RIS Partition-Assisted Non-Orthogonal Multiple Access (NOMA) and Quadrature-NOMA With Imperfect SIC

Reconfigurable intelligent surface (RIS)-assisted power-domain non-orthogonal multiple access (PD-NOMA) system has emerged as a revolutionary technology to enhance the spectrum efficiency for future wireless networks. This work introduces two novel RIS systems, namely, RIS partition-assisted (RISP) PD-NOMA (RISP-PD-NOMA) and RISP-Quadrature-NOMA (RISP-Q-NOMA), to maximize the diversity order and to improve the signal quality of all users by dedicating fixed RIS units to each user. The closed-form expressions of average sum-rate, outage probability, and diversity order are evaluated for both systems under the Rician fading channel for perfect and imperfect successive interference cancellation (SIC) operations. Further, the performance of both RISP-PD-NOMA and RISP-Q-NOMA systems is compared with RIS partitioned PD-NOMA, RIS-assisted PD-NOMA, and PD-NOMA systems. It is noticed from the analysis that under perfect SIC, RISP-PD-NOMA outperforms all other systems. However, under imperfect SIC, RISP-Q-NOMA demonstrates superior performance to other systems due to the lesser number of SIC operations that reduce detection delay and decoding complexity. Furthermore, the analytical expression for bit error rate (BER) of RISP-PD-NOMA and RISP-Q-NOMA is derived, and then the closed-form expression of average-BER is evaluated. Numerical results demonstrate that RISP-Q-NOMA is superior to RISP-PD-NOMA.


