Analysis of Individual User Data Rate in a TDMA-RIS-NOMA Downlink System: Beyond the Limitation of Conventional NOMA

Abstract

Non-orthogonal multiple access (NOMA) is playing a pivotal role in 5G technology and has the potential to be useful in future developments beyond 5G. Although the effectiveness of NOMA has largely been explored in the sum throughput maximization, the identification of individual user data rate (IDR) still remained an unexplored area. Previously, it has been shown that reconfigurable intelligent surfaces (RIS) can lead to an overall improvement in the data rate by enhancing the effective channel gain of the downlink NOMA system. When time division multiple access (TDMA) is clubbed with multiple RISs in a distributed RIS-assisted NOMA (TDMA-RIS-NOMA) downlink system, a point-to-point communication model is created between access point-to-RIS-to-user device. Due to this point-to-point communication model, optimization of the phase shifts provided by meta-atoms of each RIS is facilitated. The optimized phase shifts of meta-atoms maximize the equivalent channel gain between the access point to the user. In this scenario, the channel becomes saturated and signal-to-interference plus noise ratio (SINR) becomes a function of power coefficients only. In this study, the power coefficients are calculated to maximize the SINR of each user belonging to a NOMA cluster using a geometric progression-based power allocation method such that IDR reaches its upper bound. These observations are also verified using the recently published magic matrix-based power allocation method. There are two observations from this study: (i) the IDR is better in the case of the TDMA-RIS-NOMA downlink system than using downlink NOMA alone and (ii) irrespective of the number of meta-atoms and total cluster power, the upper bound of IDR cannot be increased beyond a certain limit for all users except the highest channel gain user. Because of the restricted upper bound for IDR, we suggest that the RIS-assisted downlink TDMA-NOMA system is more suitable for IoT applications, where minimum IDR can also suffice

1. Introduction

Non-orthogonal multiple access (NOMA) is a suitable technique to achieve the goals of the massive machine-type communication (mMTC) in the 5G wireless networks [1]. NOMA serves multiple users in the same frequency band and time slot by assigning different transmit powers [2]. At the base station (BS) of the downlink NOMA system, a composite signal is generated by superimposing the constellations of all users belonging to any particular cluster [3]. At the receiver’s end, the desired signal for each user is detected from the composite signal using the successive interference cancellation (SIC) technique [3]. By using NOMA, we can serve the massive connecting devices simultaneously, which is not possible if the network resources are shared in orthogonal ways [1]. However, in NOMA, the transmit power levels for all users in a cluster are required to be sufficiently high to mitigate the AWGN noise and the interference due to signals of other users present in the same composite signal [3]. Furthermore, the stand-alone NOMA downlink system does not have any control over the channel gain between BS and the users [4]. If the channel quality is poor between the source and destination (due to long distances or obstacles), then there is no alternate path for communication to occur [4]. One way to resolve the problem of poor channel quality is addressed in [5], where the author has considered the NOMA downlink system for the relay-to-relay cooperative communication as a solution. Another way to handle these challenges in the downlink NOMA system is by using the RIS [6].

The RIS is an array of a large number of passive reflecting meta-surfaces (called meta-atoms). The electromagnetic properties of these meta-atoms can be reconfigured to control the amplitude and phase shift of the reflected signal from each meta-atom to focus the beam at the desired location [7]. Hence, RIS can control the radio environment intelligently [8]. The main advantages of using RIS are as follows: (i) Each meta atom contributes to a separate path from RIS to the user [9]. (ii) By adjusting the phase shift provided by the meta-atom to the reflected signal, constructive interference condition is achieved among the signals received from the different meta-atoms towards the desired user in a point-to-point communication model. This improves the SNR for the received signal. Consequently, the data rate of the user also increases. (iii) Another advantage of RIS is that if the direct path from BS to the user is blocked, then the system has a better coverage area by more indirect wireless paths from BS-RIS-User [7].

From the current literature on NOMA, it has been observed that sum data rate (SR) maximization is a common problem to encounter [10,11]. But, the problem with the SR in the NOMA downlink system is that when power allocation is conducted to maximize the SR, the more and more the data-rate is increased for the highest channel gain (CG) user and the data rates of lower CG users do not increase significantly. During the literature survey, we found that in the stand-alone NOMA downlink system, although optimization work has been conducted to maximize SR, the lower CG users could not achieve the upper bound of their individual data rate (IDR).

The drawback of a stand-alone NOMA downlink cluster is that lower CG users could not achieve the upper bound of their IDRs. This drawback can be overcome in the RIS-assisted TDMA-NOMA point-to-point communication model by allocating a sufficient number of meta-atoms such that the SINR of each user becomes saturated to reach the upper bound of IDR.

There are two popular resource allocation strategies: one is concentrating on SR maximization [11] and the other is focusing on fairness Index equalization [12,13,14] for the IDRs of all users present in the NOMA downlink cluster. But the previous allocation strategy does not hit always the upper bound of IDRs for lower CG users and the later allocation strategy does not allow the highest CG user to increase its data rate beyond the lowest IDR of the cluster. So, the resource allocation strategy should be able to hit the upper bound of IDRs for lower CG users and then allow to increase in the IDR of the highest CG user.

