Joint evaluation of imperfect SIC and fixed power allocation scheme for wireless powered D2D-NOMA networks with multiple antennas at base station

Abstract

In this study, device-to-device (D2D) communication is exploited to adopt the proximity transmission among two nearby devices. The base station in this scenario transmits both information and energy to both D2D users to deploy wireless power transfer feature in green communications. In addition, to enhance the overall spectrum utilization of considered system, downlink non-orthogonal multiple access (NOMA) network is considered to evaluate performance in such network. In particular, a downlink from the base station (BS) equipping multiple antennas is designed to robust performance. We further determine the worse case of considered NOMA at the receiver in cellular network as considering impact of imperfect successive interference cancellation (SIC). A two-user downlink NOMA scenario is considered, where a far user directly communicates with the BS without SIC, whereas a near user is assigned to serve D2D link with imperfects on SIC. The ergodic capacity of D2D NOMA is further analyzed under both perfect and imperfect SIC. The performance gap among two considered users in D2D NOMA network is demonstrated through simulation and analysis.

Appendices

Appendix 1

Proof of Proposition 1

Based on definition, the outage probability for the first user in the D2D NOMA network can be obtained by examining each component as follows

$$\begin{aligned} {A_1}&= \Pr \left( {{\gamma _{1,1}}< {\varepsilon _1},{\gamma _{2,1}}< {\varepsilon _1}} \right) \nonumber \\&= \Pr \left( {{{\left| {{h_{n,1}}} \right| }^2}< \theta ,{{\left| {{h_{n,2}}} \right| }^2} < \theta } \right) \nonumber \\&= \sum \limits _{n = 1}^N {{\left( { - 1} \right) }^{n - 1}}\left( {\begin{array}{*{20}{c}} N\\ n \end{array}} \right) \frac{n}{{{\lambda _1}}}\int \limits _0^\theta {e^{\left( { - \frac{{nx}}{{{\lambda _1}}}} \right) }}dx \nonumber \\&\quad \times \sum \limits _{m = 1}^N {{{\left( { - 1} \right) }^{m - 1}}\left( {\begin{array}{*{20}{c}} N\\ m \end{array}} \right) \frac{m}{{{\lambda _2}}}\int \limits _0^\theta {{e^{\left( { - \frac{{my}}{{{\lambda _2}}}} \right) }}dy} } \nonumber \\&= \sum \limits _{n = 1}^N {\sum \limits _{m = 1}^N {\left( {\begin{array}{*{20}{c}} N\\ n \end{array}} \right) } } \left( {\begin{array}{*{20}{c}} N\\ m \end{array}} \right) {\left( { - 1} \right) ^{n + m - 2}}\left( {1 - {e^{ - \frac{{\theta n}}{{{\lambda _1}}}}}} \right) \left( {1 - {e^{ - \frac{{\theta m}}{{{\lambda _2}}}}}} \right) , \end{aligned}$$

(37)

where $\theta =\frac{{{\varepsilon }_{1}}}{\rho \left( 1-\alpha \right) \left( {{\varTheta }_{1}}-{{\varepsilon }_{1}}{{\varTheta }_{2}} \right) }$.

Then, the computation of $A_2$ can be given by

$$\begin{aligned} {A_2}&= \Pr \left( {\max \left( {{\gamma _{1,1}},{W_1}} \right)< {\varepsilon _1},{\gamma _{2,1}}> {\varepsilon _1}} \right) \nonumber \\&= \Pr \left( {{{\left| {{h_{n,1}}} \right| }^2}< \theta ,{{\left| {{g_{k,1}}} \right| }^2} < \frac{\upsilon }{{{{\left| {{h_{n,1}}} \right| }^2}}}} \right) \Pr \left( {{{\left| {{h_{n,2}}} \right| }^2} > \theta } \right) \nonumber \\&= \sum \limits _{m = 1}^N {\left( {\begin{array}{*{20}{c}} N\\ m \end{array}} \right) } {\left( { - 1} \right) ^{m - 1}}{e^{ - \frac{{\theta m}}{{{\lambda _2}}}}}\left( {\underbrace{\int \limits _0^\theta {{f_{{{\left| {{h_{n,1}}} \right| }^2}}}\left( x \right) {F_{{{\left| {{g_{k,1}}} \right| }^2}}}\left( {\frac{\upsilon }{x}} \right) dx} }_{{A_{2,1}}}} \right) , \end{aligned}$$

(38)

where $\upsilon =\frac{{{\varepsilon }_{1}}}{\eta \rho \alpha }$.