I. INTRODUCTION
S EVERAL key technologies, such as massive multipleinput multiple-output (MIMO) systems, ultra-dense networks, and millimeter wave communications, have been investigated extensively over the past few years to fulfill the stringent requirement for beyond 5 th generation (B5G) wireless network [1]. However, some of these technologies require a high degree of design complexity and result in significantly higher energy usage. Reflecting intelligent surfaces (RISs) have arisen as a revolutionary technology for enhancing the spectrum efficiency (SE) and coverage for B5G wireless networks [2] by leveraging significant advancements in reconfigurable artificial surfaces [3]. RISs are composed of a large number of reflector units that deliberately and precisely control the propagation environment to improve signal quality. Each unit of the RIS has the ability to absorb energy and change the phase of the incoming signal independently [4]. By correctly adjusting the angle of reflection of the RIS units, the reflected signal can be modified. Due to their potential benefits, RIS has been combined with many pre-existing wireless systems as a major and auxiliary module. For instance, a RIS is combined with a traditional single-input single-output (SISO) system in [5]. A RIS is employed as part of the transmitter (Tx) in [6], to accomplish various types of index modulation and to bypass the need of multi-antenna base station (BS). In [7], reflection pattern based on ON/OFF state of RIS units is considered for multiple input single output (MISO) system. Orthogonal frequency division multiplexing based single user wireless system is considered in [8], where RIS is used in ON/OFF reflection mode and the performance of the proposed system is evaluated in terms of achievable data rate. A grouping technique of RIS units is introduced in [9], where group of RIS units share the common coefficient of reflection to reduce the estimation complexity. In [10], RIS is combined with the receive quadrature spatial modulation technique, in which RIS is partitioned into two main parts to form the in-phase and quadature components of the signals. Apart from RIS, non-orthogonal multiple access (NOMA) is also considered a key technique in B5G as it supports massive connectivity, improves SE and user throughput, reduces transmission latency, and increases reliability [11]. Although several types of NOMA [12], [13], [14] have been developed in recent times, the most widely studied and popular NOMA is the power-domain NOMA (PD-NOMA). The simple implementation of PD-NOMA on the existing network without any significant changes and additional bandwidth is the reason behind the popularity. In the PD-NOMA systems, multiple users share the same frequency and time slot by allocating different power to each user. To do this, superposition coding (SC) is used at the Tx and successive interference cancellation (SIC) is used at the receiver (Rx) [12]. In-depth study has been carried out in the designing of downlink PD-NOMA system using user pairing [15], user scheduling algorithm [16], and generalized channel model [17]. The application of MIMO system has been introduced to PD-NOMA in [18]. The key attributes of PD-NOMA that improve the SE of the system are SC and SIC. Due to the weak detection capabilities of wireless devices and the impact of modulation techniques and decoding strategies, Rx can not always perform perfect SIC, resulting in imperfect SIC [19]. Under imperfect SIC, interference cannot be perfectly cancelled if the signals are not recovered accurately at a given stage of SIC. This results in decoding error, which will become more significant as the number of stages grows and this greatly reduces NOMA's performance [19], [20], [21]. As a result, compensating the impact of SIC in PD-NOMA is of major interest nowadays, which in turn improves the SIC stability, reduces the decoding complexity and detection delay. In order to reduce the a priority-oriented design, a RIS assisted PD-NOMA for SISO network is proposed in [32], where the passive beamforming weights are designed at the RIS to enhance the outage and ergodic performance of only prioritized user. To improve the rate performance, the combined channel strength based user ordering is proposed in [33]. MISO two-user PD-NOMA in combination with RIS is proposed in [34], which is more energy efficient than the RIS-assisted OMA system. In [35], RIS aided MIMO-PD-NOMA system is proposed where passive beamforming weights are used at the RIS to simultaneously serve coupled users and the system performance is analyzed in terms of outage probability (OP). RIS is combined with different multiple access techniques in [36] and the performance of the considered systems is examined in terms of OP and diversity order. However, in [37], it is shown that the sum-rate performance of RIS empowered NOMA system is superior to the RIS empowered OMA system by using active beamforming at the BS and passive beamforming at the RIS.
Recently, a new way of employing RIS is being investigated [39], [40], [41], [42], where RIS is partitioned into several subsurfaces, each consisting of fixed units of RIS. The phases of each subsurface is then modified with respect to BS-RIS and RIS to a specific user to improve the overall performance of the system. Since this new system partitions RIS into several subsurfaces, we name it the RIS partition assisted system. All the RIS partition-based NOMA systems [39], [40], [41], [42] consider the link between BS to RIS and RIS to a specific user as a Rayleigh. However, the assumption of the Rayleigh channel in the RIS system has limited legitimacy, since RIS is deployed to enhance the received signal strength by leveraging the LOS component. Furthermore, RIS primarily used as an anomalous reflector in which the LOS component is dominant [36].
