Resource allocation and BER performance analysis of NOMA based cooperative networks

Abstract

Resource sharing and management can significantly improve the performance and spectral efficiency of wireless systems. In power domain non-orthogonal multiple access (NOMA) systems, users share a common bandwidth for simultaneous data transmission and therfore, the spectral efficiency of the system will be improved at the expense of increased complexity. Provided proper power allocation, it is possible to detect symbols at receivers. In this paper, the downlink of a cellular system is considered where a base station (BS) communicates with two users using NOMA with the assistance of a decode-and-forward (DF) relay. In the proposed scheme, receivers combine received signals from direct and cooperative links, and decode symbols by employing successive interference cancellation (SIC). The bit error rate (BER) performance of the proposed scheme is analyzed and the optimal power allocation is also derived through a Min–Max optimization, and a closed form approximate solution is proposed for binary phase shift keying (BPSK) modulation. Furthermore, it is proved that the proposed receiver achieves full diversity order for all users for proper choice of power allocation. Simulation results also corroborate BER and diversity analysis.

Appendices

Appendix A

Proof of proposition 1

Since $\gamma _{min}=min \{\gamma _{SR}, \gamma _{RD} \}$, $\gamma _{min}\le \gamma _{RD}$. Hence, an upper bound for (33) can be derived as,

$$\begin{aligned} P_{1}{} & {} \le E\bigg \{Q(x\sqrt{2(\gamma _{SD}+\gamma _{min})})\bigg \} \\{} & {} \le \int _{0} ^{\infty } \int _{0} ^{\infty }\int _{0} ^{\infty }\exp (-x^2(\gamma _{SD}+\gamma _{min})) f_{\gamma _{SD}}f_{\gamma _{SR}}f_{\gamma _{RD}} \end{aligned}$$

$$\begin{aligned}{} & {} d\gamma _{SD}d\gamma _{SR} d\gamma _{RD}{=} \frac{1}{{\bar{\gamma }}_{SD}\left[ x^2{+}\frac{1}{{\bar{\gamma }}_{SD}}\right] } \int _{0} ^{\infty } \int _{0} ^{\infty }\exp ({-}x^2\gamma _{min})\nonumber \\{} & {} \qquad \times \frac{1}{\bar{\gamma _{SR}}} \exp \left( -\frac{\gamma _{SR}}{\bar{\gamma _{SR}}}\right) \frac{1}{\bar{\gamma _{RD}}} \exp \left( -\frac{\gamma _{RD}}{\bar{\gamma _{RD}}}\right) d\gamma _{SR} d\gamma _{RD}\nonumber \\{} & {} \quad = \frac{1}{\bar{\gamma _{SD}}\left[ x^2+\frac{1}{\bar{\gamma _{SD}}}\right] }~ \frac{1}{x^2\bar{\gamma _{RD}}\bar{\gamma _{SR}}+\bar{\gamma _{SR}}+\bar{\gamma _{RD}}}\nonumber \\{} & {} \qquad \times \frac{x^2\bar{\gamma _{SR}}^2+[x^2+1]\bar{\gamma _{SR}}+\bar{\gamma _{RD}}}{x^2\bar{\gamma _{SR}}+1} \end{aligned}$$

(42)

When $\bar{\gamma }$ tends to infinity, the average of proposition 1 decays with order of two.

Appendix B

Proof of proposition 2

We can claim $P_2<Y+U$, where

$$\begin{aligned} \begin{aligned}&Y=\int _{0} ^{\infty } \int _{0} ^{\infty } \int _{\frac{z\gamma _{min}}{y}} ^{\infty } \frac{1}{2} \exp (-x\gamma _{SR}) \frac{1}{2}\\&\quad \times \exp \left( -\frac{(y\gamma _{SD}-z\gamma _{min})^2}{\gamma _{SD}+\gamma _{min}}\right) \\&\quad \times f_{\gamma _{SD}} f_{\gamma _{SR}}f_{\gamma _{RD}} d\gamma _{SD} d\gamma _{RD} d\gamma _{SR}\\&\quad U=\int _{0} ^{\infty } \int _{0} ^{\infty } \int _{0} ^{\frac{z\gamma _{min}}{y}} \frac{1}{2} \exp (-x\gamma _{SR})\\&\quad f_{\gamma _{SD}} f_{\gamma _{SR}}f_{\gamma _{RD}} d\gamma _{SD} d\gamma _{SR} d\gamma _{RD} \end{aligned} \end{aligned}$$

