A Low-Complexity Coding Scheme for NOMA

This work focuses on exploiting the constructive interference among different users' data waveforms to introduce new coding and decoding techniques, which are specifically designed for nonorthogonal multiple access (NOMA) systems. In this article, a structured coding scheme is devised. In essence, the proposed technique focuses on finding a relationship between the sent users' data waveforms and then uses this relationship in the decoding process at the receiving destination. It is worth pointing out that the proposed coding and decoding techniques exhibit better performance and reduced the complexity compared with the conventional uncoded NOMA. The complexity order evaluation shows that the proposed scheme attains a reduction in the required number of the floating point operations of $\text{5}$ and $\text{6}\,N$ at the second and third users, respectively, compared with that of the uncoded NOMA. Moreover, we have derived a closed-form expression for the bit error rate, which is verified using the Monte Carlo simulation. To demonstrate the practicality of the proposed system, the obtained results are compared with those of the uncoded and convolutional coding NOMA systems. Finally, the performance of the proposed system outperformed the conventional systems by an average of 5 dB in the case of two users and an average of 15 dB in the case of three users in the same work environment.

A Low-Complexity Coding Scheme for NOMA I. INTRODUCTION N ONORTHOGONAL multiple access (NOMA) is expected to be a promising enabling technology to increase the spectral efficiency of the future wireless communication systems [1], [2]. NOMA is based on applying the superposition concept at the transmitter side, where different users' data waveforms are superimposed together by assigning different power levels for each user, whereas at the receiver side, successive interference cancellation (SIC) is utilized to successfully decode the desired data [3], [4], [5], [6], [7]. Numerous studies have recently appeared investigating the performance of NOMA-aided wireless systems. For instance, Li et al. [8] analyzed the reliability of integrating the NOMA scheme with a dual-hop cooperative system in the presence of an eavesdropper. Tian et al. [9] studied the tradeoff between spectral efficiency and power efficiency in NOMA systems and formulated an optimization problem based on the required rate and the allocated power for each user, which was solved using the bisection method. The impact of imperfect channel state information on the performance of a two-way relay network NOMA system was investigated in [10]. Furthermore, a downlink NOMA system in which a base station (BS) is communicated with multiantenna users over Nakagami-m fading channels is considered in [11], where the authors investigated the reliability of the considered network in the presence of a multiantenna eavesdropper. Do et al. [12] proposed to combine the NOMA scheme with a full-duplex unmanned aerial vehicle that was employed as a relay node to connect the BSs with the users with no direct path with these due to obstacles and shadowing fading.
However, a key drawback of NOMA is that the superimposition of data waveforms for different users will introduce extra interference, which can result in bit error rate (BER) performance degradation. This has motivated researchers to consider error detection and correction techniques to effectively improve the BER performance. A polar-coded NOMA system was proposed in [13], which recognized the NOMA from the perspective of channel polarization. Then, multidimensional trellis coded modulation (TCM) techniques were incorporated with the NOMA system to effectively improve the BER performance [14]. Moreover, a resource-pattern-aided bit-interleaved NOMA was introduced in [15], where the low-density parity check (LDPC) code was integrated with this NOMA system. Idris et al. [16] investigated the performance of the NOMA system when different coding schemes are utilized, and they concluded that the LDPC code outperforms the turbo and convolutional codes. Despite the efficiency of these coding schemes in orthogonal multiple access (OMA) communication systems, they are not designed to deal with the particularity of NOMA, i.e., having interference from the superimposed users' data. Hence, their efficiency is reduced, and their ability to detect and correct errors compared with the proposed system in this research article, which took into account the particularity of NOMA.
Although in [17], [18], [19], [20], and [21], the researchers focused on the integration of the NOMA with reflecting intelligent surfaces (RISs) to improve the system BER and secrecy performance; this was, however, achieved at the cost of added complexity to the system. Kim et al. [22] developed two algorithms for scheduling power allocation and user selection; while in our case, the need for such additional techniques has been eliminated, since the interference between users' signals is either constructive or can be mitigated using the proposed decoding method.
