{2} } \right)}}{{\left| {g_{r,1} } \right|^{2} - \theta }},\left| {g_{r,1} } \right|^{2} \ge \theta } \right) \\ & = 1 - \int\limits_{\theta }^{\infty } {\frac{1}{{\Omega_{r,1} }}} \exp \left( { - \frac{y}{{\Omega_{r,1} }}} \right)\int\limits_{{\frac{{\theta \left( {1 + \gamma y} \right)}}{y - \theta }}}^{\infty } {\frac{1}{{\Omega_{r} }}} \exp \left( { - \frac{x}{{\Omega_{r} }}} \right){\text{d}}x{\text{d}}y \\ & = 1 - \exp \left( { - \theta \left( {\frac{1}{{\Omega_{r,1} }} + \frac{1}{{\sigma_{r}^{2} }}} \right)} \right)\sqrt {\frac{{4\theta \left( {1 + \gamma \theta } \right)}}{{\gamma\Omega_{r,1}\Omega_{r} }}} K_{1} \left( {\sqrt {\frac{{4\theta \left( {1 + \gamma \theta } \right)}}{{\gamma\Omega_{r,1}\Omega_{r} }}} } \right) \\ \end{aligned}$$
(16)

Equation (16) is obtained with the aid [13] [(3.324.1)]. And substituting (15) and (16) into (11).

The closed-form representation of the outage performance of D1 is

$$\begin{aligned} P_{\text{out}}^{1} & = \left( {1 - \exp \left( { - \frac{2\theta }{{\Omega_{1} }}} \right)} \right) \\ & \quad \,\,\left\{ {1 - \exp \left( { - \theta \left( {\frac{1}{{\Omega_{r,1} }} + \frac{1}{{\Omega_{r} }}} \right)} \right)\sqrt {\frac{{4\theta \left( {1 + \gamma \theta } \right)}}{{\gamma\Omega_{r,1}\Omega_{r} }}} K_{1} \left( {\sqrt {\frac{{4\theta \left( {1 + \gamma \theta } \right)}}{{\gamma\Omega_{r,1}\Omega_{r} }}} } \right)} \right\} \end{aligned}$$

Appendix 2

Proof of Theorem 2

The outage probability of D2 is:

$$P_{\text{out}}^{2} = P\left( {\gamma_{R2} < \gamma_{\text{th2}} } \right)$$

and

$$\begin{aligned} P_{\text{out}}^{2} & = P\left( {\gamma_{R2} < \gamma_{\text{th2}} } \right) \\ & = P\left( {\left| {g_{r,2} } \right|^{2} < \varepsilon } \right) + P\left\{ {\left| {g_{r} } \right|^{2} < (\varepsilon /r)\left( {\gamma \left| {g_{r,2} } \right|^{2} { + 1}} \right)/\left( {\left| {g_{r,2} } \right|^{2} - \varepsilon } \right),\left| {g_{r,2} } \right|^{2} > \varepsilon } \right\} \\ & = \int\limits_{0}^{\varepsilon } {f_{{\left| {g_{r,2} } \right|^{2} }} \left( x \right)} {\text{d}}x + \int\limits_{\varepsilon }^{\infty } {f_{{\left| {g_{r,2} } \right|^{2} }} } \left( x \right){\text{d}}x\int\limits_{0}^{{\frac{{\varepsilon \left( {1 + \gamma x} \right)}}{{\gamma \left( {x - \varepsilon } \right)}}}} {f_{{\left| {g_{r} } \right|^{2} }} } \left( y \right){\text{d}}y \\ & = 1 - \frac{1}{{\Omega_{r,2} }}\exp \left( { - \varepsilon \left( {\frac{1}{{\Omega_{r} }} + \frac{1}{{\Omega_{r,2} }}} \right)} \right)\int\limits_{\varepsilon }^{\infty } {\exp \left( { - \frac{x - \varepsilon }{{\Omega_{r,2} }}} \right)\exp \left( { - \frac{{\varepsilon \left( {1 + \varepsilon \gamma } \right)}}{{\Omega_{r} \gamma \left( {x - \varepsilon } \right)}}} \right){\text{d}}x} \\ & = 1 - \frac{1}{{\Omega_{r,2} }}\exp \left( { - \varepsilon \left( {\frac{1}{{\Omega_{r} }} + \frac{1}{{\Omega_{r,2} }}} \right)} \right)\int\limits_{0}^{\infty } {\exp \left( { - \frac{y}{{\Omega_{r,2} }}} \right)\exp \left( { - \frac{{\varepsilon \left( {1 + \varepsilon \gamma } \right)}}{{\Omega_{r} \gamma y}}} \right){\text{d}}y} \\ & = 1 - \frac{1}{{\Omega_{r,2} }}\exp \left( { - \varepsilon \left( {\frac{1}{{\Omega_{r} }} + \frac{1}{{\Omega_{r,2} }}} \right)} \right)\int\limits_{0}^{\infty } {\exp \left( { - \frac{{4\varepsilon \left( {1 + \varepsilon \gamma } \right)}}{{\frac{{\Omega_{r} \gamma }}{4y}}} - \frac{y}{{\Omega_{r,2} }}} \right){\text{d}}y} \\ & = 1 - \exp \left( { - \varepsilon \left( {\frac{1}{{\Omega_{r} }} + \frac{1}{{\Omega_{r,2} }}} \right)} \right)\frac{1}{{\Omega_{r,2} }}\sqrt {\frac{{4\varepsilon \left( {1 + \varepsilon \gamma } \right)\Omega_{r,2} }}{{\Omega_{r} \gamma }}} \,K_{1} \left( {\sqrt {\frac{{4\varepsilon \left( {1 + \varepsilon \gamma } \right)}}{{\Omega_{r,2}\Omega_{r} \gamma }}} } \right) \\ & = 1 - \exp \left( { - \varepsilon \left( {\frac{1}{{\Omega_{r} }} + \frac{1}{{\Omega_{r,2} }}} \right)} \right)\sqrt {\frac{{4\varepsilon \left( {1 + \varepsilon \gamma } \right)}}{{\Omega_{r,2}\Omega_{r} \gamma }}}\ K_{1} \left( {\sqrt {\frac{{4\varepsilon \left( {1 + \varepsilon \gamma } \right)}}{{\Omega_{r,2}\Omega_{r} \gamma }}} } \right) \\ \end{aligned}$$
(17)

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Fan, J., He, L. (2020). Outage Performance for Relaying Aided Non-orthogonal Multiple Access. In: Jain, V., Patnaik, S., Popențiu Vlădicescu, F., Sethi, I. (eds) Recent Trends in Intelligent Computing, Communication and Devices. Advances in Intelligent Systems and Computing, vol 1006. Springer, Singapore. https://doi.org/10.1007/978-981-13-9406-5_88

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