$$\begin{aligned} {A_{2,1}}&= \int \limits _0^\theta {{f_{{{\left| {{h_{n,1}}} \right| }^2}}}\left( x \right) {F_{{{\left| {{g_{k,1}}} \right| }^2}}}\left( {\frac{\upsilon }{x}} \right) dx} \nonumber \\&= \sum \limits _{n = 1}^N {\left( {\begin{array}{*{20}{c}} N\\ n \end{array}} \right) {{\left( { - 1} \right) }^{n - 1}}\frac{n}{{{\lambda _1}}}} \int \limits _0^\theta {{e^{ - \frac{{nx}}{{{\lambda _1}}}}}\left( {1 - {e^{ - \frac{\upsilon }{{{\lambda _{k,1}}x}}}}} \right) dx} \nonumber \\&= \sum \limits _{n = 1}^N {\left( {\begin{array}{*{20}{c}} N\\ n \end{array}} \right) {{\left( { - 1} \right) }^{n - 1}}\frac{n}{{{\lambda _1}}}} \underbrace{\left( {\int \limits _0^\theta {{e^{ - \frac{{nx}}{{{\lambda _1}}}}}dx} - \int \limits _0^\theta {{e^{ - \frac{{nx}}{{{\lambda _1}}} - \frac{\upsilon }{{{\lambda _{k,1}}x}}}}dx} } \right) }_{{A_{2,2}}}. \end{aligned}$$

(39)

Unfortunately, it is very hard to achieve the closed-form (39), the approximate form can be obtained. By motivated by generalized incomplete gamma, using interesting result of $\int \limits _{0}^{x}{{{t}^{\alpha -1}}{{e}^{-at-b{{t}^{-1}}}}=}{{a}^{-\alpha }}\gamma \left( \alpha ,ax;ab \right)$ satisfying condition $a>0$. As a result, ${{A}_{2,2}}$ can be given by

$$\begin{aligned} {A_{2,2}}&= \int \limits _0^\theta {{e^{ - \frac{{nx}}{{{\lambda _1}}}}}dx} - \int \limits _0^\theta {{e^{ - \frac{{nx}}{{{\lambda _1}}} - \frac{\upsilon }{{{\lambda _{k,1}}x}}}}dx} \nonumber \\&= \frac{{{\lambda _1}}}{n}\left( {1 - {e^{ - \frac{{n\theta }}{{{\lambda _1}}}}}} \right) - \frac{{{\lambda _1}}}{n}\gamma \left( {1,\frac{{\theta n}}{{{\lambda _1}}};\frac{{\upsilon n}}{{{\lambda _1}{\lambda _{k,1}}}}} \right) . \end{aligned}$$

(40)

Plugging (40) into (39), ${{A}_{2,1}}$ is expressed

$$\begin{aligned} {A_{2,1}} = \sum \limits _{n = 0}^N {\left( {\begin{array}{*{20}{c}} N\\ n \end{array}} \right) {{\left( { - 1} \right) }^n}} \left[ {\left( {1 - {e^{ - \frac{{n\theta }}{{{\lambda _1}}}}}} \right) - \gamma \left( {1,\frac{{\theta n}}{{{\lambda _1}}};\frac{{\upsilon n}}{{{\lambda _1}{\lambda _{k,1}}}}} \right) } \right] . \end{aligned}$$

(41)

Replacing (41) into (38), ${{A}_{2}}$ can be written by

$$\begin{aligned} {A_2}&= \sum \limits _{n = 1}^N \sum \limits _{m = 1}^N {\left( {\begin{array}{*{20}{c}} N\\ m \end{array}} \right) } {{\left( { - 1} \right) }^{m + n - 2}}{e^{ - \frac{{\theta m}}{{{\lambda _2}}}}}\left[ \left( {1 - {e^{ - \frac{{n\theta }}{{{\lambda _1}}}}}} \right) \right. \nonumber \\&\quad \left. - \gamma \left( {1,\frac{{\theta n}}{{{\lambda _1}}};\frac{{\upsilon n}}{{{\lambda _1}{\lambda _{k,1}}}}} \right) \right] . \end{aligned}$$

(42)

Combining (42) and (37) into (14) $P_{{\text {cN}},1}$ it can be exhibited final result

$$\begin{aligned} P_{\mathrm{{cN}}\mathrm{{,1}}}^{}&= \sum \limits _{n = 1}^N {\sum \limits _{m = 1}^N {\left( {\begin{array}{*{20}{c}} N\\ n \end{array}} \right) } } \left( {\begin{array}{*{20}{c}} N\\ m \end{array}} \right) {\left( { - 1} \right) ^{n + m - 2}}\nonumber \\&\quad \times \left\{ \left( {1 - {e^{ - \frac{{\theta n}}{{{\lambda _1}}}}}} \right) \left( {1 - {e^{ - \frac{{\theta m}}{{{\lambda _2}}}}}} \right) + {e^{ - \frac{{\theta m}}{{{\lambda _2}}}}}\left[ \left( {1 - {e^{ - \frac{{n\theta }}{{{\lambda _1}}}}}} \right) \right. \right. \nonumber \\&\quad \left. \left. - \gamma \left( {1,\frac{{\theta n}}{{{\lambda _1}}};\frac{{\upsilon n}}{{{\lambda _1}{\lambda _{k,1}}}}} \right) \right] \right\} . \end{aligned}$$

