Performance analysis of hybrid half and full duplex user relaying non orthogonal multiple access system with advanced successive interference cancellation

Abstract

This paper describes a hybrid half duplex (HD) and full duplex (FD) user relaying cooperative non orthogonal multiple access (NOMA) system, where a HD/FD user relaying node shares the information to far users from base station (BS). The system works under residual self-interference (RSI) due to imperfect self-interference cancellation. An advanced successive interference cancellation (ASIC) scheme is introduced in the system as an alternative of successive interference cancellation (SIC) technique. The implementation of ASIC scheme in the proposed system is done by mapping the received signal into number of subgroups. Then, the conventional SIC technique is applied to each of these subgroups for decoding information signals. The performance analysis of the proposed system is comprehensively conducted through the closed form expressions, which are derived in terms of outage probability, system throughput, achievable sum rate and energy efficiency. The effectiveness and correctness of the proposed system is verified by analytical results, which clearly boost the outage performance and system throughput than traditional system in moderate to high SNR region. The robustness of the proposed system is also provided with high achievable sum rate and energy efficiency than traditional system under RSI.

Appendices

Appendix A: proof of proposition 1

For deriving the result of (19), the outage probability of user $D_1$ can be evaluated as

$$\begin{aligned} P_{D1}^{FD}&= 1 - P \biggl ( r_{1,1}>R_1 r_{1,2}>R_2 r_{1,3}>R_3 \biggr )&\nonumber \\&= 1 - P\biggl (\frac{a_1P|h_s{_r}|^2}{\frac{V}{2}+\frac{N_0}{2}}>\gamma _{th1} \frac{a_2 P |h_s{_r}|^2}{\frac{V}{2}+\frac{N_0}{2}}>\gamma _{th2} \nonumber \\ {}& \frac{a_3P|h_s{_r}|^2}{a_1P|h_s{_r}|^2+\frac{V}{2}+\frac{N_0}{2}}>\gamma _{th3}\biggr ) \nonumber \\&= 1 - P\biggl (\frac{a_1\gamma _s{_r}}{\frac{V}{2N_0}+\frac{1}{2}}>\gamma _{th1} \frac{a_2\gamma _s{_r}}{\frac{V}{2N_0}+\frac{1}{2}}>\gamma _{th2} \nonumber \\& \frac{a_3\gamma _s{_r}}{a_1\gamma _s{_r}+\frac{V}{2N_0}+\frac{1}{2}}>\gamma _{th3}\biggr ) \nonumber \\&= 1 - P\biggl ( \gamma _s{_r}> \frac{\gamma _{th1}(V+N_0)}{2a_1} \gamma _s{_r}> \frac{\gamma _{th2}(V+N_0)}{2a_2} \nonumber \\ {}& \gamma _s{_r}> \frac{\gamma _{th3}(V+N_0)}{2(a_3-a_1\gamma _{th3})}\biggr ), \end{aligned}$$

(A.1)

where the variable $\gamma _s{_r}$ has exponential distribution with parameter $\lambda _s{_r}^\gamma$. Therefore,

$$\begin{aligned} P_{D1}^{FD}&= 1 - \biggl (e^{-\frac{\gamma _{th1}(V+N_0)}{2a_1\lambda _s{_r}^\gamma }}\times e^{-\frac{\gamma _{th2}(V+N_0)}{2a_2\lambda _s{_r}^\gamma }} \nonumber \\&\times e^{-\frac{\gamma _{th3}(V+N_0)}{2(a_3-a_1\gamma _{th3})\lambda _s{_r}^\gamma }}\biggr ) \nonumber \\&= 1 - e^{-\frac{(V+N_0)}{2\lambda _s{_r}^\gamma }\biggl [\frac{\gamma _{th1}}{a_1},\frac{\gamma _{th2}}{a_2},\frac{\gamma _{th3}}{a_3-a_1\gamma _{th3}}\biggr ]} \nonumber \\&= 1 - e^{-\frac{(V+N_0)}{2\lambda _s{_r}^\gamma }\theta _1}, \end{aligned}$$

(A.2)

where $\theta _1 = \max (\tau _1,\tau _2,\tau _3)$, $\tau _1 = \frac{\gamma _{th1}}{a_1}$, $\tau _2 = \frac{\gamma _{th2}}{a_2}$, and $\tau _3 = \frac{\gamma _{th3}}{(a_3-a_1\gamma _{th3})}$.