1.1. Motivation and Contribution

Figure 1 shows the motivation behind the proposed work in this paper. Although the effectiveness of NOMA has largely been explored in the sum throughput maximization, the identification of IDR still remained an unexplored area. Previously, it has been shown that RIS can lead to an overall improvement in the data rate by enhancing the effective channel gain of the downlink NOMA system. When time division multiple access (TDMA) is clubbed with multiple RISs in a distributed RIS-assisted NOMA (TDMA-RIS-NOMA) downlink system, a point-to-point communication model is created between the access point-to-RIS-to-user device. Due to this point-to-point communication model, the optimization of the phase shifts provided by meta-atoms of each RIS is facilitated. The optimized phase shifts of meta-atoms maximize the equivalent channel gain between the access point to the user. In this scenario, the channel becomes saturated and signal-to-interference plus noise ratio (SINR) becomes a function of power coefficients only. In this study, the power coefficients are calculated to maximize the SINR of each user belonging to a NOMA cluster using a geometric progression-based power allocation method such that IDR reaches its upper bound. Our findings set an upper limit for IDR and are useful for planning the resource allocation to achieve the desired quality of service (QoS) requirements in a NOMA downlink cluster.

To the best of the authors’ knowledge and belief, no resource allocation strategy tries to achieve the upper bound on the IDRs of lower CG users and simultaneously increases the IDR of the highest CG in stand-alone NOMA and RIS-NOMA systems. This IDR maximization becomes important when the access point (AP) is radiating low transmit power, as in the case of internet-of-things (IoT) networks [15].

Motivated by the above-mentioned points, this paper presents the TDMA-RIS-NOMA downlink system, where exploiting the upper bound of IDRs is possible with easy beam-forming and channel gain controlling. The main contributions from this paper are listed below:

• A point-to-point beam-forming strategy is exploited in the TDMA-RIS-NOMA downlink cluster such that at any instant, one RIS is serving only one user device for downlink communication.
• A geometric progression (GP)-based power coefficient allocation scheme is derived to find out the power coefficients of all the users belonging to an RIS-assisted downlink NOMA cluster for maximizing the IDRs of the users.
• Distribution of data rate among the users of an RIS-NOMA downlink cluster is analyzed and an expression is derived to calculate the upper bound of IDRs.
• Based on the upper bound of IDRs and QoS requirements, a clustering algorithm is proposed where we decide the cluster bandwidth and cluster size in the downlink NOMA-RIS system for different applications.

1.2. Paper Organization

The rest of the paper contains the system description in Section 2 followed by problem formulation in Section 3. The proposed scheme is elaborated in Section 4, then Section 5 contains the proposed algorithm. The complexity and convergence analysis of the proposed power allocation algorithm is done in Section 6. Section 7 explains the simulation results and briefly discusses the results obtained. Finally, the conclusion is written in Section 8,

2. System Description

We consider a downlink TDMA-RIS-NOMA point-to-point communication system that contains one AP with a single antenna; ‘M’ devices, each equipped with a single antenna; and ‘n’ RIS made up of ‘N’ meta-atoms, each as shown in Figure 2.

There are ‘n’ numbers of RIS present in the proposed system model, as shown in Figure 1. We assume that the distances among the RISs are large enough to mitigate the possibility of interference from one RIS to another RIS. Here, the TDMA system is followed with ‘m’ time slots. During one time slot, only one device is active in a TDMA group to access the corresponding RIS. Hence, a point-to-point communication model is formed between AP-RIS-user devices. One RIS is dedicated to one TDMA group.

The entire communication period $T_C$ is divided into multiple equal duration time slots of ‘t’ milliseconds, as in a TDMA system and shown in Figure 3. During any time slot ‘n’ devices are active from ‘n’ different TDMA groups. These active ‘n’ devices make a downlink NOMA cluster. There are m devices in a TDMA group. Hence, there are ‘m’ NOMA clusters. Each slot is assigned to a different cluster of user devices. The cluster formation is conducted as per the QoS (data-rates) requirements.

When a new time slot starts, only one user device is communicating through one RIS. The number of user devices active in any time slot is equal to the number of RIS deployed (n) in the system. During the present time slot, these n active devices receive a common broadcasted composite NOMA signal through different wireless channels (i.e., AP-to-RIS-to-user device). This received NOMA composite signal is decoded at the receiver (user device). The realization of time synchronization and frequency distribution among different resource blocks are conducted as described in [16] for NB-IoT systems.

Let $D_{il}$ denote the ith device of the lth cluster, which is scheduled in lth time slot. Further, N is the number of meta-atoms allocated to the device ‘$D_{il}$’ such that these N meta-atoms are tuned to reflect the incoming signal from AP towards the user device ‘$D_{il}$’. Let $h_{irl}$ denote the channel coefficient between $D_{il}$ and rth meta-atoms of ith RIS allocated to $D_{il}$. The $g_{irl}$ indicates the channel gain between rth meta-atom of ith RIS allocated to $D_{il}$ and AP. Here, $i\in \{1,2,\dots ,n\}$$r\in \{1,2,3,\dots ,N\}$.

