I Introduction
The demanding for having high spectrum efficiency (SE) and energy efficiency (EE) has been rapidly increasing in nextgeneration (NG) networks [2]. A promising access technique, nonorthogonal multiple access (NOMA), has been proposed, which is capable of simultaneously serving multiple users at different qualityofservice requirements [3, 4, 5]. Multiple users share the same time/frequency/code resource block by with the aid of superposition coding (SC) at the transmitter and successive interference cancellation (SIC) at the receiver by capitalizing the difference of users’ channel state information (CSI) [6, 7, 8].
Since multiple antennas (MAs) offer extra diversity by its spacial domain, MA techniques are of significant importance. The application of MA techniques assisted NOMA networks has attached significant attention. In classic multipleinput multipleoutput (MIMO) designs, the base (BS) equipped with transmitting antennas (TAs) is capable of transmitting maximal interferencefree beams. However, since multiple users are paired to perform NOMA in each cluster, the BS equipped with TAs has to compulsorily serve users simultaneously, where denotes the number of users in each cluster. Hence, how to design interferencefree beamforming becomes an interesting problem, which is valuable to examine. A zeroforcingbased (ZFbased) design was proposed in [9]
, where the active beamforming at the BS is an identity matrix. However, there are two drawbacks, where 1) the number of receiving antennas (RAs) has to be higher than that of the BS to ensure the existence of a solution; and 2) due to the property of singular value decomposition, the antenna gain of the ZFbased design can be obtained as
, where denotes the number of RAs. Then a signalalignmentbased (SAbased) design is proposed to release the constraint of the number of RAs [10], which relies on the gain shifting of user’s channel matrixes. By doing so, the smallscale channel gains of users in each cluster are identical, but correspond to different distances, and hence users can be treated as users. It is demonstrated that multiuser NOMA networks may be not practical due to high computational complexity [11]. Numerous applications related to MIMONOMA were proposed, i.e. MIMONOMA enhanced physical layer security networks [12] and MIMONOMA enhanced simultaneous wireless information and power transfer (SWIPT) networks [13]. However, as mentioned above, there were many constrains on the number of RAs in the previous designs.Recently, reconfigurable intelligent surface (RIS) technique stands as the next generation relay technique, also namely relay 2.0, received considerable attention due to its high EE [14, 15, 16, 17]. The RIS elements are capable of independently shifting the signal phase and absorbing the signal energy, and hence the reflected signals can be boosted or diminished for wireless transmission [18, 19, 20]. By doing so, numerous application scenarios have been considered, e.g. RISaided coverage enhancement. The RIS elements are normally deployed on the building or on the wall [21]. A novel threedimension design for aerial RIS network was proposed in [22], where RIS elements are employed at aerial platforms, and hence a fullangle reflection can be implemented. Currently, RIS networks are simply separated into two categories [23], i.e. anomalous reflector or diffuse scatterer for mmWave and sub6G networks, respectively. The coverage distance is reduced in mmWave networks [24], and hence more users are located in coverageholes compared to conventional networks. Thus, reflected signals can be aligned by RISs for serving users located in the coverageholes [25].
NOMA and RIS techniques can be naturally integrated for enhancing both SE and EE. The RISs can be deployed for the celledge users in the NOMA networks, where the reflected signal cannot be received at the cellcenter users [26]. An onebit coding scheme was invoked in the RISaided NOMA networks, where imperfect SIC scenario was evaluated in [27]. Since both the BS and RISs are predeployed, and hence the lineofsight (LoS) links between the BS and RISs are expected for improving desired signal power [28]. The Rician fading channels were utilized for illustrating the channel gain of both the BSRIS and RISuser links in [29]. A SISONOMA network was proposed in [30], where a prioritized design was proposed for further enhancing the network’s SE and EE. However, previous contributions mainly focus on the signal enhancement based (SEB) designs, where signals are boosted at the user side or at the BS side.
