Enlarged Achievable Rate Region of NOMA by CSC Without SIC

Abstract

Non-orthogonal multiple access (NOMA) is a multiple access scheme using superposition coding (SC) and successive interference cancellation (SIC). Recently, a lossless NOMA implementation without SIC was proposed using correlated SC (CSC), namely, the CSC/non-SIC NOMA scheme. A key feature of the CSC/non-SIC NOMA scheme is that the correlation coefficient of transmitted signals can be adjusted at the base station. This adjustability of the correlation coefficient is fully exploited in this study. We demonstrate that the achievable rate region of the CSC/non-SIC NOMA scheme is larger than that of the conventional independent SC (ISC)/SIC NOMA scheme. In addition, we show that the CSC/non-SIC NOMA scheme outperforms the orthogonal multiple access (OMA) scheme, even when the channel gains are equal.

1. Introduction

One of the most promising multiple access (MA) techniques, non-orthogonal multiple access (NOMA), has been extensively investigated for fifth-generation (5G) mobile networks [1,2]. Independent superposition coding (ISC) and successive interference cancellation (SIC), originally proposed for broadcast channels [3,4], have been implemented within the NOMA framework [5,6]. Sharing channel resources can lead to improved spectral efficiency compared to orthogonal multiple access (OMA) [7]. Reduced access delays and massive connectivity can be advantages of NOMA over OMA [8]. State-of-the-art advances in NOMA include studies on the max-min fairness for uplink finite blocklength (FBL)-NOMA [9], the total transmit power minimization for a simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS)-aided full-duplex NOMA scheme [10], and the performance analysis of NOMA-assisted ambient backscatter communication (AmBC) with RIS [11]. Advanced studies on NOMA include a novel cooperative NOMA based on precoded faster-than-Nyquist (FTN) signaling [12], a joint design of RIS and rate splitting (RS) [13], and an uplink Internet of Things (IoT) network in NOMA [14]. Optimal power control was studied based on individual QoS constraints [15], while energy harvesting for machine-to-machine (M2M) communications has also been investigated [16]. Recently, the significance of SIC in NOMA has been emphasized [17,18].

Moreover, NOMA implementation without SIC with a tolerable loss of the achievable rate region, namely, the non-SIC NOMA scheme, has also been investigated [19]. Specifically, a discrete-input lattice-based NOMA scheme was considered for non-SIC decoding. A non-SIC decoder that treats other user signals as interference was proposed as an alternative to a conventional SIC decoder. In addition, a lossless NOMA implementation without SIC was proposed using correlated superposition coding (CSC), namely, the CSC/non-SIC NOMA scheme [20]. A key feature of the CSC/non-SIC NOMA scheme is that the correlation coefficient of the transmitted signals can be adjusted at the base station. In this study, we enlarged the achievable rate region of NOMA by fully exploiting the adjustability of the correlation coefficient. We also demonstrate that the CSC/non-SIC NOMA scheme outperforms the OMA scheme, even when the channel gains are equal. Interference of CSC is unknown to receivers in downlink NOMA networks. In addition, CSC employs a similar approach to that of cooperative jamming for secure unmanned aerial vehicle (UAV) communications, in that both schemes introduce friendly interference [21]. Also, in Appendix A, we explains why non-SIC is needed.

This paper’s contribution is the increased attainable rate. The contributions of this study are summarized as follows. First, we expanded the achievable rate region of NOMA; that is, we proved that the achievable rate region of the CSC/non-SIC NOMA scheme is larger than that of the conventional ISC/SIC NOMA scheme. Second, when the channel gains are equal, the achievable rate region of OMA is the same as that of conventional ISC/SIC NOMA. However, we demonstrate that even when the channel gains are equal, the achievable rate region of the CSC/non-SIC NOMA scheme is larger than that of the OMA scheme. There are no studies that analyze the rate regions of the CSC/non-SIC NOMA scheme.

The remainder of this paper is organized as follows. In Section 2, the system and channel model are described. The main results are given in Section 3. The numerical results are presented and discussed in Section 4. Finally, the conclusions are presented in Section 5.

Notation: The superscript * represents a complex conjugate. $E[·]$ denotes the expectations. $CN(\mu ,{\sigma }^2)$ represents the distribution of a circularly symmetric complex Gaussian (CSCG) random variable (RV) with mean $\mu$ and variance ${\sigma }^2$. $I(·|·)$ refers to the conditional mutual information. $h(·|·)$ represents the entropy [22].