(43)

It complete the proof.□

Appendix 2

Proof of Proposition 2

We first calculate ${{C}_{1}}$ as

$$\begin{aligned} {C_1} = \sum \limits _{n = 1}^N {\left( {\begin{array}{*{20}{c}} N\\ n \end{array}} \right) {{\left( { - 1} \right) }^{n - 1}}\left( {1 - {e^{ - \frac{{n\theta }}{{{\lambda _1}}}}}} \right) \left[ {1 - \sum \limits _{m = 1}^N {\left( {\begin{array}{*{20}{c}} N\\ m \end{array}} \right) {{\left( { - 1} \right) }^{m - 1}}{e^{ - \frac{{m\chi }}{{{\lambda _2}}}}}} } \right] }, \end{aligned}$$

(44)

where $\chi =\max \left( \frac{{{\varepsilon }_{2}}}{\rho \left( 1-\alpha \right) \left( {{\varTheta }_{2}}-{{\varTheta }_{1}}\mu {{\varepsilon }_{2}} \right) },\theta \right)$

Next, the closed-form of is more complicated and it can be formulated by

$$\begin{aligned} {C_2}&= \Pr \left\{ {\left( {{\gamma _{2,2}}< {\varepsilon _2} \cup \max \left( {{\gamma _{2,1}},{W_2}} \right) < {\varepsilon _1}} \right) ,{\gamma _{1,1}} > {\varepsilon _1}} \right\} \nonumber \\&= \sum \limits _{n = 1}^N {\left( {\begin{array}{*{20}{c}} N\\ n \end{array}} \right) {{\left( { - 1} \right) }^{n - 1}}{e^{ - \frac{{n\theta }}{{{\lambda _1}}}}}\left[ {{C_{2,1}} + {C_{2,2}} - {C_{2,3}}} \right] } , \end{aligned}$$

(45)

where ${{C}_{2,1}}=\int \limits _{0}^{\frac{{{\varepsilon }_{2}}}{\rho \left( 1-\alpha \right) \left( {{\varTheta }_{2}}-{{\varTheta }_{1}}\mu {{\varepsilon }_{2}} \right) }}{{{f}_{{{\left| {{h}_{n,2}} \right| }^{2}}}}\left( x \right) dx},$${{C}_{2,2}}=\int \limits _{0}^{\theta }{{{f}_{{{\left| {{h}_{n,2}} \right| }^{2}}}}\left( y \right) {{F}_{{{\left| {{g}_{k,2}} \right| }^{2}}}}\left( \frac{\upsilon }{y} \right) dy}$, ${{C}_{2,3}}=\int \limits _{0}^{\varDelta }{{{f}_{{{\left| {{h}_{n,2}} \right| }^{2}}}}\left( z \right) {{F}_{{{\left| {{g}_{k,2}} \right| }^{2}}}}\left( \frac{\upsilon }{z} \right) dz}$, $\varDelta =\min \left( \frac{{{\varepsilon }_{2}}}{\rho \left( 1-\alpha \right) \left( {{\varTheta }_{2}}-{{\varTheta }_{1}}\mu {{\varepsilon }_{2}} \right) },\theta \right)$

It can be solved by examining each component. We have following results

$$\begin{aligned} {C_{2,1}}&= \int \limits _0^{\frac{{{\varepsilon _2}}}{{\rho \left( {1 - \alpha } \right) \left( {{\varTheta _2} - {\varTheta _1}\mu {\varepsilon _2}} \right) }}} {{f_{{{\left| {{h_{n,2}}} \right| }^2}}}\left( x \right) dx} \nonumber \\&= \sum \limits _{m = 1}^N {\left( {\begin{array}{*{20}{c}} N\\ m \end{array}} \right) {{\left( { - 1} \right) }^{m - 1}}\left( {1 - {e^{ - \frac{{m{\varepsilon _2}}}{{\rho {\lambda _2}\left( {1 - \alpha } \right) \left( {{\varTheta _2} - {\varTheta _1}\mu {\varepsilon _2}} \right) }}}}} \right) }. \end{aligned}$$

(46)