Ia Motivations and Contributions
Previous contributions mainly focus on the SEB designs, whilst there is a paucity of investigations on the signalcancellationbased (SCB) design of the RISaided networks. Inspired by the concepts of the signal cancellation [31, 32], we propose a novel SCB design concept, which provides the desired degree of flexibility for the RISaided networks. In the MIMONOMA networks, one of key challenge is to eliminate the intercluster interference. Hence, in order to illustrate the potential benefits provided by RISs, a RISaided SCB design in MIMONOMA networks is proposed for comprehensively analyzing the performance of the networks.
Against to above background, our contributions can be summarized as follows:

We propose a novel SCB design in RISaided MIMONOMA networks, where the intercluster interference can be eliminated for enhancing the network’s performance. The impacts of both the diffuse scattering and anomalous reflector scenarios are exploited. The impact of the proposed design on the attainable performance is characterized.

We first derive the minimal required number of RISs for implementing the proposed SCB design. For the idealRIS (IRIS) cases, our analytical results illustrate that the intercluster interference can be beneficially eliminated. We then evaluate the impact of nonidealRIS (NIRIS) cases by finite resolution techniques.

Explicitly, we derive closedform expressions of both the OP and of the ER for the proposed SCB design. The exact closedform expressions of the OP and of the ER are derived in both the diffuse scattering and anomalous reflector scenarios. The diversity orders and highSNR slopes are derived based on the OP and ER. The results confirm that the diversity order is obtained as the number of RAs.

Our analytical results illustrate that the proposed SCB design is capable of 1) releasing the constraint of the number of RAs; 2) perfectly eliminating the intercluster interference in the IRIS cases. The simulation results confirm our analysis, illustrating that: 1) the outage floors and ergodic ceilings occur for the NIRIS cases; 2) the proposed SCB design does not rely on a high number of RISs for increasing both the OP and ER. 3) the proposed SCB design is capable of outperforming the classic ZFbased and SAbased designs.
IB Organization and Notations
In Section II, a SCB design is investigated in RISaided MIMONOMA networks. The minimal required number of RISs is analyzed. Then the passive beamforming weight at RISs is proposed for implementing SCB design. Analytical results are presented in Section III to show the performance of the proposed SCB design. In Section IV, the numerical results are provided for verifying our analysis. Then Section V concludes this article. denotes the probability, and denotes the expectation. , , denote the transpose and conjugate transpose, as well as rank of the matrix . denotes the Frobenius norm. Table I lists some critical notations used in this article.
The target rate of user in cluster .  

Fading factor between the BS and RISs.  
Fading factor between the RISs and users.  
The number of TAs.  
The number of RAs.  
The number of users in each cluster.  
The number of RISs.  
The channel matrix of the BSRIS link.  
The channel matrix between the RISs and user in cluster .  
The channel matrix between the BSs and user in cluster .  
The effective matrix of RISs. 
TABLE OF NOTATIONS
Ii System Model
Let us focus our attention on a MIMONOMA network in downlink communication, where a BS equipped with TAs is simultaneously communicating with users each equipped with RAs by utilizing the powerdomain NOMA techniques. For simplicity, users are separated into clusters, and users are paired in each cluster. We have intelligent surfaces at a proper location with for implementing wireless communication. By properly controlling the amplitude coefficients and phase shifts of each RIS element, the signals can be beneficially manipulated. The system model is illustrated in Fig. 1.
Iia System Description of RISaided MIMONOMA Networks
Since the LoS link does not exist in the BSuser link, the smallscale fading matrix between the BS and user in cluster is defined by Rayleigh fading channels, which can be expressed as
(1) 
where is a
element matrix. Note that the probability density function (PDF) of the Rayleigh fading channel gains is given by
(2) 
The largescale fading between the BS and user in cluster is given by
(3) 
where denotes the path loss exponent between the BS and user in cluster .