Assume ${\mathbf{g}}_{\mathbf{il}}$ and ${\mathbf{h}}_{\mathbf{il}}$ are the corresponding channel vectors for AP to RIS and RIS to the device $D_{il}$, respectively. The ${\mathbf{g}}_{\mathbf{il}}$ and ${\mathbf{h}}_{\mathbf{il}}$ have N total RIS elements, which generate the maximum channel gain (CG). If the symbol ${\mathsf{\Theta }}_{\mathbf{l}}$ denotes the reflectivity-matrix of RIS for all $D_{il}$s, then equivalent channel gain $h_{e{q}_{il}}$ is given by

$$h_{e{q}_{il}}={\mathbf{h}}_{\mathbf{il}}^{\mathbf{T}}{\mathsf{\Theta }}_{\mathbf{l}}{\mathbf{g}}_{\mathbf{il}}=\alpha \sum _{r=1}^N\left(h_{irl}\right)\left(g_{irl}\right)e^{j{\theta }_{rl}},$$

(1)

where ${\mathsf{\Theta }}_{\mathbf{il}}=\alpha \times diag(e^{j{\theta }_{1l}},e^{j{\theta }_{2l}},\dots ,e^{j{\theta }_N})$ is $N\times N$ diagonal matrix for ith device $D_{il}$. ${\mathbf{h}}_{\mathbf{il}}$ and ${\mathbf{g}}_{\mathbf{il}}$ are the $N\times 1$ matrix containing the channel gain coefficients. The ${\mathbf{h}}_{\mathbf{il}}^{\mathbf{T}}$ denotes the transpose of matrix ${\mathbf{h}}_{\mathbf{il}}$.

The parameter $\alpha \in [0,1]$ is responsible for altering the amplitude of the reflected signal from the meta-atom.

The optimal value of phase shifts is calculated using the point-to-point downlink model ${\theta }_{rl}=-\angle (h_{irl}\times g_{irl})\forall r\in [1,N]$, which causes the in-phase addition of all the paths [17,18].

We assume that the positions of RISs and AP are fixed so the distance between RIS and AP is also fixed. The direct path between all the devices and AP is completely blocked and channel gain for direct paths is equal to zero. This yields,

$$h_{e{q}_{il}}=\alpha \sum _{r=1}^N\left(\right|h_{irl}|\times |g_{irl}\left|\right)\forall i\in [1,n],l\in [1,m]$$

(2)

For maximizing the amplitude of reflected signal, we put $\alpha$ in (1) as unity. The $|h_{e{q}_{il}}{|}^2$ is computed as,

$$|h_{e{q}_{il}}{|}^2={\left(\sum _{r=1}^N|h_{irl}|\times |g_{irl}|\right)}^2\forall i\in [1,n],l\in [1,m]$$

(3)

We assume the Rician fading channel model [12] for channel gain calculation between AP to IRS and IRS to the mobile user.

$$h_{irs}=\sqrt{\frac{\kappa }{1+\kappa }}{X}_{r_{LOS}}+\sqrt{\frac{1}{1+\kappa }}{X}_{r_{NLOS}}$$

(4)

$$g_{irs}=\sqrt{\frac{\kappa }{1+\kappa }}{Y}_{r_{LOS}}+\sqrt{\frac{1}{1+\kappa }}{Y}_{r_{NLOS}}$$

(5)

The value of $\kappa$ parameter is taken equal to 3 and $X_{r_{NLOS}}$ and $Y_{r_{NLOS}}$ are non-line-of-sight components of the corresponding paths and characterized by Rayleigh distributed random variables with unit variance and zero mean in (4) and (5). $X_{r_{LOS}}$ and $Y_{r_{LOS}}$ are line-of-sight components of the corresponding paths and their value is taken as unity in this study. Direct paths from AP to devices are assumed to be missing due to signal blockage or longer distance, and communication occurs only via RIS to devices. A common frequency band is shared among the devices belonging to a NOMA cluster.

In this paper, our primary focus is to maximize the IDR when channel gains are already known. Hence, we assumed the perfect channel state information (CSI) and calculated the expression for IDRs. Although, the channel estimation is a complex task for the TDMA-RIS-assisted NOMA system, some channel estimation techniques [19,20] may be used to estimate the channels from AP-to-RIS and from RIS-to-user devices. The positions of RIS and AP are fixed.

The composite signal received at the $D_{il}$ is given by,

$$y_{il}=h_{e{q}_{il}}\sum _{v=1}^n\sqrt{P_T{a}_{vl}}{x}_{vl}+w_{il},$$

(6)

where $y_{il}$ is the received signal at $D_{il}$, $w_{il}$ represents the additive white Gaussian noise (AWGN) with zero mean and ${\sigma }^2$ variance for all $D_{il}$s. Then the IDR of any user device inside the cluster can be given as,

$$R_{il}=B{log}_2\left(1+{\gamma }_{il}\right),\forall i\in [1,n],l\in [1,m],$$

(7)

where $R_{il}$ is the individual data-rate (IDR) achieved by the device $D_{il}$ in the system and ${\gamma }_{il}$ is the SINR corresponding to device $D_{il}$. The SINR of ith user in a downlink NOMA cluster is given by

$${\gamma }_{il}=\frac{P_T{a}_{il}{\left|h_{e{q}_{il}}\right|}^2}{P_T\left({\sum }_{v=1}^{i-1}{a}_{vl}\right){\left|h_{e{q}_{il}}\right|}^2+{\sigma }^2},$$

(8)

where $P_T$ is total power allocated to the cluster, $a_{il}$ is the power coefficient of ith device in the lth cluster. ${\sigma }^2$ is the variance of AWGN noise that represents the noise power.

