Reconfigurable Intelligent Surface-Assisted NOMA With Coordinate Reflector Interleaving Under Rician Fading Channel

The concept of reconfigurable intelligent surface-assisted (RIS) for assisting non-orthogonal multiple access (NOMA) transmission, where the phases can be adjusted to decrease error rates and increase capacity, has been gaining popularity as a promising candidate for 6G communication and beyond. This paper considers RIS-NOMA downlink transmission under the Rician fading channel. A coordinate reflector interleaving (CRI) is proposed, where two groups of reflector elements are introduced, such as in-phase element and quadrature element groups. Therefore, it can reduce system complexity because of fewer successive interference cancellations (SIC) among users. As a result, a computer simulation shows that CRI-RIS-NOMA preserves a lower BER compared to conventional RIS-NOMA. Moreover, the superiority of CRI-RIS-NOMA over RIS-NOMA is that it can work at any power allocation for two users. This paper comprehensively studies theoretical derivation for 2 and 4 users to verify the computer-simulated BER. An upper-bound analysis of CRI-RIS-NOMA was also studied to observe the impact of RIS elements for high SNR cases.

SECTION I. Introduction

Wireless communications have advanced from first-generation to 6G, and the demand for reliable and high spectral efficiency is increasing. One main challenge to overcome beyond 6G communication is random channel and base station (BS) coverage. Reconfigurable intelligent surfaces (RIS) recently gained much interest due to its promising candidate to assist BS in beyond 6G communication [1]. RIS is a planar surface consisting of several reflecting elements that actively modify an incoming signal from BS and then reflect it to users [2]. In contrast with relay, RIS preserves a low-cost deployment because every element can be tuned independently to increase the signal-to-noise ratio (SNR) and consumes less energy [3]. The other benefit of RIS is that it can be integrated with existing multiple access techniques, providing high spectral efficiency and low error rates [4].

Numerous research studies have integrated RIS with existing multiple-access techniques, such as RIS-assisted non-orthogonal multiple access (NOMA). Specifically, power domain (PD) NOMA is a technique where multiple-user symbols are superimposed by adding specific power allocation (PA) with superposition coding (SC) [5]. Thus, the users with lower power allocation had to perform successive interference cancellation (SIC) to obtain the information [6]. Therefore, NOMA can fulfill ever-increasing demands for high spectral efficiency performance and enormous bandwidth in 6G and potentially beyond [7]. Because of the NOMA benefit, RIS-NOMA is proposed to decrease error rates and increase spectral efficiency simultaneously [8]. However, RIS-NOMA is highly complex for many users due to its number of SIC processes. Moreover, with more than 2 users, the PA will become a complex problem.

The main objective of this paper is to reduce error rates that occur in RIS-NOMA communication. This paper introduces a novel coordinate reflector interleaving (CRI) to reduce SIC process in RIS-NOMA. In particular, CRI-based RIS-NOMA exploits signal space diversity (SSD) [9] by employing M-ary amplitude shift keying (MASK), which rotated as far as $\pi /4$ . Furthermore, by exploiting the principle of signal separation in quadrature spatial modulation (QSM) [10], coordinate interleaving (CI) is applied on the transmitter and receiver to reduce detection complexity. In contrast with the existing constellation rotation NOMA (con-NOMA), which relies on SSD without RIS [11], [12].

A. Related Works

Based on the superiority of RIS to increase SNR for NOMA-based transmission, several works have been done to propose RIS-NOMA. Aside from RIS-NOMA to reduce BER for two users in [8], numerous algorithms have been proposed to enhance energy efficiency and sum rates. In [13], consider MISO-RIS-NOMA transmission and propose an algorithm to enhance energy efficiency by optimizing the BS and RIS precoding matrix. Numerous works to integrate RIS with NOMA technique have been done before to increase capacity have been done before. In [14], consider RIS single-input single-output (SISO) and proposed RIS beamforming for two NOMA users. In [15] improved the previous beamforming optimization by proposing a joint optimization of BS and RIS beamforming to maximize signal-to-interference-plus-noise ratio (SINR). User fairness is also considered in the works. In [16], consider a downlink RIS-NOMA scenario and propose user clustering to enhance energy efficiency. In [17] proposed a unique solution to enhance the sum rate for NOMA users. Consider a multiple-input single-output (MISO) BS-based transmission; near users are multiplexed with SDMA, while far users are served with RISs. In [18] propose a joint transmission coordinated multi-point (JT-CoMP) method to increase the ergodic capacity of the far users without reducing the capacity of the near users. In contrast with other previous work, [19] proposed capacity fairness for all NOMA users in SISO-RIS-NOMA and MIMO-RIS-NOMA.

According to SSD fundamentals in [9], several works to exploit constellation rotation have been done to reduce system complexity. In [20] proposed a constellation rotation for both near and far users in the NOMA downlink scenario. The main objective is to reduce BER and obtain an optimal rotation for two users. In [21] proposed a phase rotation for one of NOMA users. Then, SSD is applied to avoid overlapping signal constellation between users. Furthermore, in [22] extend the phase rotation system model for one of NOMA users. They proposed an algorithm to determine an optimal rotation value based on power allocation. Moreover, a close-form SER approximation was proposed for 4-QAM-based modulation. In [23], phase rotation NOMA is enhanced and applicable for generalized users to improve achievable data rates. Furthermore, an optimization rate is proposed to choose the suitable angle of the constellation rotation. Finally, in [24], an in-phase constellation rotation NOMA is proposed to enhance and reduce the BER of users and derive an SER approximation.

B. Motivation and Contribution

Motivated by RIS and NOMA techniques, this study presents a novel CRI-RIS-NOMA to decrease BER. The CRI-RIS-NOMA can work under similar channel gain (resulting in similar power allocation).

