Effects of Hardware Impairment on the Cooperative NOMA EH Relaying Network Over Nakagami-m Fading Channels

Abstract

In this paper, we analyze the performance of Non Orthogonal Multiple Access (NOMA) Energy harvesting relaying system in presence of hardware impairments at both the transmitter and receiver over a Nakagami-m fading channel. The proposed system with destinations are allocated different power levels in which the source node communicates with the destinations via an energy harvesting (EH) relay employing a power-splitting relaying architecture and a direct link to the near destination. Moreover, the amplify-and-forward protocol at the EH relay is investigated to evaluate the performance of system. Additionally, the expressions for outage probability are derived, and these analyses are verified by a Monte Carlo simulation. Furthermore, these results are also compared to an orthogonal multiple access (OMA) system. Finally, the effects of various parameters, such as power allocation levels, position of EH relay node, channel coefficients, and hardware impairment levels on the outage performance and throughput of proposed NOMA-EH system and OMA system are investigated. These results demonstrate the advantage of NOMA-EH system as user fairness since multiple destinations compared to the OMA-EH system.

Appendices

Appendix 1: Proof of Theorem 1

This appendix derives the outage probability $OP_1$ at Destination 1 in (22–25) from $P_1$ and $P_2$ in Eq. (19).

Let consider $P_1$: According to Eq. (20), the components $\left( {\frac{{{\alpha _1}}}{{{\gamma _1}}} - \varepsilon {\alpha _2} - K_1^2} \right)$ and $\left( {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} - K_1^2} \right)$ can be negative or positive. Thus, we consider some cases to calculate the OP at ${\mathcal {D}}_1$ as follows:

+ If $\mathrm{{max}}\left\{ {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1},\frac{{{\alpha _1}}}{{{\gamma _1}}} - \varepsilon {\alpha _2}} \right\} \le K_1^2$, Eq. (20) can be rewritten as

$$\begin{aligned} {P_1} = \Pr \left[ {{{\left| {{h_1}} \right| }^2}< \frac{{d_1^\lambda \sigma _{{D_1}}^2}}{{\underbrace{{P_s}\left( {\frac{{{\alpha _1}}}{{{\gamma _1}}} - \varepsilon {\alpha _2} - K_1^2} \right) }_{< 0}}},{{\left| {{h_1}} \right| }^2}< \frac{{d_1^\lambda \sigma _{{D_1}}^2}}{{\underbrace{{P_s}\left( {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} - K_1^2} \right) }_{ < 0}}}} \right] = 0 \end{aligned}$$

(29)

+ If $\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} \le K_1^2 \le \frac{{{\alpha _1}}}{{{\gamma _1}}} - \varepsilon {\alpha _2}$, Eq. (20) can be rewritten as

$$\begin{aligned} {P_1} = \Pr \left[ {{{\left| {{h_1}} \right| }^2}> \frac{{d_1^\lambda \sigma _{{D_1}}^2}}{{\underbrace{{P_s}\left( {\frac{{{\alpha _1}}}{{{\gamma _1}}} - \varepsilon {\alpha _2} - K_1^2} \right) }_{ > 0}}},{{\left| {{h_1}} \right| }^2}< \frac{{d_1^\lambda \sigma _{{D_1}}^2}}{{\underbrace{{P_s}\left( {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} - K_1^2} \right) }_{ < 0}}}} \right] = 0 \end{aligned}$$

(30)

+ If $\frac{{{\alpha _1}}}{{{\gamma _1}}} \le K_1^2 \le \frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1}$, Eq. (20) can be rewritten as

$$\begin{aligned} {P_1} = \Pr \left[ {{{\left| {{h_1}} \right| }^2}< \frac{{d_1^\lambda \sigma _{{D_1}}^2}}{{\underbrace{{P_s}\left( {\frac{{{\alpha _1}}}{{{\gamma _1}}} - \varepsilon {\alpha _2} - K_1^2} \right) }_{ < 0}}},{{\left| {{h_1}} \right| }^2}> \frac{{d_1^\lambda \sigma _{{D_1}}^2}}{{\underbrace{{P_s}\left( {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} - K_1^2} \right) }_{ > 0}}}} \right] = 0 \end{aligned}$$