(43)

For calculating Y we use this inequality.

$$\begin{aligned} \begin{aligned}&\int _{\frac{z\gamma _{min}}{y}} ^{\infty } \exp \left( -\frac{(y\gamma _{SD}-z\gamma _{min})^2}{\gamma _{SD}+\gamma _{min}}\right) \exp \left( -\frac{\gamma _{SD}}{\bar{\gamma _{SD}}}\right) d\gamma _{SD}\le \\&\quad \exp \left( -\gamma _{min}\left[ z^2-\frac{(y+z)^4}{16(y^2+\frac{1}{\bar{\gamma _{SD}}})}\right] \right) \times \frac{\bar{\gamma _{SD}}}{(y^2\bar{\gamma _{SD}}+1)^2}\\&\quad \times \left[ 1+y^2\bar{\gamma _{SD}}+\frac{(y+z)^2}{4}\sqrt{(1+y^2\bar{\gamma _{SD}})\gamma _{min}\bar{\gamma _{SD}}\pi }\right] \end{aligned}\nonumber \\ \end{aligned}$$

(44)

The upper bound of this inequality is drawn using,

$$\begin{aligned} \begin{aligned}&\frac{\left( y\gamma _{SD}-\gamma _{min}\right) ^2}{\gamma _{SD}+\gamma _{min}}\\&\quad \le y^2\gamma _{SD}+z^2\gamma _{min}-\frac{(y+z)^2}{2}\sqrt{\gamma _{SD}\gamma _{min}} \end{aligned} \end{aligned}$$

(45)

After applying this inequality and integrating over different regions, we arrive at

$$\begin{aligned} \begin{aligned}&Y\le \frac{1}{4\bar{\gamma _{SR}}\bar{\gamma _{RD}}(y^2\bar{\gamma _{SD}}+1)}~\bigg [ \frac{\bar{\gamma _{SR}}}{x\bar{\gamma _{SR}}+1}\times \\&\quad \frac{16(y^2\bar{\gamma _{SD}}+1)\bar{\gamma _{SR}}\bar{\gamma _{RD}}}{\bar{\gamma _{SR}}\bar{\gamma _{RD}}(\bar{\eta _{1}\gamma _{SD}}+16\eta _{2})+16(y^2\bar{\gamma _{SD}}+1)[\bar{\gamma _{SR}}+\bar{\gamma _{RD}}]}\\&\quad + \frac{16(y^2\bar{\gamma _{SD}}+1)\bar{\gamma _{SR}}\bar{\gamma _{RD}}}{\eta _{1} \bar{\gamma _{SD}}\bar{\gamma _{SR}}+16(y^2\bar{\gamma _{SD}}+\eta _{2}\bar{\gamma _{SR}}+1)}\bigg ]\\&\quad +\frac{(y+z)^2\pi \sqrt{(1+y^2\bar{\gamma _{SD}})\bar{\gamma _{SD}}}}{32\bar{\gamma _{SR}}(y^2\bar{\gamma _{SD}}+1)^2}\frac{1+\bar{\gamma _{RD}}}{\bar{\gamma _{RD}}}\\&\quad \times \left[ \frac{(y^2\bar{\gamma _{SD}}+1)\bar{\gamma _{SD}}\bar{\gamma _{RD}}}{\bar{\gamma _{SD}}\bar{\gamma _{RD}}(\frac{\eta _{1}}{16}\bar{\gamma _{SD}}+y^2\eta _{2})+(y^2\bar{\gamma _{SD}}+1)[\bar{\gamma _{RD}}+\bar{\gamma _{SD}}]}\right] ^\frac{2}{3} \end{aligned} \end{aligned}$$

(46)

where $\eta _{1}=16y^2(x+z^2)-(y+z)^4$ and $\eta _{2}=x+z^2$.