In contrast with [8] and [19] which incorporate RIS and NOMA in the proposed system, this study proposed CRI-RIS-NOMA to reduce system complexity. In addition, this study derived analytical exact for 4 users NOMA with upper-bound approximation. Moreover, this study considers Rician fading scenario because Rayleigh fading is impractical in wireless telecommunication. This study contribution is summarized as follows:

• A novel technique CRI-RIS-NOMA is proposed to reduce number of SIC counts in the system. In contrast with traditional RIS-NOMA [8], BS will segregate cell users to utilize the in-phase or quadrature part of the signal constellation. Furthermore, by exploiting CI technique, users can detect the incoming signal with lower complexity.

• This study presents a complete derivation for 4 users CRI-RIS-NOMA with same modulation level. Based on derived Q-function, this study presents a novel analytical SER derivation for 2 and 4 user cases. In addition, this work considers Rician fading channel for generalized theoretical expression.

• Finally, to verify the exact analytical expression, this study presents an upper-bound approximation for a high transmit SNR case. Furthermore, this study analyzed the theoretical SER, which is proportional to high SNR and RIS elements.

This study considers Rician flat fading channel with given channel gain and Rician factor. Based on RIS capability to maximize SNR for cell users for 6G and beyond [25]. The proposed CRI-RIS-NOMA can assist BS transmission in sub-6GHz or THz communication. The derived analytical BER of this study is mainly focused on general case, regardless of the transmission band.

C. Paper Organization

Firstly, the system model of CRI-RIS-NOMA is presented in Section II. The transmitter model and the example of CRI-RIS-NOMA are presented in Sub-Section II-A. The Rician channel model and user detector are presented in Sub-Section II-B The analytical BER of CRI-RIS-NOMA is derived in Section III. The theoretical approximation for 2 users case is derived in Sub-Section III-A and IV users case in Sub-section III-B. The numerical results are presented in Section IV to investigate the analytical and simulation results. Finally, the conclusion of this paper is presented in Section V.

SECTION II. System Model

This study considers RIS-NOMA system and proposes CRI, as shown in Fig. 1. BS transmits information toward K user concurrently over Rician fading channel. It is presumed that BS had full knowledge of users’ channel information.

FIGURE 1.

Block diagram of CRI-RIS-NOMA for generalized cell user.

A. Transmitter Model

In CRI-RIS-NOMA, bits input are modulated with an M-ary amplitude shift keying (MASK) mapper. Contrary to conventional MASK, the constellation signal diagram is rotated as far as $\pi /4$ counter-clockwise. Suppose the user’s index in the BS coverage is $k\in \left [{1,2,\cdots, K}\right]$ , then $S_{k}$ is the modulated symbol of every user over $\pi /4$ MASK modulator. Therefore, it can be written as: $$\begin{equation*} S_{k}=S_{kI}+S_{kQ}, \tag{1}\end{equation*}$$ where $S_{kI}$ , $S_{kQ}$ are in-phase $\Re \left [{\cdot }\right]$ and quadrature $\Im \left [{\cdot }\right]$ components of $S_{k}$ , respectively. The MASK modulation mapper is expressed as: $$\begin{equation*} S_{kI} = S_{kQ} = (2m-1-M) \sqrt {\frac {3}{2M^{2}-2}}, \tag{2}\end{equation*}$$ where m is the input symbol ($m\in \left [{1,2,\cdots, M }\right]$ ) and M is the modulation order of the MASK modulator. Assuming there are two users in a cell ($K={2}$ ), the modulated symbols for users 1 and 2 are denoted by $S_{1}$ and $S_{2}$ , respectively. The CI is applied on both users to obtain $x_{1}$ and $x_{2}$ , it can be expressed as: $$\begin{align*} x_{1}&=\sqrt {P_{1}} S_{1,I}+j \sqrt {P_{2}} S_{2,I}, \\ x_{2}&=\sqrt {P_{1}} S_{1,Q}+j \sqrt {P_{2}} S_{2,Q}. \tag{3}\end{align*}$$

In Eq. (3), $P_{k}$ is the allocated transmitted power for a user k. $P_{k}$ is $P_{T}\alpha _{k}$ , where $\alpha _{k}$ is an PA coefficient for $k^{\text {th}}$ user with its value $0 < \alpha _{k} < 1$ , $\alpha _{2}=1-\alpha _{1}$ . $P_{T}$ is the total transmit power on the BS. Thus, an allocated transmit power for user k is $0 < P_{k} < P_{T}$ . Assuming $\alpha _{2}\geq \alpha _{1}$ then, the channel gain of user 2 is lower or equal to user 1. The superposition signal from BS is denoted by $x = x_{1} + x_{2}$ and x for user 1 and 2 is equal to $2x_{1}$ . Example: Utilizing Eq. (2), define $\sqrt {P_{k}} = 1$ , $S_{1} = 1 + j$ and $S_{2} = -1 - j$ , then CI is applied accordingly from Eq. (3). Therefore, the users symbol are $x_{1} = \Re \{S_{1}\} + \Re \{S_{2}\}$ and $x_{2} = \Im \{S_{1}\} + \Im \{S_{2}\}$ . Thus, $x_{1} = 1 - j$ and $x_{2} = 1 - j$ are obtained and indicates that superposition coding x is equal to $2x_{1}$ .