The foregoing discussion prompts us to investigate the performance of the RIS partition assisted PD-NOMA (RISP-PD-NOMA) system for multiuser scenarios under Rician fading channel. More precisely, this work, for the first time, offers closed-form expression for the average sum-rate (ASR), OP, diversity order, and average bit error rate (ABER) of RISP-PD-NOMA system under perfect as well as imperfect SIC operations. The proposed RISP-PD-NOMA system significantly differs from the systems introduced in [39], [40], [41], and [42] in the following ways: 1) In [39] and [40], the RIS surface assigned to one user is not linked with the other users. However, in the real world, this arrangement is difficult to achieve as users may be randomly located in the vicinity of RIS. Whereas in the proposed work, all RIS subsurfaces are exposed/coupled to all users. Therefore, the system model considered in this work is more general and practical. Furthermore, the direct link channels are not incorporated in [39] and [40], whereas the proposed work incorporates the direct link channels from BS to a respective user. 2) In [39] and [40], only the error performance is investigated for two and multi-user scenarios, respectively. Whereas the proposed work evaluates various performance metrics of RISP-PD-NOMA system, viz., ASR, OP, diversity order, and ABER. 3) In [41], the two main key aspects of the PD-NOMA system, namely SC and SIC, are not used in the RIS-partition based cluster. Whereas the proposed system uses SC at the Tx and SIC at the Rx for signal transmission and detection, respectively. Furthermore, in [41], only OP is evaluated. Whereas in the proposed work assesses the closed-form expressions of ASR, OP, diversity order, and ABER for RISP-PD-NOMA system. 4) In [41], only the transmission of phase shift keying (PSK) symbols is possible, whereas the proposed RISP-PD-NOMA system can work with any known modulation scheme, for example, M −ary PSK, M −ary quadrature amplitude modulation, etc. 5) The BER in [42] reaches a constant value at a higher signal to noise ratio (SNR) due to subsurface interference, resulting in zero diversity gain. However, the proposed RISP-PD-NOMA system exhibits non-zero diversity order. Furthermore, the direct link channels are not considered in [42], while in the proposed work, direct link channels are incorporated. Since the RIS partition-assisted system considered in [39] and [41] divides the RIS into two and several subsurfaces where each subsurface is used to serve a single user, these systems will hereafter be called the RIS-division PD-NOMA [39] (RISD-PD-NOMA) and RISD-NOMA [41] systems, respectively.
Apart from evaluating the performance of generalized RISP-PD-NOMA system, this work proposes a novel RIS partition assisted Q-NOMA (RISP-Q-NOMA) system. The proposed RISP-Q-NOMA system utilizes two quadrature carriers appropriately to improves the overall performance and reduce SIC iterations as compared to RISP-PD-NOMA system. In simple words, this work proposes and examines RISP-PD-NOMA and RISP-Q-NOMA systems for multiple users considering both perfect and imperfect SIC at the Rx. Moreover, the following are the novelty and contributions of this work: 1) This paper proposes two novel systems, viz., RISP-PD-NOMA and RISP-Q-NOMA, for the multi-user downlink network under the Rician fading channel with the goal of maximizing the diversity order. In the proposed work, RIS units are partitioned into several subsurfaces and each RIS subsurface linked to all users. To improve the signal quality of all users and make optimum use of RIS, the phases of each subsurface are modified with respect to BS to RIS and RIS to user channel links. 2) Both direct and reflection links are considered in the system and channel models, and consequently, the closed-form expressions for ASR, OP, and diversity order of both RISP-PD-NOMA and RISP-Q-NOMA are determined under perfect and imperfect SIC conditions. 3) The RISP-Q-NOMA system significantly reduces the number of SIC iterations that reduces the detection delay and decoding complexity compared to RISP-PD-NOMA system. The RISP-PD-NOMA and RISP-Q-NOMA are further compared with RISD-PD-NOMA [39], RISD-NOMA [41], RIS-NOMA [31], and PD-NOMA [17] in terms of ASR and OP. 4) The analytical expression for BER of RISP-PD-NOMA and RISP-Q-NOMA is derived, then the closed-form expression of ABER is determined under the Rician fading channel. All the derived mathematical expressions are verified through Monte-Carlo simulation. These rigorous mathematical analyses and comparisons reveal that the RISP-PD-NOMA and RISP-Q-NOMA systems always outperform RISD-PD-NOMA, RISD-NOMA, RIS-NOMA, and PD-NOMA systems, under both perfect and imperfect SIC operations in terms of ASR and OP. Moreover, in terms of ABER, the performance of RISP-Q-NOMA system is better than that of RISP-PD-NOMA system due to less number of SIC iterations.
The rest of the manuscript is organized as follows. The system and channel models are introduced in Section II. Section III includes evaluation of various performance matrices of RISP-PD-NOMA and RISP-Q-NOMA systems. The numerical and simulation results are discussed in Section IV. Finally, Section V concludes the paper. Throughout the work, c,d represent the expectation operator, absolute value, conjugate, probability function, probability density function (PDF), gamma function, lower incomplete gamma function, factorial, gaussian Q function, error complementary function, gauss hypergeometric function, real and imaginary part of random variable X, and meijer-G function, respectively.

