Outage Probability of CR-NOMA Schemes with Multiple Antennas Selection and Power Transfer Approach

Abstract

In this paper, the outage performance of CR-NOMA schemes in decode-and-forward (DF) relay systems Device-to-Device (D2D) with antenna selection is investigated. We propose the power beacon, which can feed energy to the relay device node to further support the transmission from the source to the destination. To this end, closed-form expressions for the outage probabilities at user are derived. An asymptotic analysis at a high signal-to-noise ratio (SNR) is carried out to provide additional insights into the system performance. Furthermore, computer simulation results are presented to validate the accuracy of the attained analytical results.

Appendices

Appendix A: Proof of Theorem 1

From (13), $A$ can be formulated by

$$ \begin{gathered} A = \Pr \left( {\left| {h_{1n*} } \right|^{2} > \frac{\xi }{{P_{BS}^{\max } }},P_{BS}^{\max } < \frac{H}{{\left| {h_{p1*} } \right|^{2} }}} \right) + \Pr \left( {\left| {h_{1n*} } \right|^{2} > \frac{\xi }{{\frac{H}{{\left| {h_{p1*} } \right|^{2} }}}},P_{BS}^{\max } > \frac{H}{{\left| {h_{p1*} } \right|^{2} }}} \right) \hfill \\ = \underbrace {{\Pr \left( {\left| {h_{1n*} } \right|^{2} > \frac{\xi }{{P_{BS}^{\max } }},P_{BS}^{\max } < \frac{H}{{\left| {h_{p1*} } \right|^{2} }}} \right)}}_{{A_{1} }} + \underbrace {{\Pr \left( {\left| {h_{1n*} } \right|^{2} > \frac{\xi }{{\frac{H}{{\left| {h_{p1*} } \right|^{2} }}}},P_{BS}^{\max } > \frac{H}{{\left| {h_{p1*} } \right|^{2} }}} \right)}}_{{A_{2} }}. \hfill \\ \end{gathered} $$

(A.1)

Next, $A_{1}$ it can be first calculated as

$$ \begin{gathered} A_{1} = \Pr \left( {\left| {h_{1n*} } \right|^{2} > \frac{\xi }{{P_{BS}^{\max } }},\left| {h_{p1} } \right|^{2} < \frac{H}{{P_{BS}^{\max } }}} \right) = \Pr \left( {\left| {h_{1n*} } \right|^{2} > \frac{\xi }{{P_{BS}^{\max } }}} \right)\left[ {1 - \Pr \left( {\left| {h_{p1} } \right|^{2} \ge \frac{H}{{P_{BS}^{\max } }}} \right)} \right] \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} = \sum\limits_{n = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n - 1} } \exp \left( { - \frac{n\xi }{{P_{BS}^{\max } \lambda_{1n} }}} \right) \times \left[ {1 - \sum\limits_{n = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n - 1} } \exp \left( { - \frac{nH}{{P_{BS}^{\max } \lambda_{p1} }}} \right)} \right]. \hfill \\ \end{gathered} $$

(A.2)

In a similar way, $A_{2}$ it can be first calculated as

$$ \begin{gathered} A_{2} = \Pr \left( {\left| {h_{1n*} } \right|^{2} > \frac{{\left| {h_{p1*} } \right|^{2} \xi }}{H},\left| {h_{p1*} } \right|^{2} > \frac{H}{{P_{BS}^{\max } }}} \right) = \int_{{\frac{H}{{P_{BS}^{\max } }}}}^{\infty } {\left( {1 - F_{{\left| {h_{1n*} } \right|^{2} }} \left( {\frac{\xi x}{H}} \right)} \right)f_{{\left| {h_{p1*} } \right|^{2} }} \left( x \right)dx} \hfill \\ = \sum\limits_{n = 1}^{N} {\sum\limits_{m = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( \begin{gathered} N \hfill \\ m \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n + m - 2} } \frac{m}{{\lambda_{p1} }}} \int_{{\frac{H}{{P_{BS}^{\max } }}}}^{\infty } {\exp \left( { - \left( {\frac{n\xi }{{H\lambda_{1n} }} + \frac{m}{{\lambda_{p1} }}} \right)x} \right)dx} \hfill \\ = \sum\limits_{n = 1}^{N} {\sum\limits_{m = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( \begin{gathered} N \hfill \\ m \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n + m - 2} } } \frac{{mH\lambda_{1n} }}{{n\xi \lambda_{p1} + mH\lambda_{1n} }} \times \exp \left( { - \left( {\frac{n\xi }{{H\lambda_{1n} }} + \frac{m}{{\lambda_{p1} }}} \right)\frac{H}{{P_{BS}^{\max } }}} \right). \hfill \\ \end{gathered} $$