After calculating the average over different parameters,

$$\begin{aligned} \begin{aligned}&U=\int _{0} ^{\infty } \int _{0} ^{\infty } \int _{0} ^{\frac{z\gamma _{min}}{y}} \frac{1}{2} \exp (-\gamma _{SR} x)\\&\qquad \times \frac{1}{\gamma _{SR}} \exp (-\frac{\gamma _{SR}}{\bar{\gamma _{SR}}}) \frac{1}{\gamma _{RD}} \exp \left( -\frac{\gamma _{RD}}{\bar{\gamma _{RD}}}\right) \\&\qquad \times \frac{1}{\gamma _{SD}} \exp \left( -\frac{\gamma _{SD}}{\bar{\gamma _{SD}}}\right) d\gamma _{SD} d\gamma _{RD} d\gamma _{SR}=\\&\qquad \frac{z\bar{\gamma _{SR}}\bar{\gamma _{RD}}}{2\xi (\bar{\gamma _{SR}}x+1)}, \end{aligned} \end{aligned}$$

(47)

where $\xi =z\bar{\gamma _{SR}}\bar{\gamma _{RD}}+\bar{\gamma _{SD}}[xy\bar{\gamma _{SR}}\bar{\gamma _{RD}}+y\bar{\gamma _{RD}}+y\bar{\gamma _{SR}}]$.

If $\eta _{1}\ne 0$, we can claim that $P_2$ decays with order of two, when $\bar{\gamma }$ tends to infinity.

Appendix C

Proof of proposition 3

Based on (20) and (31), we should note that the possible values for x,y and z are:

$$\begin{aligned} x, y, z:~~~\alpha +\beta , \alpha -\beta , \beta \end{aligned}$$

There are two general classes which require satisfaction of proposition 2 condition. The first class is attributed to the terms in which $y=z$. For this class, the condition of proposition 2 will be reduced to $16xy^2$. Since the coefficients of x and y are positive the condition will be always satisfied.

For the second class, there are two terms which require the satisfaction of the condition $16y^2(x+z^2)-(y+z)^4>0$. The coefficients of the first term are $x=\alpha +\beta$, $y=\alpha -\beta$, $z=\alpha +\beta$. We consider a scenario in which $\alpha =k \beta$. After some calculations, the inequality of proposition 2 can be written as:

$$\begin{aligned} k^3-(1+2\beta )k^2-k+1+\beta >0 \end{aligned}$$

(48)

To find an approximate solution for this inequality, we limit the search region by considering the equation below,

$$\begin{aligned} k^3-(1+2\beta )k^2-k>0 \end{aligned}$$

(49)

If the variable k satisfies (49), the equation (48) will be satisfied too. By solving this inequality, the valid value for k can be calculated as ($k>0$):

$$\begin{aligned} k>\frac{(2\beta +1)+\sqrt{(2\beta +1)^2+4}}{2} \end{aligned}$$

(50)

Based on power assignment constraint ($\alpha ^2+\beta ^2=1$), $\beta$ can be computed as,

$$\begin{aligned} \beta =\sqrt{\frac{1}{1+k^2}} \end{aligned}$$

By applying a recursive calculation with initial value of $\beta =1$, the approximate solution of (49) will be,

$$\begin{aligned} k=2.2547,~ \beta =0.4054 \end{aligned}$$

The coefficients of the second term are $x=\alpha -\beta$, $y=\alpha -\beta$, $z=\alpha +\beta$. By substituting the coefficients, the resulting inequality will be,

$$\begin{aligned} k^3-(3-2\beta )k^2+3k-1+\beta >0 \end{aligned}$$

(51)

Since $k>1$, $k-1+\beta$ is positive; we can use the inequality below,

$$\begin{aligned} k^3-(3-2\beta )k^2+2k>0 \end{aligned}$$

(52)

The delta function of this inequality is,

$$\begin{aligned} \delta =(3-2\beta )^2-8 \end{aligned}$$

Hence, if $k>2.2547$, both of (48) and (51) inequalities will be true.