(31)

+ If $K_1^2 < \min \left\{ {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1},\frac{{{\alpha _1}}}{{{\gamma _1}}} - \varepsilon {\alpha _2}} \right\}$, Eq. (20) can be rewritten as

$$\begin{aligned} \begin{array}{l} {P_1} = \Pr \left[ {{{\left| {{h_1}} \right| }^2}> \frac{{d_1^\lambda \sigma _{{D_1}}^2}}{{\underbrace{{P_s}\left( {\frac{{{\alpha _1}}}{{{\gamma _1}}} - \varepsilon {\alpha _2} - K_1^2} \right) }_{> 0}}},{{\left| {{h_1}} \right| }^2}> \frac{{d_1^\lambda \sigma _{{D_1}}^2}}{{\underbrace{{P_s}\left( {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} - K_1^2} \right) }_{> 0}}}} \right] \\ \qquad = \Pr \left[ {{{\left| {{h_1}} \right| }^2} > \frac{{d_1^\lambda \sigma _{{D_1}}^2}}{{{P_s}}}\underbrace{\mathrm{{max}}\left\{ {{{\left( {\frac{{{\alpha _1}}}{{{\gamma _1}}} - \varepsilon {\alpha _2} - K_1^2} \right) }^{ - 1}},{{\left( {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} - K_1^2} \right) }^{ - 1}}} \right\} }_\varpi } \right] \\ \qquad = 1 - \Pr \left[ {{{\left| {{h_1}} \right| }^2} < \varpi \frac{{d_1^\lambda \sigma _{{D_1}}^2}}{{{P_s}}}} \right] \end{array} \end{aligned}$$

(32)

By applying Eq. (1), $P_1$ is given by

$$\begin{aligned} {P_1} = \exp \left( { - \frac{{\varpi md_1^\lambda \sigma _{{D_1}}^2P_s^{ - 1}}}{\varOmega }} \right) \sum \limits _{k = 0}^{m - 1} {\left[ {\frac{1}{{k!}}{{\left( {\frac{{\varpi md_1^\lambda \sigma _{{D_1}}^2}}{\varOmega }} \right) }^k}} \right] } \end{aligned}$$

(33)

Let consider $P_2$: It can be seen that the sign of $\left( {\frac{{{\alpha _2}{A_2}}}{{{\gamma _2}}} - {A_3}} \right)$ is is the same as the sign of $\left( {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} - K_{A{F_1}}^2} \right)$, and the sign of $\left( {\frac{{{\alpha _1}{A_2}}}{{{\gamma _1}}} - {A_5}} \right)$ is the same as the sign of $\left( {\frac{{{\alpha _1}}}{{{\gamma _1}}} - \varepsilon {\alpha _2} - K_{A{F_1}}^2} \right)$ due to ${A_2} = \frac{{\theta \mu {P_s}}}{{d_0^\lambda d_3^\lambda \sigma _{{D_1}}^2\left( {1 + K_{SR}^2} \right) }}$ , ${A_3} = \frac{{\theta \mu {P_s}\left( {{\alpha _1} + K_{AF1}^2} \right) }}{{d_0^\lambda d_3^\lambda \sigma _{{D_1}}^2\left( {1 + K_{SR}^2} \right) }}$, and ${A_5} = \frac{{\theta \mu {P_s}\left( {\varepsilon {\alpha _2} + K_{AF1}^2} \right) }}{{d_0^\lambda d_3^\lambda \sigma _{{D_1}}^2\left( {1 + K_{SR}^2} \right) }}$.