It is noted that ${{C}_{2,2}}={{B}_{2,2}}$

$$\begin{aligned} {C_{2,3}}&= \int \limits _0^\varDelta {{f_{{{\left| {{h_{n,2}}} \right| }^2}}}\left( y \right) {F_{{{\left| {{g_{k,2}}} \right| }^2}}}\left( {\frac{\upsilon }{y}} \right) dy} \nonumber \\&= \sum \limits _{m = 1}^N {\left( {\begin{array}{*{20}{c}} N\\ m \end{array}} \right) {{\left( { - 1} \right) }^{m - 1}}} \left[ {\left( {1 - {e^{ - \frac{{m\varDelta }}{{{\lambda _2}}}}}} \right) - \gamma \left( {1,\frac{{m\varDelta }}{{{\lambda _2}}};\frac{{m\upsilon }}{{{\lambda _{k,2}}{\lambda _2}}}} \right) } \right] \end{aligned}$$

(47)

Plugging (46), (47) and (21) into (45) the new expression of ${{C}_{2}}$ can be rewritten as

$$\begin{aligned} {C_2}&= \sum \limits _{n = 1}^N {\sum \limits _{m = 1}^N {\left( {\begin{array}{*{20}{c}} N\\ m \end{array}} \right) \left( {\begin{array}{*{20}{c}} N\\ n \end{array}} \right) {{\left( { - 1} \right) }^{n + m - 2}}{e^{ - \frac{{n\theta }}{{{\lambda _1}}}}}} } \nonumber \\&\quad \times \left\{ \left( {1 - {e^{ - \frac{{m{\varepsilon _2}}}{{\rho {\lambda _2}\left( {1 - \alpha } \right) \left( {{\varTheta _2} - {\varTheta _1}\mu {\varepsilon _2}} \right) }}}}} \right) \right. \nonumber \\&\quad + \left[ {\left( {1 - {e^{ - \frac{{m\theta }}{{{\lambda _2}}}}}} \right) - \gamma \left( {1,\frac{{m\theta }}{{{\lambda _2}}};\frac{{m\upsilon }}{{{\lambda _{k,2}}{\lambda _2}}}} \right) } \right] \nonumber \\&\quad \left. { - \left[ {\left( {1 - {e^{ - \frac{{m\varDelta }}{{{\lambda _2}}}}}} \right) - \gamma \left( {1,\frac{{m\varDelta }}{{{\lambda _2}}};\frac{{m\upsilon }}{{{\lambda _{k,2}}{\lambda _2}}}} \right) } \right] } \right\} . \end{aligned}$$

(48)

We continue to replace (44) and (48) into (26), $P_{{\text {cN}},2}^{{\text {ipSIC}}}$ can be expressed by

$$\begin{aligned} P_{{\mathrm{cN}}{{,\mathrm 2}}}^{\mathrm{{ipSIC}}}&= \sum \limits _{n = 1}^N {\left( {\begin{array}{*{20}{c}} N\\ n \end{array}} \right) {{\left( { - 1} \right) }^{n - 1}}\left( {1 - {e^{ - \frac{{n\theta }}{{{\lambda _1}}}}}} \right) \left[ {1 - \sum \limits _{m = 1}^N {\left( {\begin{array}{*{20}{c}} N\\ m \end{array}} \right) {{\left( { - 1} \right) }^{m - 1}}{e^{ - \frac{{m\chi }}{{{\lambda _2}}}}}} } \right] } \nonumber \\&\quad + \sum \limits _{n = 1}^N {\sum \limits _{m = 1}^N {\left( {\begin{array}{*{20}{c}} N\\ m \end{array}} \right) \left( {\begin{array}{*{20}{c}} N\\ n \end{array}} \right) {{\left( { - 1} \right) }^{n + m - 2}}{e^{ - \frac{{n\theta }}{{{\lambda _1}}}}}} } \nonumber \\&\quad \times \left\{ \left( {1 - {e^{ - \frac{{m{\varepsilon _2}}}{{\rho {\lambda _2}\left( {1 - \alpha } \right) \left( {{\varTheta _2} - {\varTheta _1}\mu {\varepsilon _2}} \right) }}}}} \right) \right. \nonumber \\&\quad + \left[ {\left( {1 - {e^{ - \frac{{m\theta }}{{{\lambda _2}}}}}} \right) - \gamma \left( {1,\frac{{m\theta }}{{{\lambda _2}}};\frac{{m\upsilon }}{{{\lambda _{k,2}}{\lambda _2}}}} \right) } \right] \nonumber \\&\quad \left. { - \left[ {\left( {1 - {e^{ - \frac{{m\varDelta }}{{{\lambda _2}}}}}} \right) - \gamma \left( {1,\frac{{m\varDelta }}{{{\lambda _2}}};\frac{{m\upsilon }}{{{\lambda _{k,2}}{\lambda _2}}}} \right) } \right] } \right\} . \end{aligned}$$

(49)

It is the end of the proof.□