In the proposed system, while $K = 2$ , receivers are not required to do SIC to decode their information. This is because of the CI principle where user 1 symbol is mapped to a real component while user 2 is to an imaginary one. For a generalized number of users where $K>2$ , CI implementation for each user is divided into a user pairing group. Superposition users symbol x can be expressed as follows: $$\begin{align*} x_{k}=\begin{cases} \displaystyle \sqrt {P_{k}}s_{kI}+j\sqrt {P_{k+1}}s_{k+1,I} & \mathrm {if} ~k ~\mathrm {is ~odd,}\\ \displaystyle \sqrt {P_{k-1}}s_{k-1,Q}+j\sqrt {P_{k}}s_{kQ} & \mathrm {if}~ k~ \mathrm {is ~even.} \end{cases} \tag{4}\end{align*}$$

In Eq. (4), it can be seen that users are divided into an even group and an odd number group. Sum of $x_{k}$ for $k={1,2,\cdots,K}$ represents the transmitted base-band signal. It is important to take a note that $x_{k} = x_{k+1}$ for odd $k^{th}$ user grouping, then $k={1,3,\cdots,K-1}$ and written as: $$\begin{align*} x=2 \sum _{\substack{k=1 \\ K {~\text {is odd }}}}^{K-1} x_{k}. \tag{5}\end{align*}$$ Then, for the generalized users’ cases, a SIC corresponds within their grouping, and SIC is not performed for the user $(K-1)^{th}$ and K user. Therefore, SIC process sequences are modeled as $$\begin{align*} \mathrm {SIC~ for}&s_{K-1,I}, s_{K-3,I},\ldots,s_{k+2,I}\mathrm {\quad if} ~k~ \mathrm {is ~odd,} \\ \mathrm {SIC ~for}&s_{K,I}, s_{K-2,I},\ldots,s_{k+2,I}\mathrm {\qquad if} ~k~ \mathrm {is~ even}. \tag{6}\end{align*}$$

B. Channel Model and Detector

RIS contains a finite number of elements of N with i as an index of reflecting element, $i\in \left [{1,2,\cdots,N}\right]$ . The channel from single antenna BS to RIS is denoted by $\boldsymbol {H}$ while the channel from RIS to cell users is denoted by $\boldsymbol {G}$ . In this study, Rician fading channel is considered, and it can be written as: $$\begin{equation*} \boldsymbol {G}=\left ({\underbrace {\sqrt {\frac {\varkappa }{2(1+\varkappa)}}}_{\boldsymbol {\mu }} \boldsymbol {G}^{\mathrm {LoS}}+ \underbrace {\sqrt {\frac {1}{2(1+\varkappa)}}}_{\boldsymbol {\sigma }} \boldsymbol {G}^{\mathrm {NLoS}}}\right), \tag{7}\end{equation*}$$ where $\varkappa$ denotes Rician factor, the channel gain ratio of LoS and NLoS can be obtained from $\varkappa = \mu ^{2}/\sigma ^{2}$ . In Eq. (7), $\mu _{k}\in \boldsymbol {\mu }$ denotes channel gain for LoS channel, whilst $\sigma _{k}\in \boldsymbol {\sigma }$ is channel gain for NLoS channel of user k. Furthermore, the received signal of user k can be expressed as: $$\begin{equation*} y_{k}= \boldsymbol {H} \boldsymbol {\Phi } \boldsymbol {G}^{\text {T}} x + n =\left [{{ \sum _{i=1}^{N} h_{i} e^{j \phi _{i}} g_{i,k} }}\right] x_{k} + n_{k}, \tag{8}\end{equation*}$$ where $\left [{\boldsymbol {H} \in \mathbb {C}^{1 \times N}, \boldsymbol {\Phi } \in \mathbb {C}^{N \times N},\boldsymbol {G}\in \mathbb {C}^{K \times N} }\right]$ . Additionally, $n_{k}$ denotes an AWGN noise with mean 0 and variance $N_{0}$ . The $\boldsymbol {\Phi }$ denotes an RIS phase shifter that is defined as: $$\begin{equation*} \boldsymbol {\Phi } = \text {diag}\{ \Lambda _{1} e^{j\phi _{1}},\cdots,\Lambda _{i} e^{j\phi _{i}},\cdots,\Lambda _{N} e^{j\phi _{N}} \}, \tag{9}\end{equation*}$$ where $\Lambda _{i}$ is an RIS element magnitude ($|\boldsymbol {\Phi }|=1$ ) and $\phi _{i}$ is a phase response of RIS elements ($\angle (\boldsymbol {\Phi })$ ). Assume $h_{i}$ is channel from BS to $i^{\text {th}}$ RIS element and the $g_{i,k}$ is channel from $i^{\text {th}}$ RIS element to $k^{\text {th}}$ user. Magnitude and channel phase of $h_{i}$ and $g_{i,k}$ are denoted by $A_{i} e^{j\psi _{i}}$ and $\beta _{i,k}e^{j\theta _{i,k}}$ , respectively. Therefore, a value of the RIS phase is defined as $\phi _{i}=\psi _{i}+\theta _{i,k}$ . [14], [26].

Here, RIS elements are utilized to minimize channel phase. With given condition, the instantaneous SNR at receiver is written as: $$\begin{equation*} \gamma _{ins~k}=\frac {\sqrt {P_{k}} | \sum _{i=1}^{N} \Lambda _{i} A_{i} \beta _{i,k}e^{j(\phi _{i}-\psi _{i}-\theta _{i,k})} |^{2}} {N_{0} }. \tag{10}\end{equation*}$$ On the receiver side, the maximum likelihood (ML) is performed to detect $b_{k}$ . Therefore, it can be written as: $$\begin{equation*} \hat {b}_{k} =\underset {b_{k}}{\arg \min }\left |{y_{k}-\sqrt {P_{k}}\underbrace {\sum _{i=1}^{N}\left ({h_{i} e^{j \phi _{i}} g_{i,k}}\right)}_{Z_{k}} S_{k} }\right |^{2}, \tag{11}\end{equation*}$$ where $\boldsymbol {Z}_{k}$ is the channel vector of user k and $\hat {b}_{k}$ denotes a detected symbol.