+ If ${\mathrm{{max}}}\left\{ {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1},\frac{{{\alpha _1}}}{{{\gamma _1}}} - \varepsilon {\alpha _2}} \right\} \le K_{\mathrm{{A}}{\mathrm{{F}}_1}}^2$, $P_2$ can be expressed as

$$\begin{aligned} {P_2}&= \Pr \left[ {{\left| {{h_0}} \right| }^2}{{\left| {{h_3}} \right| }^2}< \underbrace{\frac{1}{{\frac{{{\alpha _2}{A_2}}}{{{\gamma _2}}} - {A_3}}}}_{< 0}\left( {{A_4}{{\left| {{h_3}} \right| }^2} + 1} \right) ,\nonumber \right. \\&\left. \quad {{\left| {{h_0}} \right| }^2}{{\left| {{h_3}} \right| }^2}< \underbrace{\frac{1}{{\frac{{{\alpha _1}{A_2}}}{{{\gamma _1}}} - {A_5}}}}_{ < 0}\left( {{A_4}{{\left| {{h_3}} \right| }^2} + 1} \right) \right] = 0 \end{aligned}$$

(34)

+ If $\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} \le K_{A{F_1}}^2 \le \frac{{{\alpha _1}}}{{{\gamma _1}}}$ , $P_2$ can be expressed as

$$\begin{aligned} {P_2}&= \Pr \left[ {{\left| {{h_0}} \right| }^2}{{\left| {{h_3}} \right| }^2}< \underbrace{\frac{1}{{\frac{{{\alpha _2}{A_2}}}{{{\gamma _2}}} - {A_3}}}}_{ < 0}\left( {{A_4}{{\left| {{h_3}} \right| }^2} + 1} \right) ,\nonumber \right. \\&\left. \quad {{\left| {{h_0}} \right| }^2}{{\left| {{h_3}} \right| }^2}> \underbrace{\frac{1}{{\frac{{{\alpha _1}{A_2}}}{{{\gamma _1}}} - {A_5}}}}_{ > 0}\left( {{A_4}{{\left| {{h_3}} \right| }^2} + 1} \right) \right] = 0 \end{aligned}$$

(35)

+ If $\frac{{{\alpha _1}}}{{{\gamma _1}}} - \varepsilon {\alpha _2} \le K_{A{F_1}}^2 \le \frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1}$ , $P_2$ can be expressed as

$$\begin{aligned} {P_2}&= \Pr \left[ {{\left| {{h_0}} \right| }^2}{{\left| {{h_3}} \right| }^2}> \underbrace{\frac{1}{{\frac{{{\alpha _2}{A_2}}}{{{\gamma _2}}} - {A_3}}}}_{ > 0}\left( {{A_4}{{\left| {{h_3}} \right| }^2} + 1} \right) ,\nonumber \right. \\&\left. \quad {{\left| {{h_0}} \right| }^2}{{\left| {{h_3}} \right| }^2}< \underbrace{\frac{1}{{\frac{{{\alpha _1}{A_2}}}{{{\gamma _1}}} - {A_5}}}}_{ < 0}\left( {{A_4}{{\left| {{h_3}} \right| }^2} + 1} \right) \right] = 0 \end{aligned}$$

(36)

+ If $K_{A{F_1}}^2 < \min \left\{ {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1},\frac{{{\alpha _1}}}{{{\gamma _1}}} - \varepsilon {\alpha _2}} \right\}$ , $P_2$ can be expressed as