3. Problem Formulation

As discussed earlier at the end of the introduction section, the drawback of resource allocation for SR maximization in a downlink NOMA cluster is that all the lower CG users could not achieve the upper bound of IDRs and the data-rate of the highest CG gain user increases very rapidly. The drawback of other fairness index-based resource allocation techniques for data rate equalization is that the IDR of the highest CG user can not be increased beyond the lowest IDR in a NOMA downlink cluster. These two limitations can be overcome by the RIS-assisted NOMA downlink system. Hence, we see a problem statement about power and meta atom resource allocation in a downlink RIS-NOMA cluster such that all the lower CG users are achieving their upper bound of IDRs and the remaining resources are allocated to the highest CG user to enhance its IDR. The power coefficients and meta-atoms are the resources that need to be cleverly distributed among the users of the downlink RIS-NOMA cluster to achieve this goal.

The problem is defined below under the constraints C1 to C3.

$$\begin{array}{c}max\left\{{\displaystyle \sum _{i=1}^n}{R}_{il}\right\}\forall i\in [1,n],l\in [1,m]\hfill \\ \{{\varphi }_{il},a_{il}\}\hfill \end{array}$$

(9)

Subjected to:

$$\begin{gathered} C1:{\mathsf{\Theta }}_{rl}\in [0,360) \mathrm{in} \mathrm{degrees} \\ C2:{\sum }_{i=1}^n{a}_{il}=1\forall l\in [1,m] \\ C3:0<a_{il}<1\forall i\in [1,n]\mathrm{and}\forall l\in [1,m] \\ C4:0<N_{i{l}_{min}}<N\forall i\in [1,n],l\in [1,m] \end{gathered}$$

Constraints Involved in Reflectivity Matrix (Phase Shifts at All the Meta-Atoms) and Power Allocation:

Constraint 1 (C1): Ideally, phase shifts may take any value in the range $[0,360)$ degrees.

Constraint 2 (C2): The sum of all power coefficients in a cluster is equal to 1.

$$\sum _{i=1}^n{a}_{il}=1\forall i\in [1,n],l\in [1,m]$$

Constraint 3 (C3): All power coefficients are positive fractions, i.e.,

$$0<a_{il}<1\forall i\in [1,n],l\in [1,m]$$

Constraint 4 (C4): Sufficient meta-atoms should be available in each RIS to achieve the condition of channel saturation., i.e.,

$$0<N_{i{l}_{min}}<N\forall i\in [1,n],l\in [1,m]$$

Here, $N_{i{l}_{min}}$ are the minimum number of meta-atoms required to bring the equivalent channel gain in the saturation condition with respect to IDR of the user.

4. Proposed Scheme

The proposed scheme executes the task in two phases: (i) suppressing the noise by allocating an adequate number of meta-atoms (one RIS for one user) to each user of a TDMA-RIS-NOMA cluster, (ii) finding the power coefficients to maximize the SINR.

4.1. Suppressing the Noise Using RIS

If a sufficient number of meta atoms are available for each user belonging to a RIS-NOMA cluster, then meta atoms can be used as a tool to make the data rate less dependent on channel gain, total cluster power, and AWGN noise power for downlink RIS-NOMA communication. The mathematical derivation to prove the above-mentioned hypothesis is presented in this subsection. By putting $h_{e{q}_{il}}=h_{e{q}_{il}}^{sat}$ in (8), ${\gamma }_{il}$ becomes equal to ${\gamma }_{i{l}_{sat}}$, as shown below:

$${\gamma }_{i{l}_{sat}}=\frac{P_T{a}_{il}{\left|h_{e{q}_{il}}^{sat}\right|}^2}{P_T\left({\sum }_{v=1}^{i-1}{a}_{vl}\right){\left|h_{e{q}_{il}}^{sat}\right|}^2+{\sigma }^2},$$

(10)

where $h_{e{q}_{il}}^{sat}$ represents the equivalent channel gain for $D_{il}$ in saturation state, i.e., increasing channel gain beyond this point does not increase the SINR anymore for $D_{il}$. Dividing numerator and denominator both by $P_T{\left|h_{e{q}_{il}}^{sat}\right|}^2$ we get

$${\gamma }_{i{l}_{sat}}=\frac{a_{il}}{{\sum }_{v=1}^{i-1}{a}_{vl}+\frac{{\sigma }^2}{P_T{\left|h_{e{q}_{il}}^{sat}\right|}^2}},$$

(11)

$${\gamma }_{i{l}_{sat}}=\frac{a_{il}}{{\sum }_{v=1}^{i-1}{a}_{vl}+\frac{1}{{\rho }_i{\left|h_{e{q}_{il}}^{sat}\right|}^2}},$$

(12)

where ${\rho }_i=\frac{P_T}{{\sigma }^2}$. If ${\rho }_i{\left|h_{e{q}_{il}}^{sat}\right|}^2\ge 1$ and $\frac{1}{{\rho }_i{\left|h_{e{q}_{il}}^{sat}\right|}^2}<<{\sum }_{v=1}^{i-1}{a}_{vl}$ then (12) reduces to:

$${{\gamma }_{il}}_{sat}=\frac{a_{il}}{{\sum }_{v=1}^{i-1}{a}_{vl}}\forall i\in \{2,3,\dots ,n\}$$

(13)

$$R_{i{l}_{max}}=B{log}_2\left(1+\frac{a_{il}}{{\sum }_{v=1}^{i-1}{a}_{vl}}\right)\forall i\in \{2,3,\dots ,n\},$$

(14)

For i = 1, interference signal is zero; hence, (8) becomes

$${\gamma }_{1l}=a_{1l}{\rho }_{1l}{\left|h_{e{q}_{1l}}\right|}^2$$

(15)

$$R_{1l}=B{log}_2\left(1+a_{1l}{\rho }_{1l}{\left|h_{e{q}_{1l}}\right|}^2\right)fori=1.$$

(16)

The data rate function $R_{1l}$ for the highest channel gain user is an increasing function. Hence, the upper bound on data rate for user 1 is decided by maximum power-handling capabilities of the user device. It is clear from the (12) that when an appropriate number of meta-atoms is allocated to a user in a downlink NOMA cluster, its SINR reaches the level of saturation, as calculated in (13). Beyond this limit, if we increase the number of meta atoms serving to the user, it will not change the SINR for a given set of power coefficients. Hence, the data rate of the individual user becomes saturated with respect to $N_{i,l}$ and we may say that the channel saturation condition is achieved. At this point, maximum throughput is obtained for the user belonging to the RIS-assisted NOMA downlink cluster.

