Non-Orthogonal Multiple Access with Spatial Modulation in Downlink Coordinated Multipoint Transmission

TL;DR

This paper introduces a novel joint transmission NOMA with spatial modulation in downlink coordinated multipoint systems, demonstrating improved cell edge user capacity and ergodic sum capacity over existing schemes, especially with increasing cell numbers.

Contribution

It proposes a new JT-COMP NOMA-SM scheme that enhances capacity and mitigates inter-cell interference, outperforming existing NOMA and VP-NOMA methods under imperfect conditions.

Findings

Proposed system achieves higher CEU capacity than other schemes.

ESC of the proposed system surpasses competing methods.

SM mitigates inter-cell interference beyond Shannon bounds.

Abstract

In this paper, a joint transmission coordinated multi-point based non-orthogonal multiple access (JT-COMP NOMA) combined with spatial modulation (SM), termed as JT-COMP NOMA-SM, is proposed to enhance capacity. User capacity and ergodic sum capacity (ESC) of M number coordinated multi-point base stations (COMP BSs) within N number of cells are analyzed by considering imperfect successive interference cancellation (SIC) and imperfect channel state information (CSI). The performances of the proposed syatem are compared with non-orthogonal multiple access (NOMA), and joint transmission coordinated multi-point combined with virtual user pairing based non-orthogonal multiple access (JT-COMP VP-NOMA) by both simulation and analysis. The results show that the proposed system has the same cell center user (CCU) capacity compared to JT-COMP VP-NOMA and a higher cell edge user (CEU) capacity than the other schemes. ESC of the proposed system outperforms the other schemes due to enhancing CEU capacity. Imperfect SIC and imperfect CSI may degrade capacity. The proposed system can maintain CEU capacity better than the other schemes if the number of cells is increased. It happens because SM works beyond Shannon upper bounds which can mitigate inter-cell interference (ICI).

Figures

Equations

$${\tau }_j$$

$$C_j$$

$${\tau }_k$$

$$C_k$$

$$C_{\text{JT-CoMP NOMA}}^{erg}=C_k+\sum _{j=1}^M{C}_j,$$

$$C_j$$

$$={\log }_2\left(\alpha \rho \sum _{i=1}^M|{\hat{h}}_{ij}{|}^2+\rho \sum _{i=M+1}^N|{\hat{h}}_{ij}{|}^2+\rho \sum _{i=1}^N{\sigma }_{{\epsilon }_{ij}}+\rho {\rm Y}+1\right)$$

$$-{\log }_2\left(\alpha \rho \sum _{\begin{array}{c}i=1\\ i\ne j\end{array}}^M|{\hat{h}}_{ij}{|}^2+\rho \sum _{i=M+1}^N|{\hat{h}}_{ij}{|}^2+\rho \sum _{i=1}^N{\sigma }_{{\epsilon }_{ij}}+\rho {\rm Y}+1\right).$$

$$C_j^{exact}$$

$$={\int }_0^{\infty }{\log }_2\left(x+a\right)f_{X_j}(x)dx$$

$$-{\int }_0^{\infty }{\log }_2\left(y+a\right)f_{Y_j}(y)dy,$$

$$C_j^{\text{exact}}$$

$$-{\int }_0^{\infty }{\log }_2\left(y+a\right)\sum _{\begin{array}{c}i=1\\ i\ne j\end{array}}^M{f}_{Y_{ij}}(y)\prod _{\begin{array}{c}h=1\\ h\ne i\\ h\ne j\end{array}}^N\frac{k_{hj}}{k_{hj}-k_{ij}}dy.$$

$$C_j^{exact}$$

$$\times \prod _{\begin{array}{c}h=1\\ h\ne i\end{array}}^N\frac{k_{hj}}{k_{hj}-k_{ij}}dx$$

$$-{\int }_0^{\infty }{\log }_2\left(y+a\right)\sum _{\begin{array}{c}i=1\\ i\ne j\end{array}}^M{k}_{ij}\exp (-k_{ij}y)$$

$$\times \prod _{\begin{array}{c}h=1\\ h\ne i\\ h\ne j\end{array}}^N\frac{k_{hj}}{k_{hj}-k_{ij}}dy.$$

$$C_j^{exact}$$

$$\times \prod _{\begin{array}{c}h=1\\ h\ne i\end{array}}^N\frac{k_{hj}}{k_{hj}-k_{ij}}$$