$$\begin{aligned} {P_2}&= \Pr \left[ {{{\left| {{h_0}} \right| }^2}{{\left| {{h_3}} \right| }^2}> \frac{1}{{\frac{{{\alpha _2}{A_2}}}{{{\gamma _2}}} - {A_3}}}\left( {{A_4}{{\left| {{h_3}} \right| }^2} + 1} \right) ,{{\left| {{h_0}} \right| }^2}{{\left| {{h_3}} \right| }^2}> \frac{1}{{\frac{{{\alpha _1}{A_2}}}{{{\gamma _1}}} - {A_5}}}\left( {{A_4}{{\left| {{h_3}} \right| }^2} + 1} \right) } \right] \nonumber \\&= \Pr \left[ {{{\left| {{h_0}} \right| }^2}{{\left| {{h_3}} \right| }^2}> \left( {{A_4}{{\left| {{h_3}} \right| }^2} + 1} \right) \underbrace{\max \left( {\frac{1}{{\frac{{{\alpha _2}{A_2}}}{{{\gamma _2}}} - {A_3}}},\frac{1}{{\frac{{{\alpha _1}{A_2}}}{{{\gamma _1}}} - {A_5}}}} \right) }_\phi } \right] \nonumber \\&= \Pr \left[ {{{\left| {{h_0}} \right| }^2} > \phi \left( {{A_4} + \frac{1}{{{{\left| {{h_3}} \right| }^2}}}} \right) } \right] \nonumber \\&= 1 - \Pr \left[ {X < \phi \left( {{A_4} + \frac{1}{Y}} \right) } \right] \nonumber \\&= 1 - \int \limits _0^{ + \infty } {{f_Y}\left( y \right) {F_X}\left( {\phi {A_4} + \frac{\phi }{y}} \right) dy} \end{aligned}$$

(37)

where $X = {\left| {{h_0}} \right| ^2}$ , $Y = {\left| {{h_3}} \right| ^2}$ .

By substituting Eqs. (1) and (2) to (A11), Equation in (A11) can be expressed as

$$\begin{aligned} {P_2}&= 1 - \int \limits _0^{ + \infty } {\frac{{{n^n}{y^{n - 1}}}}{{\varGamma \left( n \right) \varOmega _2^n}}{e^{ - \frac{{ny}}{{{\varOmega _2}}}}}\left( {1 - {e^{ - \frac{{m\left( {\phi {A_4} + \frac{\phi }{y}} \right) }}{{{\varOmega _0}}}}}\sum \limits _{k = 0}^{m - 1} {\left[ {\frac{1}{{k!}}{{\left( {\frac{{m\left( {\phi {A_4} + \frac{\phi }{y}} \right) }}{{{\varOmega _0}}}} \right) }^k}} \right] } } \right) dy} \nonumber \\&= 1 - \underbrace{\int \limits _0^{ + \infty } {\frac{{{n^n}{y^{n - 1}}}}{{\varGamma \left( n \right) \varOmega _2^n}}{e^{ - \frac{{ny}}{{{\varOmega _2}}}}}dy} }_{{I_1}} + \underbrace{\int \limits _0^{ + \infty } {\frac{{{n^n}{y^{n - 1}}}}{{\varGamma \left( n \right) \varOmega _2^n}}{e^{ - \frac{{ny}}{{{\varOmega _2}}} - \frac{{m\left( {\phi {A_4} + \frac{\phi }{y}} \right) }}{{{\varOmega _0}}}}}\sum \limits _{k = 0}^{m - 1} {\left[ {\frac{1}{{k!}}{{\left( {\frac{{m\left( {\phi {A_4} + \frac{\phi }{y}} \right) }}{{{\varOmega _0}}}} \right) }^k}} \right] } dy} }_{{I_2}}\nonumber \\&= 1 - {I_1} + {I_2} \end{aligned}$$

(38)

Using Eq. [24, Eq.(3.351)], $I_1$ can be formulated as

$$\begin{aligned} {I_1} = \frac{{{n^n}}}{{\varGamma \left( n \right) \varOmega _2^n}}\int \limits _0^{ + \infty } {{y^{n - 1}}{e^{ - \frac{{ny}}{{{\varOmega _2}}}}}dy} = \frac{{{n^n}\left( {n - 1} \right) !{{\left( {\frac{n}{{{\varOmega _2}}}} \right) }^{ - n}}}}{{\varGamma \left( n \right) \varOmega _2^n}} = \frac{{\left( {n - 1} \right) !}}{{\varGamma \left( n \right) }} = 1 \end{aligned}$$

(39)

