Performance analysis of downlink NOMA–EH relaying network in the presence of residual transmit RF hardware impairments

Abstract

In this paper, we investigate the effects of residual-transmit radio-frequency impairments on the downlink non-orthogonal multiple access system with perfect successive interference cancellation using an energy-harvesting (EH) relay in which nodes in the network are equipped with a single antenna. The source node communicates with two users with the help of the amplify-and-forward EH relay node. In addition, both the power-splitting relaying (PSR) and time-switching relaying (TSR) protocols are examined. To evaluate the performance of the proposed system, closed-form expressions of the outage performance and the available throughput are derived over Rayleigh fading channels. Moreover, the accuracy of analytical results is verified by a Monte Carlo simulation. The results show the effects of various parameters, such as power allocation factors, the relay node location, data rate, transmit hardware impairment level, and power allocation factors on the outage performance (${\mathcal {OP}}$) and throughput with two users for both PSR and TSR architectures in the presence of hardware impairments. Furthermore, these results are compared with performance of orthogonal multiple access (OMA) EH relaying system.

Appendices

Appendix 1: Proof of Theorem 1

This appendix derives $OP_1^q$ in (26) at ${\mathcal {U}}_1$ in a cooperative NOMA–EH network employing perfect SIC with the residual transmit hardware impairment. According to (25), the factor ${\alpha _1} - {\gamma _1}\left( {{\alpha _2} + {K^2}} \right)$ can be positive or negative; thus, we consider the following two cases. If ${K^2} \ge \frac{{{\alpha _1}}}{{{\gamma _1}}} - {\alpha _2}$ the outage probability at ${\mathcal {U}}_1$ of PSR and TSR protocols is expressed as

$$\begin{aligned} OP_1^q = \Pr \left[ {{g_0} > \frac{{{\gamma _1}a_2^q + \frac{{{\gamma _1}}}{{{g_1}}}}}{{\underbrace{a_1^q\left( {{\alpha _1} - {\gamma _1}\left( {{\alpha _2} + {K^2}} \right) } \right) }_{ < 0}}}} \right] = 1 \end{aligned}$$

(32)

If ${K^2} \ge \frac{{{\alpha _1}}}{{{\gamma _1}}} - {\alpha _2}$, from (25) let denotes $X = {g_0}$ and $Y = {g_1}$, the outage probability at ${\mathcal {U}}_1$ with PSR and TSR protocols is expressed as

$$\begin{aligned} OP_1^q&= \Pr \left[ {X < \frac{{{\gamma _1}}}{{a_1^q\left( {{\alpha _1} - {\gamma _1}\left( {{\alpha _2} + {K^2}} \right) } \right) }}\left( {a_2^q + \frac{1}{Y}} \right) } \right] \nonumber \\ {}&= \int \limits _0^{ + \infty } {{F_X}\left( {\frac{{{\gamma _1}}}{{a_1^q\left( {{\alpha _1} - {\gamma _1}\left( {{\alpha _2} + {K^2}} \right) } \right) }}\left( {a_2^q + \frac{1}{y}} \right) } \right) {f_Y}\left( y \right) dy} \nonumber \\&= \int \limits _0^{ + \infty } {{\lambda _1}{e^{ - {\lambda _1}y}}\left( {1 - {e^{ - \frac{{{\lambda _0}{\gamma _1}}}{{a_1^q\left( {{\alpha _1} - {\gamma _1}\left( {{\alpha _2} + {K^2}} \right) } \right) }}\left( {a_2^q + \frac{1}{y}} \right) }}} \right) dy} \nonumber \\&= 1 - {\lambda _1}\exp \left( { - \frac{{a_2^q{\lambda _0}{\gamma _1}}}{{a_1^q\left( {{\alpha _1} - {\gamma _1}\left( {{\alpha _2} + {K^2}} \right) } \right) }}} \right) \nonumber \\&\qquad \times \,\int \limits _0^{ + \infty } {{e^{ - \left( {{\lambda _1}y + \frac{{{\lambda _0}{\gamma _1}}}{{a_1^q\left( {{\alpha _1} - {\gamma _1}\left( {{\alpha _2} + {K^2}} \right) } \right) }}\frac{1}{y}} \right) }}dy} \end{aligned}$$

(33)

By using [21, Eq. (3.324.1)], the OP of ${\mathcal {U}}_1$ in this case can be expressed as in (26). This ends the proof of Theorem 1.

Appendix 2: Proof of Theorem 2

This appendix derives $OP_2^q$ in (29) at the user ${\mathcal {U}}_2$ in a cooperative NOMA–EH network employing perfect SIC with the residual transmit hardware impairment. According to (28), the factor ${\alpha _1} - {\gamma _1}\left( {{\alpha _2} + {K^2}} \right)$ can be positive or negative; thus, we consider the following cases.