$$-\frac{1}{\ln (2)}\sum _{\begin{array}{c}i=1\\ i\ne j\end{array}}^M\left(\ln (a)-\exp (a{k}_{ij})\text{Ei}(-a{k}_{ij})\right)$$

$$\times \prod _{\begin{array}{c}h=1\\ h\ne i\\ h\ne j\end{array}}^N\frac{k_{hj}}{k_{hj}-k_{ij}}.$$

$$C_k^{exact}$$

$$={\int }_0^{\infty }{\log }_2\left(x+b\right)f_{X_k}(x)dx$$

$$-{\int }_0^{\infty }{\log }_2\left(y+b\right)f_{Y_k}(y)dy,$$

$$C_k^{exact}$$

$$\times \prod _{\begin{array}{c}h=1\\ h\ne i\end{array}}^N\frac{l_{hk}}{l_{hk}-l_{ik}}$$

$$-\frac{1}{\ln (2)}\sum _{\begin{array}{c}i=1\end{array}}^M\left(\ln (b)-\exp (b{m}_{ik})\text{Ei}(-b{m}_{ik})\right)$$

$$\times \prod _{\begin{array}{c}h=1\\ h\ne i\end{array}}^N\frac{m_{hk}}{m_{hk}-m_{ik}}.$$

$$C_{\text{JT-CoMP NOMA}}^{exact}=C_k^{exact}+\sum _{j=1}^M{C}_j^{exact},$$

$$C_{k}=\begin{cases}\left(1-P_{e,k}\right)\left\lfloor log_{2}\left(\sum\limits_{i=1}^{2}T_{Xi}\right)\right\rfloor&\mbox{, for $k=2$},\\ \left(1-P_{e,k}\right)\left\lfloor log_{2}\left(T_{X_{k}}\right)\right\rfloor&\mbox{, for $3\leq k\leq M$.}\end{cases}$$

$$C_{SM}^{erg}=\sum _{k=2}^M{C}_k.$$

$$C_{SM}^{exact}=\sum _{k=2}^M{C}_k^{exact}.$$

$$C_{\text{JT-CoMP NOMA-SM}}^{erg}$$

$$=\left(\sum _{j=1}^M{C}_j+C_{k=1}\right)+\sum _{k=2}^M{C}_k$$

$$=C_{\text{JT-CoMP NOMA}}^{erg}+C_{SM}^{erg}.$$

$$C_{\text{JT-CoMP NOMA-SM}}^{exact}=C_{\text{JT-CoMP NOMA}}^{exact}+C_{SM}^{exact}.$$

$$f_{X_{ij}}(x)$$

Full text

††journal: Physical Communication††journal: Physical Communicationt1t1footnotetext: Wireless and Emerging Networks System (WENS) Lab., Dept. of IT Convergence Engineering, Kumoh National Institute of Technology (KIT), 39177, Gumi, South Korea.

1 Introduction

Non-orthogonal multiple access (NOMA) is a multiple access technique considered for further improving the spectral efficiency compared to orthogonal multiple access (OMA) [1-4]. NOMA supports multiple users multiplexed over a particular resource by allocating lower power to cell center user (CCU) than cell edge user (CEU). In addition, CCU uses successive interference cancellation (SIC) to decode own signal by canceling CEU signal, while CEU performs direct decoding without any SIC [3,4].

In downlink multi-cell network, inter-cell interference (ICI) is one of the critical issues because it may degrade the system performance, such as user throughput and cell capacity. Coordinated multi-point (CoMP) is introduced as an ICI mitigation technique that may escalate capacity of NOMA [5-11]. In a conventional CoMP, one CEU is served by joint transmissions of multiple coordinated BSs simultaneously to improve its performance [5-7]. In [7], it exploited conventional CoMP based NOMA scheme by considering both imperfect channel state information (CSI) and imperfect SIC simulated in multi-cell scenarios. However, conventional CoMP has a limitation that only can handle one CEU within $M$ number of coordinated multi-point base stations (CoMP BSs).