Since the LoS links are expected for both the BSRIS and RISuser links, the channel matrixes are defined by Rician fading channels as follows:
(4) 
where is a Rician fading channel gains, which can be modeled as follows:
(5) 
where denotes the Rician factor of the BSRIS link. and denote the LoS and NonLoS (NLoS) components, respectively.
Similar to (4), the smallscale fading matrix between the RISs and user in cluster is defined as
(6) 
where is matrix whose elements represent Rician fading channel gains with fading parameter . Similarly, the channel gains of the RISuser link is given by
(7) 
where denotes the Rician factor of the BSRIS link. and denote the LoS and NLoS components, respectively.
In this article, and represent the distances of the BSRIS and RISuser links. In order to limit the intercluster interference, the received signal power levels of both the BSuser and reflected links are expected to the same order. Recently, anomalous reflecting and diffuse scattering scenarios are presented for the mmWave and sub6G networks, respectively [33]. On the one hand, the size of RISs is comparable with the wavelength for the sub6G networks, and hence the RISs are expected to be diffusers. Such as radar networks, the path loss of reflected links are expected as productdistance law [34]. On the other hand, the wavelength is sufficiently small compared with the size of RISs, hence the theory of geometric optics is capable of modeling the path loss, where the sumdistance law of a specular reflection holds in the anomalous reflector scenarios based on the generalized Snell’s law. We then study the feasibility of two scenarios, where the RISs act as anomalous reflectors or diffuse scatterers.
1) Diffuse Scattering Scenario: When the size of RISs is comparable with the wavelength, where the size of RIS elements is usually set to wavelength, the RISs are considered as diffuse scatters [33]. Therefore, the largescale fading between the BS and user through RISs can be expressed as
(8) 
where and denote the path loss exponent of the BSRIS and RISuser links, respectively. We then discuss the minimal required number of RISs for the diffuse scattering scenario in the following Lemma.
Lemma 1.
Assuming that the smallscale fading environments of BSRIS, RISuser and BSuser links are strong enough, i.e. , where the elements in fading matrixes are all one. Hence, the following constraint in diffuse scattering scenario needs to be met for implementing the proposed SCB design:
(9) 
Proof.
In order to limit the intercluster interference, the power level of the reflected signals need to be higher than that of the BSuser links. Thus, the following constraint needs to be met [35]:
(10) 
After some algebraic manipulations, the results in (9) can be readily obtained. Thus, the proof is complete. ∎
2) Anomalous Reflector Scenario: We the turn our attention to the anomalous reflector scenario, where the frequency of wave is usually sufficiently high, the largescale fading can be considered as anomalous reflectors [33]. Therefore, the largescale fading between the BS and user through RISs can be expressed as
(11) 
If , the largescale fading can be simplified to
(12) 
Lemma 2.
Let us assume that the smallscale fading environments of BSRIS, RISuser and BSuser links are strong enough, i.e. , where the elements in fading matrixes are all one. Hence, the following constraint in anomalous reflector scenario needs to be met for implementing the proposed SCB design:
(13) 
Then we conclude the feasibility of two alternatives scenarios in Table II, and we set m and m. represents the minimal required number of RISs.
RIS Mode  
Diffuse scattering  
Anomalous reflector  
FEASIBILITY ANALYSIS
Remark 1.
The results in TABLE II demonstrate that the proposed SCB design is only applicable for the case of and in the diffuse scattering scenario.
Remark 2.
The results in TABLE II demonstrate that the proposed SCB design can be beneficially implemented in the anomalous reflector scenarios for the case that both the BSRIS and RISuser links experience Rayleigh fading channels.
IiB Passive Beamforming Designs
We first pay our attention to the signal model, and the information bearing vector at the BS can be expressed as:
(14) 
where denotes the signal intended for user in cluster . represents the power allocation factor for user . Based on the NOMA protocol, we have .