II. PRELIMINARIES
This section first discusses the system and channel model for RIS assisted downlink multi-user network. The system and channel model for multiuser RIS partition based downlink network will then be covered in detail. In the end, the design structure of RISP-PD-NOMA and RISP-Q-NOMA systems will be presented.

A. RIS Assisted Downlink Model
Consider a multi-user downlink wireless network in which K users are served by a single BS and a RIS consisting of N reflector units, as shown in Fig. 1. The h dk , h l , and g l,k represent the fading channel from BS to user k (U k ), BS to l th (l = 1, 2, . . . N) reflecting unit of RIS, and from l th reflecting unit of RIS to U k , respectively, ∀k ∈ K, K = {1, 2, . . . , K}. The fading channel between BS-U k and BS-RIS-U k link is assumed to be Rayeligh and Rician, respectively.
For flat fading and slowly varying channel model, the signal received at U k , after reflection by a RIS, can be expressed as follows [3]: where φ l is the tunable phase applied by the l th element of the RIS, x is the transmitted message symbol, n k is the zero mean complex additive white gaussian noise (AWGN) with variance N o . Moreover, h dk , h l , and g l,k can also be written as , and g l,k = L RU k |g l,k | exp(−jθ g l,k ), where |h dk |, |h l |, and |g l,k | represent the channel amplitude, θ h dk , θ h l , and θ g l,k represent the channel phase, L dk , L BR , and L RU k denote the large scale fading path loss from BS to U k , BS to RIS, and RIS to U k , respectively. The received signal given in (1) can be rewritten in a matrix form as follows: where , and Φ = diag q 1 e jφ1 q 2 e jφ2 . . . ..q N e jφN represents the diagonal matrix that contains the amplitude and phase shift induced by the l th reflecting element of the RIS. More precisely, q l ∈ (0, 1], φ l ∈ [−π, π) and without the loss of generality q l assumed to be 1. In RIS based transmission systems, it is assumed that RIS has perfect knowledge of channel phases of BS-RIS-U k link, which represents the best case scenario for system operation and serves as a performance benchmark for real-world applications [3], [6]. In this configuration, the RIS can adjust its phases with respect to a single user at a given point of time [43]. In such a RIS assisted downlink system, the optimal choice of φ l that maximizes the instantaneous signal to noise ratio (SNR) of U k is given by To serve the multiple users and to enhance the signal quality of all users at the same time, RIS partitioning technique is proposed in this work that will be discussed in the next subsection.