Considering $I_2$ as follows

$$\begin{aligned} {I_2}&= \int \limits _0^{ + \infty } {\frac{{{n^n}{y^{n - 1}}}}{{\varGamma \left( n \right) \varOmega _2^n}}{{\mathop {\mathrm{e}}\nolimits } ^{ - \frac{{ny}}{{{\varOmega _2}}} - \frac{{m\left( {\phi {A_4} + \frac{\phi }{y}} \right) }}{{{\varOmega _0}}}}}\sum \limits _{k = 0}^{m - 1} {\left[ {\frac{1}{{k!}}{{\left( {\frac{{m\left( {\phi {A_4} + \frac{\phi }{y}} \right) }}{{{\varOmega _0}}}} \right) }^k}} \right] } dy} \nonumber \\&= \frac{{{n^n}{e^{ - \frac{{m\phi {A_4}}}{{{\varOmega _0}}}}}}}{{\varGamma \left( n \right) \varOmega _2^n}}\sum \limits _{k = 0}^{m - 1} {\left[ {\frac{{{m^k}}}{{k!\varOmega _0^k}}} \right] } \int \limits _0^{ + \infty } {{y^{n - 1}}{{\left( {\phi {A_4} + \frac{\phi }{y}} \right) }^k}{{\mathop {\mathrm{e}}\nolimits } ^{ - \frac{{ny}}{{{\varOmega _2}}} - \frac{{m\phi }}{{{\varOmega _0}y}}}}dy} \end{aligned}$$

(40)

Applying the Newton binomial as ${\left( {a + b} \right) ^k} = \sum \limits _{i = 0}^k {\left( \begin{array}{l} k\\ i \end{array} \right) } {a^{k - i}}{b^i}$, and using [24, Eq.(3.471.9)] $I_2$ can be expressed as

$$\begin{aligned} {I_2}&= \frac{{{n^n}{e^{ - \frac{{m\phi {A_4}}}{{{\varOmega _0}}}}}}}{{\varGamma \left( n \right) \varOmega _2^n}}\sum \limits _{k = 0}^{m - 1} {\sum \limits _{i = 0}^k {\left( \begin{array}{l} k\\ i \end{array} \right) \frac{{{m^k}}}{{k!\varOmega _0^k}}{{\left( {\phi {A_4}} \right) }^{k - i}}{\phi ^i}} } \int \limits _0^{ + \infty } {{y^{n - i - 1}}{{\mathop {\mathrm{e}}\nolimits } ^{ - \frac{{ny}}{{{\varOmega _2}}} - \frac{{m\phi }}{{{\varOmega _0}y}}}}dy} \nonumber \\&= 2\frac{{{e^{ - \frac{{m\phi {A_4}}}{{{\varOmega _0}}}}}}}{{\varGamma \left( n \right) }}\sum \limits _{k = 0}^{m - 1} {\sum \limits _{i = 0}^k {\left( \begin{array}{l} k\\ i \end{array} \right) \frac{{{m^{k + \frac{{n - i}}{2}}}{n^{n - \frac{{n - i}}{2}}}}}{{k!\varOmega _0^{k + \frac{{n - i}}{2}}\varOmega _2^{n - \frac{{n - i}}{2}}}}{{\left( {\phi {A_4}} \right) }^{k - i}}{\phi ^{i + \frac{{n - i}}{2}}}} } {K_{n - i}}\left( {2\sqrt{\frac{{mn\phi }}{{{\varOmega _0}{\varOmega _2}}}} } \right) \end{aligned}$$

(41)

where $K_i[.]$ is the Bessel function. In fact, because of the value of hardware impairment level of $K_{A{F_1}}^2 > K_1^2$ , due to the combination of result $P_1$ and $P_2$ we obtain $OP_1$ at ${\mathcal {D}}_1$ as in (22–25). This ends the Proof of Theorem 1.

