On the leakage-rate performance of untrusted relay-aided NOMA under co-channel interference

Abstract

This study seeks an answer for the impact of the co-channel interference on the leakage rate of illegitimate relay-assisted power-domain non-orthogonal multiple access (NOMA) networks. The power-domain NOMA-based uni-/bi-directional information exchange through an illegitimate half-/full-duplex relay is adopted in the system model. Moreover, a limited number of co-channel interferers and friendly jammers are affected by the illegitimate relay in the network. Extensive computer simulations, analytical, and asymptotic results reveal that as the illegitimate relay is affected by a limited number of friendly jammers and co-channel interferers, its achievable rate cannot be enhanced and saturates at high signal-to-noise ratio. Results also show that the co-channel interference causes system coding gain losses on the users’ outage performance. In addition, the illegitimate relay remains active between $-\,10$ and 25 dB for uni-/bi-directional information exchange. After 25 dB, the illegitimate relay saturates and experiences outages. Additionally, the order of decoding also impacts the illegitimate relay’s performance.

Appendices

Appendix A: Proof of Proposition 1

Revisiting (2) and also considering the logarithm properties, below expressions can be achieved.

$$\begin{aligned}&F_{LR_{x_{2}}}^{HD(OWR)}\left( \gamma _{th}^{HD}\right) \nonumber \\&\quad =P _{r}\left( \frac{\sum _{v=i+1}^{L-1}\beta _{v}\gamma _{x}}{\beta _{i}\gamma _{x}+\gamma _{F}+\gamma _{J}+1}\le \underbrace{2^{2R}-1}_{\gamma _{th}^{\textrm{HD}}}\right) \nonumber \\&\quad ={\textrm{P}}_{r}\left( \gamma _{x}\le \frac{\gamma _{th}^{\textrm{HD}}\left( \gamma _{F}+\gamma _{J}+1\right) }{\left( \sum _{v=i+1}^{L-1}\beta _{v}-\beta _{i}\gamma _{th}^{HD}\right) }\right) \nonumber \\&\quad =1-P _{r}\left( \gamma _{x}\ge \frac{\gamma _{th}^{\textrm{HD}}\left( \gamma _{F}+\gamma _{J}+1\right) }{\left( \sum _{v=i+1}^{L-1}\beta _{v}-\beta _{i}\gamma _{th}^{HD}\right) }\right) \nonumber \\&\quad =1-P _{r}\left( 1-F_{\gamma _{x}}\left( \frac{\gamma _{th}^{\textrm{HD}}\left( \gamma _{F}+\gamma _{J}+1\right) }{\left( \sum _{v=i+1}^{L-1}\beta _{v}-\beta _{i}\gamma _{th}^{HD}\right) }\right) \right) \nonumber \\&\quad =1-{\mathbb {E}}_{\gamma _{J}}\left[ e^{-\gamma _{th}^{HD}\left( \frac{\gamma _{F}+\gamma _{J}+1}{\left( \sum _{v=i+1}^{L-1}\beta _{v}-\beta _{i}\gamma _{th}^{HD}\right) }\right) }|_{\gamma _{F},\gamma _{J}} \right] \nonumber \\&\quad =1-e^{-\gamma _{th}^{HD}\left( \frac{1}{P _{s}^{1}\Omega _{h}\left( \sum _{v=i+1}^{L-1}\beta _{v}-\beta _{i}\gamma _{th}^{HD}\right) }\right) }\nonumber \\&\qquad \times \int _{0}^{\infty }e^{-\gamma _{F}\left( \frac{\gamma _{th}^{HD}}{P _{s}^{1}\Omega _{h}\left( \sum _{v=i+1}^{L-1}\beta _{v}-\beta _{i}\gamma _{th}^{HD}\right) }\right) }f_{\gamma _{F}}\left( \gamma _{F}\right) d\gamma _{F}\nonumber \\&\qquad \times \int _{0}^{\infty }e^{-\gamma _{J}\left( \frac{\gamma _{th}^{HD}}{P _{s}^{1}\Omega _{h}\left( \sum _{v=i+1}^{L-1}\beta _{v}-\beta _{i}\gamma _{th}^{HD}\right) }\right) }f_{\gamma _{J}}\left( \gamma _{J}\right) d\gamma _{J} . \end{aligned}$$

(56)