(A.3)

From (A.2) and (A.3), $A$ can write such as

$$ \begin{gathered} A = \sum\limits_{n = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n - 1} } \exp \left( { - \frac{n\xi }{{P_{BS}^{\max } \lambda_{1n} }}} \right) \times \left[ {1 - \sum\limits_{n = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n - 1} } \exp \left( { - \frac{nH}{{P_{BS}^{\max } \lambda_{p1} }}} \right)} \right] \hfill \\ + \sum\limits_{n = 1}^{N} {\sum\limits_{m = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( \begin{gathered} N \hfill \\ m \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n + m - 2} } } \frac{{mH\lambda_{1n} }}{{n\xi \lambda_{p1} + mH\lambda_{1n} }} \times \exp \left( { - \left( {\frac{n\xi }{{H\lambda_{1n} }} + \frac{m}{{\lambda_{p1} }}} \right)\frac{H}{{P_{BS}^{\max } }}} \right). \hfill \\ \end{gathered} $$

(A.4)

We plug (A.4) into (13), it can be achieved $OP_{1}^{{}}$ as the proposition.

This is the end of the proof.

Appendix B: Proof of Theorem 2

From (15), $B_{1}$ can be formulated by

$$ \begin{aligned} B_{1} &=\, \Pr \left( {\gamma_{{BS - D_{2} *}}^{{\left( {x_{2} } \right)}} \ge \varepsilon_{2} } \right) = \Pr \left( {\left| {h_{2n*} } \right|^{2} \ge \frac{{\varepsilon_{2} \omega_{0} }}{{P_{BS}^{\max } \left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)}},P_{BS}^{\max } < \frac{H}{{\left| {h_{p1*} } \right|^{2} }}} \right) \\& - \, \Pr \left( {\left| {h_{2n*} } \right|^{2} \ge \frac{{\varepsilon_{2} \omega_{0} }}{{\frac{H}{{\left| {h_{p1*} } \right|^{2} }}\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)}},P_{BS}^{\max } > \frac{H}{{\left| {h_{p1*} } \right|^{2} }}} \right) \hfill \\& = \, \underbrace {{\Pr \left( {\left| {h_{2n*} } \right|^{2} \ge \frac{{\varepsilon_{2} \omega_{0} }}{{P_{BS}^{\max } \left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)}},P_{BS}^{\max } < \frac{H}{{\left| {h_{p1*} } \right|^{2} }}} \right)}}_{{B_{1a} }}\\& - \, \underbrace {{\Pr \left( {\left| {h_{2n*} } \right|^{2} \ge \frac{{\varepsilon_{2} \omega_{0} \left| {h_{p1*} } \right|^{2} }}{{H\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)}},P_{BS}^{\max } > \frac{H}{{\left| {h_{p1*} } \right|^{2} }}} \right)}}_{{B_{1b} }}. \end{aligned} $$

(B.1)

Next, $B_{1a}$ it can be first calculated as

$$ \begin{gathered} B_{1a} = \Pr \left( {\left| {h_{2n*} } \right|^{2} \ge \frac{{\varepsilon_{2} \omega_{0} }}{{P_{BS}^{\max } \left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)}}} \right) \times \left[ {1 - \Pr \left( {\left| {h_{p1*} } \right|^{2} \ge \frac{H}{{P_{BS}^{\max } }}} \right)} \right] \hfill \\ = \sum\limits_{n = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n - 1} } \exp \left( { - \frac{{n\varepsilon_{2} \omega_{0} }}{{P_{BS}^{\max } \left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{2n} }}} \right) \times \left[ {1 - \sum\limits_{n = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n - 1} } \exp \left( { - \frac{nH}{{P_{BS}^{\max } \lambda_{p1} }}} \right)} \right]. \hfill \\ \end{gathered} $$