2. System Model

Consider a cellular downlink NOMA network with one base station and two users. The complex channel coefficient between the mth user and the base station is denoted by $h_m$, $m=1,2$ [1]. The channel gains are sorted as $\left|h_1\right|\ge \left|h_2\right|$ (if there is no other word, we will use this inequality). The base station sends the superimposed signal $z=\sqrt{P_A{\beta }_1}{c}_1+\sqrt{P_A{\beta }_2}{c}_2$, for CSC (we use $x=\sqrt{P{\alpha }_1}{s}_1+\sqrt{P{\alpha }_2}{s}_2$, for ISC), where $c_m$ is the signal for the mth user for CSC (we use $s_m$ for ISC), ${\beta }_m$ is the power allocation coefficient for CSC (we use ${\alpha }_m$ for ISC), ${\beta }_1+{\beta }_2=1$, $({\alpha }_1+{\alpha }_2=1)$, and $P_A$ is the total average allocated power, for a given total average transmitted power P at the base station. The power of signal $c_m$ (or $s_m$) for the mth user is normalized as the unit power, $E\left[{\left|c_m\right|}^2\right]=E\left[{\left|s_m\right|}^2\right]=1$. ${\rho }_{1,2}=E\left[c_1{c}_2^{*}\right]$ is the correlation coefficient. Note that ${\rho }_{1,2}$ is complex. However, it is assumed to be real for complexity reduction and WLOG. Signals are generated as follows: we generate the signals with each element independently and identically distributed (i.i.d.) according to a jointly Gaussian distribution with correlation coefficient ${\rho }_{1,2}$. It should be noted that for ISC, ${\rho }_{1,2}=0$. Due to the correlation of the CSC, the power of the superimposed signal z is not equal to that of x. Thus, given P at the base station, $P_A$ is effectively scaled as follows:

$$P_A={\displaystyle \frac{P}{\left(1+2{\rho }_{1,2}\sqrt{{\beta }_1{\beta }_2}\right)}},$$

(1)

where the given total average transmitted power P of z is calculated as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & P=E\left[{\left|z\right|}^2\right]\hfill \\ \hfill & =P_A\left(1+2{\rho }_{1,2}\sqrt{{\beta }_1{\beta }_2}\right).\hfill \end{array}\end{array}$$

(2)

For ISC, ${\rho }_{1,2}=0$ and $P_A=P$, as in the conventional ISC/SIC NOMA system. Observations $r_m$ and $y_m$ at the mth user for ISC x and CSC z, respectively, are given as follows:

$$\begin{array}{c}{r}_m=h_{m}x+n_m,\\ y_m=h_{m}z+n_m,\end{array}$$

(3)

where $n_m$ is the complex additive white Gaussian noise (AWGN) of the mth user, $n_m\sim CN(0,{\sigma }^2)$. The conditional achievable rate region of the conventional ISC/SIC NOMA scheme given $\left|h_1\right|$ and $\left|h_2\right|$ is expressed as follows [2]:

$$R_{1\left|\right|h_1|}={log}_2\left(1+{\displaystyle \frac{|h_1{|}^{2}P{\alpha }_1}{{\sigma }^2}}\right),$$

(4)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & R_{2|\left|h_2\right|}={log}_2\left(1+{\displaystyle \frac{{\left|h_2\right|}^{2}P{\alpha }_2}{{\left|h_2\right|}^{2}P{\alpha }_1+{\sigma }^2}}\right)\hfill \\ \hfill & ={log}_2\left({\displaystyle \frac{{\left|h_2\right|}^{2}P+{\sigma }^2}{{\left|h_2\right|}^{2}P{\alpha }_1+{\sigma }^2}}\right).\hfill \end{array}\end{array}$$

(5)

For any channel fading model (such as Rayleigh, Rician, or Nakagami-m fading), most performance measures such as the ergodic sum rate, outage probability, and achievable rate region are calculated based on Equations (4) and (5). For example, the achievable rate region of the conventional ISC/SIC NOMA scheme is given by

$$R_1=E\left[R_{1|\left|h_1\right|}\right],$$

(6)

and

$$R_2=E\left[R_{2|\left|h_2\right|}\right],$$

(7)

Therefore, if we enlarge the conditional achievable rate region given $\left|h_1\right|$ and $\left|h_2\right|$, we can also enlarge the achievable rate region for any fading channel model.