4.2. Finding the Power Coefficients to Maximize the SINR

To maximize the ratio in (13), the numerator is maximized and the denominator is minimized. Equation (13) can also be written as:

$${{\gamma }_{il}}_{sat}=\frac{a_{il}}{{\sum }_{v=1}^{i-1}{a}_{vl}}\forall i\in \{2,3,\dots ,n\}$$

(17)

$${{\gamma }_{il}}_{sat}=\frac{a_{il}}{1-{\sum }_{x=i+1}^n{a}_{xl}}\forall i\in \{2,3,\dots ,n\}.$$

(18)

For the lowest channel gain user in a cluster $i=n$, putting $i=n$ in (18), we get:

$${{\gamma }_{nl}}_{sat}=\frac{a_{nl}}{1-a_{nl}}.$$

(19)

If $a_{nl}$ is maximized in (19), then $1-a_{nl}$ is minimized and the ratio, i.e., ${\gamma }_{nl}$ is maximized. Similarly, for $i=n-1$ (18) becomes,

$${{\gamma }_{n-1,l}}_{sat}=\frac{a_{n-1,l}}{1-a_{nl}-a_{n-1,l}}.$$

(20)

In (20), $a_{nl}$ is already maximized and $a_{n-1,l}$ needs to be maximized. From the (19) and (20), it can be concluded that, if we maximize the power coefficients in the order from $a_{n,l},a_{n-1,l},a_{n-2,l},\dots ,a_{1,l}$, i.e., the lowest channel gain to second-lowest channel gain to the successive higher channel gain and so on, then it can be assured to achieve the maximum SINRs for all the users of an RIS-assisted NOMA cluster.

For performing the SIC process successfully at the receiver, it is important to distribute the total cluster power such that the lowest channel gain user must have more than $50\%$ of total cluster power. Similarly, second lowest channel gain user must have more than $25\%$ of the total cluster power and so on for successive lower channel gain users [11]. In the first iteration of power coefficient allocation, if minimum $50\%$ of cluster power is allocated to $D_{nl}$ ($a_{nl}=\frac{1}{2}$), $25\%$ of cluster power is allocated to $D_{n-1,l}$ ($a_{n-1,l}=\frac{1}{4}$), $12.5\%$ of cluster power is allocated to $D_{n-2,l}$ ($a_{n-2,l}=\frac{1}{8}$) and so on. Then remaining power in the fraction after allocation till the highest channel gain user would be

$$1-\frac{1}{2}-\frac{1}{4}-\frac{1}{8}-...-\frac{1}{2^n}=\frac{1}{2^n}$$

.

Now, this remaining power needs to be redistributed among the n users of the cluster in the second iteration. If half of the remaining power is allocated to lowest channel gain user, i.e., $\frac{1}{2}\times \frac{1}{2^n}$, then $a_{n,l}=\frac{1}{2}+\frac{1}{2^{n+1}}$. After this allocation, remaining power becomes $\frac{1}{2^n}-\frac{1}{2^{n+1}}=\frac{1}{2^{n+1}}$. Half of this remaining power is distributed to the next lowest channel gain user ($D_{n-2,l}$). The process of allocating half of the remaining power is repeated till the remaining power is negligible as compared to $P_{Toll}$, where $P_{Toll}$ is minimum detectable power at the NOMA receiver. If $\frac{P_{Toll}}{P_T{\left|h_{e{q}_{min}}\right|}^2}<\frac{1}{2^{2n}}$, then the geometric progression series for calculating the power coefficient can be given as:

$$a_n=\frac{1}{2}+\frac{1}{2^{(n+1)}}+\frac{1}{2^{(2n+1)}}+\frac{1}{2^{(3n+1)}}...$$

(21)

$$a_{n-1}=\frac{1}{2^2}+\frac{1}{2^{(n+2)}}+\frac{1}{2^{(2n+2)}}+\frac{1}{2^{(3n+2)}}...$$

(22)

$$...............$$

$$a_1=\frac{1}{2^n}+\frac{1}{2^{\left(2n\right)}}+\frac{1}{2^{\left(3n\right)}}+\frac{1}{2^{\left(4n\right)}}...$$

(23)

So, if $\frac{P_{Toll}}{P_T{\left|h_{e{q}_{min}}\right|}^2}<\frac{1}{2^{2n}}$ and ${\rho }_i{\left|h_{il}{g}_{il}\right|}^2\ge 1$ then, the generalized expression for ith user’s power coefficient calculation, when cluster size is equal to n, can be written as:

$$a_i=\frac{1}{2^{(n-i+1)}}+\frac{1}{2^{(2n-i+1)}}+\frac{1}{2^{(3n-i+1)}}+....\forall i\in [1,n].$$

(24)

For $n\ge 3$ the first five terms of the series can be taken into consideration and subsequent terms may be neglected because they do not contribute significantly in the magnitude of the power coefficient.

