Co-Channel Interference Effects on Downlink Power-Domain Non-Orthogonal Multiple Access

Abstract

This paper investigates the fundamental performance limits of downlink power-domain non-orthogonal multiple access network in the presence of co-channel interference. The investigation considers that base-station makes multiple information exchanges with a designated mobile terminal via the power-domain non-orthogonal multiple access technique. The investigation also considers that the designated mobile terminal is under the effect of a limited number of co-channel interference, which are independent and identical distributed random variables subject to Rayleigh fading. Analytical, asymptotic, and Monte-Carlo based intensive computer simulation results reveal that co-channel interference causes coding gain losses on free-interference case, utilized as a benchmark, in high signal-to-noise-ratios. Outage probability, error probability, ergodic rate, and throughput performance metrics are considered for the performance analysis. Optimization techniques and also asymptotic analysis are provided.

Appendices

Proof of Proposition 1

Considering (2) and the logarithm properties, following expressions, which are related to CDF expression, can be written.

$$\begin{aligned} F_{R_{x_{i}}}\left( \gamma _{\mathrm{th}}\right)&={\mathrm{P}}_{r}\left( \frac{\beta _{i}\gamma _{x}}{\sum _{m=1, m\ne i}^{M-1}\beta _{m}\gamma _{x}+\gamma _{J}+1}\le \underbrace{2^{R}-1}_{\gamma _{th}}\right) \nonumber \\&={\mathrm{P}}_{r}\left( \gamma _{x}\le \left( \gamma _{th}\left( \frac{\gamma _{J}+1}{\beta _{i}-\gamma _{th}\sum _{m=1, m\ne i}^{M-1}\beta _{m}}\right) \right) \right) \nonumber \\&=1-{\mathrm{P}}_{r}\left( \gamma _{x}\ge \left( \gamma _{th}\left( \frac{\gamma _{J}+1}{\beta _{i}-\gamma _{th}\sum _{m=1, m\ne i}^{M-1}\beta _{m}}\right) \right) \right) \nonumber \\&=1-{\mathrm{P}}_{r}\left( 1-F_{\gamma _{x}}\left( \gamma _{th}\left( \frac{\gamma _{J}+1}{\beta _{i}-\gamma _{th}\sum _{m=1, m\ne i}^{M-1}\beta _{m}}\right) \right) \right) \nonumber \\&=1-e^{-\gamma _{th}\left( \frac{1}{\text { P}_{s}\varOmega _{h}\left( \beta _{i}-\gamma _{th}\sum _{m=1, m\ne i}^{M-1}\beta _{m}\right) }\right) }\nonumber \\&\times \int _{0}^{\infty }e^{-\gamma _{J}\left( \frac{\gamma _{th}}{\text { P}_{s}\varOmega _{h}\left( \beta _{i}-\gamma _{th}\sum _{m=1, m\ne i}^{M-1}\beta _{m}\right) }\right) }f_{\gamma _{J}}\left( \gamma _{J}\right) d\gamma _{J} \end{aligned}$$

(17)

Note that sum of N i.i.d Rayleigh distribution becomes a Gamma distribution [44]. As such, substituting $f_{\gamma _{J}}\left( \gamma _{J}\right) =\frac{\gamma ^{N-1}}{(\text { P}_{J}\varOmega _{g_{k}})^{N}(N-1)!}e^{-\frac{\gamma _{J}}{\text { P}_{J}\varOmega _{g_{k}}}}$ [44] into (17) and solving the expression by means of [40, Eq. ($3.351^{3}$)], the final CDF can be obtained as in (4). By considering the similar methodologies as in (17), remaining CDF expression, which is (3), can also be obtained. Obtained results are presented as in Proposition 1.

Proof of Proposition 2

The ER can be formulated as: $ER_{x_{i}}= \frac{1}{ln2}\int _{0}^{\infty }\frac{1-F_{ \gamma _{x_{i}}}\left( \gamma _\mathrm{th}\right) }{1+\gamma _\mathrm{th}}d\gamma _\mathrm{th}$ [27, Eq. (38)]. Substituting (4) into ER formulation and also doing some basic algebraic manipulation, following expression can be obtained.

$$\begin{aligned} ER_{x_{i}}&=\frac{\mathrm{1}}{\mathrm{ln2}}\int _{0}^{\infty }\left( \frac{\text { P}_{J}\varOmega _{g_{k}}\gamma _{th}}{\text { P}_{s}\varOmega _{h}\left( \beta _{i}-\gamma _{th}\sum _{m=1, m\ne i}^{M-1}\beta _{m}\right) }+1\right) ^{-N}\nonumber \\&\quad \times \left( 1+\gamma _{th}\right) ^{-1} e^{-\gamma _{th}\left( \frac{1}{\text { P}_{s}\varOmega _{h}\left( \beta _{i}-\gamma _{th}\sum _{m=1, m\ne i}^{M-1}\beta _{m}\right) }\right) }d\gamma _{th} \end{aligned}$$