The mechanism of CSC is explained as follows: we generate the signals with each element independently and identically distributed (i.i.d.) according to a jointly Gaussian distribution with correlation coefficient ${\rho }_{1,2}$, and the base station transmits the superimposed signal.

$R_{m|\left|h_n\right|}^{\left(\mathrm{a}\mathrm{given}\mathrm{NOMA}\mathrm{scheme}\right)}$ represents the conditional achievable data rate given $\left|h_n\right|$ for a given NOMA scheme in the superscript and the mth user over the nth channel with $\left|h_n\right|$ in the subscript. (The default for the NOMA scheme is ISC/SCI.) Using this notation, the conditional achievable data rate $R_{1|\left|h_1\right|}^{(\mathrm{CSC}/\mathrm{non}-\mathrm{SIC})}$ given $\left|h_1\right|$ for CSC z is given as [20]

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & R_{1|\left|h_1\right|}^{(\mathrm{CSC}/\mathrm{non}-\mathrm{SIC})}=I(y_1;c_1)\hfill \\ \hfill & \hspace{1em}\hspace{1em}\hspace{1em}=h\left(y_1\right)-h\left(y_1\right|c_1)\hfill \\ \hfill & \hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{c}\hfill =h(h_{1}z+n_1)-h(h_{1}z+n_1|c_1)\end{array}\hfill \\ \hfill & \hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{c}\hfill =h(h_{1}z+n_1)-h(h_1\sqrt{P_A{\beta }_2}{c}_2+n_1|c_1)\end{array}\hfill \\ \hfill & \hspace{1em}\hspace{1em}\hspace{1em}={log}_2\pi e({\left|h_1\right|}^{2}P+{\sigma }^2)\hfill \\ \hfill & \hspace{1em}\hspace{1em}\hspace{1em}-{log}_2\pi e({\left|h_1\right|}^2{P}_A{\beta }_2\left(1-{\rho }_{2,1}^2\right)+{\sigma }^2)\hfill \\ \hfill & \hspace{1em}\hspace{1em}\hspace{1em}={log}_2\left({\displaystyle \frac{{\left|h_1\right|}^{2}P+{\sigma }^2}{{\left|h_1\right|}^2{P}_A{\beta }_2\left(1-{\rho }_{2,1}^2\right)+{\sigma }^2}}\right),\hfill \end{array}\end{array}$$

(8)

where in the fourth equality, $h_1\sqrt{P_A{\beta }_1}{c}_1$ is removed, and the fifth equality uses the conditional variance. Similarly, the conditional achievable data rate $R_{2|\left|h_2\right|}^{(\mathrm{CSC}/\mathrm{non}-\mathrm{SIC})}$ given $\left|h_2\right|$ for CSC z is expressed as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill R_{2|\left|h_2\right|}^{(\mathrm{CSC}/\mathrm{non}-\mathrm{SIC})}={log}_2\left({\displaystyle \frac{{\left|h_2\right|}^{2}P+{\sigma }^2}{{\left|h_2\right|}^2{P}_A{\beta }_1(1-{\rho }_{1,2}^2)+{\sigma }^2}}\right).\end{array}\end{array}$$

(9)

3. Main Results

In this section, we present the main results.

Theorem 1.

For the weakest channel gain user, the following equation is obtained:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill R_{2|\left|h_2\right|}=R_{2|\left|h_2\right|}^{(\mathrm{CSC}/\mathrm{non}-\mathrm{SIC})},\mathrm{for}0\le {\alpha }_1\le (1-{\rho }_{1,2}^2).\end{array}\end{array}$$

(10)

Proof of Theorem 1. 

By inspection, from Equations (5) and (9), with $P{\alpha }_1=P_A{\beta }_1(1-{\rho }_{1,2}^2),$ the equality holds true, and we explain the interval $0\le {\alpha }_1\le (1-{\rho }_{1,2}^2)$. To obtain the maximum value of ${\alpha }_1=(1-{\rho }_{1,2}^2)$, we set ${\beta }_1=1$, i.e., the maximum value. Then,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & P_A={\displaystyle \frac{P}{1+2{\rho }_{1,2}\sqrt{{\beta }_1{\beta }_2}}}\hfill \\ \hfill & ={\displaystyle \frac{P}{1+2{\rho }_{1,2}\sqrt{1·0}}}\hfill \\ \hfill & =P,\hfill \end{array}\end{array}$$