SECTION III. Performance Analysis

A. 2 Users Case

The detection of two-user CRI-RIS is expressed with a maximum likelihood (ML) detector at a single-antenna receiver. Therefore, analytical symbol error rates (SER) over AWGN channel can be expressed as [27]: $$\begin{equation*} P_{k}(e | \boldsymbol {Z}_{k})=2\frac {M-1}{M} Q\left ({{\sqrt {12 \frac {\alpha _{k} |\Omega _{k}|^{2}}{M^{2}-1}\frac {P_{T}}{N_{0}}}}}\right), \tag{12}\end{equation*}$$ where $|\Omega _{k}|^{2}$ is channel gain of user k, $P_{T}$ is the total transmit power, and M is the modulation level. Here, $Q \left ({\cdot }\right)$ denotes an Q-function with $Q(x)=\frac {1}{\sqrt {2 \pi }} \int _{x}^{\infty } \exp \left ({-\frac {u^{2}}{2}}\right) du$ . The average analytical SER over Rician fading can be calculated by employing moment generating function (MGF) [28], and it is denoted by: $$\begin{equation*} \mathcal {M}(j\omega)=\left ({\frac {1}{1-2 j \omega \sigma ^{2}}}\right)^{\frac {n}{2}} \text {exp}\left ({\frac {j \omega s^{2}}{1-2 j \omega \sigma ^{2}}}\right), \tag{13}\end{equation*}$$ where $j\omega$ is moment of MGF, s denotes mean, and $\sigma ^{2}$ denotes variance. Further, we follow the summation of N Rician random variables. Following the central limit theorem (CLT), the channel gain of user k served with $N \gg 1$ RIS elements can be expressed as $N|\Omega _{k}|$ . Based on the SNR in Eq. (10), the mean and variance of Rician random variable $A_{i}$ and $\beta _{i,k}$ are written as follows: $$\begin{align*} \!VAR \{A_{i} \beta _{i,k} \} &\!= \varepsilon _{k} \!=\! 2 \sigma ^{2}\!+\!\mu ^{2}\!-\!\frac {\pi \sigma ^{2}}{2} \mathcal {L}_{1 / 2}^{2} \left ({-\frac {\mu ^{2}}{2\sigma ^{2}}}\right), \tag{14a}\\ \mathop {\mathrm {\mathbb {E}}}\nolimits \{A_{i} \beta _{i,k} \}&= \upsilon _{k} =\sqrt {\frac {\pi }{2}} \mathcal {L}_{1 / 2}\left ({-\frac {\mu ^{2}}{2 \sigma ^{2}}}\right) \sigma, \tag{14b}\end{align*}$$ where $\mathcal {L}_{\cdot }\left ({\cdot }\right)$ denotes Laguerre polynomial. Given the mean and variance of Rician random variable, then MGF of Eq. (13) can be equivalently written as $\mathcal {M}(j\omega)=\left ({\frac {1}{1-2 j \omega \varepsilon _{k}}}\right)^{\frac {n}{2}} \text {exp}\left ({\frac {j \omega \upsilon _{k}^{2}}{1-2 j \omega \varepsilon _{k}}}\right)$ . Therefore, with the impact of Rician factor [29], the MGF of CRI-RIS-NOMA can be written as: $$\begin{align*} &\hspace {-1pc}\mathcal {M}\left ({-\frac {N \ell \gamma ^{P}_{k}|\Omega _{k}|^{2}}{\sin ^{2} \xi }}\right) \\ &=\frac {(1+\varkappa) \sin ^{2} \xi }{(1+\varkappa) \sin ^{2} \xi + N \ell \bar {\gamma }^{P}_{k} \varepsilon _{k}^{2} } \\ &\quad \times \exp \left ({-\frac {\varkappa N^{2} ~\ell \bar {\gamma }^{P}_{k}\upsilon _{k}^{2} }{(1+\varkappa) \sin ^{2} \xi + N \ell \bar {\gamma }^{P}_{k}\varepsilon _{k}^{2} }}\right), \tag{15}\end{align*}$$ where $\bar {\gamma }^{P}_{k} = P_{k}/N_{0}$ , $\ell = 3/(M^{2}-1)$ and $|\Omega _{k}|^{2}=1$ . Finally, the analytical SER expression of the proposed system for 2 users ($|\Omega _{k}|\in \left [{|\Omega _{1}|, |\Omega _{2}|}\right]$ ) MASK can be written as: $$\begin{align*} \bar {P}^{s}_{k} &= \frac {2(M-1)}{M\pi } \int _{0}^{\pi /2} \frac {(1+\varkappa) \sin ^{2} \xi }{(1+\varkappa) \sin ^{2} \xi + N \ell \bar {\gamma }^{P}_{k}|\Omega _{k}|^{2} \varepsilon _{k}^{2} } \\ &\quad \times \exp \left ({-\frac {\varkappa N^{2} ~\ell \bar {\gamma }^{P}_{k}|\Omega _{k}|^{2}\upsilon _{k}^{2} }{(1+\varkappa) \sin ^{2} \xi + N \ell \bar {\gamma }^{P}_{k}|\Omega _{k}|^{2}\varepsilon _{k}^{2}}}\right) d\xi. \tag{16}\end{align*}$$