Note that $f\gamma _{x}\left( \gamma \right) =\frac{1}{P _{s}^{1}\Omega _{h}}e^{-\gamma \left( \frac{1}{P _{s}^{1}\Omega _{h}}\right) }$. Also note that sum of M independent and identically distributed (i.i.d) exponentially distributed RVs follows the Gamma distribution [44]. As such, the PDF expressions of $\gamma _{F}$ and $\gamma _{J}$ can be written as: $f_{\gamma _{F}}\left( \gamma _{F}\right) =\left( \frac{1}{P _{J}\Omega _{f_{j}}}\right) ^{N}\frac{\gamma _{F}^{N-1}}{(N-1)!}e^{-\frac{\gamma _{F}}{P _{F}\Omega _{f_{k}}}}$ and $f_{\gamma _{J}}\left( \gamma _{J}\right) =\left( \frac{1}{P _{J}\Omega _{m_{j}}}\right) ^{M}\frac{\gamma _{J}^{M-1}}{(M-1)!}e^{-\frac{\gamma _{J}}{P _{J}\Omega _{m_{j}}}}$ [53]. In this regard, substituting $f_{\gamma _{F}}\left( \gamma _{F}\right)$ and $f_{\gamma _{J}}\left( \gamma _{J}\right)$ into (56) and solving the integrals with the help of [46, Eq. (3.310.11)] and [46, Eq. (3.351.3)], the final CDF expression can be obtained as in (17). Likewise, following the similar methodologies as in (56), the other CDFs, $F_{{\text {LR}_{\text {x}_{1}}}}^{\mathrm{HD(OWR)}}$, $F_{{\text {LR}_{\text {x}_{2}}}}^\mathrm{FD(OWR)}$, $F_{{\text {LR}_{\text {x}_{1}}}}^{\mathrm{FD(OWR)}}$, $F_{\text {LR}_{\text {x}_{2}}}^{\mathrm{HD(TWR)}}$, $F_{{\text {LR}_{\text {y}_{2}}}}^{\mathrm{HD(TWR)}}$, $F_{{\text {LR}_{\text {x}_{1}}}}^{\mathrm{HD(TWR)}}$, $F_{{\text {LR}_{\text {y}_{1}}}}^\mathrm{HD(TWR)}$, $F_{{\text {LR}_{\text {x}_{2}}}}^{\mathrm{FD(TWR)}}$, $F_{\text {LR}_{\text {y}_{2}}}^{\mathrm{FD(TWR)}}$, $F_{{\text {LR}_{\text {x}_{1}}}}^{\mathrm{FD(TWR)}}$, and $F_{{\text {LR}_{\text {y}_{1}}}}^{\mathrm{FD(TWR)}}$ can be calculated as in (18), (19), (20), (21), (22), (23), (24), (25), (26), (27), and (28), respectively.

Appendix B: Proof of Proposition 2

Utilizing (2) and the intercept probability formulation, which is $P _{r}\left[ C_{e}> R\right]$ [45, Eq. (5)], and also logarithm properties, following results can be obtained.

$$\begin{aligned} F_{IP_{x_{2}}}^{HD(OWR)}&=\mathrm Pr\left( \frac{\sum _{i\ne v, v=1}^{L-1}\beta _{v}\gamma _{x}}{\beta _{i}\gamma _{x}+\gamma _{F}+\gamma _{J}+1}> \gamma _{th}^{HD}\right) \nonumber \\&=\mathrm P_{r}\left( \gamma _{x}>\frac{\gamma _{th}^{HD}\left( +\gamma _{F}+\gamma _{J}+1\right) }{\left( \sum _{v=i+1}^{L-1}\beta _{v}-\gamma _{th}^{HD}\beta _{i}\right) }\right) \nonumber \\&=\int _{0}^{\infty }\int _{0}^{\frac{\gamma _{th}^{HD}\left( \gamma _{F}+\gamma _{J}+1\right) }{\left( \sum _{v=i+1}^{L-1}\beta _{v}-\gamma _{th}^{HD}\beta _{i}\right) }}\nonumber \\&\quad \times f\gamma _{x}\left( \gamma _{x}\right) f\gamma _{F}\left( \gamma _{F}\right) f\gamma _{J}\left( \gamma _{J}\right) d\gamma _{x}d\gamma _{F}d\gamma _{J}\nonumber \\&=e^{-\gamma _{th}^{HD}\left( \frac{1}{P _{s}^{1}\Omega _{h}\left( \sum _{v=i+1}^{L-1}\beta _{v}-\gamma _{th}^{HD}\beta _{i}\right) }\right) }\nonumber \\&\quad \times \int _{0}^{\infty }e^{-\gamma _{F}\left( \frac{\gamma _{th}^{HD}}{P _{s}^{1}\Omega _{h}\left( \sum _{v=i+1}^{L-1}\beta _{v}-\gamma _{th}^{HD}\beta _{i}\right) }\right) }\nonumber \\&\quad \times f_{\gamma _{F}}\left( \gamma _{F}\right) d\gamma _{F}\nonumber \\&\quad \times \int _{0}^{\infty }e^{-\gamma _{J}\left( \frac{\gamma _{th}^{HD}}{P _{s}^{1}\Omega _{h}\left( \sum _{v=i+1}^{L-1}\beta _{v}-\gamma _{th}^{HD}\beta _{i}\right) }\right) }\nonumber \\&\quad \times f_{\gamma _{J}}\left( \gamma _{J}\right) d\gamma _{J} . \end{aligned}$$

(57)

Substituting $f_{\gamma _{F}}\left( \gamma _{F}\right)$ and $f_{\gamma _{J}}\left( \gamma _{J}\right)$ into (57) and utilizing [46, Eq. (3.310.11)] and [46, Eq. (3.351.3)] for solving the integral expressions, the final CDF can be achieved as in (29). Note that considering the methodology as in (57), the other related CDFs, presented in Proposition 2, can be obtained.