(11)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & P{\alpha }_1=P_A{\beta }_1(1-{\rho }_{1,2}^2)\hfill \\ \hfill & P{\alpha }_1=P·1·(1-{\rho }_{1,2}^2)\hfill \\ \hfill & {\alpha }_1=(1-{\rho }_{1,2}^2).\hfill \end{array}\end{array}$$

(12)

Therefore, we obtain the maximum value ${\alpha }_1=(1-{\rho }_{1,2}^2)$ so that the interval $0\le {\alpha }_1\le (1-{\rho }_{1,2}^2)$ can be calculated. □

Thus, we did not increase the user-2 rate because the rates of user-2 were the same, and we only increased the user-1 rate.

Our objective is to prove the following inequality:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill R_{1|\left|h_1\right|}^{(\mathrm{CSC}/\mathrm{non}-\mathrm{SIC})}>R_{1|\left|h_1\right|}.\end{array}\end{array}$$

(13)

Theorem 2.

If the condition

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\rho }_{1,2}>{\rho }_{1,2}^{\left(\mathrm{Enlarge}\right)},\end{array}\end{array}$$

(14)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\rho }_{1,2}^{\left(\mathrm{Enlarge}\right)}& =1+2{\left({\left|h_1\right|}^{2}P/{\sigma }^2\right)}^{-1}\hfill \\ \hfill & -\sqrt{4{\left({\left|h_1\right|}^{2}P/{\sigma }^2\right)}^{-1}+4{\left({\left|h_1\right|}^{2}P/{\sigma }^2\right)}^{-2}},\hfill \end{array}\end{array}$$

(15)

is satisfied, then we have the following inequality:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill R_{1|\left|h_1\right|}^{(\mathrm{CSC}/\mathrm{non}-\mathrm{SIC})}>R_{1|\left|h_1\right|}.\end{array}\end{array}$$

(16)

Proof of Theorem 2. 

From Theorem 1, with $P{\alpha }_1=P_A{\beta }_1(1-{\rho }_{1,2}^2),$ we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & R_{1|\left|h_1\right|}^{(\mathrm{CSC}/\mathrm{non}-\mathrm{SIC})}>R_{1|\left|h_1\right|}\hfill \\ \hfill & ={log}_2\left({\displaystyle \frac{{\left|h_1\right|}^2{P}_A{\beta }_1\left(1-{\rho }_{1,2}^2\right)+{\sigma }^2}{{\sigma }^2}}\right).\hfill \end{array}\end{array}$$

(17)

Let $f\left({\beta }_1\right)$ be as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill f\left({\beta }_1\right)=R_{1|\left|h_1\right|}^{(\mathrm{CSC}/\mathrm{non}-\mathrm{SIC})}-{log}_2\left({\displaystyle \frac{{\left|h_1\right|}^2{P}_A{\beta }_1\left(1-{\rho }_{1,2}^2\right)+{\sigma }^2}{{\sigma }^2}}\right).\end{array}\end{array}$$

(18)

Solving the equation $f\left({\beta }_1\right)=0$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill f\left({\beta }_1\right)=a{\beta }_1^2+b{\beta }_1+c=0,\end{array}\end{array}$$

(19)

where

$$\begin{array}{c}\hfill \begin{array}{c}\hfill a=1,\end{array}\end{array}$$

(20)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill b=-1,\end{array}\end{array}$$

(21)

and

$$c={\left({\displaystyle \frac{-2{\sigma }^2{\rho }_{1,2}\left(1-{\rho }_{1,2}^2\right)\pm \sqrt{B}}{2\left(P{\left|h_1\right|}^2{\left(1-{\rho }_{1,2}^2\right)}^2-4{\sigma }^2{\rho }_{1,2}^2\right)}}\right)}^2.$$

(22)

where $B=4{\sigma }^4{\rho }_{1,2}^2{\left(1-{\rho }_{1,2}^2\right)}^2+4\left(P{\left|h_1\right|}^2{\left(1-{\rho }_{1,2}^2\right)}^2-4{\sigma }^2{\rho }_{1,2}^2\right){\sigma }^2{\rho }_{1,2}^2$. The reason to obtain the double root is that the two curves want to meet. To obtain the double root of $f\left({\beta }_1\right)=0$, we should have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\beta }_1={\displaystyle \frac{-b}{2a}}={\displaystyle \frac{1}{2}}.\end{array}\end{array}$$