Without loss of generality, we focus on user in cluster . Thus, the receiving signal at user in cluster is given by
(15) 
where denotes the transmit power at the BS, denotes the precoding matrix,
denotes both the effective phase shifts and amplitude coefficients by RISs. More specifically, denotes the amplitude coefficient of RIS element , . denotes the phase shift by RIS element . Finally, the additive white Gaussian noise (AWGN) is denoted by
, which is a zeromean complex circularly symmetric Gaussian variable with variance
. In order to implement the proposed SCB design, the CSIs are assumed to perfectly known at the RIS controller [36].In practice, user in cluster applies a detection vector to its received signals, therefore the user’s observations is given by:
(16) 
In order to provide a general framework, we assume that the active beamformer weights at the BS obey:
(17) 
where represents a identity matrix. The detection vectors of users can be expressed as all one vector as:
(18) 
We then turn our attention to the passive beamforming design at the RISs, where the phase shifts and reflection amplitude coefficients are jointly manipulated. In this article, the passive beamforming design at RISs mainly focuses on interference cancellation, and hence we first remove the th column of the matrix as follows:
(19) 
where denotes the th column of the matrix . Since the elements in the active beamforming weight and detection vectors are all one, the effective intercluster interference of all users can be expressed as follows:
(20) 
where denotes an all one vector.
In order to design the passive beamforming weight at RISs, we first define an effective matrix in the diffuse scattering scenarios by stacking the channel gains of all users as follows:
(21) 
where is a element matrix. Then we define an effective RIS vector , which is an effective vector. Hence, in order to limit the intercluster interference at each user in the diffuse scattering scenarios, the following constraint needs to be met:
(22) 
To achieve the ambitions design objective, the solution of RISs can be given as follows:
(23) 
Similarly, let us define an effective matrix for the anomalous reflector scenarios as follows:
(24) 
Then, the RISs can be designed as follows:
(25) 
Note that we have , thus for the case of , there exists no solution for passive beamforming at RISs, which satisfy the constraints of and , .
Remark 3.
Lemma 3.
Remark 4.
In this article, users are paired to perform NOMA in each cluster, and thus based on the proposed passive beamforming design at RISs, the signaltointerferenceplusnoise ratio (SINR) of user in cluster for the IRIS cases in both the diffuse scattering and anomalous reflector scenarios can be expressed as
(28) 
where denotes the effective channel gain of user in cluster , which will be evaluated in the next section.
IiC RIS Designs for Finite Resolutions
In most previous research, perfect RIS assumption was assumed, and hence the amplitude coefficients and phase shifts are continuous. However, in practice, the amplitude coefficients and phase shifts rely on the diodes employed at RISs [36]. Thus the discrete phase shifts were considered [37]. Both perfect and imperfect phase shifters were considered in [38, 39]. In this article, the amplitude coefficients and phase shifts may be not continuous due to the hardware limitations. We then consider an alternative lowcost implementation for applying multibit control to RISs, i.e. each diagonal element of is selected from a set of discrete finite resolutions. It is also worth mentioning that several amplitude dividers may be employed at the RIS elements, whereas the current hardware design of RISs only contains several phase shifters. As such, the RIS performs a linear mapping based on an equivalent amplitudecoefficient vector as well as phaseshift vector. It is assumed that the phase shifts and amplitude coefficients at RISs take a finite number of discrete values. The number of bits is used to indicate the number of phase shift and amplitude coefficient levels with , and hence the discrete phase shift and amplitude coefficient values can be uniformly mapped to the interval and , respectively. Thus, the set of discrete amplitude coefficient as well as phase values at each RIS element can be given by
(29) 
and
(30) 
where and . The effective phase shifts and amplitude coefficients matrix need to be selected from the above two sets in (29) and (30). Based on the proposed design at RISs, the interference residue at user in cluster can be transformed into
(31) 
Hence, the SINR of user in cluster for the NIRIS cases can be given by
(32) 
Iii Performance Evaluation
In this section, new channel statistics, OPs, ERs, SE and EE are illustrated in the IRIS cases.