B. RIS Partition Assisted Downlink Model
In the proposed system, to partially enhance the signal quality of all users at the same time, RIS units are partitioned into K subsurfaces as shown in Fig. 2. The k th subsurface, denoted as S k , consists of N k RIS units, maximizes the channel gain of BS-S k -U k by adjusting its phase shifts and making the BS-S k -U k channel phases equal to zero. Therefore, for the proposed RIS partition system, the signal received at k th user Rx from BS-to-RIS-to-U k and from BS directly, can be jointly expressed as follows: h p , and g p,k are the channel link between BS to U k , BS to p th reflecting element of RIS, and p th reflecting element of RIS to U k , respectively, Θ p,s is the tunable phase applied by p th reflecting element with respect to BS-S s -U s channel link. Furthermore, similar to [41], this work does not consider the effect of the phases of h dk since all RIS units are not assigned and aligned to a single user. Therefore, the phases of the RIS units belong to s th subsurface is adjusted as where p ∈ {φ s−1 + 1, φ s−1 + 2, .., φ s }, θ hp , and θ gp,s represent the channel phase from BS to p th reflecting element of RIS and p th reflecting element of RIS to U s , respectively. It is clear from (4) that for the proposed RIS partition system, the phases of first N 1 units of RIS are modified with respect to BS-S 1 -U 1 channel link, the phases of next N 2 units are modified with respect to BS-S 2 -U 2 channel link and so on, and the phases of last N K units are modified with respect to BS-S K -U K channel link. Therefore, for RIS partition system, the modified reflection coefficient matrix can be written as where diag e jΘ1,1 . .
where ω k = h dk + r k + m k . The r k and m k can be given by Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
It is clear from (7) and (8) that k th user only gets N k units of RIS that makes the BS-S k -U k (∀k ∈ K) channel phases equals to zero, while the remaining units have residual phases due to allotment of remaining units to other users. For example, the phases of S 2 subsurface of RIS are modified with respect to BS-S 2 -U 2 . Therefore, only N 2 units of RIS make the channel phases of BS-S 2 -U 2 equals to zero, while the remaining S i (i = 2) subsurfaces create residual phases for U 2 . Please note that the proposed work only requires phase information of the BS-RIS-user channel link, not magnitude information.
To further simplify y k , it is multiplied by where statistics ofn k = n k e k remains same as e k , since |e k | 2 = 1. Furthermore, without the loss of generality, this work assumes perfect channel state information (CSI) at the Rx side. Now, the symbol transmission and detection will be carried out based on the PD-NOMA system used.
Remark 1: In [39], RIS subsurface assigned to one user is not linked to another user, therefore there is no residual link i.e., m k = 0. However, in the proposed work, all RIS subsurfaces are linked to all users, yielding m k = 0.

C. Channel Model
In this subsection, the channel model adopted for the proposed systems will be discussed. As already mentioned in the previous subsection that h dk is characterized as Rayleigh fading channel and both h p and g p,k are characterized by Rician fading channel model, the distribution of h dk , h p , and g p,k can be expressed as follows [44]: where λ represents the Rician factor, R are the small-scale channel fading coefficient variance of h dk , h p , and g p,k , respectively. Similar to [33], variance of the small-scale channel fading coefficient is set to 1. Now, the distribution of |h p | and |g p,k | can be given by [45 where Rice(X, Y ) denotes the Rician distribution with X is the mean of the real part of the Gaussian RV and Y is the standard deviation of real/imaginary part of the Gaussian RV.
In (7), both |h p | and |g p,k | are independently distributed Rician RV, therefore the mean and variance of their product can be obtained, [ Var According to central limit theorem (CLT), r k converges to a Gaussian RV for N K. Furthermore, the mean and variance of r k can be written as follows: Since where (m k ) and (m k ) are given by where x p,k and y p,k are the real and quadrature part of g p,k , since g p,k can be written in a form of g p,k ∼ (x p,k + jy p,k ).
RU k . Similar to r k , the mean and variance of m k , which is the summation of large number of complex Gaussian RV can be determined by From (6), (14), and (16), it is clear that ω k can be written as Therefore, According to [48,Sec. 2.2.2], the PDF of z k = |ω k | 2 , can be approximated using the first term of a Laguerre expansion and can be written as follows: where a k + 1 and b k represent the shape parameter and scale parameter, respectively, with

D. RIS Partition PD-NOMA System
For RISP-PD-NOMA system, the superimposed message symbol transmitted by BS to RIS can be written as follows: where x k and α k represents the transmitted complex symbol and allocated power to U k subjected to the constraint on E |x k | 2 = 1 and K k=1 α k = 1, for ∀k ∈ K, and P is the total transmission power. Similar to (9), the received signal at U k can be expressed as where ω k is given in (6) andn PD k is a complex AWGN with mean zero and variance N o . Without the loss of generality, we assume that σ 2 RU1 > . . . > σ 2 RU k > . . . > σ 2