(23)

Finally, by substituting ${\beta }_1=1/2$ into $f\left({\beta }_1\right)=0$, after some algebraic calculations, we obtain Equation (15). □

4. Numerical Results and Discussions

First, we depict (Figure 1) the conditional enlarged achievable rate region given $\left|h_1\right|$ and $\left|h_2\right|$, which was proven analytically in Section 3. The channel gains $\left|h_1\right|$ and $\left|h_2\right|$ are assumed to be $\sqrt{2}$ and $0.1$, respectively, and the average total transmitted signal-to-noise power ratio (SNR) is $P/{\sigma }^2=50$. Using Theorem 2, we obtain the minimum value ${\left({\rho }_{1,2}^{\left(\mathrm{Enlarge}\right)}\right)}^2=0.67$ of the correlation coefficient squared by the CSC design. We chose ${\rho }_{1,2}^2=0.68$ because we wanted to accurately verify the analysis. $I_1$ and $I_2$ are the achievable data rates of user-1 and user-2, respectively. This is in good agreement with the results of the analysis. Specifically, these simulations have were as follows: Prepare the vector $\overline{\alpha }=0.0001\times \left[0,1,2,\dots 9999,10000\right]$. Calculate the vectors ${\overline{R}}_{1|\left|h_1\right|}^{(\mathrm{CSC}/\mathrm{non}-\mathrm{SIC})}$, ${\overline{R}}_{2|\left|h_2\right|}^{(\mathrm{CSC}/\mathrm{non}-\mathrm{SIC})}$, ${\overline{R}}_{1|\left|h_1\right|}$, and ${\overline{R}}_{2|\left|h_2\right|}$. Plot $\left({\overline{R}}_{1|\left|h_1\right|}^{(\mathrm{CSC}/\mathrm{non}-\mathrm{SIC})},{\overline{R}}_{2|\left|h_2\right|}^{(\mathrm{CSC}/\mathrm{non}-\mathrm{SIC})}\right)$ and $\left({\overline{R}}_{1|\left|h_1\right|},{\overline{R}}_{2|\left|h_2\right|}\right)$. As shown Figure 1, the conditional enlarged achievable rate region of the CSC/non-SIC NOMA scheme is much larger than that of the conventional ISC/SIC NOMA scheme over the entire power allocation region. In addition, as we mentioned in Section 2, if we enlarge the conditional achievable rate region given $\left|h_1\right|$ and $\left|h_2\right|$, then we can enlarge the achievable rate region for any fading channel model. Thus, we now present the simulation results of an enlarged achievable rate region under a fading channel model, that is, the Rayleigh fading channel model. For this, it is assumed that the channels are Rayleigh faded with $E\left[{\left|h_1\right|}^2\right]=2$ and $E\left[{\left|h_2\right|}^2\right]=0.01$. In this simulation, the number of independently generated channel gains $\left|h_1\right|$ and $\left|h_2\right|$ are 10000. The correlation coefficient squared is ${\rho }_{1,2}^2={\left({\rho }_{1,2}^{\left(\mathrm{Enlarge}\right)}\right)}^2+0.01$. Figure 2 depicts the enlarged achievable rate region under the Rayleigh fading channel model. It can be seen that the simulation results are entirely consistent with our analysis. Thus, the achievable rate region of the CSC/non-SIC NOMA scheme is larger than that of the conventional ISC/SIC NOMA scheme. Therefore, we enlarged the achievable rate region of the conventional ISC/SIC NOMA scheme.

In fact, the upper bounds are on those that are really achievable. In most of the papers considering SIC applied in NOMA, it is assumed that the SIC process is ideal in the sense that all the data symbol decisions used in interference cancellation are correct, which is a perfect SIC system. In reality, however, this is not true. This is an additional reason why the results presented (at least for SIC-based methods) are the upper bound of what can be achieved in practice. We emphasize that the results in this paper are the upper bound of real performance.