To handle multiple CEUs, several multiple CoMP-user techniques have been developed [8-11]. In [11], a joint transmission coordinated multi-point combined with virtual user pairing based non-orthogonal multiple access (JT-CoMP VP-NOMA) was proposed to handle multiple CEUs paired with one CCU. Evaluating JT-CoMP VP-NOMA in [11], it assumed a cluster contains a number of both cells and users have been predetermined. JT-CoMP VP-NOMA may not be efficient while the cluster contains a lot of cells and/or each cell contains a lot of CEUs [12]. Considering each cell contains two users, it may degrade CEU capacity because both bandwidth and BS power are divided by one CCU and $M$ number of CEUs, although each CEU receives signals from $M$ number of CoMP BSs. However, the distribution of users in a cell is random in practice; the number of CCU is equal to CEU or the number of CCU may be more/less than CEU.

Regarding the issues, spatial modulation (SM) technique is considered may handle multiple CCUs and/or multiple CEUs efficiently [13-17]. Considering conventional CoMP can handle $M$ number of CCUs and one CEU within $M$ number of CoMP BSs, SM is utilized to handle the other users that cannot be served by using conventional CoMP [16,17]. In this paper, conventional CoMP is used because power allocated for CEU can be optimized while the number of CoMP BSs is increased. By combining conventional CoMP and SM, it is expected may enhance user capacity. An opportunity for further improve user capacity appears, by coordinating SM antennas to transmit a specific user, because CoMP and SM techniques are combined simultaneously.

The main contributions of this paper are listed as follows:

• $\bullet$

This paper proposes JT-COMP NOMA-SM to handle random user distribution based on conventional CoMP. It is expected may improve user capacity by mitigating ICI compared to both NOMA and JT-COMP VP-NOMA.

• $\bullet$

By using CoMP technique, this paper proposes coordinated SM (CoSM) for further improve user performance.

• $\bullet$

This paper conducts a mathematical analysis of the proposed system.

The rest of this paper is organized as follows: Section 2 presents system model of the proposed JT-CoMP NOMA-SM. Section 3 conducts a closed-form of ergodic sum capacity (ESC) for the proposed system. Then, section 4 provides simulation result and analysis. Finally, section 5 concludes the overall of this paper.

2 System Model

This paper considers $M$ number of CoMP BSs within $N$ number of cells. To keep in line with previous work for comparison, each cell which is coordinated contains one BS, one CCU, and one CEU [11]. Each $i^{th}$ cell represents coverage of each $i^{th}$ BS $(i\in 1,2,...,N)$. Considering $N$ number of cells, $M$ number of cells $(2\leq M\leq N$ and $M\subset N)$ are coordinated while the other cells do not apply coordination. Considering $M$ number of cells, the $j^{th}$ CCU is represented with $CCU_{j}$ $(j\in 1,2,...,M)$, whereas the $k^{th}$ CEU is expressed with $CEU_{k}$ $(k\in 1,2,...,M)$. In this paper, user positions are generated based on the random user algorithm. Furthermore, $r_{ij}$ and $r_{ik}$ represent the $j^{th}$ CCU and the $k^{th}$ CEU distances from the $i^{th}$ BS ground, respectively. The maximum cell radius from the BS ground is normalized by $R=1$. In this case, BS antenna high and BS distances to the other BSs are assigned by $0.01R$ and $2R$, respectively. Moreover, $d_{ij}$ and $d_{ik}$ can be calculated, as the $j^{th}$ CCU and the $k^{th}$ CEU distance from the $i^{th}$ BS antenna respectively, using the concept of trigonometry [7,11].

Considering $M$ number of CoMP BSs, the proposed JT-CoMP NOMA-SM can be divided into JT-CoMP NOMA and SM systems as shown in Fig. 1. In JT-CoMP NOMA system, $M$ number of CCUs ($CCU_{1}$, $CCU_{2}$,…,$CCU_{M}$) and a CEU are assigned as NOMA users. In this paper, the first CEU ($CEU_{1}$) is assigned as a CoMP user, while all CCUs do not apply CoMP technique. In SM system, other CEUs ($CEU_{2},...,CEU_{M}$), which are not included in the JT CoMP NOMA system, are assigned as SM users.

2.1 JT-CoMP NOMA system

In this system, a conventional JT-CoMP NOMA is applied for $M$ number of CoMP BSs within $N$ number of cells. This paper assumes CoMP user, i.e. $CEU_{k}$ for $k=1$, is transmitted by multiple coordinated BSs simultaneously. It paired with non-CoMP users, i.e. $CCU_{j}$ for $1\leq j\leq M$, in each cell.