B. 4 Users Case

For $K=4$ case,1 we can divide users who perform SIC ($c \in [{1,2}]$ ) and non-SIC users as ($n \in [{3,4}]$ ). Suppose decision boundary of ML detector with given received signal $y_{n}$ for user n is denoted by $\Re \{\boldsymbol y_{n}\} < 0$ , $\Re \{\boldsymbol y_{n}\}\geq 0$ and $\Im \{\boldsymbol y_{n}\} < 0$ , $\Im \{\boldsymbol y_{n}\}\geq 0$ . Therefore, the error probability of ML detector under condition channel for all n users, $\boldsymbol {Z}_{n} = \sum _{i=1}^{N} h_{i} e^{j \phi _{i}} g_{i,n}$ can be written as: $$\begin{align*} P_{n}\left ({e \mid \boldsymbol {Z}_{n}}\right)&=\frac {1}{2} P_{r}\left ({\Re \{\boldsymbol B_{n}\} \geq \sqrt {\frac {P_{T}}{2}} L_{A} \boldsymbol {Z}_{n} \boldsymbol {Z}_{n}^{\mathbf {H}}}\right) \\ &\quad + \frac {1}{2} P_{r}\left ({\Re \{\boldsymbol B_{n}\} \geq \sqrt {\frac {P_{T}}{2}} L_{B} \boldsymbol {Z}_{n} \boldsymbol {Z}_{n}^{\mathbf {H}}}\right), \tag{17}\end{align*}$$ where L denotes a decision boundary based on power allocation with $L_{A} = \sqrt {\alpha _{n}}+\sqrt {\alpha _{c}}$ and $L_{B} = \sqrt {\alpha _{n}}-\sqrt {\alpha _{c}}$ . $\Re \{\boldsymbol B_{n}\}$ denotes AWGN with zero mean and $N_{0}/2$ variance. Further, the BEP of users n in the form of a Q-function can be written as: $$\begin{align*} P_{n, \Re }(e | |\Omega _{n}|)&=\frac {M-1}{M} \left [{ Q\left ({{\sqrt {12 \frac {L_{A}^{2} |\Omega _{n}|^{2}}{M^{2}-1}\frac {P_{T}}{N_{0}}}} }\right) }\right. \\ &\quad +\left.{ Q\left ({{\sqrt {12 \frac {L_{B}^{2} |\Omega _{n}|^{2}}{M^{2}-1}\frac {P_{T}}{N_{0}}}} }\right) }\right]. \tag{18}\end{align*}$$ Consider ML decision rules and $\Im \{\boldsymbol B_{n}\}$ , the error probability of user n is similar to Eq. (18). The error probability of it can be formulated as $P_{n}(e) = (P_{n,\Re }(e)+P_{n,\Im }(e))/2$ . Finally, the MGF in Eq. (13) is utilized to estimate an SER of users n. Therefore, with given BEP in the form of Q-function in Eq. (18), it can be rewritten as follows: $$\begin{align*} \bar {P}^{s}_{n} &= \frac {2(M-1)}{M\pi } \left ({\int _{0}^{\pi /2}\mathcal {M}\left ({-\frac {N \ell \gamma ^{A}_{n}|\Omega _{k}|^{2}}{\sin ^{2} \xi }}\right) }\right. \\ &\quad + \left.{ \mathcal {M}\left ({-\frac {N \ell \gamma ^{B}_{n}|\Omega _{k}|^{2}}{\sin ^{2} \xi }}\right)}\right), \tag{19}\end{align*}$$ where $\bar {\gamma }^{A}_{n}$ denotes $(\sqrt {\alpha _{n}}+\sqrt {\alpha _{c}})^{2} P_{T}/N_{0}$ , while $\bar {\gamma }^{B}_{n}$ denotes $(\sqrt {\alpha _{n}}-\sqrt {\alpha _{c}})^{2} P_{T}/N_{0}$ .

In contrast with user n, user c had to perform SIC. As a result, the analysis for user c can be divided by two conditions: BEP under the condition of detected symbol user n correctly and erroneously. Therefore, BEP for user c can be written as: $$\begin{equation*} P_{c}(e) = P_{c}(e | correct_{n}) + P_{c}(e | error_{n}). \tag{20a}\end{equation*}$$

Proposition.

The symbols of user n are presumed to be detected correctly and erroneously, further subtracted from the received signal at c. Therefore, BEP of user c can be written as in Eq. (20b), shown at the bottom of the page. $$\begin{align*} P_{c, \Re }(e)&=\frac {M-1}{M} \left [{ 2 \times Q\left ({{\sqrt {12 \frac {(\sqrt {\alpha _{c}})^{2} |\Omega _{c}|^{2}}{M^{2}-1}\frac {P_{T}}{N_{0}}}} }\right) +Q\left ({{\sqrt {12 \frac {(\sqrt {\alpha _{n}}-\sqrt {\alpha _{c}})^{2} |\Omega _{c}|^{2}}{M^{2}-1}\frac {P_{T}}{N_{0}}}} }\right) }\right. \\ &\quad \left.{ - Q\left ({{\sqrt {12 \frac {(\sqrt {\alpha _{n}}\!+\!\sqrt {\alpha _{c}})^{2} |\Omega _{c}|^{2}}{M^{2}-1}\frac {P_{T}}{N_{0}}}} }\right) \!-\! Q\left ({{\sqrt {12 \frac {(2\sqrt {\alpha _{n}}\!-\!\sqrt {\alpha _{c}})^{2} |\Omega _{c}|^{2}}{M^{2}-1}\frac {P_{T}}{N_{0}}}} }\right) + Q\left ({{\sqrt {12 \frac {(2\sqrt {\alpha _{n}}+\sqrt {\alpha _{c}})^{2} |\Omega _{c}|^{2}}{M^{2}-1}\frac {P_{T}}{N_{0}}}} }\right) }\right]. \qquad \quad \tag {20b} \\ \bar {P}^{s}_{c} &= \frac {2(M-1)}{M\pi } \left ({2 \times \int _{0}^{\pi /2} \mathcal {M}\left ({-\frac {N \ell \gamma ^{C}_{c}|\Omega _{c}|^{2}}{\sin ^{2} \xi }}\right) + \mathcal {M}\left ({-\frac {N \ell \gamma ^{D}_{c}|\Omega _{c}|^{2}}{\sin ^{2} \xi }}\right) -\mathcal {M}\left ({-\frac {N \ell \gamma ^{E}_{c}|\Omega _{c}|^{2}}{\sin ^{2} \xi }}\right) }\right. \\ &\quad \left.{ -\mathcal {M}\left ({-\frac {N \ell \gamma ^{F}_{c}|\Omega _{c}|^{2}}{\sin ^{2} \xi }}\right) +\mathcal {M}\left ({-\frac {N \ell \gamma ^{G}_{c}|\Omega _{c}|^{2}}{\sin ^{2} \xi }}\right) }\right), \tag{20c}\end{align*}$$

Proof:

See Appendix A for correct detection and Appendix B for incorrect detection of user n symbols.