(B.2)

From (B.1), $B_{1b}$ it can be first calculated as

$$ \begin{gathered} B_{1b} = \Pr \left( {\left| {h_{2n*} } \right|^{2} \ge \frac{{\varepsilon_{2} \omega_{0} \left| {h_{p1*} } \right|^{2} }}{{H\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)}},\left| {h_{p1*} } \right|^{2} > \frac{H}{{P_{BS}^{\max } }}} \right) = \int_{{\frac{H}{{P_{BS}^{\max } }}}}^{\infty } {\left( {1 - F_{{\left| {h_{2n*} } \right|^{2} }} \left( {\frac{{\varepsilon_{2} \omega_{0} x}}{{H\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)}}} \right)} \right)f_{{\left| {h_{p1*} } \right|^{2} }} \left( x \right)dx} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} = \sum\limits_{n = 1}^{N} {\sum\limits_{m = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( \begin{gathered} N \hfill \\ m \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n + m - 2} } \frac{m}{{\lambda_{p1} }}} \times \int_{{\frac{H}{{P_{BS}^{\max } }}}}^{\infty } {\exp \left( { - \left( {\frac{{n\varepsilon_{2} \omega_{0} }}{{H\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{2n} }} + \frac{m}{{\lambda_{p1} }}} \right)x} \right)dx} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} = \sum\limits_{n = 1}^{N} {\sum\limits_{m = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( \begin{gathered} N \hfill \\ m \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n + m - 2} } } \frac{{mH\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{2n} }}{{n\varepsilon_{2} \omega_{0} \lambda_{p1} + mH\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{2n} }} \times \exp \left( { - \left( {\frac{{n\varepsilon_{2} \omega_{0} }}{{H\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{2n} }} + \frac{m}{{\lambda_{p1} }}} \right)\frac{H}{{P_{BS}^{\max } }}} \right). \hfill \\ \end{gathered} $$

(B.3)

From (B.2) and (B.3), $B_{1}$ can written such as

$$ \begin{gathered} B_{1} = \sum\limits_{n = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n - 1} } \exp \left( { - \frac{{n\varepsilon_{2} \omega_{0} }}{{P_{BS}^{\max } \left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{2n} }}} \right) \times \left[ {1 - \sum\limits_{n = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n - 1} } \exp \left( { - \frac{nH}{{P_{BS}^{\max } \lambda_{p1} }}} \right)} \right] \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} + \sum\limits_{n = 1}^{N} {\sum\limits_{m = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( \begin{gathered} N \hfill \\ m \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n + m - 2} } } \frac{{mH\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{2n} }}{{n\varepsilon_{2} \omega_{0} \lambda_{p1} + mH\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{2n} }} \times \exp \left( { - \left( {\frac{{n\varepsilon_{2} \omega_{0} }}{{H\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{2n} }} + \frac{m}{{\lambda_{p1} }}} \right)\frac{H}{{P_{BS}^{\max } }}} \right). \hfill \\ \end{gathered} $$

(B.4)

From (15), $B_{2}$ can be formulated by

$$ B_{2} = \Pr \left( {\min \left( {\gamma_{{D_{1} *}}^{{\left( {x_{2} } \right)}} ,\gamma_{{D_{1} - D_{2} *}}^{{\left( {x_{2} } \right)}} } \right) < \varepsilon_{2} } \right) = 1 - \underbrace {{\Pr \left( {\gamma_{{D_{1} *}}^{{\left( {x_{2} } \right)}} \ge \varepsilon_{2} } \right)}}_{{B_{2a} }}\underbrace {{\Pr \left( {\gamma_{{D_{1} - D_{2} *}}^{{\left( {x_{2} } \right)}} \ge \varepsilon_{2} } \right)}}_{{B_{2b} }}. $$