Iiia New Channel Statistics
In this subsection, new channel statistics are derived for the proposed RISaided SCB design in a MIMONOMA network, which will be used for evaluating the network’s performance.
Lemma 4.
Assuming that the fading channels of the BSuser links follow Rayleigh distribution. The elements of channel gains are independently and identically distributed (i.i.d.). The distribution of the effective channel gain of user in cluster for the IRIS cases can be given by
(33) 
where
represents the Gamma distribution.
Proof.
Please refer to Appendix A. ∎
Then the PDF of the effective channel gain can be expressed as
(34) 
IiiB OP and ER
We first focus on analyzing the OP of both the diffuse scattering and anomalous scattering scenarios. In this article, user needs to decode the signals of the farer users by SIC technique, i.e. to users, and hence the OP of user in cluster is defined by
(35) 
where denotes the target rate of user in cluster , and .
Then the OP of user in cluster is given in the following Theorem.
Theorem 1.
We then focus on the diversity orders of user in cluster , which can be obtained for evaluating the slope of OP.
Proposition 1.
From Theorem 1, the diversity orders for the IRIS cases can be determined by expanding the lower incomplete Gamma function, and the diversity order of user in cluster of the proposed RISaided SCB design can be given by
(39) 
Proof.
Please refer to Appendix B. ∎
Remark 5.
Based on results in (39), it is indicated that the diversity orders of all the NOMA users can be approximated to the number of RAs for the IRIS cases when the number of RISs is high enough.
We then turn our attention to the ER of user in cluster , which is a salient metric for performance analysis, and hence the approximated ER expressions for user in cluster is given in the following Theorem.
Theorem 2.
When the number of RISs is sufficiently high, and with , the ER of user in cluster can be expressed in the closedform as follows:
(40) 
where .
Proof.
Please refer to Appendix C. ∎
Furthermore, the SINR of user in cluster may approach in the highSNR regimes [40] based on the SINR analysis in (28). Hence, the expected rate of user for can be written as , which is a constant.
The highSNR slope is defined as the asymptotic slope of the logarithmic plot of ER against the transmit power in dBm, which is a key parameter determining the ER in the highSNR regimes, and hence the highSNR slope can be expressed as
(41) 
where .
Proposition 2.
Remark 6.
Based on the results in (42), one can know that the highSNR slopes of user in each cluster of the IRIS cases are one, which is not a function of the number of RISs.
Remark 7.
Remark 8.
Based on the SINR analysis in (32), the diversity orders and highSNR slopes of all the NOMA users are 0 in the NIRIS cases.
We then compare the OP of the proposed SCB enhanced MIMONOMA network and its OMA counterparts in the following Corollary, i.e. TDMA. The OMA counterparts adopted in this article is that by dividing the users in equal time slots.
Corollary 1.
Let us assume that multiple users are divided in equal time slots in the OMA counterparts, the closedform OP expression of user in cluster of the IRIS cases is given by:
(43) 
where , and .
IiiC SE and EE
Here, we focus on the SE of cluster , which can be formulated based on the ER analysis in the previous subsection.
Proposition 3.
In the proposed SCB design, the SE of cluster can be given by
(45) 
Since RISs are passive equipment, where only RIS controller needs power supply [41, 42, 43], hence we model the total dissipation power of the proposed SCB design as
(46) 
where and denote the power consumption and the efficiency of power amplifier at the BS, respectively. and denote the power consumption of each user and each RIS controller, respectively. Hence, the EE of the proposed design is given by the following Proposition.
Iv Numerical Results
In this section, numerical results are provided for the performance evaluation of the proposed SCB design. Monte Carlo simulations are provided for verifying the accuracy of our analytical results. The transmission bandwidth of the proposed network is set to MHz. In practice, the power of the AWGN is related to the bandwidth, which can be modeled as dBm. For simplicity, the number of TAs is set to , and the number of users is set to . Based on NOMA protocol, the paired users share the power with the power allocation factors and . The fading factors are set to . The distance of the BSRIS links is set to m, and the distance of the RISuser links are set to m and m and those of the BSuser links are set to m and m. The path loss exponents of the BSuser links are set to while those of the BSRIS and RISuser links are set to . The target rates are and bits per channel use (BPCU), unless otherwise clarified.