(18)

After some algebraic manipulations and utilizing $e^{-x}=\left( 1-x\right),$ $x\rightarrow 0,$ [40], (18) can be written as in (19).

$$\begin{aligned} ER_{x_{i}}&=\frac{\mathrm{1}}{\mathrm{ln2}}\int _{0}^{\infty }\left( 1+\gamma _{th}\left( \frac{\text { P}_{J}\varOmega _{g_{k}}-\text { P}_{s}\varOmega _{h}\sum _{m=1, m\ne i}^{M-1}\beta _{m}}{\text { P}_{s}\varOmega _{h}\beta _{i}}\right) \right) ^{-N}\left( 1-\gamma _{th}\left( \frac{\sum _{m=1, m\ne i}^{M-1}\beta _{m}}{\beta _{i}}\right) \right) ^{N-1}\nonumber \\&\quad \times \left( 1+\gamma _{th}\right) ^{-1}\left( 1-\gamma _{th}\left( \frac{\text { P}_{s}\varOmega _{h}\sum _{m=1,m\ne i}^{M-1}\beta _{m}+1}{\text { P}_{s}\varOmega _{h}\beta _{i}}\right) \right) d\gamma _{th} \end{aligned}$$

(19)

Utilizing partial fraction decomposition technique, (19) can be presented as:

$$\begin{aligned} ER_{x_{i}}&=\frac{\mathrm{1}}{\mathrm{ln2}}\int _{0}^{\infty }\left( 1+\gamma _{th}\left( \frac{\text { P}_{J}\varOmega _{g_{k}}-\text { P}_{s}\varOmega _{h}\sum _{m=1, m\ne i}^{M-1}\beta _{m}}{\text { P}_{s}\varOmega _{h}\beta _{i}}\right) \right) ^{-N}\left( 1-\gamma _{th}\left( \frac{\sum _{m=1, m\ne i}^{M-1}\beta _{m}}{\beta _{i}}\right) \right) ^{N-1}\nonumber \\&\quad \times \Bigg [\frac{A}{\left( 1+\gamma _{th}\right) }+\frac{B}{\left( 1-\gamma _{th}\left( \frac{\text { P}_{s}\varOmega _{h}\sum _{m=1,m\ne i}^{M-1}\beta _{m}+1}{\text { P}_{s}\varOmega _{h}\beta _{i}}\right) \right) ^{-1}}\Bigg ]d\gamma _{th} \end{aligned}$$

(20)

where $A=\displaystyle \lim _{\gamma _{th}\rightarrow -1}\frac{\partial }{\partial \gamma _{th}}\left( \gamma _{th}+1\right) [\varDelta ],$ $B=\displaystyle \lim _{\gamma _{th}\rightarrow \frac{\text { P}_{s}\varOmega _{h}\beta _{i}}{\text { P}_{s}\varOmega _{h}\sum _{m=1,m\ne i}^{M-1}\beta _{m}+1}}\frac{\partial }{\partial \gamma _{th}}\left( 1-\gamma _{th}\left( \frac{\text { P}_{s}\varOmega _{h}\sum _{m=1,m\ne i}^{M-1}\beta _{m}+1}{\text { P}_{s}\varOmega _{h}\beta _{i}}\right) \right) ^{-1}[\varDelta ],$ and $\varDelta =\Bigg [\frac{1}{\left( 1+\gamma _{th}\right) \left( 1-\gamma _{th}\left( \frac{\text { P}_{s}\varOmega _{h}\sum _{m=1,m\ne i}^{M-1}\beta _{m}+1}{\text { P}_{s}\varOmega _{h}\beta _{i}}\right) \right) ^{-1}}\Bigg ].$ Considering distributive property and [45, Eq. (9,10)] and also [39, Eq. (13)], the final $ER_{x_{i}}\left( \gamma _{\mathrm{th}}\right)$ expression can be obtained as in (9). Likewise, starting with (5) and also utilizing [45, Eq. (10)] and [39, Eq. (13)] $ER_{x_{N}}\left( \gamma _{\mathrm{th}}\right)$ can be obtained as in (10). $\alpha$ term is set to 1 in [39, Eq. (13)].