In this paper, each $j^{th}$ CCU is allocated the same resource as CoMP user, where total bandwidth of each BS is normalized by $B=1$. Total power of each BS is normalized by $P=1$. Power allocation factor for CCU and CEU are assigned by $\alpha$ and $\beta$, respectively. This paper assumes $\alpha=0.1$, while $\beta$ can be calculated with 1-$\alpha$. Then, $P_{j}=\alpha P$ and $P_{k}$=$\beta P$ are assigned as power allocation for CCU and CoMP user respectively, where $k=1$ is assigned for each $k$ symbol mentioned in JT-CoMP NOMA system.

In this paper, imperfect SIC and imperfect CSI are considered in the simulation parameters as a representation of real condition [7,11]. For simplicity, links between each antenna in a BS to a user are assumed to have the same channel characteristic. Imperfect CSI is modeled with channel estimation error, where a priory of a variance of the error estimation is known. The channel estimation error for the link between the $i^{th}$ BS to the $j^{th}$ CCU can be modeled by $h_{\varepsilon ij}$=$h_{ij}$-${\hat{h}}_{ij}$ [7,11]. In addition, channel estimation error for the link between the $i^{th}$ BS to CoMP user can be written as $h_{\varepsilon ik}$=$h_{ik}$-${\hat{h}}_{ik}$. It is assumed channel over each link is independent Rayleigh flat fading with channel coefficients $h_{\varepsilon ij}\sim CN(0,\sigma_{\varepsilon ij})$, and $h_{\varepsilon ik}\sim CN(0,\sigma_{\varepsilon ik})$. Furthermore, received signals for the $j^{th}$ CCU and CoMP user need consider a channel estimation gain $|\hat{h}_{ij}|^{2}$ and $|\hat{h}_{ik}|^{2}$, respectively. Therefore, the channel estimation characteristic for the $j^{th}$ CCU and CoMP user are assumed to be distributed independently with mean zero, which can be modeled as $\hat{h}_{ij}\sim CN(0,\hat{\sigma}_{ij}=d_{ij}^{-v}-\sigma_{\varepsilon ij})$, and $\hat{h}_{ik}\sim CN(0,\hat{\sigma}_{ik}=d_{ik}^{-v}-\sigma_{\varepsilon ik})$, respectively. In this case, $\hat{\sigma}_{ij}$ ̵̂and $\hat{\sigma}_{ik}$ represent estimation variance for the link from the $i^{th}$ BS antenna to the $j^{th}$ CCU and CoMP user, respectively, where $v$ represents the path-loss exponent. For simplicity, the variance of each channel estimation error is assumed fixed.

2.2 SM system

Considering $M$ number of CoMP BSs, conventional CoMP already applied to all CCUs ($CCU_{1}$,$CCU_{2}$,…,

$CCU_{M}$) and one CEU ($CEU_{1}$) in this case, then the other users ($CEU_{2}$,…,$CEU_{M}$) are not allocated bandwidth by BS. Furthermore, SM users, i.e. $CEU_{k}$ for $2\leq k\leq M$, are handled by using SM technique.

For simplicity, each CoMP BS is assumed to have two antennas $(T_{Xi}=2)$. $T_{Xi}$ represents the number of the $i^{th}$ BS antenna. Considering SM based space shift keying (SSK), it activates an antenna to transmit frequency for NOMA users while another antenna is utilized as an index modulation [15,16]. It is important to be noted that generalized SSK (GSSK) is required by putting more BS antennas while SM users require more capacity or $K$ number of users are considered [17,19].

In this paper, both the first BS and the second BS coordinate to transmit information to one of the SM users (CoSM user), i.e. $CEU_{k}$ for $k=2$, by using the antenna index. In addition, the other SM users are called as non-coordinated SM (non-CoSM) users, i.e. $CEU_{k}$ for $3\leq k\leq M$. Each $k^{th}$ non-CoSM gets information from the $i^{th}$ BS, where $i=k$, by using its antenna index. Table 1 explains that the number of antenna index combinations will increase if the first and the second BS perform CoSM. As a result of combination, CoSM user has four antenna combinations that each antenna may convey two input bits, while non-CoSM users only has two antenna combinations. It means each antenna of $i^{th}$ BS, for $3\leq i\leq M$, only able to convey bit ”0” or ”1” delivered to non-CoSM users.

In the receiver side, sometimes users cannot estimate information modulated through its antenna index perfectly. It may decrease user capacity, which is represented by a probability of error ($P_{e}$) [20-21]. This paper assumes $P_{e}$ only comes from the error estimation of binary phase-shift keying (BPSK) constellation symbol, while SM users are assumed may detect its antenna index perfectly.