(B.5)

$B_{2a}$ can be formulated such as $B_{1}$, we have

$$ \begin{gathered} B_{2a} = \sum\limits_{n = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n - 1} } \exp \left( { - \frac{{n\varepsilon_{2} \omega_{0} }}{{P_{BS}^{\max } \left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{1n} }}} \right) \times \left[ {1 - \sum\limits_{n = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n - 1} } \exp \left( { - \frac{nH}{{P_{BS}^{\max } \lambda_{p1} }}} \right)} \right] \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} + \sum\limits_{n = 1}^{N} {\sum\limits_{m = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( \begin{gathered} N \hfill \\ m \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n + m - 2} } } \frac{{mH\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{1n} }}{{n\varepsilon_{2} \omega_{0} \lambda_{p1} + mH\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{1n} }} \times \exp \left( { - \left( {\frac{{n\varepsilon_{2} \omega_{0} }}{{H\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{1n} }} + \frac{m}{{\lambda_{p1} }}} \right)\frac{H}{{P_{BS}^{\max } }}} \right). \hfill \\ \end{gathered} $$

(B.6)

From (B.5), $B_{2b}$ it can be first calculated as

$$ \begin{gathered} B_{2b} = \Pr \left( {\gamma_{{D_{1} - D_{2} }}^{{\left( {x_{2} } \right)}} \ge \varepsilon_{2} } \right) = \Pr \left( {\min \left\{ {P_{{D_{1} }}^{EH} ,P_{{D_{1} }}^{\max } ,H/\left| {h_{p2} } \right|^{2} } \right\} \ge \omega_{0} \varepsilon_{2} } \right) \hfill \\ = \Pr \left( {P_{{D_{1} }}^{EH} \ge \frac{{\omega_{0} \varepsilon_{2} }}{{\left| {h_{3} } \right|^{2} }}} \right)\Pr \left( {P_{{D_{1} }}^{\max } \ge \frac{{\omega_{0} \varepsilon_{2} }}{{\left| {h_{3} } \right|^{2} }}} \right)\Pr \left( {\frac{H}{{\left| {h_{p2} } \right|^{2} }} \ge \frac{{\omega_{0} \varepsilon_{2} }}{{\left| {h_{3} } \right|^{2} }}} \right) \hfill \\ = \exp \left( { - \frac{{\omega_{0} \varepsilon_{2} }}{{P_{{D_{1} }}^{\max } \lambda_{3} }}} \right)\underbrace {{\Pr \left( {\left| {h_{3} } \right|^{2} \ge \frac{{\omega_{0} \varepsilon_{2} \left( {1 - \chi } \right)}}{{2\theta P_{B} \chi \left| {h_{b} } \right|^{2} }}} \right)}}_{{B_{2b}^{\left( 1 \right)} }} \times \underbrace {{\Pr \left( {\left| {h_{3} } \right|^{2} \ge \frac{{\omega_{0} \varepsilon_{2} \left| {h_{p2} } \right|^{2} }}{H}} \right)}}_{{B_{2b}^{\left( 2 \right)} }}. \hfill \\ \end{gathered} $$

(B.7)

From (B.7), $B_{2b}^{\left( 1 \right)}$ it can be first calculated as

$$ \begin{gathered} B_{2b}^{\left( 1 \right)} = \Pr \left( {\left| {h_{3} } \right|^{2} \ge \frac{{\omega_{0} \varepsilon_{2} \left( {1 - \chi } \right)}}{{2\theta P_{B} \chi \left| {h_{b} } \right|^{2} }}} \right) = \int_{0}^{\infty } {\exp \left( { - \frac{{\omega_{0} \varepsilon_{2} \left( {1 - \chi } \right)}}{{2\theta P_{B} \chi \lambda_{3} x}}} \right)\frac{1}{{\lambda_{b} }}\exp \left( { - \frac{x}{{\lambda_{b} }}} \right)dx} \hfill \\ = \sqrt {\frac{{2\omega_{0} \varepsilon_{2} \left( {1 - \chi } \right)}}{{\theta P_{B} \chi \lambda_{3} \lambda_{b} }}} K_{1} \left( {\sqrt {\frac{{2\omega_{0} \varepsilon_{2} \left( {1 - \chi } \right)}}{{\theta P_{B} \chi \lambda_{3} \lambda_{b} }}} } \right). \hfill \\ \end{gathered} $$

(B.8)

It is worth noting that the last equation follows from the fact that.