Second, we show that when the channel gains are equal, that is, $\left|h_1\right|=\left|h_2\right|$, the achievable rate region of OMA is the same as that of conventional ISC/SIC NOMA, which is as follows: For OMA, we assume a time division multiple access (TDMA) scheme. Then, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & R_{1|\left|h_1\right|}^{\left(\mathrm{OMA}\right)}+R_{2|\left|h_2\right|}^{\left(\mathrm{OMA}\right)}\hfill \\ \hfill & =\tau {log}_2\left(1+{\displaystyle \frac{{\left|h_1\right|}^{2}P}{{\sigma }^2}}\right)+\left(1-\tau \right){log}_2\left(1+{\displaystyle \frac{{\left|h_1\right|}^{2}P}{{\sigma }^2}}\right)\hfill \\ \hfill & ={log}_2\left(1+{\displaystyle \frac{{\left|h_1\right|}^{2}P}{{\sigma }^2}}\right),\hfill \end{array}\end{array}$$

(24)

where $\tau$ is the fraction of time required for $0\le \tau \le 1$. For conventional ISC/SIC NOMA, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & R_{1|\left|h_1\right|}+R_{2|\left|h_2\right|}\hfill \\ \hfill & ={log}_2\left(1+{\displaystyle \frac{{\left|h_1\right|}^{2}P{\alpha }_1}{{\sigma }^2}}\right)+{log}_2\left(1+{\displaystyle \frac{{\left|h_1\right|}^{2}P{\alpha }_2}{{\left|h_1\right|}^{2}P{\alpha }_1+{\sigma }^2}}\right)\hfill \\ \hfill & ={log}_2\left(1+{\displaystyle \frac{{\left|h_1\right|}^{2}P}{{\sigma }^2}}\right).\hfill \end{array}\end{array}$$

(25)

It should be noted that Equations (24) and (25) are the same straight line. Therefore, when the channel gains are equal, the achievable rate region of OMA is the same as that of conventional ISC/SIC NOMA. Thus, if the condition ${\rho }_{1,2}>{\rho }_{1,2}^{\left(\mathrm{Enlarge}\right)}$ in Equation (14) is satisfied, CSC/non-SIC NOMA outperforms OMA and conventional ISC/SIC NOMA. To demonstrate that the CSC/non-SIC NOMA scheme outperforms the OMA scheme even when the channel gains are equal, we compare the conditional achievable rate region of the CSC/non-SIC NOMA scheme with that of the OMA scheme, which is the same as that of the conventional ISC/SIC NOMA scheme. Conditions $\left|h_1\right|$ and $\left|h_2\right|$ are assumed to be $\left|h_1\right|=\left|h_2\right|=1$ and ${\rho }_{1,2}^2=0.58$, which is slightly larger than the minimum value ${\left({\rho }_{1,2}^{\left(\mathrm{Enlarge}\right)}\right)}^2=0.57$. As shown in Figure 3, the CSC/non-SIC NOMA scheme outperforms the OMA scheme, even when the channel gains are equal.

Similarly to the case of unequal channel gains, we added Monte Carlo simulation results to verify that CSC/non-SIC NOMA outperforms OMA and conventional ISC/SIC NOMA. This confirms the results of the analytical expressions. In addition, simulation results demonstrate that the CSC/non-SIC NOMA scheme outperforms the OMA scheme under the Rayleigh fading channel model when the channel gains are equal. To this end, it is assumed that the channels are Rayleigh faded with $E\left[{\left|h_1\right|}^2\right]=1$ and $\left|h_1\right|=\left|h_2\right|$. As shown in Figure 4, the CSC/non-SIC NOMA scheme outperforms the OMA scheme under the Rayleigh fading channel model when the channel gains are equal. Moreover, we demonstrate that when the channel gains are almost equal, that is, $E\left[{\left|h_1\right|}^2\right]=1$ and $E\left[{\left|h_2\right|}^2\right]=1$, the CSC/non-SIC NOMA scheme outperforms the conventional ISC/SIC NOMA scheme under the Rayleigh fading channel model. In Figure 5, we depict the enlarged achievable rate region under the Rayleigh fading channel model when the channel gains are almost equal. Thus, the simulation results agree well with the results of our analyses.

5. Conclusions

In this study, we proved that the achievable rate region of the CSC/non-SIC NOMA scheme is larger than that of the conventional ISC/SIC NOMA scheme. We analytically showed the enlarged achievable rate region using NOMA and demonstrated results obtained via Monte Carlo simulation. In addition, we showed that the CSC/non-SIC NOMA scheme outperforms OMA and the conventional ISC/SIC NOMA scheme, even when the channel gains are equal. As a direction for future research, it would be significant to study UAV-related CSC/non-SIC NOMA.