Based on proposition Eq. (20a), the BEP of user c in the form of Q-function is obtained with $P_{c}(e) = (P_{c,\Re }(e)+P_{c,\Im }(e))/2$ . Then, exact BEP of CRI-NOMA is expressed by (20b). Given BEP, to estimate $\bar {P}^{s}_{n}$ , the MGF from Eq. (13) is utilized for transforming Eq. (20b) to SER over Rician fading channel. Then, the approximation of SER $\bar {P}^{s}_{c}$ can be written as Eq. (20c), shown at the bottom of the page, where $\bar {\gamma }^{C}_{c}$ denotes $(\sqrt {\alpha _{c}})^{2}$ , $\bar {\gamma }^{F}_{c}$ denotes $(2\sqrt {\alpha _{n}}-\sqrt {\alpha _{c}})^{2}$ , $\bar {\gamma }^{G}_{c}$ denotes $(2\sqrt {\alpha _{n}}+\sqrt {\alpha _{c}})^{2}$ , and $\bar {\gamma }^{E}_{n} = \bar {\gamma }^{A}_{n}$ ; $\bar {\gamma }^{D}_{n} = \bar {\gamma }^{B}_{n}$ .

C. SER Upperbound Analysis

This study derived and simulated an SER upper-bound theoretical approximation through computer simulation. Moreover, it analyses the impact of RIS elements number N and approximation for high transmit SNR.

To approximate upper-bound analysis, recalling Eq. (16), all $\xi$ = $\pi /2$ . Therefore, the upper bound expression $\bar {P}^{u}_{k}$ of the proposed system for two users is written as: $$\begin{align*} \bar {P}^{s}_{k}& \leq \frac {2(M-1)}{M\pi } \frac {1+\varkappa }{1+\varkappa + N \varepsilon _{k}^{2} \ell \bar {\gamma }^{P}_{k}|\Omega _{k}|^{2}} \\ &\quad \times \exp \left ({-\frac {\varkappa N^{2} ~\upsilon _{k}^{2} \ell \bar {\gamma }^{P}_{k}|\Omega _{k}|^{2}}{1+\varkappa + N \varepsilon _{k}^{2} \ell \bar {\gamma }^{P}_{k}|\Omega _{k}|^{2}}}\right). \tag{21}\end{align*}$$ Eq. (21) shows that the impact of Rician factor $\varkappa$ affects SER upper bound analysis. From the second term of exponent, the $\varkappa$ acts as a multiplier of $N^{2}$ . Thus, it can be concluded that $\varkappa \to \infty$ then $\bar {P}^{u}_{k} \to \infty$ as well for LoS case. In contrast, with $\varkappa = 0$ , the $\bar {P}^{u}_{k}$ is only affected by the power of $N^{2}$ .

Based on the proposed upper bound results, with the assumption of $N\bar {\gamma }^{P}_{k} \gg 1$ , the approximation of upper bound is proportional to: $$\begin{equation*} \bar {P}^{s}_{k} \propto \exp \left ({-\frac {\varkappa N^{2} ~\upsilon _{k}^{2} \ell \bar {\gamma }^{P}_{k}|\Omega _{k}|^{2}}{1+\varkappa + N \varepsilon _{k}^{2} \ell \bar {\gamma }^{P}_{k}|\Omega _{k}|^{2}}}\right). \tag{22}\end{equation*}$$ Here, the impact of the number of RIS is exponentially increasing with the factor of $N^{2}$ and Ricean factor $\varkappa$ . Therefore, it can be seen from Fig. 2 that the proposed system will achieve lower BER with the increasing element of N. In addition, Fig. 2 shows an upper bound result of the proposed system. It can be seen that the characteristics of upper bound analysis are closely tight with exact SER approximation as $\frac {P_{T}}{N_{0}} \to \infty$ . A higher RIS element number N leads to a tight upper bound and exact analysis for low $P_{T}$ .

FIGURE 2.

Simulation result ($K=2$ ) of user 1 with the different number of elements and upper bound analysis.

SECTION IV. Numerical Results

This study compares CRI-RIS-NOMA BER performance to conventional 2 users RIS-NOMA in [8] and [19]. Then, given power allocation, this study extends 2 to 4 users to compare the proposed system’s superiority to RIS-NOMA. Finally, SER performance results are validated with the exact and upper-bound analytical derived in the previous section. This study considers a different channel gain for cell users and follows the previous classical NOMA assumption in [8], [30], and [31]. Assuming user 1 is cell center user with strongest channel gain, then $K-1$ users will follow the rules $|\Omega _{1}|^{2} = 0, |\Omega _{2}|^{2} = |\Omega _{1}|^{2} - X, \cdots, |\Omega _{K}|^{2} = |\Omega _{K-1}|^{2} {-} X$ , with X being a -5 dB in value. Therefore, the channel gain of all users is defined as follows2: $|\Omega _{1}|^{2} = 0 \text {dB}; |\Omega _{2}|^{2} = -5 \text {dB};|\Omega _{3}|^{2} = -10 \text {dB};|\Omega _{4}|^{2} = -15 \text {dB};$ . In this study, the conversion of SER to BER is calculated as $\approx \text {SER}/log2(M)$ and $M = 4$ .