3 Ergodic Sum Capacity (ESC)

3.1 ESC of JT-CoMP NOMA system

In this section, ESC of JT-CoMP NOMA system is determined through a mathematical analysis based on Shannon formula. Considering $N$ number of cells, this paper analyzes ESC for $M$ number of CoMP BSs, while the other BSs do not apply coordination.

By modifying received SINR in [7], then SINR of the $j^{th}$ CCU for $1\leq j\leq M$ can be derived as

[TABLE]

In equation (1), CCU suffers ICI from non-CoMP BSs, and ICI from CoMP BSs caused by power allocated to CCU. Considering JT-CoMP NOMA technique, CCU may cancel CoMP user signal by using SIC when the other BSs transmit the same data to CoMP user. Furthermore, imperfect CSI and imperfect SIC should be considered while users cannot estimate its channel and CCU cannot decode its signal perfectly. Then, SINR of the $j^{th}$ CCU for $1\leq j\leq M$ can be derived as

[TABLE]

The achievable data rate of the $j^{th}$ CCU for $1\leq j\leq M$ can be written as

[TABLE]

Similarly, by modifying received SINR for CoMP user in [7], SINR of the $k^{th}$ CEU for $k=1$ can be written as

[TABLE]

Based on equation (4), the desired signal of CoMP user will increase if the number of CoMP BSs is increased, while it has the same interference pattern with CCU as shown in equation (1). Considering imperfect CSI, equation (4) can be written as

[TABLE]

The achievable data rate of the $k^{th}$ CEU for $k=1$ can be written as

[TABLE]

Moreover, ESC of JT-CoMP NOMA system can be expressed as follow

[TABLE]

where $k=1$.

The probability density function (PDF) of CCU capacity, i.e. $C_{j}^{exact}$, can be determined based on equation (3). By using $\log_{n}(1+x/y)=\log_{n}\left[(x+y)/y\right]$ and $\log_{n}(x/y)=\log_{n}(x)-\log_{n}(y)$, then equation (3) can be simplified as

[TABLE]

Furthermore, PDF of equation (8) for $C_{j}^{exact}$ is given by

[TABLE]

where $E(.)$ is denoted as the expectation operator and $a=\rho\sum_{i=1}^{N}\sigma_{\varepsilon_{ij}}+\rho\gamma+1$. By using equation (23) and (24), the PDF of equation (9) can be determined as

[TABLE]

By substituting equation (21) and equation (22) into equation (10), then $C_{j}^{exact}$ can be written as

[TABLE]

By using $\int_{0}^{\infty}exp(-Ax)ln(B+x)dx$=$\frac{1}{A}\left[ln(B)exp(AB)\right]$ and $log_{2}(x)=\frac{ln(x)}{ln(2)}$, then achievable data rate $C_{j}^{exact}$ can be written as

[TABLE]

The PDF of equation (6) for $C_{k}^{exact}$ is given as

[TABLE]

where $b=\rho\sum_{i=1}^{N}\sigma_{\varepsilon_{ik}}+1$. Similarly, by using equation (21), (22), (23) and (24), achievable data rate for $C_{k}^{\text{exact}}$ can be written as

[TABLE]

Moreover, ESC of JT-CoMP NOMA system can be expressed as follow

[TABLE]

where $k=1$.

3.2 ESC of SM system

In this section, ESC of SM system in $M$ number of CoMP BSs is determined based on the SSK scheme. Probability of error for each $k^{th}$ SM user, i.e. $P_{e,k}$ for $2\leq k\leq M$, is calculated by comparing the number of error information received by SM users, and the number of original information transmitted from its BS. Considering Table 1, given $P_{e,k}$ in Appendix (A.3) and SSK capacity in [15], then achievable data rate for SM users can be determined as

[TABLE]

Therefore, ESC for SM system can be written as

[TABLE]

By substituting equation (38) and equation (39) into equation (16), then exact ESC of SM system can be assigned as

[TABLE]

3.3 ESC of JT-CoMP NOMA-SM system

Considering $N$ number of cells, this section calculates ESC for $M$ number of CoMP BSs, while the other BSs do not apply coordination as explained in Section 2. ESC of the proposed system can be determined by summing the total of both CCU and CEU capacity as follow

[TABLE]

Similarly, exact ESC of the proposed system can be written as

[TABLE]

4 Simulation Result and Analysis

This section analyzes performances of the proposed system such as CCU capacity, CEU capacity, and ESC compared to NOMA and JT-COMP VP-NOMA. For simplicity, these are simulated and analyzed based on a simplified three-CoMP BS ($M=3$) within a twelve-cell model ($N=12$).