Appendix B: proof of proposition 2

The outage probability of $D_2$ is written as

$$\begin{aligned} P_{D2}^{FD}&= 1 - P\bigl (r_{1,2}>R_2 r_{2,2}>R_2\bigr ) \nonumber \\&= 1 - P\biggl (\frac{c_2P|h_r{_2}|^2}{N_0}>\gamma _{th2} \frac{a_2P|h_s{_r}|^2}{\frac{V}{2}+\frac{N_0}{2}}>\gamma _{th2}\biggr ) \nonumber \\&= 1 - P\biggl ({c_2\gamma _r{_2}}>\gamma _{th2} \frac{a_2\gamma _s{_r}}{\frac{V}{2N_0}+\frac{1}{2}}>\gamma _{th2}\biggr ) \nonumber \\&= 1 - P\biggl ({\gamma _r{_2}}>\frac{\gamma _{th2}}{c_2}{\gamma _s{_r}}>\frac{\gamma _{th2}(V+N_0)}{2a_2}\biggr ) \nonumber \\&= 1 - e^{-\frac{\gamma _{th2}}{c_2\lambda _r{_2}^\gamma }}\times e^{-\frac{\gamma _{th2}(V+N_0)}{2a_2\lambda _s{_r}^\gamma }} \nonumber \\&= 1 - e^{\gamma _{th2} \biggl [\frac{1}{c_2\lambda _r{_2}^\gamma }+\frac{V+N_0}{2a_2\lambda _s{_r}^\gamma } \biggr ]} \nonumber \\&= 1 - e^{-\gamma _{th2}\theta _2}. \end{aligned}$$

(B.1)

where the variable $\gamma _r{_2}$ is an exponentially distributed random variable with parameter $\lambda _r{_2}^\gamma$, and $\theta _2 = (\phi _1+\phi _2)$, $\phi _1 = \frac{1}{c_2\lambda _r{_2}^\gamma }$, and $\phi _2 = \frac{V+N_0}{2a_2\lambda _s{_r}^\gamma }$.

Appendix C: proof of proposition 3

The outage probability of user $D_3$ for deriving the result of (25) can be written as

$$\begin{aligned} P_{D3}^{FD}&= 1 - P\bigl (r_{1,3}>R_3 r_{3,3}>R_3\bigr ) \nonumber \\&= 1 - P\biggl (\frac{c_3P|h_r{_3}|^2}{c_2P|h_r{_3}|^2+N_0}>\gamma _{th3} \nonumber \\ {}& \frac{a_3P|h_s{_r}|^2}{a_1P|h_s{_r}|^2+\frac{V}{2}+\frac{N_0}{2}}>\gamma _{th3}\biggr ) \nonumber \\&= 1 - P\biggl (\frac{c_3\gamma _r{_3}}{c_2\gamma _r{_3}+1}>\gamma _{th3} \nonumber \\ {}& \frac{a_3\gamma _s{_r}}{a_1\gamma _s{_r}+\frac{V}{2N_0}+\frac{1}{2}}>\gamma _{th3}\biggr ) \nonumber \\&= 1 - P\biggl ({\gamma _r{_3}}>\frac{\gamma _{th3}}{(c_3-c_2\gamma _{th3})} \nonumber \\ {}& {\gamma _s{_r}}>\frac{\gamma _{th3}(V+N_0)}{2(a_3-a_1\gamma _{th3})}\biggr ) \nonumber \\&= 1 - \biggl [e^{-\frac{\gamma _{th3}}{(c_3 - c_2\gamma _{th3})\lambda _r{_3}^\gamma }} \nonumber \\& \times e^{-\frac{\gamma _{th3}(V+N_0)}{2(a_3-a_1\gamma _{th3})\lambda _s{_r}^\gamma }}\biggr ] \nonumber \\&= 1 - e^{-\gamma _{th3}\biggl [\frac{1}{(c_3 - c_2\gamma _{th3})\lambda _r{_3}^\gamma }+\frac{V+N_0}{2(a_3-a_1\gamma _{th3})\lambda _s{_r}^\gamma }\biggr ]} \nonumber \\&= 1 - e^{-\gamma _{th3}\theta _3}. \end{aligned}$$

(C.1)

where the variable $\gamma _r{_3}$ is an exponentially distributed random variable with parameter $\lambda _r{_3}^\gamma$, and $\theta _3 = (\psi _1+\psi _2)$, $\psi _1 = \frac{1}{(c_3-c_2\gamma _{th3})\lambda _r{_3}^\gamma }$, and $\psi _2 = \frac{V+N_0}{2(a_3-a_1\gamma _{th3})\lambda _s{_r}^\gamma }$.