4.3. Theoretical Upper Bounds of Normalized IDRs ($SU{F}_{max}$)

If we normalize the IDR with respect to bandwidth, then

$$IDR=B{log}_2\left(1+SINR\right)$$

(25)

$$IDR=B\times SUF$$

(26)

$$SUF=IDR/B={log}_2\left(1+SINR\right)$$

(27)

$SUF$ is obtained. For lower CG users, $SU{F}_{max}$ is given by

$$SU{F}_{i{l}_{max}}={log}_2\left(1+\frac{a_{il}}{{\sum }_{v=1}^{i-1}{a}_{vl}}\right)\forall i\in \{2,3,\dots ,n\}$$

(28)

Putting $n=3,4,5,6$ in (24) gives $a_i$s for different cluster sizes. $a_i$s are calculated for lower CG users, then maximum normalized IDR can be found with the help of (28). Theoretically, the upper bounds of normalized IDRs are calculated and tabulated in

Table 1. Normalized IDR theoretical analysis in bits per second per Hertz for GP-based power allocation inside the RIS-assisted NOMA Downlink cluster.

IDR Theoretical Analysis in Bits per Second per Hertz for GP-Based Optimal Power Allocation in Downlink RIS-NOMA Clusters
Cluster Size	User 1	User 2	User 3	User 4	User 5	User 6
n = 3	variable	1.585	1.222	-	-	-
n = 4	variable	1.585	1.222	1.099	-	-
n = 5	variable	1.585	1.222	1.099	1.047	-
n = 6	variable	1.585	1.222	1.099	1.047	1.023

.

The second column of the Table 1 indicates the normalized data rate of the Highest CG user, which depends on equivalent channel gain and noise power. Hence, its value is not specified in the table and we have written ‘variable’ there.

4.4. Cluster Size Calculation Based on Maximum IDRs and QoS Requirements

Table 1 shows the upper bound of normalized IDR with respect to cluster bandwidth. This normalized IDR is also known as the spectrum utilization factor (SUF). The maximization of SUF is presented in Table 1 for all lower CG users. If we multiply the SUF with bandwidth allocated to one NOMA cluster we find the IDR for user. This IDR must be greater than or equal to QoS requirements

$$Qo{S}_{min}\le B\times SU{F}_{max}$$

(29)

We can use (29) to determine the maximum cluster size for any application. The values of $SU{F}_{max}$ are taken from Table 1 for different cluster members. For example, narrow band IoT (NB-IoT) devices’ bandwidth and minimum data rate requirement is less, as described in [15,21]. For downlink NB-IoT devices, $B=180$ kHz and if $Qo{S}_{min}=215$ kbps for an application, then $SU{F}_{max}$ values obtained ensure that we can take the cluster size as 2 or 3, only because when $n\ge 4$, then (29) is not satisfied for the lowest CG user.

5. Proposed Algorithm for Cluster Formation, RIS Allocation, Power Allocation and IDR Maximization in Downlink TDMA-RIS-NOMA System

Channel Saturation-Based Algorithm for RIS and Power Allocation for Maximizing IDR

In Algorithm 1, steps 1 and 2 define the input and output parameters, respectively. Steps 3 to 8 are used for cluster formation as shown in Figure 4. Here, the time slot scheduling of the user devices is done according to their QoS requirements. Steps 9 to 13 describe the meta-atom allocation ($N_{i{l}_{min}}$) for channel saturation. Then power coefficients for maximum IDR are calculated using step 14. Finally, Step 16 terminates the algorithm, when resource allocation is complete for all the clusters.

The advantages of the proposed algorithm are as follows:

• Control over channel gain becomes very easy and straight forward because of the point-to-point communication model.
• Clustering and decoding order may be changed instantly according to Qos requirements.
• TDMA-RIS-NOMA point-to-point downlink environment is the ideal situation to exploit the maximum controllability over IDRs of a NOMA downlink cluster.
• Upper bound of IDRs is achieved.

Algorithm 1 Channel Saturation-based Algorithm for Cluster Formation, RIS Allocation, Power Allocation and IDR Maximization in Downlink TDMA-RIS-NOMA System
1:	Inputs:$M,n,N,\{q_1,q_2,\dots ,q_M\},h_{irl},g_{irl}$
2:	Outputs:$N_{{il}_{min}},a_{il},R_{{il}_{max}}\forall i\in [1,n]andl\in [1,m]$
3:	Arrange all User Devices in descending order according to their Qos requirements: $q_1>q_2>...>q_M$
4:	Decide cluster size with the help of (29).
5:	Create clusters such that max and min QoS devices are included in the same group.
6:	for$l=1:1:m$do
7:	$CLUSTER\left(l\right)=\{q_{1l},q_{2l},\dots ,q_{(n-1)l},q_{nl}\}=$
	$\{q_{\left(l\right)},q_{(l+m)},q_{(l+2m)},\dots ,$
	$q_{(M-l-2m)},q_{(M-l-m)},q_{(M-l+1)}\}$
8:	end for
9:	if$\frac{P_{Toll}}{P_T{\left|h_{e{q}_{il}}\right|}^2}\ge \frac{1}{2^{2n}}and{N_{il}}_{min}\le N$then
10:	$N_{{il}_{min}}=N_{{il}_{min}}+1$.
11:	Allocate one more meta atom with max channel gain and calculate new effective channel gain $h_{e{q}_{il}}$ from (2).
12:	end if
13:	To saturate the effective channel gain $h_{e{q}_{il}}$, allocate all the meta-atoms (N) of a RIS to the user device to ensure $\frac{P_{Toll}}{P_T{\left|h_{e{q}_i}\right|}^2}<\frac{1}{2^{2n}}$ condition is achieved.
14:	Find the values of power coefficient $a_{il}$ for maximizing IDR of each user using (24).
15:	Repeat steps 9 to 14 for all integer values of $i\in [1,n]$ and $l\in [1,m]$.
16:	End of the algorithm.