To the best of our knowledge, the research work that considered coding with NOMA has not addressed the fact that superimposing multiple users' data waveforms in the NOMA system will introduce extra noise to each user's signal. Hence, these algorithms can work properly up to a certain number of users and then will not be able to detect and correct errors. Moreover, most of the coded NOMA research work has focused only on performance enhancement achieved by the utilized coding scheme without investigating the complexity of these schemes, which does have a direct impact on the latency of the network. Also, as shown in the literature review the imperfect SIC induces extra noise in NOMA systems, which can dramatically degrade the performance of such systems. Motivated by these, we proposed a new low-complexity coding scheme, which is specifically designed for the NOMA scheme. The introduced scheme will dramatically reduce the utilization of the SIC in the decoding process, which will reduce the resulting interference from the imperfect SIC.
The main contributions of this work can be summarized as follows.
1) Unlike the existing work, we propose a new coding scheme specifically designed for NOMA systems. In essence, in this coding scheme, we exploit the interference resulting from superimposing multiple users' data waveforms. 2) Theoretical BER formulas for the introduced coded NOMA system are derived for two-user and three-user cases, and these formulas are validated using simulation results.
3) The complexity evaluation of the proposed scheme is performed, which shows that our scheme exhibits less complexity order than that of the conventional NOMA. Thus, it is fair to say that the time and computational requirements for the introduced scheme are less than that of the existing algorithms. Note that although the proposed coding scheme reduces the transmission rate to half, the performance of this system can be significantly improved. Moreover, for NOMA with the twouser case, the SIC process can be completely removed from the system, which is a huge advantage of our proposed scheme. For a NOMA system with more users, our scheme will also greatly reduce the complexity, since the number of SIC processes is less than conventional NOMA systems.
The proposed system may open the door for other ideas to develop a new coded NOMA system. For instance, it is possible to develop the current system using a more efficient coding scheme than the one deployed herein and take advantage of the constructive interference. Furthermore, it is possible to establish a relationship between users' data that always allows data retrieval without the need for SIC operations, which will further reduce the complexity. Last but not the least, since the transmitted signal in NOMA systems always contains data of all users, this feature can be exploited and employed in cooperative coded NOMA communication.
The rest of this article is organized as follows. The proposed coded NOMA system is introduced in Section II. Then, the closed-form formula of the BER performance and the complexity analysis of that system are presented in Section III. Moreover, Section IV illustrates the obtained results of the proposed system along with the discussion of these results. Finally, Section V concludes this article.

II. PROPOSED CODED NOMA SYSTEM
The left-hand side of Fig. 1 presents the transmitter of the proposed coded NOMA system and it starts with [n, k] elementary encoders for two users structured coded NOMA system with code rate of R = k n , where n is codeword length and k is the length of message bits. The first encoder will act like a repetitive code with [2,1], and the codeword of the user 1 will be where u 1 is the message bit of the far user. In other words, the codeword of the first encoder will be [0 0] if the transmitted message bit is 0 and [1 1] if the transmitted bit is 1. The second encoder will encode the user 2 data u 2 to C 2 . The first bit of C 2 will be u 2 and the flipped version of it is the parity check or redundancy bit. Hence, C 2 is given by whereū 2 = 1 − u 2 . In particular, every 0 of the second user's message will be mapped to [0 1] and every 1 will be [1 0] as the output of the second encoder. It is very clear that the design of the encoders is very simple but the main idea of the design is to build a relationship between their outputs so that one element must always be identical in the codewords of the encoders, and it can be expressed mathematically as The overlapping set in (3) plays the key role to take advantage of the users' interference, which ensures that 1 bit from each user is always in the same phase as the other user and this will lead to constructive interference [23]. It is worth noting that this is one of the most important strengths of the proposed system, because other systems will have an equal proportion of constructive and destructive interference, and this leads to low efficiency in detecting and correcting errors.
After encoding, the binary phase-shift keying (BPSK) is used for the two users to modulate C 1 and C 2 to X 1 and X 2 , respectively, where every 0 is mapped to -1 and every 1 to 1. Then, the superposed signal is given as where E c is the coded bit energy and it is equal to E b R, where E b is the uncoded bit energy, a 1 and a 2 are the allocated power coefficients for the first and second users, respectively, and a 1 + a 2 = 1.