Fig. 2 shows that CEU capacity of the proposed system is the highest than the other schemes. CEU capacity of the proposed system increases by 41.3% compared to NOMA at $\rho=20$ dB. It is interesting that JT-CoMP VP-NOMA shows the worst CEU capacity. It happens because JT-CoMP VP-NOMA divides its power to one CCU and three CEUs ($M$=3), while the proposed system uses conventional CoMP and SM simultaneously can optimize its bandwidth and power. On the other hand, the proposed system has the same CCU capacity compared to JT-CoMP VP-NOMA as shown in Fig. 3. It is caused CCU of the proposed system has the same interference pattern as JT-CoMP VP-NOMA. By using JT-CoMP technique of both two schemes compared, CCU can mitigate most of CoMP user signals by using SIC while imperfect SIC is considered.

In this paper, each variance of the error estimation parameter $\sigma_{\varepsilon ij}$ and $\sigma_{\varepsilon ik}$ is assumed to have the same value expressed by $\sigma_{\varepsilon}$. Considering $\sigma_{\varepsilon}=0.01$, Fig. 4 shows that the proposed system has 38.3% and 58.5% higher ESC at $\rho=20$ dB than NOMA and JT-CoMP VP-NOMA, respectively. Based on Fig. 2 and Fig. 3, it is clear that increasing ESC of the proposed system is caused due to enhancing CEU capacity, while CCU capacity can be maintained.

Fig. 5 shows that considering $\gamma=-15$ dB and $\gamma=-25$ dB can degrade 2.7% and 0.28% ESC of the proposed system at $\rho=20$ dB respectively, compared to perfect SIC. On the other hand, by considering $\sigma_{\varepsilon}=0.01$ and $\sigma_{\varepsilon}=0.02$, it may degrade 6.56% and 16.65% ESC of the proposed system respectively at $\rho=20$ dB compared to perfect SIC as shown in Fig. 6. Furthermore, Fig. 5 and Fig. 6 prove that both imperfect SIC and imperfect CSI can lead to additional interference may degrade capacity.

For $N=3$ to $N=12$, CEU capacity of the proposed system decreases 14.42% while NOMA and JT-CoMP VP-NOMA degrade 16.57% and 32.93% respectively as shown in Fig. 7. It proves that the proposed system can maintain CEU capacity better than the other schemes if the number of cells is increased. It is caused SM users do not suffer ICI because SM works beyond Shannon upper bounds. In addition, utilizing CoSM may improve around 1 bit/s/Hz higher CEU capacity than without using CoSM.

5 Conclusion

In this paper, JT-CoMP NOMA-SM is proposed to enhance capacity. The results show that proposed system has the same CCU capacity compared to JT-CoMP VP-NOMA. Considering $\rho=20$ dB, the proposed system has 41.3% higher CEU capacity compared to NOMA. Furthermore, the proposed system has 38.3% and 58.5% higher ESC than NOMA and JT-CoMP VP-NOMA, respectively. Imperfect SIC and imperfect CSI may degrade capacity. The proposed system can maintain CEU capacity better than the other schemes if the number of cells is increased. By utilizing CoSM, CEU capacity increases around 1 bit/s/Hz higher than without using CoSM. Moreover, the proposed system outperforms the other schemes.

For future work, a different method of ICI avoidance and SM techniques can be exploited to achieve higher spectral efficiency.

Appendix A

A.1. Derivation of $f_{X_{j}}(x)$ and $f_{Y_{j}}(y)$ for CCU capacity

Let suppose X_{j}$$\triangleq$$\alpha\rho\sum_{i=1}^{M}|\hat{h}_{ij}|^{2}+\rho\sum\limits_{i=M+1}^{N}|\hat{h}_{ij}|^{2} and Y_{j}$$\triangleq$$\alpha\rho\sum_{i=1,i\neq j}^{M}|\hat{h}_{ij}|^{2}+\rho\sum\limits_{i=M+1}^{N}|\hat{h}_{ij}|^{2}. Considering each $j$ symbol mentioned in this section represents the $j^{th}$ CCU for $1\leq j\leq M$. then PDF of $X_{j}$ can be determined by using the sum of the number of cells. In this case, random variables are assumed independent and identically distributed exponential with different parameters. In addition, $Y_{j}$ can be determined by using the sum of the $i^{th}$ cell, where $i\neq j$. The parameters of exponential random variables are assumed to be different because the distance from each $i^{th}$ BS to the $j^{th}$ CCU is different. The PDF for each exponential random variable can be determined as