RUK
such that the k th is closer to the RIS as compared to k + 1, k + 2, . . . , K users. The K th user will decode its information by treating all other users' information as an interference and no SIC is required at the Rx of U K . On the other hand, k th user will first decode the information of the K − k users in order to obtain their own information. More precisely, k th user performs the SIC for U k+1 , U k+2 , . . . U K users' symbol. This results in total of K(K − 1)/2 SIC iterations throughout the system. For perfect SIC, the information of subsequent K − k users are decoded and eliminated from the total received signal. Moreover, for imperfect SIC, the subsequent K − k users have some uncancelled signal parts, which can be seen as a interference for U k . Since the proposed work assumes that the perfect CSI is available at the Rx side, only the imperfect SIC leads to the decoding error. For this case, the interference at U k can be written as follows [19]: where x D andx D represent the actual transmitted symbol and estimated symbol, respectively, and x D −x D denotes the decoding error. Therefore, signal to interference noise ratio (SINR) of k th user's signal at the V th user Rx is where ρ = P/W N o is the SNR, W is the transmission bandwidth in Hz, = E |x D −x D | 2 [19], 0 ≤ ≤ 1, = 0 and = 1 denote perfect and no interference cancellation, respectively. By measuring a large number of samples, the residual interference can be accurately approximated as a Gaussian distribution due to CLT.

E. RIS Partition Q-NOMA System
For RISP-Q-NOMA system, the superimposed message signal transmitted from BS can be expressed as follows: where K is even and term √ 2 presented in x QN is used to make symbol energy equal to 1, since Q-NOMA uses π/4-rotated symbols and E[|x k | 2 ] = 1/2. In RISP-Q-NOMA system, the in-phase and quadrature components of superimposed message signal contain the information of odd users and even users, respectively.
Similar to RISP-PD-NOMA, the received signal at U k for RISP-Q-NOMA is given bȳ wheren QN kI is real-valued AWGN noise with mean zero and variance N o /2. Similarly, if U k is the even user then the Rx of U k only needs the quadrature component ofȳ QN k , i.e., wheren QN kQ is the real-valued AWGN having zero mean and N o /2 variance. Since the information of odd and even users are multiplexed in the in-phase and quadrature component of superimposed message signal, respectively, no SIC is required at K th and (K − 1) th user. However, k th user performs the SIC for This results in a total of K(K −2)/4 SIC iterations throughout the system. For RISP-Q-NOMA system, the SINR of k th user's signal at V th users Rx, can be expressed as follows: where binary factor β D ∈ {0, 1}. When both k and D are even, β D = 1, for (k, D) ∈ {2, 4, ..K}, otherwise zero. Similarly, when both k and D are odd, β D = 1, for (k, D ∈ {1, 3, ..K − 1}), zero otherwise. Remark 2: Due to multiplexing multiple users' information into in-phase and quadrature components of superimposed message signal, the RISP-Q-NOMA system requires half number of SIC iterations as compared to RISP-PD-NOMA system.

III. PERFORMANCE ANALYSIS
In this section, we formulate the analytical expression of ASR, OP, and diversity order for RISP-PD-NOMA and RISP-Q-NOMA systems for both perfect and imperfect SIC operations. Furthermore, the closed-form expressions for BER and ABER are also derived for both the proposed systems.

A. Average Sum-Rate Analysis
The achievable data rate of k th user in bps/Hz for RISP-PD-NOMA system can be obtained by using (21) as Due to proper allocation of power levels among users' signals based on their channel variance, min V ∈{1,2,...,k} SINR PD V,k = SINR PD k,k . After calculating the data rate of each user, the sum-rate of RISP-PD-NOMA system can be determined by Similarly, using (26), the achievable date rate and sum-rate for RISP-Q-NOMA system are Factor 1/2 is multiplied in (30) due to real symbol transmission in RISP-Q-NOMA. 1) ASR of RISP-PD-NOMA System: ASR of PD-NOMA can be evaluated by averaging (28) using (17) , and C PD can be expressed as where where c PD k1 = (1 + A PD k )/B PD k and c PD k2 = 1/B PD k . Solving (33) and (34), we get where τ ∈ {1, 2}. Proof: Refer to Appendix A for the proof. The closed-form expression of ASR for RISP-PD-NOMA system can be obtained by substituting (35) into (32).
2) ASR of RISP-Q-NOMA: ASR of RISP-Q-NOMA can be evaluated by averaging (30) using (17). Similar to (32), we can write the ASR of RISP-Q-NOMA as follows: , (36) can be written where, where (38) and (39), we get Proof: Refer to Appendix A for the proof. The closed-form expression of ASR can be obtained by substituting (40) into (37).
Remark 3: To acquire the minimum data rate supported by any user while maintaining a certain quality of service, error correction coding can also be used at the Rx side [49].

B. Outage Probability
The mathematical framework to determine the OP for the proposed RISP-PD-NOMA and RISP-Q-NOMA systems is given in this subsection.

1) OP of RISP-PD-NOMA:
The OP of the k th user for RISP-PD-NOMA can be obtained as where R is the target data rate in bps/Hz, U PD Now, the overall OP of RISP-PD-NOMA system can calculated as The closed-form expression for OP can be obtained by substituting (42) into (43).