The right-hand side of Fig. 1 shows the receiver of the proposed system. The received signal of two users can be expressed as are the far and near user received signal vectors, respectively, N is the length of transmitted data, y i 1 and y i 2 are the ith received signal of user 1 and user 2 , respectively, and N 1 = [n 1 1 , n 2 1 , . . ., n N 1 ] and N 2 = [n 1 2 , n 2 2 , . . ., n N 2 ] are the additive white Gaussian noise (AWGN) vectors, respectively, where n i 1 and n i 2 are the ith noise element that has zero mean and variance σ 2 of user 1 and user 2 , respectively.
With our proposed coding scheme, a simple decoding process can be used to recover the user 1 (û 1 ) as we can see in Fig. 1 and Y e 1 = [y 2 1 , y 4 1 , . . .]; then a simple decoding process can be used to recover the ith bit of user 1 's message aŝ Using the same mechanism, by dividing the received signal of the near user into odd and even subsequences, ith bit of user 2 's received message can be retrieved as follows: Note that our proposed scheme can also be extended to more than two users. For instance, the decoders of a coded NOMA system with three users can be as follows: two encoders are used the same as in the case of a two-user system and a third encoder will be added with the same mechanism as the first encoder, given by In this matter, the first encoder is used for the first and third users to represent the proposed system. Further, the power allocation of the users will be divided among three users and the relationship between the power allocation parameters of the three users will be, a 1 + a 2 + a 3 = 1.
For more users case, the proposed system can be extended similarly.

III. PERFORMANCE ANALYSIS
A. BER for Two Users 1) BER of Two Uncoded BPSK NOMA Users: In this section, we start from the analysis of uncoded BPSK BER for two users' NOMA system and then extend to our proposed coded NOMA system. It is possible to graphically and numerically represent all possible points of two users' BPSK modulated NOMA system, as shown in Table I and Fig. 2, respectively. This table includes all the numerical possibilities of decoding processes of the proposed coded NOMA and uncoded NOMA system for various values of a 1 and a 2 , assuming that there is no noise, in order to clarify the effect of users' interference with each other. In this figure, all the possible constellations of two users are shown, taking into account the superposition coding and noise-free scenario, where U 1 and U 2 refer to the first and second user various constellations, respectively, and the superposed points defined as U 1 (u 1 1 , u 2 1 ) and U 2 (u 2 2 , u 2 2 ) for the far and near user, respectively. The graphic also clearly shows the boundaries that, if crossed, will cause errors in the detection and retrieval of transmitted data.
In addition to a graphical illustration, this can be represented mathematically; for an uncoded NOMA system with BPSK modulation, the received signal can be written in a simple form as Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  I  DECODING TABLE OF THE PROPOSED CODED AND UNCODED NOMA SYSTEM FOR VARIOUS POWER ALLOCATION COEFFICIENTS where n 1 and n 2 follow a Gaussian distribution N (0, σ 2 ). Let us assume that the user 1 sent a bit of 1, and Fig. 2 and Table I clearly illustrate that the error will occur when the noise exceeds the boundary. Hence, according to the decision boundary given in Fig. 2, we can obtain the probability that u 1 is in error as since n 1 is Gaussian distributed N (μ, σ 2 ) with zero mean and variance σ 2 . Because BPSK was used, only the real part of n influences decision making, so the value of n real will be following the distribution of N (0, σ 2 /2) [24]. The above- be calculated by integrating the Gaussian PDF (probability density function) as By substituting v σ with t, then dt = σdv and (14) will be [25] Note that (15) and (16) 2 dt. Hence, P u1 can be rewritten as where A = √ a 1 + √ a 2 , B = √ a 1 − √ a 2 , and E b /σ 2 = SNR; then, (17) can be written as By following the same procedure, we can calculate the probability of error in u 2 . The error of the received codeword would occur only if the u 1 is in error and if the SIC process has an error. Since recovering u 2 is depending on u 1 and on the error that occurs on the u 2 data itself, thus, we have where P e 2 is the SIC error and can be estimated easily in the same way (18) was calculated. It is clear that the error will happen only if the level of the received modulated signal is less than the level of the noise that impacts the second user, and it can be evaluated as Thus, the total error probability for the second user in (19) can be written as The concord of the theoretical and practical results of (18) and (21) proves the correctness of the mathematical calculations of different power allocation coefficients, and this is very clear, as can be seen in Fig. 3.