[TABLE]

[TABLE]

where $k_{ij}$ represents each parameter of exponential random variables. Then, the PDF of $X_{j}$ and $Y_{j}$ for $1\leq j\leq M$ within $N$ number of cells can be written as

[TABLE]

[TABLE]

Considering $M$ number of CoMP BSs within $N$ number of cells, then $k_{ij}$ can be assigned as

[TABLE]

A.2. Derivation of $f_{X_{k}}(x)$ and $f_{Y_{k}}(y)$ for CEU capacity

Let suppose X_{k}$$\triangleq$$\rho\sum_{i=1}^{N}|\hat{h}_{ik}|^{2} and Y_{k}$$\triangleq$$\alpha\rho\sum_{i=1}^{M}|\hat{h}_{ik}|^{2}+\rho\sum\limits_{i=M+1}^{N}|\hat{h}_{ik}|^{2}. In this section, each $k$ symbol mentioned represents the $k^{th}$ CEU for $k=1$. Similarly, PDF for each exponential random variable can be determined as

[TABLE]

[TABLE]

where $l_{ik}=\frac{1}{(\alpha+\beta)\rho\hat{\sigma}_{ik}}$. In addition, $m_{ik}$ depends on the number of CoMP BSs. Then, PDF of $X_{k}$ and $Y_{k}$ for $k=1$ can be written as (28) and (29), respectively.

[TABLE]

[TABLE]

Considering $M$ number of CoMP BSs within $N$ number of cells, then $m_{ik}$ can be assigned as

[TABLE]

A.3. Derivation probability of error

In this case, BS transmits a signal for SM users mapped in BPSK constellation symbol. Considering information of SM users is superimposed through power $P$, then the transmitted signal is given as

[TABLE]

where $S_{k}$ represents coded signals of the $k^{th}$ SM user. Considering $BS_{1}$ and $BS_{2}$ make coordination each other to transmit information to CoSM user, then received signal for the $k^{th}$ CoSM user, i.e. $y_{k}$ for $k=2$, is given as

[TABLE]

Furthermore, the other BSs ($BS_{3},...,BS_{M}$) transmit information symbols to non-CoSM users ($CEU_{3},...,CEU_{M}$) independently. Therefore, the received signal for each $k^{th}$ non-CoSM user, i.e. $y_{k}$ for $3\leq k\leq M$, is given as

[TABLE]

Considering $\hat{y}_{k}$ represents the desired signal at SM users, decoded information of $\hat{y}_{k}$ may be different with $y_{k}$ due to error transmission. Given decoded signal in [21], then error detection for each $k^{th}$ SM user can be written as

[TABLE]

The PDF of error probability for CoSM user, i.e. $P_{e,k}^{exact}$ for $k=2$, can be calculated by solving

[TABLE]

where $\delta_{1k}^{*}=\rho\hat{h}_{1k}$ and $\delta_{2k}^{*}=\rho\hat{h}_{2k}$. In case each random variable of Rayleigh fading channel is distributed independently, then PDF of $\delta_{ik}^{*}$ can be derived as

[TABLE]

where $\hat{\delta}_{ik}^{*}=\rho Ei\left(\hat{\sigma}_{ik}\right)$, for $k=2$ and $1\leq i\leq 2$. By using an alternative representation of $Q$ function defined in [22], the probability of error for CoSM user is given as

[TABLE]

where $MGF_{\delta_{ik}^{*}}(s)=\frac{1}{1+s\hat{\delta}_{ik}^{*}}$, for $k=2$ and $1\leq i\leq 2$. By modifying probability of error in [23], then $P_{e,k}^{exact}$ for $k=2$ is given as

[TABLE]

Similarly, the exact probability of error for each $k^{th}$ non-CoSM user, i.e. $P_{e,k}^{exact}$ for $3\leq k\leq M$, can be written as

[TABLE]

where $\hat{\delta}_{ik}=\rho Ei\left(\hat{\sigma}_{ik}\right)$ for $3\leq k\leq M$ and $i=k$.