Figure 4. Clustering mechanism used in the proposed downlink TDMA-RIS-NOMA.

Figure 4. Clustering mechanism used in the proposed downlink TDMA-RIS-NOMA.

6. Complexity and Convergence of Proposed Algorithm

We have compared the complexity in terms of flop count with the stand-alone NOMA system described by Ali et al. [11]. The detailed analysis of the complexity of Ali et al. is conducted in [22]. The flops are basically defined as any mathematical operation (like addition, subtraction, multiplication, and division) performed during the execution of algorithm [23]. For the purpose of comparison, we have compared the proposed algorithm’s complexity with Ali et al. in

Table 2. Complexity comparison in flop count.

Power Allocation Method	Ali et al. [11]	Proposed Algorithm
No. of Additions	$n-1$	$4n$
No. of Subtractions	$(n-1)n/2$	0
No. of Multiplications	$\left(\right(n-1)n/2)+(n-2)$	$5n-1$
No. of Divisions	$n(n+1)/2$	0
Total Number of Flops	$(3n^2+3n-6)/2$	$9n-1$

.

In the proposed algorithm, the complexity is calculated for the power allocation block only, i.e., step 14. We have calculated the initial 5 terms appearing in the GP for each power coefficient by putting $\frac{1}{2}=0.5$ in (24) and, hence, the number of additions required is equal to 4 for calculating any power coefficient. There are n members in a cluster so we need to calculate n power coefficients. The total add operations are equal to $4n$ during the power allocation. Similarly, the total multiplication operations required for power allocation are equal to $5n-1$. Total number of flops required is $4n+5n-1=9n-1$

The power coefficients are calculated with the help of (24). In (24) the initial 5 terms only contribute significantly and subsequent terms are ignored as the value of the power coefficient converges until the third decimal place with the initial 5 terms only. Furthermore, the convergence of the proposed algorithm is proofed by the number of meta atoms Vs. IDR (in bits per second per Hertz) curve (Figure 5) presented in the simulation result section. As the number of meta-atoms are increased, the IDR converges to its upper limit value. these upper limits of IDR are calculated and tabulated in Section 4.1, Table 1.

7. Simulation Results and Discussion

Table 3. Simulation Parameters.

Description of the Parameter	Symbol	Value
Total number of devices present in the system	M	12
Number of devices included in a cluster (cluster size)	n	3/4/6
Total number of meta-atoms in the RIS	N	10 to 300
Minimum distance between RIS and devices	$d_{min}$	50 m
Fixed distance between RIS and Access point	$d_{fix}$	50 m
Distance between ith device $D_{il}$ to RIS	$D_{il}$	50 m to 300 m
Carrier frequency	$f_c$	3 GHz
Bandwidth for each device	B	180 kHz
Transmit power from AP	$P_T$	23 dBm
AWGN power spectral density in dBm	w	−130 dBm
Minimum data-rate requirement for each device	$R_{min}$	100 kbps
Noise Figure in dB	$F_n$	10 dB

describes the simulation parameters. The AWGN noise power is calculated according to the ${\sigma }^2=-130+10{log}_{10}\left(B\right)+F_n$, where $F_n$ is in dB, ${\sigma }^2$ is in dBm, B is in Hz [17].

Monte-Carlo simulations are performed for 100,000 samples and the average value of normalized IDR in bits per sec per Hertz is calculated for different cluster sizes. For RIS-NOMA simulations, the total transmit power ($P_T$) is taken as 23 dBm [24]. For simulation of stand-alone NOMA without RIS system, the total transmit power is taken as 46 dBm [11].

Table 1 and

Table 4. Normalized IDR Monte-Carlo simulation analysis in bits per sec per Hertz for GP-based optimal power allocation in downlink TDMA-RIS-NOMA clusters.

Cluster Size	User1	User2	User3	User4	User5	User6
n = 3	14.2965	1.5849	1.2224			-
n = 4	12.5822	1.5848	1.2223	1.0995		-
n = 5	10.7713	1.5844	1.2222	1.0995	1.0473	-
n = 6	9.9945	1.5838	1.2219	1.0993	1.0472	1.0230

present the theoretical upper bounds of IDRs and the Monte-Carlo simulation results to verify the theoretical benchmark, respectively, for proposed GP-based power allocation in the RIS-NOMA downlink cluster with different cluster sizes. The results shown here validate the proposed hypothesis that each lower CG user (user 2 to user ‘n’) present in a NOMA downlink cluster cannot achieve the IDR higher than its upper bound benchmark, irrespective of higher channel gain or higher cluster power provided to it. The empty places in these tables are intentionally left blank to show that the cluster is already full and no more users can be accommodated in the cluster than the cluster size.

Similarly,

Table 5. Normalized IDRs’ theoretical upper bound in bits per sec per Hertz for different cluster sizes in Downlink RIS-NOMA, when magic matrix-based power allocation [22] is adopted.