Proof of Proposition 3

The CDF based EP expression, which is presented in (21) [27, Eq. (27)], is utilized for the EP analysis for the analytical derivations.

$$\begin{aligned} \bar{P_{e}} = \frac{a}{2}\sqrt{\frac{b}{\pi }} \int _0^\infty \frac{\exp \left( -bx\right) }{\sqrt{x}}F(x) dx \end{aligned}$$

(21)

Note that $a=b=1$ represents the BPSK and $a=b=2$ represents the QPSK modulations. The BPSK modulation is utilized for the analytical derivations. Substituting (4) into (21), following expression can be obtained.

$$\begin{aligned}&\bar{P_{e}x_{i}^{\infty }} = \frac{1}{2\sqrt{\pi }}\Bigg [\int _0^\infty x^{-\frac{1}{2}}\exp \left( -x\right) dx\nonumber \\&\quad -\int _0^\infty x^{-\frac{1}{2}}\left( 1+\gamma _{th}\left( \frac{\text { P}_{J}\varOmega _{g_{k}}-\text { P}_{s}\varOmega _{h}\sum _{m=1, m\ne i}^{M-1}\beta _{m}}{\text { P}_{s}\varOmega _{h}\beta _{i}}\right) \right) ^{-N}\nonumber \\&\quad \times \left( 1-\gamma _{th}\left( \frac{\sum _{m=1, m\ne i}^{M-1}\beta _{m}}{\beta _{i}}\right) \right) ^{N-1}\nonumber \\&\quad \times \left( 1-\gamma _{th}\left( \frac{\text { P}_{s}\varOmega _{h}\sum _{m=1, m\ne i}^{M-1}\beta _{m}+1}{\text { P}_{s}\varOmega _{h}\beta _{i}}\right) \right) \mathrm exp(-\gamma _{th})d\gamma _{th} \end{aligned}$$

(22)

The first integral expression in (22) can be solved with the help of [40, Eq. ($3.326.2^{10}$)] as: $\sqrt{\pi }.$ Utilizing partial fraction decomposition technique, which is given in (23), [45, Eq. (9,10,11)] and [39, Eq. (13)], the second integral expression in (22) can be obtained as in (11). However, after several attempts, unable to obtained the asymptotic curves by using the obtained result in (11). Therefore, the second integral expression of (22) is plotted numerically. Likewise, utilizing (5), [45, Eq. (10,11)], and setting $\alpha$ term as $\frac{1}{2},$ the $Pe_{x_{M}}\left( \gamma _{\mathrm{th}}\right)$ can be obtained as in (12).

$$\begin{aligned}&\int _0^\infty \gamma _{th}^{-\frac{1}{2}}e^{-\gamma _{th}}\left( 1+\gamma _{th}\left( \frac{\text { P}_{J}\varOmega _{g_{k}}-\text { P}_{s}\varOmega _{h}\sum _{m=1, m\ne i}^{M-1}\beta _{m}}{\text { P}_{s}\varOmega _{h}\beta _{i}}\right) \right) ^{-N}\nonumber \\&\quad \Bigg [\sum _{s=1}^{N}\frac{C_{s}}{\left( 1-\gamma _{th}\left( \frac{\sum _{m=1, m\ne i}^{M-1}\beta _{m}}{\beta _{i}}\right) \right) ^{-s+1}}\nonumber \\&\quad +\frac{D}{\left( 1-\gamma _{th}\left( \frac{\text { P}_{s}\varOmega _{h}\sum _{m=1, m\ne i}^{M-1}\beta _{m}+1}{\text { P}_{s}\varOmega _{h}\beta _{i}}\right) \right) ^{-1}}\Bigg ]d\gamma _{th} \end{aligned}$$

(23)

where $C_{s}=\displaystyle \lim _{\gamma _{th}\rightarrow \frac{\beta _{i}}{\sum _{m=1,m\ne i}^{M-1}\beta _{m}}}\frac{\partial ^{N-s+1}}{(N-s+1)!\partial \gamma _{th}^{N-s+1}}\left( 1-\gamma _{th}\left( \frac{\sum _{m=1,m\ne i}^{M-1}\beta _{m}}{\beta _{i}}\right) \right) ^{-N+1}[\varTheta ],$ $D=\displaystyle \lim _{\gamma _{th}\rightarrow \frac{\text { P}_{s}\varOmega _{h}\beta _{i}}{\text { P}_{s}\varOmega _{h}\sum _{m=1,m\ne i}^{M-1}\beta _{m}+1}}\frac{\partial }{\partial \gamma _{th}}\left( 1-\gamma _{th}\left( \frac{\text { P}_{s}\varOmega _{h}\sum _{m=1,m\ne i}^{M-1}\beta _{m}+1}{\text { P}_{s}\varOmega _{h}\beta _{i}}\right) \right) ^{-1}[\varTheta ]$

and $\varTheta =\Bigg [\frac{1}{\left( 1-\gamma _{th}\left( \frac{\sum _{m=1, m\ne i}^{M-1}\beta _{m}}{\beta _{i}}\right) \right) ^{-N+1}\left( 1-\gamma _{th}\left( \frac{\text { P}_{s}\varOmega _{h}\sum _{m=1, m\ne i}^{M-1}\beta _{m}+1}{\text { P}_{s}\varOmega _{h}\beta _{i}}\right) \right) ^{-1}}\Bigg ]$. Utilizing distributive properties and [45, Eq. (9,10,11)] and also [39, Eq. (13)], the final $\bar{P_{e}x_{i}}$ expression can be obtained as in (11). Note that $\alpha$ term is set to $\frac{1}{2}$ in [39, Eq. (13)].