Fig. 3 shows BER performance of the proposed system over different $\alpha _{1}$ , compared to RIS-NOMA. The transmit power is − 6dB, $\varkappa = 7 \text {dB}$ , $N = 12$ . Compared to RIS-NOMA, the proposed system can perform on every PA. The figure shows that for $\alpha$ is higher than 0.5, BER of RIS-NOMA user 2 stays above 10⁻¹. In addition, BER performance for user 1 keeps decreasing for a higher value of $\alpha$ .

FIGURE 3.

The simulation result of different $\alpha$ with same modulation $M = 4$ and transmit power $P = -6 \text {dB}$ .

Fig. 4 shows BER performance of the proposed system over a different number of elements $N =\,\,[16,\,\,32,\,\,64]$ . In addition, the parameter is set as follows: $\alpha _{1} = 0.28$ ; $\varkappa = 3 \text {dB}$ . The figures show that BER is lower for more RIS elements. Targeting BER of User 2 below 10⁻³ is achieved with transmit power $P_{T} =-1\text {dB}$ for $N=16$ , $P_{T} =-7\text {dB}$ for $N=32$ , $P_{T} =-13\text {dB}$ for $N=32$ .

FIGURE 4.

BER simulation and analytical result for different number of elements N with user 1 power allocation $\alpha _{1}=0.17$ .

In Fig. 5, we set the parameter as follows: $N=64$ ; $\varkappa = 3\text {dB}$ ; $\alpha$ for a CRI-RIS user 1 to 4 is $\alpha =[{2/52, 2/52, 24/52, 24/52}];$ while for the NOMA users are $\alpha =[{1/156, 5/156, 25/156, 125/156}]$ , respectively. As expected, The PA of CRI-RIS-NOMA for user 1 and user 2 can be set as the same because CRI-RIS-NOMA can work over different PAs. The figure shows clearly that proposed system outperformed RIS-NOMA. Additionally, an impact of CI of real and imaginary components of the modulated signal makes the BER of User 3 near User 1, while User 4 is near User 2. Targeting BER at $\approx 10^{-4}$ , user 1 CRI-RIS-NOMA can achieve it at $P_{T} = -12 \text {dB}$ , while RIS-NOMA at $P_{T} = 0 \text {dB}$ . User 2 CRI-RIS-NOMA achieve it at $P_{T} = -6 \text {dB}$ , RIS-NOMA at $P_{T} = 2 \text {dB}$ . User 3 CRI-RIS-NOMA achieve it at $P_{T} = -10 \text {dB}$ , RIS-NOMA at $P_{T} = 4 \text {dB}$ . User 4 CRI-RIS-NOMA achieve it at $P_{T} = -4 \text {dB}$ , RIS-NOMA at $P_{T} = 4 \text {dB}$ .

FIGURE 5.

The result of generalized user proposed system $(N=64)$ , compare to RIS-NOMA.

Fig. 6 shows result of different Rician factors with the same number of elements $N = 64$ and $\alpha = 0.35$ . As expected, higher Rician factor $\varkappa$ leads to lower BER because of stronger LoS channel gain. However, the impact of higher $\kappa$ Targeting BER under 10⁻⁴, User 2 can achieve it with $P_{T} = -11 \text {dB}$ at $\varkappa = 3 \text {dB}$ , $P_{T} = -13 \text {dB}$ at $\varkappa = 5 \text {dB}$ and $P_{T} = -14 \text {dB}$ at $\varkappa = 7 \text {dB}$ .

FIGURE 6.

The BER result of different Rician factor $\kappa$ with $N = 64$ .

SECTION V. Conclusion

This study proposes a novel CRI-RIS-NOMA to reduce SIC count. As a result, CRI-RIS-NOMA outperforms RIS-NOMA in terms of simulated BER. Furthermore, the proposed analytical BER validates simulated results for 2 and 4 users CRI-RIS-NOMA. Compared to conventional RIS-NOMA, CRI-RIS-NOMA exploits CI technique; thus, it gives a lower system complexity by reducing NOMA’s SIC to half with several cell users. The trade-off of CRI-RIS-NOMA is that it has a relatively higher BER than RIS-aided orthogonal multiple access (RIS-OMA) due to power allocation for each user. As a gain, the proposed system has a higher sum rate than RIS-OMA. Further investigation for sub-6GHz or THz transmission with a sophisticated channel model (e.g., Saleh-Valenzuela model) appears essential for CRI-RIS-NOMA application in 6G communication. In addition, the future research direction is to integrate spatial modulation (SM) with CRI-RIS-NOMA to decrease system complexity even more.