$\int_{0}^{\infty } {\exp \left( { - \frac{\delta }{4x} - \varphi x} \right)dx} = \sqrt {\frac{\delta }{\varphi }} K_{1} \left( {\sqrt {\delta \varphi } } \right)$ in [25, Eq. (3.324)].

From (B.7), $B_{2b}^{\left( 2 \right)}$ it can be first calculated as

$$ B_{2b}^{\left( 2 \right)} = \Pr \left( {\left| {h_{3} } \right|^{2} \ge \frac{{\omega_{0} \varepsilon_{2} \left| {h_{p2} } \right|^{2} }}{H}} \right) = \frac{1}{{\lambda_{p2} }}\int_{0}^{\infty } {\exp \left( { - \left( {\frac{{\omega_{0} \varepsilon_{2} }}{{H\lambda_{3} }} + \frac{1}{{\lambda_{p2} }}} \right)x} \right)dx} = \frac{{H\lambda_{3} }}{{\omega_{0} \varepsilon_{2} \lambda_{p2} + H\lambda_{3} }}. $$

(B.9)

From (B.6), (B.7), (B.8) and (B.9), $B_{2}$ can written such as

$$ \begin{gathered} B_{2} = 1 - \left\{ {\sum\limits_{n = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n - 1} } \exp \left( { - \frac{{n\varepsilon_{2} \omega_{0} }}{{P_{BS}^{\max } \left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{1n} }}} \right)\left[ {1 - \sum\limits_{n = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n - 1} } \exp \left( { - \frac{nH}{{P_{BS}^{\max } \lambda_{p1} }}} \right)} \right]} \right. \hfill \\ \left. \begin{gathered} \begin{array}{*{20}c} {} & {} \\ \end{array} + \sum\limits_{n = 1}^{N} {\sum\limits_{m = 1}^{N} {\left( \begin{gathered} N \hfill \\ n \hfill \\ \end{gathered} \right)\left( \begin{gathered} N \hfill \\ m \hfill \\ \end{gathered} \right)\left( { - 1} \right)^{n + m - 2} } } \frac{{mH\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{1n} }}{{n\varepsilon_{2} \omega_{0} \lambda_{p1} + mH\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{1n} }} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} \times \exp \left( { - \left( {\frac{{n\varepsilon_{2} \omega_{0} }}{{H\left( {\beta_{2} - \varepsilon_{2} \beta_{1} } \right)\lambda_{1n} }} + \frac{m}{{\lambda_{p1} }}} \right)\frac{H}{{P_{BS}^{\max } }}} \right) \hfill \\ \end{gathered} \right\} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} \times \frac{{H\lambda_{3} }}{{\omega_{0} \varepsilon_{2} \lambda_{p2} + H\lambda_{3} }}\exp \left( { - \frac{{\omega_{0} \varepsilon_{2} }}{{P_{{D_{1} }}^{\max } \lambda_{3} }}} \right) \times \sqrt {\frac{{2\omega_{0} \varepsilon_{2} \left( {1 - \chi } \right)}}{{\theta P_{B} \chi \lambda_{3} \lambda_{b} }}} K_{1} \left( {\sqrt {\frac{{2\omega_{0} \varepsilon_{2} \left( {1 - \chi } \right)}}{{\theta P_{B} \chi \lambda_{3} \lambda_{b} }}} } \right). \hfill \\ \end{gathered} $$

(B.10)

We plug (B.4), (B.10) into (15), it can be achieved $OP_{2}$ as the proposition.

This is the end of the proof.