Cluster Size	User 1	User 2	User 3	User 4	User 5	User 6
n = 3	Variable	1.2224	1.0995	-	-	-
n = 4	Variable	1.2224	1.1926	1.0875	-	-
n = 5	Variable	1.2224	1.0995	1.0931	1.0224	-
n = 6	Variable	1.5850	1.1155	1.0544	1.0265	1.0131

shows the theoretical upper bound of IDRs for the magic matrix-based power allocation [22] in RIS-NOMA downlink cluster and

Table 6. Normalized IDRs’ Monte-Carlo simulation analysis in bits per sec per Hertz for Magic matrix-based power allocation [22] in downlink RIS NOMA clusters.

Cluster Size	User 1	User 2	User 3	User 4	User 5	User 6
n = 3	15.0917	1.2224	1.0995	-	-	-
n = 4	12.8800	1.2223	1.1926	1.0874	-	-
n = 5	11.5115	1.2221	1.0994	1.0930	1.0223	-
n = 6	9.9145	1.5837	1.1151	1.0542	1.0264	1.0130

verifies the theoretical benchmark set in Table 5 by showing Monte-Carlo simulation-based results in deep agreement with the benchmark of Table 5. The results obtained in Table 4 and Table 6 again validate the proposed hypothesis about the upper bound of IDR for each lower CG user belonging to a NOMA downlink cluster.

Table 7. Normalized IDR analysis in bits per sec per Hertz of Magic matrix-based power allocation [22] in downlink NOMA clusters without IRS.

Cluster Size	User 1	User 2	User 3	User 4	User 5	User 6
n = 3	5.2786	1.1676	1.0169	-	-	-
n = 4	4.3149	1.1361	1.1206	0.9919	-	-
n = 5	3.5750	1.0935	1.0118	1.0180	0.9224	-
n = 6	2.4561	1.2822	0.9790	0.9571	0.9449	0.9073

and

Table 8. Normalized IDR analysis in bits per sec per Hertz of Optimal power allocation [11] in downlink NOMA clusters without IRS.

Optimal Power Allocation [11] in Downlink NOMA Clusters without IRS
Cluster Size	User 1	User 2	User 3	User 4	User 5	User 6
n = 3	5.5925	0.9613	0.9274	-	-	-
n = 4	4.7940	0.9456	0.9468	0.9147	-	-
n = 5	3.9800	0.9146	0.9286	0.9346	0.9031	-
n = 6	3.1276	0.8584	0.8888	0.9117	0.9220	0.8960

highlight the simulation results for stand-alone NOMA downlink cluster without RIS for different cluster sizes, when Magic matrix-based power allocation [22] and optimal power allocation [11] have been performed, respectively. Now, if we compare the results obtained in Table 4 and Table 8, then it clearly shows that IDRs for all the users are better in RIS-NOMA as compared to stand-alone NOMA without RIS. The same conclusion can also be drawn by comparing the results of Table 6 and Table 7. This comparison shows that RIS-NOMA is better than NOMA without RIS for all cases of power allocation. A reduction of 23Dbm in total power is achieved in the RIS-NOMA system while improving the data-rates as compared to their non-RIS counter part.

Results of Table 4 also show the optimality of the proposed GP-based power allocation technique for all users of the RIS-NOMA cluster because IDRs in Table 4 are highest among Table 5, Table 6, Table 7 and Table 8.

As shown in Figure 5, the saturation point is reached for all the lower CG users in the downlink NOMA clusters as the effective CG is increased. The highest CG user for RIS-NOMA cluster is not saturated with respect to effective CG. This fact is shown in the Figure 6 where the number of meta-atoms is increased to enhance the effective CG. The curves for different cluster sizes are plotted in this graph. The graph shows that the IDR of the highest CG user increases as the effective CG increases.

The simulation results confirmed the analysis conducted in Section 4. The data rate increases as the number of meta-atoms increases before the saturation point. After this point, as we increase the number of meta-atoms, the change in the data rate of the corresponding user is negligible and we can say that the user has reached the upper limit of the IDR.

The data rate of the highest CG user is not saturated with respect to its effective CG. The rest of the users belonging to the same NOMA cluster are saturated to a predefined value $R_{i{l}_{max}}$, which is dependent on the power coefficients of the users. Simulation results for $R_{{il}_{max}}$ are in agreement with (14).

8. Conclusions

This study finds the upper limit of the IDR for all lower channel gain users in the TDMA-RIS-NOMA scenario. The proposed GP-based power allocation method maximizes the power allocation coefficients and, therefore, improves the SINR to provide an upper limit of the individual data rates. Although the TDMA-RIS-NOMA scenario has a high chance of achieving the saturated channel due to its inherent characteristics of serving one user at a time, the proposed method could also be useful in non-TDMA scenarios to provide an upper limit of the IDR. The proposed method also helps in deciding the maximum size of the cluster based on the spectrum utilization factor and predefined quality of service requirements. The maximum IDR obtained from the analytical method for all lower channel gain users has also been verified with the Monte-Carlo-based simulation that showed a minimum dissimilarity. This provides an additional benefit over the previous studies that only focus on maximizing the overall sum data rate. This was the pilot study showing the effectiveness of using a GP-based power allocation method for obtaining maximum IDR and, can be useful in lower power requirement applications like NB-IoT systems. In the future, we intend to verify the study findings in complex scenarios by estimating the channel state information and investigating the effect of imperfect CSI and imperfect SIC on the maximum achievable individual data rate.