Appendix 2: Proof of Theorem 2

This appendix derives OP in (27) at ${\mathcal {D}}_2$ of the proposed system with the EH relaying node using AF protocol. According to (26) shown that $\left( {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} - K_{AF}^2} \right)$ can be negative or positive, thus we consider two cases as follows:

+ If $\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} \le K_{AF}^2$ , Eq. (26) can be expressed as

$$\begin{aligned} OP_2^{AF}&= \Pr \left[ {{{\left| {{h_0}} \right| }^2} > \underbrace{\frac{{{b_2}\left( {1 + K_{R{D_2}}^2} \right) }}{{{b_1}\left( {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} - K_{AF}^2} \right) }} + \left( {1 + K_{SR}^2} \right) \frac{1}{{{b_1}\left( {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} - K_{AF}^2} \right) {{\left| {{h_2}} \right| }^2}}}}_{ < 0}} \right] = 1 \end{aligned}$$

(42)

+ If $\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} > K_{AF}^2$, Eq. (26) can be expressed as

$$\begin{aligned} OP_2^{AF}&= \Pr \left[ {{{\left| {{h_0}} \right| }^2} > \frac{{{b_2}\left( {1 + K_{R{D_2}}^2} \right) }}{{{b_1}\left( {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} - K_{AF}^2} \right) }} + \frac{{1 + K_{SR}^2}}{{{b_1}\left( {\frac{{{\alpha _2}}}{{{\gamma _2}}} - {\alpha _1} - K_{AF}^2} \right) {{\left| {{h_2}} \right| }^2}}}} \right] \nonumber \\&= 1 - \Pr \left[ {X < {B_1} + \frac{{{B_2}}}{Z}} \right] \nonumber \\&= 1 - \int \limits _0^{ + \infty } {{f_Z}\left( z \right) {F_X}\left( {{B_1} + \frac{{{B_2}}}{z}} \right) dz} \end{aligned}$$

(43)

where $Z = {\left| {{h_2}} \right| ^2}$. By substituting (1) and (2) to (B2), Equation in (B2) can be expressed as

$$\begin{aligned} OP_2^{\mathrm{{AF}}}&= 1 - \int \limits _0^{ + \infty } {\frac{{{n^n}{z^{n - 1}}}}{{\varGamma \left( n \right) \varOmega _2^n}}{{\mathop {\mathrm{e}}\nolimits } ^{ - \frac{{nz}}{{{\varOmega _2}}}}}\left( {1 - {{\mathop {\mathrm{e}}\nolimits } ^{ - \frac{{m\left( {{B_1} + \frac{{{B_2}}}{z}} \right) }}{{{\varOmega _0}}}}}\sum \limits _{k = 0}^{m - 1} {\left[ {\frac{1}{{k!}}{{\left( {\frac{{m\left( {{B_1} + \frac{{{B_2}}}{z}} \right) }}{{{\varOmega _0}}}} \right) }^k}} \right] } } \right) dz}\nonumber \\&= 1 - \underbrace{\int \limits _0^{ + \infty } {\frac{{{n^n}{z^{n - 1}}}}{{\varGamma \left( n \right) \varOmega _2^n}}{{\mathop {\mathrm{e}}\nolimits } ^{ - \frac{{nz}}{{{\varOmega _2}}}}}dy} }_{{I_3}} + \underbrace{\int \limits _0^{ + \infty } {\frac{{{n^n}{z^{n - 1}}}}{{\varGamma \left( n \right) \varOmega _2^n}}{{\mathop {\mathrm{e}}\nolimits } ^{ - \frac{{nz}}{{{\varOmega _2}}} - \frac{{m\left( {{B_1} + \frac{{{B_2}}}{z}} \right) }}{{{\varOmega _0}}}}}\sum \limits _{k = 0}^{m - 1} {\left[ {\frac{1}{{k!}}{{\left( {\frac{{m\left( {{B_1} + \frac{{{B_2}}}{z}} \right) }}{{{\varOmega _0}}}} \right) }^k}} \right] } dz} }_{{I_4}}\nonumber \\&= 1 - {I_3} + {I_4} \end{aligned}$$

(44)

$I_3$ and $I_4$ components are formulated similar to the calculate $I_1$ and $I_2$ in “Appendix 1”. Thus, we obtain OP at ${\mathcal {D}}_2$ as in (27). This ends the Proof of Theorem 2.