2) OP of RISP-Q-NOMA:
The OP of RISP-Q-NOMA system can be calculated as follows: Using [50,Eq. (8.350)], the OP of k th user is given by Thereby, total OP for RISP-Q-NOMA system can be determined by By substituting (45) into (46), the closed-form expression for OP can be obtained.

C. Diversity Order
Diversity order (Λ k ) of k th user is the asymptotic slope of the OP curve of the respective user with respect to high SNR values. Mathematically, it is given by

1) Diversity Order of RISP-PD-NOMA:
In order to obtain the diversity order of the proposed RISP-PD-NOMA system, we first rewrite (41) as To solve the above integral, we convert the exponential function given in (48) in a form of Maclaurin series as exp (48) can be expressed as By solving (49), the OP of k th user can be written as Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. At high ρ, only the first term (t = 0) in the infinite series of (50) is dominant. Therefore, Substituting (51) into (47) and performing some mathematical manipulations, the diversity order of k th user is given by It is clear from (52) that the diversity order of k th user depends on shape parameter (a k + 1). Furthermore, the total diversity order of RISP-PD-NOMA can be calculated by Λ PD = min(Λ PD k ).

2) Diversity Order of RISP-Q-NOMA:
The diversity order of k th user for RISP-Q-NOMA can be calculated using (44) and following the same procedure as given in (48)-(52), we have The total diversity order of RISP-Q-NOMA system can be calculated by Λ QN = min(Λ QN k ). Remark 4: The diversity order for RISD-PD-NOMA can be obtained by substituting both (m k ) = 0 and (m k ) = 0 in the shape parameter equation. As m k contributes to the a k + 1 and consequently in the diversity order, resulting in an increase in the diversity order of RISP-PD-NOMA and RISP-Q-NOMA systems as compared to RISD-PD-NOMA system.
As already mentioned that the main aim of the proposed work is to obtain the highest diversity order. Figure 3 shows the total diversity order of RISP-PD-NOMA, RISP-Q-NOMA, and RISD-PD-NOMA systems for N = 16 and K = 2. All the parameters that are used to evaluate the total diversity order are given in Section IV. It is evident from the figure that the maximum diversity gain achieved by RISP-PD-NOMA/RISP-Q-NOMA and RISD-PD-NOMA are 11.39 and 6.25, respectively. This performance gap is due to the absence of additional channel link, i.e., m k , in RISD-PD-NOMA. Furthermore, it can also be observed from the figure that maximum diversity gain is attained when an equal number of RIS units are assigned to each subsurface/user, i.e., when N 1 = 8 and N 2 = N − N 1 = 8. A similar observation can also be obtained for K users, where the total diversity order will be maximum when N/K units of RIS are allocated to each user. Further, the same is concluded in [41,. Therefore, all the systems will be compared assuming equal partitioning in Section IV. Furthermore, since the CLT is used to derive various performance metrics, the analysis is valid only for N K. Remark 5: Equal partitioning is best among all other partitioning to guarantee the user fairness.

D. BER Analysis of RISP-PD-NOMA System
In this subsection, we will derive analytical expressions for BER of each user for RISP-PD-NOMA system employing binary phase shift keying (BPSK). To ensure that no symbol from the newly obtained constellation after SC should cross the decision boundary of ML decoding [51], [52], the ., K} must be followed in RISP-PD-NOMA system. In order to find the closed-form expression for the BER of the RISP-PD-NOMA system, it is assumed that the system has 4 users. Let x d k k represents the transmitted symbol of user k, where k ∈ {1, 2, 3, 4} and d k ∈ {1, 2}. For BPSK modulation scheme x 1 k = 0, x 2 k = 1, and these two BPSK symbols are modulated at −1 and 1, respectively. Since BPSK modulation scheme is used for all users, this results in 16 points joint constellation as shown in Fig. 4, that are represented by (η 1 η 2 η 3 η 4 ) joint BPSK symbols (x sc ). The binary bits η 1 , η 2 , η 3 , and η 4 represent the transmitted symbol of U 1 , U 2 , U 3 , and U 4 , respectively. For example, if x 1 1 = 0, then the possible joint BPSK symbols can be written as x sc | x 1 1 ∈ {0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111}. Furthermore, the constellation point of joint BPSK symbols and the points obtained after the SIC process can be defined as where For example, if x 1 1 = x 1 2 = 0 and x 2 3 = x 2 4 = 1, then the joint BPSK symbol x sc = 0011 and δ11 11 Furthermore, for x sc = 0011, if U 3 correctly and incorrectly decode the symbol of U 4 , then δ11 10  1) BER Analysis of U 4 : As per power allocation strategy given in Subsection II-D, more power is assigned to U 4 as compared to U 1 , U 2 , and U 3 . At the Rx of U 4 , the symbols of lower power level users are considered as an interference. Therefore, no SIC operation is required at the Rx side. The total BER of U 4 can be written as where Pr(x d1 , and x d4 4 belong to independent BPSK symbols, Pr denotes the probability of incorrect decoding of U 4 bits i.e., x where Δ k f1f2f3f4 = (n PD k ≥ |ω k |δ f1f2f3f4 ) andn PD k ∼ N (0, N o /2). Solving (56), the total BER for U 4 in terms of Q-function can be written as where Ω k f1f2f3f4 = 2ρ|ω k | 2 δ 2 f1f2f3f4 . 2) BER Analysis of U 3 : The BER at U 3 is formulated differently than the BER at U 4 as it performs SIC for U 4 and treats other users' signal as an interference. Therefore, BER of U 3 can be written as where Pr(x represents the probability of incorrect decoding of U 3 bits i.e., x 3) BER Analysis of U 2 : Since more power is assigned to U 4 and U 3 as compared to U 2 , it will perform the SIC for U 4 and U 3 and treats the symbol of U 1 as an interference. Therefore, the total BER at U 2 can be written as where Pr x