In the case $\frac{{{\alpha _1}}}{{{\gamma _1}}} - {\alpha _2} \le {K^2} \le \frac{{{\alpha _2}}}{{{\gamma _2}}}$, $OP_2^q$ at ${\mathcal {U}}_2$ can be expressed as

$$\begin{aligned} OP_2^q = 1 - \Pr \left[ \begin{array}{l} {g_0}> \underbrace{\frac{{{\gamma _1}}}{{b_1^q\left( {{\alpha _1} - {\gamma _1}\left( {{\alpha _2} + {K^2}} \right) } \right) }}\left( {b_2^q + \frac{1}{{{g_2}}}} \right) }_{ > 0},\\ {g_0}< \underbrace{\frac{{{\gamma _2}}}{{b_1^q\left( {{\alpha _2} - {\gamma _2}{K^2}} \right) }}\left( {b_2^q + \frac{1}{{{g_2}}}} \right) }_{ < 0} \end{array} \right] = 1 \end{aligned}$$

(34)

In the case ${K^2} \ge \max \left\{ {\frac{{{\alpha _1}}}{{{\gamma _1}}} - {\alpha _2},\frac{{{\alpha _2}}}{{{\gamma _2}}}} \right\}$ from (28), $OP_2^q$ can be expressed as

$$\begin{aligned} OP_2^q = 1 - \Pr \left[ \begin{array}{l} {g_0} > \underbrace{\frac{{{\gamma _1}}}{{b_1^q\left( {{\alpha _1} - {\gamma _1}\left( {{\alpha _2} + {K^2}} \right) } \right) }}\left( {b_2^q + \frac{1}{{{g_2}}}} \right) }_{< 0},\\ {g_0}< \underbrace{\frac{{{\gamma _2}}}{{b_1^q\left( {{\alpha _2} - {\gamma _2}{K^2}} \right) }}\left( {b_2^q + \frac{1}{{{g_2}}}} \right) }_{ < 0} \end{array} \right] = 1 \end{aligned}$$

(35)

In the case $\frac{{{\alpha _2}}}{{{\gamma _2}}} \le {K^2} \le \frac{{{\alpha _1}}}{{{\gamma _1}}} - {\alpha _2}$ from (28), $OP_2^q$ can be expressed as

$$\begin{aligned} OP_2^q = 1 - \Pr \left[ \begin{array}{l} {g_0}> \underbrace{\frac{{{\gamma _1}}}{{b_1^q\left( {{\alpha _1} - {\gamma _1}\left( {{\alpha _2} + {K^2}} \right) } \right) }}\left( {b_2^q + \frac{1}{{{g_2}}}} \right) }_{ > 0},\\ {g_0}< \underbrace{\frac{{{\gamma _2}}}{{b_1^q\left( {{\alpha _2} - {\gamma _2}{K^2}} \right) }}\left( {b_2^q + \frac{1}{{{g_2}}}} \right) }_{ < 0} \end{array} \right] = 1 \end{aligned}$$

(36)

In the case ${K^2} \le \min \left\{ {\frac{{{\alpha _2}}}{{{\gamma _2}}},\frac{{{\alpha _1}}}{{{\gamma _1}}} - {\alpha _2}} \right\}$, from (28),let denotes $X = {g_0}$ and $Z = {g_2}$, $OP_2^q$ can be expressed as

$$\begin{aligned} OP_2^q&= 1 - \Pr \left[ \begin{array}{l} X> \frac{{{\gamma _1}}}{{b_1^q\left( {{\alpha _1} - {\gamma _1}\left( {{\alpha _2} + {K^2}} \right) } \right) }}\left( {b_2^q + \frac{1}{Z}} \right) ,\\ X> \frac{{{\gamma _2}}}{{b_1^q\left( {{\alpha _2} - {\gamma _2}{K^2}} \right) }}\left( {b_2^q + \frac{1}{Z}} \right) \end{array} \right] \nonumber \\&= 1 - \Pr \left[ {X> \frac{1}{{b_1^q}}\left( {b_2^q + \frac{1}{Z}} \right) \underbrace{\max \left\{ {\frac{{{\gamma _1}}}{{\left( {{\alpha _1} - {\gamma _1}\left( {{\alpha _2} + {K^2}} \right) } \right) }},\frac{{{\gamma _2}}}{{\left( {{\alpha _2} - {\gamma _2}{K^2}} \right) }}} \right\} }_\omega } \right] \nonumber \\&= 1 - \Pr \left[ {X > \frac{\omega }{{b_1^q}}\left( {b_2^q + \frac{1}{Z}} \right) } \right] = \Pr \left[ {X < \frac{\omega }{{b_1^q}}\left( {b_2^q + \frac{1}{Z}} \right) } \right] \end{aligned}$$

(37)

According to CDF and PDF in (1a) and (1b), (37) can be given by

$$\begin{aligned} OP_2^q&= \int \limits _0^{ + \infty } {{F_X}\left( {\frac{\omega }{{b_1^q}}\left( {b_2^q + \frac{1}{z}} \right) } \right) {f_Z}\left( z \right) dz} \nonumber \\&= \int \limits _0^{ + \infty } {{\lambda _2}{e^{ - {\lambda _2}z}}\left( {1 - \exp \left( { - {\lambda _0}\frac{\omega }{{b_1^q}}\left( {b_2^q + \frac{1}{z}} \right) } \right) } \right) dz} \nonumber \\&= 1 - \int \limits _0^{ + \infty } {{\lambda _2}{e^{ - {\lambda _2}z}}\exp \left( { - {\lambda _0}\frac{\omega }{{b_1^q}}\left( {b_2^q + \frac{1}{z}} \right) } \right) dz} \nonumber \\&= 1 - {\lambda _2}{e^{ - \frac{{\omega {\lambda _0}b_2^q}}{{b_1^q}}}}\int \limits _0^{ + \infty } {{e^{ - {\lambda _2}z - \frac{{{\lambda _0}\omega }}{{zb_1^q}}}}dz} \end{aligned}$$

(38)

By using [21, Eq. (3.324.1)], the OP of ${\mathcal {U}}_2$ in this case can be expressed as in (29). This ends the proof of Theorem 2.