1) Minimal Required Number of RISs: In Fig. 2, we evaluate the minimal required number of RISs for implementing the proposed SCB design. On the one hand, observe that in the anomalous reflector scenario, the minimal required number of RISs is only impacted by the number of TAs, RAs as well as the number of users. On the other hand, we can see that as the path loss exponent increases, the minimal required number of RISs increases for the diffuse scattering scenarios. This phenomenon indicates that the LoS links of both the BSRIS and RISuser links are required for implementing the proposed SCB design in the diffuse scattering scenarios, whereas the LoS links are not necessary for the anomalous reflector scenarios. Observe that for the case of as well as , the minimal required RISs are identity, which indicates that the diffuse scattering scenario is more susceptible to the path loss exponent of both the BSRIS and RISuser links.
2) Impact of the Number of RAs: In Fig. 3, we evaluate the OP of the proposed SCB design in the RISaided MIMONOMA networks. We can see that as the number of RAs equipped at each user increases, the OP decreases. There are two reasons, where 1) since the precoding matrix is an identity matrix, and the detection vector is an all one vector, the RISs are capable of beneficially eliminating the intercluster interference in the IRIS cases; 2) the received signal power can be significantly increased as more RAs are employed. One can observe that the slopes of the curves are approximated to the number of RAs, which validates our Remark 5.
3) Impact of the Number of Bits: In Fig. 4, we evaluate the OP of the paired NOMA users in the different number of bits, where the OP of the paired NOMA users in the IRIS cases is provided as the benchmark schemes. We can see that as the transmit power increases, the OP floors occur. Observe that as the number of bits increases from 3bit to 6bit, the OP can be beneficially decreased. This is due to the fact that the higher number of bits is capable of increasing the resolution of each RIS element. It is also worth noting that 6bit resolution is enough for obtaining the nearminimal OP, which indicates that the minimized OP is obtainable by appropriate setting the number of bits. Based on the simulation results, the diversity orders of the NIRIS cases are zero, which verifies the insights gleaned from Remark 8.
4) Comparing ZFbased and SAbased Designs: In Fig. 5, combined with the insights inferred from [9, 10], we compare the OP of the paired NOMA users in the proposed RISaided SCB design, ZFbased design as well as SAbased design. In order to provide further engineering insights, we consider that the phase shifts and amplitude coefficients of RISs can be perfectly manipulated. On the one hand, one can observe that the proposed RISaided design is capable of outperforming both the classic ZFbased and SAbased designs. On the other hand, the diversity order of the proposed RISaided design is , which is higher than both the classic ZFbased and SAbased designs, and hence illustrate the benefits of the proposed RISaided SCB design. The detail of diversity orders is concluded in TABLE III.
Mode  TAs  MRN of RAs  Antenna Gain 

ZFbased [9]  
SAbased [10]  
RISaided SCB design 
COMPARISON BETWEEN RISAIDED SCB, ZFBASED AND SABASED DESIGNS. “MRN” DENOTES “MINIMAL REQUIRED NUMBER”.
5) Comparing RISaided NOMA and OMA networks: In Fig. 6, we evaluate the OP of both the RISaided NOMA and OMA networks. The OPs of NOMA and OMA networks are derived by and , respectively. As can be seen from Fig. 6, the OP of the RISaided NOMA networks is lower than that of the RISaided OMA networks, which implies that RISaided NOMA network is capable of providing better network performance than its OMA counterpart. Observed that for both the 3bit and 4bit resolutions, an optimal point exists due to the fact that there is a cross point of curves in the proposed SCB design. This indicates that the RISaided hybrid NOMA/OMA networks may be a good solution.