Appendix A BEP Correct Detection

In this case, conditional probability for user c on $\boldsymbol B_{c}$ includes the prior probability of correct detection of user n. Therefore, the conditional probability of user c, under condition symbol user n detected correctly can be written as: $$\begin{align*} &\hspace {-.5pc}P_{c}\left ({\left.{e}\right |_{\text {correct }_{n}, \Re \{\boldsymbol B_{c}\}}}\right) \\ &= \frac {1}{2} P_{r}\left ({\Re \{\boldsymbol B_{c}\} \geq - \sqrt {P_{T}}\left ({\sqrt {a_{n} / 2}+\sqrt {a_{c} / 2}}\right) \boldsymbol {Z}_{c} \boldsymbol {Z}_{c}^{\mathbf {H}}}\right) \\ &\quad \times P_{r}\left ({\Re \{\boldsymbol B_{c}\} < -\sqrt {P_{T}}\left ({\sqrt {a_{c} / 2}}\right) \boldsymbol {Z}_{c} \boldsymbol {Z}_{c}^{\mathbf {H}} }\right. \\ &\quad \left.{ \qquad ~\mid \Re \{\boldsymbol B_{c}\} \geq -\sqrt {P_{T}}(\sqrt {a_{n} / 2} +\sqrt {a_{c} / 2}) \boldsymbol {Z}_{c} \boldsymbol {Z}_{c}^{\mathbf {H}} }\right) \\ & \quad +\frac {1}{2} P_{r}\left ({\Re \{\boldsymbol B_{c}\} \geq -\sqrt {P_{T}}\left ({\sqrt {a_{n} / 2}-\sqrt {a_{c} / 2}}\right) \boldsymbol {Z}_{c}\boldsymbol {Z}_{c}^{\mathbf {H}}}\right). \\{}\tag{23}\end{align*}$$ To derive the error probability of user c with MASK modulation, conditional probability $Pr(x \mid y) = \frac {Pr(x\cap y) }{Pr(y)}$ is utilized. Furthermore, bit error probability (BEP) can be written as: $$\begin{align*} &\hspace {-1pc}P_{c, \Re }(e | correct_{n}) \\ &=\frac {M-1}{M} \left [{ 2 \times Q\left ({{\sqrt {12 \frac {(\sqrt {\alpha _{c}})^{2} |\Omega _{c}|^{2}}{M^{2}-1}\frac {P_{T}}{N_{0}}}} }\right) }\right. \\ &\quad -\left.{ Q\left ({{\sqrt {12 \frac {(\sqrt {\alpha _{n}}+\sqrt {\alpha _{c}})^{2} |\Omega _{c}|^{2}}{M^{2}-1}\frac {P_{T}}{N_{0}}}} }\right) }\right]. \tag{24}\end{align*}$$ The BEP of user c can be easily obtained by substitute $\Re \{\boldsymbol B_{c}\}$ with $\Im \{\boldsymbol B_{c}\}$ , resulting in the same result. Furthermore, conditional BEP under condition symbol user n detected correctly can be written as $P_{c}(e|correct_{n}) = (P_{n,\Re }(e|correct_{n})+P_{n,\Im }(e|correct_{n}))/2$ . Thus, this is complete proof.

Appendix B BEP Error Detection

In this case, conditional probability for user c on $\boldsymbol B_{c}$ includes the prior probability of incorrect detection of user n. Employing the decision boundary of ML, we can write the conditional probability of user c under condition symbol user n detected erroneously as: $$\begin{align*} &\hspace {-.9pc}P_{c}\left ({\left.{e}\right |_{\text {error }_{n}, \Re \{\boldsymbol B_{c}\}}}\right) \\ & =\frac {1}{2} P_{r}\left ({\Re \{\boldsymbol B_{c}\} < -\sqrt {P_{T}}\left ({\sqrt {a_{n} / 2}+\sqrt {a_{c} / 2}}\right) \boldsymbol {Z}_{c} \boldsymbol {Z}_{c}^{\mathbf {H}}}\right) \\ &\quad \times P_{r}\left ({\Re \{\boldsymbol B_{c}\} < - \sqrt {P_{T}} \left ({2 \sqrt {a_{n} / 2}+\sqrt {a_{c} / 2}}\right)\boldsymbol {Z}_{c} \boldsymbol {Z}_{c}^{\mathbf {H}} }\right. \\ &\quad \left.{ \mid \Re \{\boldsymbol B_{c}\} < - \sqrt {P_{T}} \left ({\sqrt {a_{n} / 2}+\sqrt {a_{c} / 2}}\right) \boldsymbol {Z}_{c} \boldsymbol {Z}_{c}^{\mathbf {H}}}\right) \\ &\quad +\frac {1}{2} P_{r}\left ({\Re \{\boldsymbol B_{c}\} < - \sqrt {P_{T}}\left ({\sqrt {a_{n} / 2}-\sqrt {a_{c} / 2}}\right) \boldsymbol {Z}_{c} \boldsymbol {Z}_{c}^{\mathbf {H}}}\right) \\ &\quad \times P_{r}\left ({\Re \{\boldsymbol B_{c}\} \geq -\sqrt {P_{T}}\left ({2 \sqrt {a_{n} / 2}-\sqrt {a_{c} / 2}}\right) \boldsymbol {Z}_{c} \boldsymbol {Z}_{c}^{\mathbf {H}} }\right. \\ &\quad \left.{ \mid \Re \{\boldsymbol B_{c}\} < -\sqrt {P_{T}}\left ({\sqrt {a_{n} / 2}-\sqrt {a_{c} / 2}}\right) \boldsymbol {Z}_{c} \boldsymbol {Z}_{c}^{\mathbf {H}}}\right) \tag{25}\end{align*}$$ Furthermore, we employ similar conditional probability $Pr(x \mid y) = \frac {Pr(x\cap y) }{Pr(y)}$ to calculate BEP. Finally, we can write the Q-function of user c under the condition of incorrect detection of bit user n, is written as follows: $$\begin{align*} &\hspace {-1pc}P_{c, \Re }(e | error_{n}) \\ &=\frac {M-1}{M} \left [{ Q\left ({{\sqrt {12 \frac {(\sqrt {\alpha _{n}}-\sqrt {\alpha _{c}})^{2} |\Omega _{c}|^{2}}{M^{2}-1}\frac {P_{T}}{N_{0}}}} }\right) }\right. \\ &\quad \left.{ -Q\left ({{\sqrt {12 \frac {(2\sqrt {\alpha _{n}}-\sqrt {\alpha _{c}})^{2} |\Omega _{c}|^{2}}{M^{2}-1}\frac {P_{T}}{N_{0}}}} }\right) }\right. \\ &\quad \left.{ + Q\left ({{\sqrt {12 \frac {(2\sqrt {\alpha _{n}}+\sqrt {\alpha _{c}})^{2} |\Omega _{c}|^{2}}{M^{2}-1}\frac {P_{T}}{N_{0}}}} }\right) }\right]. \tag{26}\end{align*}$$ Furthermore, conditional BEP under condition symbol user n detected incorrectly can be written as $P_{c}(e|error_{n}) = (P_{n,\Re }(e|error_{n})+P_{n,\Im }(e|error_{n}))/2$ . The proof is completed.