E. BER Analysis of RISP-Q-NOMA
For BPSK modulation scheme, the BER of four user RISP-Q-NOMA system is similar to the BER of two-user RISP-PD-NOMA system, due to multiplexing of users' information into its in-phase and quadrature components. Therefore, using the result given in [53], the BER of U 4 , U 3 , U 2 , and U 1 for RISP-Q-NOMA system can be written as follows: Pr e U3 (z) = Pr e U2 (z) = Pr e U1 (z) = Pr e U2 (z) = In the next subsection, the ABER of RISP-PD-NOMA and RISP-Q-NOMA systems are formulated using (17).
2) ABER of U 3 : Similar to ABER of U 4 , the ABER of U 3 can be determined as On substituting the values of L υ 3 into (69), the closed form expression of ABER for U 3 can be obtained. The ABER of U 1 and U 2 can also be found by following the same procedure as given for U 4 and U 3 .
G. ABER Analysis of RISP-Q-NOMA System 1) ABER of U 4 : The ABER of U 4 can be calculated by averaging (62) using (17) as Using the result of Appendix B, the ABER of U 4 for RISP-Q-NOMA system can be written as where ξ 4 υ denotes the υ th element of ξ = [1,1], and L 1 4 = 2ρδ 2 0101 , L 2 4 = 2ρδ 2 0101 . Similar to ABER of U 4 , the ABER of U 1 , U 2 , and U 3 can also be found by following the same procedure.
Figures 5(a) and 5(b) illustrate the comparison of different systems in terms of ASR for K = 2 and N = 16. It can be seen from the figures that the performance of RISP-Q-NOMA-P and RISP-Q-NOMA-IM is same for all values of P , since RISP-Q-NOMA does not require any SIC operation for K = 2. Furthermore, the performance of RISP-PD-NOMA-P and RISP-Q-NOMA-P/RISP-Q-NOMA-IM is always better than that of RISD-PD-NOMA-P [39], RISD-NOMA [41], RIS-NOMA-P [31], and PD-NOMA-P [17]. For example, ASR of RISP-PD-NOMA-P and RISP-Q-NOMA-P/RISP-Q-NOMA-IM are 19.5 bps/Hz and 18.8 bps/Hz, respectively, at P = 50 dBm, while RISD-PD-NOMA-P, RISD-NOMA, RIS-NOMA-P, and PD-NOMA-P can provide ASR of 17.8 bps/Hz, 7.8 bps/Hz, 12 bps/Hz, and 12 bps/Hz,  respectively. The performance gap between the RISP-PD-NOMA-P and RISD-PD-NOMA-P is due to the presence of a residual channel link m k , which improves the average SNR of RISP-PD-NOMA-P. Under perfect SIC, RISP-PD-NOMA displays approximately 3 dBm better performance than RISP-Q-NOMA. While under imperfect SIC, RISP-Q-NOMA always outperform the RISP-PD-NOMA, RISD-PD-NOMA, RIS-NOMA, and PD-NOMA systems for all range of P , as shown in Fig. 5(b). Therefore, in applications like IoT, it is better to use RISP-PD-NOMA system to enhance the data rate and RISP-Q-NOMA system to reduce the delay and impact of imperfect SIC. Figure 5(  The reason behind this improvement is the lesser number of SIC operations required for RISP-Q-NOMA system. More specifically, for K = 4, RISP-Q-NOMA requires half the number of SIC operations as compared to RISP-PD-NOMA. Therefore, to enhance the data rate under imperfect SIC operation, it is better to use RISP-Q-NOMA. Furthermore, in the given figure, subsurface assignment 1 (S-A1) and S-A2 correspond to S 1 /U 1 , S 2 /U 2 , S 3 /U 3 , S 4 /U 4 and S 1 /U 4 , S 2 /U 3 , S 3 /U 2 , S 4 /U 1 , respectively. Therefore, it is interesting to note from the figure the overall system performance is not affected by the change in the subsurface assignment. It can also be noticed from Figs. 5(a), 5(b), and 6 (a) that the analytical and simulation curve of ASR are closely matched since no bound or inequality is employed in Subsection III-A.
The OP comparison of RISP-PD-NOMA and RISP-Q-NOMA with RISD-PD-NOMA, RISD-NOMA, RIS-NOMA, and PD-NOMA is presented in Fig. 6(b) under perfect ( = 0) and imperfect SIC ( = 0.1), respectively, for K = 2, N = 16, and R = 1 bps/Hz. It is interesting to note from the figure that the simulation and analytical curves of OP are perfectly matching. The reason behind this matching is the accurate channel modeling for large N . Further, it can also be observed from the figure that the under perfect SIC, the performance of RISP-PD-NOMA is superior to all other systems. For example, RISP-PD-NOMA-P requires 2 dBm, 9 dBm, and 45 dBm lower transmit power as compared to RISP-Q-NOMA-P, RISD-PD-NOMA-P, and RIS-NOMA-P, respectively, to get the OP ≤ 10 −3 . Whereas, RISD-NOMA, saturates at 25 dBm and for PD-NOMA-P is 5 × 10 −3 . Moreover, under imperfect SIC, the performance of RISP-Q-NOMA is better than that of all other systems. For example, RISP-Q-NOMA-IM require 2 dBm and 11 dBm lower transmit power as compared to RISP-PD-NOMA-IM and RISD-PD-NOMA-IM, respectively, for OP< 10 −4 . While RIS-NOMA-IM and PD-NOMA-IM are not able to attain OP< 10 −4 up to 50 dBm. Figure 6(c) shows the OP of RISP-PD-NOMA and RISP-Q-NOMA for different N with K = 2 and = 0. It can be observed from the figure that under perfect SIC operation, RISP-PD-NOMA always performs superior to RISP-Q-NOMA for all values of N . Therefore, it can be concluded that for K approaches N and for N > K, the performance trend between RISP-PD-NOMA and RISP-Q-NOMA remains unchanged. It is also evident from the figure that as N increases, the gap between the analytical and simulation results decreases. For N = 16, simulation results closely matches the analytical results. This is because the CLT approximation employed in Subsection II-C is valid only for large N or N K. Figure 7(a) illustrates OP curves of RISP-PD-NOMA and RISP-Q-NOMA for K = 4, N = 32, and R = 0.7 bps/Hz with = 0 for perfect SIC and = 0.1 for imperfect SIC. It can be noticed from Fig. 7(a) that under perfect SIC condition, RISP-PD-NOMA outperforms the RISP-Q-NOMA, whereas trend is reversed under imperfect SIC. For example, RISP-PD-NOMA-P and RISP-PD-NOMA-IM require 0.8 dBm lower and 2 dBm higher transmit power as compared to RISP-Q-NOMA-P and RISP-Q-NOMA-IM, respectively, to achieve the OP < 10 −4 . Therefore, to reduce the impact of imperfect SIC, it is better to use RISP-Q-NOMA. Remark 7: RISP-PD-NOMA and RISP-Q-NOMA always outperform RISD-PD-NOMA because of the increase in average SNR that comes from m k = 0. Figures 7(b) and 7(c) show the ABER curves of U 1 , U 3 and U 2 , U 4 , respectively. It can be seen from the figures that the analytical curves match exactly with the simulation curves since binary modulation scheme is employed. It can also be deduced from Figs. 7(b) and 7(c) that the smallest ABER over the entire P region is corresponding to U 3 and U 4 , for both systems. However, RISP-Q-NOMA outperforms the RISP-PD-NOMA over the entire P range. For example, to get ABER = 10 −4 , U 3 and U 4 of RISP-Q-NOMA requires 4 dBm and 3.5 dBm lower transmit power as compared to U 3 and U 4 of RISP-PD-NOMA, respectively. Furthermore, the error performance of U 1 for both systems are almost similar at higher P due to the domination of term Q(Ω 1 1000 ) that is present in the BER equation of both systems. It is interesting to note from Figs. 7(b) and 7(c) that the error performance of U 4 is better than U 1 , U 2 , and U 3 because of highest power allocation, for both RISP-PD-NOMA and RISP-Q-NOMA systems. Therefore, due to less number of SIC operations, RISP-Q-NOMA system outperforms the RISP-PD-NOMA system in terms of ABER.
Remark 8: Under perfect SIC, RISP-PD-NOMA outperforms all other systems, whereas under imperfect SIC, the performance of RISP-Q-NOMA is better than that of all other systems.

V. CONCLUSION
In this work, two RIS partition-assisted PD-NOMA systems, namely RISP-PD-NOMA and RISP-Q-NOMA, have been proposed with the aim to maximize the diversity order.