6) Impact of the Number of Bits on ER: We then evaluate the ER of the paired NOMA users versus the transmit power with the different number of bits in Fig. 7. Observe that as the transmit power increases, the ER ceilings occur in the NIRIS cases. One can also observe that as the number of bits increases, the ER gaps between the IRIS and NIRIS cases are getting smaller. One can also observe that for the case of , the ER of both the NIRIS and IRIS cases are nearly identical, which indicates that the 5bit finite resolution is high enough for the proposed RISaided SCB design. It is also worth noting that the ER of user 1 in cluster is constant with the different number of bits, which also verifies the insights gleaned from Remark 7. Observed that for the 3bit and 4bit resolutions, there exists an optimal number of RISs for maximizing the SE. This phenomenon also indicates that the RISs can be activated appropriately for enhancing the network’s SE in the NIRIS cases.
7) Spectrum Efficiency: In Fig. 8, we evaluate the SE of cluster of the proposed RISaided SCB design. On the one hand, observe that the SE improves as the transmit power increases. However, as the increase of the number of RISs, observe that the slope of the SE is negative of the 3bit cases, which indicates that there exists an optimal value of the number of RISs that maximizes the SE. It is also worth noting that the SE of the 4bit cases are nearly identical compared to the 3bit cases, which indicates that the RISaided SCB design with 4bit finite resolutions becomes more competitive.
8) Energy Efficiency: We then evaluate the EE of the proposed SCB design in Fig. 9. Several observations can be concluded as: 1) One can observe that the slopes of EE are negative in the NIRIS cases. This is due to the fact that in the proposed SCB design, the main goal of passive beamforming is to eliminate the intercluster interference, and hence the maximum EE is fixed, which leads to the low EE when the number of RISs is too high; 2) The EE increases as the number of bits increase, this is due to the fact that the intercluster interference residue decreases when there are more potential solutions at RISs; and 3) Based on the fact that the EE curve decreases, it is readily to observe that there exists an optimal value of the number of RISs.
V Conclusions
We first reviewed previous contributions related to the RISaided SEB designs in this article, and we then proposed a novel RISaided SCB design. In order to provide a general design of RIS networks, we adopted a MIMONOMA network, where the passive beamforming weights of the RISs were designed. The channel statistics, OPs, ERs, SE as well as EE were derived in closedform for characterizing the system performance. Compared to the previous ZFbased and SAbased designs, the proposed RISaided SCB design releases the constraint of the number of RAs. An important future direction is to minimize the required number of RISs and RAs by jointly designing both the active beamforming, passive beamforming, and detection vectors.
Appendix A: Proof of Lemma 4
Recall that the effective channel vector contains elements, and by utilizing the proposed detection vector at the users, we can obtain the following channel gain:
(A.1) 
Based on the result derived in (A.1), and exploiting the fact that the elements of
are i.i.d., the mean and variance of the effective channel gain can be given by using the property of random variables as follows
(A.2) 
and
(A.3) 
Thus, the effective channel gain can be rewritten as
(A.4) 
Hence, the effective channel gain can be obtained in a more elegant form in (33).
Appendix B: Proof of Proposition 1
In order to glean the diversity order of the proposed SCB design, the lower incomplete Gamma function can be expanded as follows [44]:
(B.1) 
When replacing by its power series expansion, the OP can be further transformed into
(B.2) 
Thus, by applying the definition of diversity order, the results in (39) can be gleaned, and the proof is complete.
Appendix C: Proof of Theorem 2
The proof starts by expressing the ER of user in cluster as follows:
(B.1)  
Based on expression in (B.1), the lower incomplete Gamma function ought to be expanded with a constant for obtaining the closedform expressions, hence we expand the lower incomplete Gamma function as follows:
(B.3) 
Thus, the ER can be written as
(B.4) 
Hence, the tractable approximated results can be derived as
(B.5) 
Hence, the ER of user in cluster is obtained in (40), and the proof is complete.
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