2) BER of the Proposed Two Coded NOMA Users:
For the proposed coded NOMA system, the received signal of each bit in Y 1 is given as Let us assume that X 1 is modulated as 11 (i.e., the input of the first encoder is considered to be 1) and then X 2 can take the modulated values of {1, −1} or {−1, 1} (i.e., the input of the second encoder is considered as either 1 or 0); therefore, the above-mentioned (22) and (23) can be written as follows to consider all possible combinations: The simple decoding process that estimates the transmitted u 1 can be carried out by adding the equations (24) and (25) and comparing the result with zero aŝ and the sum of the above-mentioned equations will be Thus, the error will occur only if the noise level exceeds the value of (2 √ a 1 E c ), and this is also shown in Fig. 4 that shows the visual representation of all possible constellations of two users with the proposed NOMA-coded BPSK system, which was extracted from Table I. In other words, the probability that the far user will receive data in error P e1 can be calculated as Similarly, the error probability for u 2 can also be estimated. Let us assume that the X 2 take the modulated values of {1, −1} (i.e., the input of the second encoder is considered to be 1), and The decoding decision on u 2 can be estimated by subtracting (29) and (30) and then comparing the outcome with zero as follows:û and the output of this subtraction will be Hence, the error probability P e2 for the second user can be evaluated as To prove the correctness of the theoretical results in (28) and (33) with the simulation results, Fig. 5 shows a great match in the results. Furthermore, it is noticeable in (33) that the process of decoding the data of u 2 does not require an SIC process, which is why it depends on the power allocation that assigns to the u 2 itself.
However, this is not the case if more users are added to the system, and this instance will be explained in the next section.
B. BER for Three Users 1) BER for Three Uncoded BPSK NOMA Users: By following the same steps used in calculating the BER performance of the two-user NOMA system in Section III-A1, it is possible to derive a closed-form BER performance analysis of three Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. uncoded BPSK NOMA users easily, as shown in the following: where P e 1 , A, B, C, and D are the probability of error for the first user, where P e 2 , E, and F are the probability of error for the second user, √ a 2 − √ a 3 , and √ a 2 + √ a 3 , respectively.
where P e 3 is the probability of error for the third user. To examine the above-mentioned three equations, (34), (35), and (36), Fig. 6 shows the theoretical and simulated BER performance versus SNR of three-user BPSK-modulated NOMA system for various values of the power allocation coefficients and has shown an exact match between the simulated and theoretical results.
2) BER of the Proposed Three Coded NOMA Users: An additional user of the proposed system will be considered in this section. Let us assume that user 3 will use the first encoder for the encoding process of u 3 , see (9). In case, the proposed system is applied to three-user scenario, and the input of the encoder of the far user is considered to be 1, the received signal of each bit in Y 1 is given as As is known by now, in the proposed system, the decoding decision based on the addition of the two signals and the sum will lead to Thus, the P e 1 can be evaluated as respectively. The decoding process for u 2 can be achieved by subtracting the received 2 bits of each codeword and will lead to Therefore, the P e 2 can be estimated as It is worth noting that the decoding of the second user did not require the SIC process, but it was done immediately after the interference with the first and third user was eliminated by a simple subtraction process, and this is what (42) illustrated.
In the case of the third user, since the same encoder will be used for u 1 and u 3 , the decoding process will require a one-time use of the SIC process because the interference with the second user will be eliminated by the simple addition, and the result of this summation will be Since √ a 1 > √ a 3 (i.e., the signal of u 1 is dominant over the signal of u 3 ) the SIC need to be applied to remove u 1 from the received signal at u 3 side. Thus, P e3 can be evaluated as follows: To validate (40), (42), and (44), see Fig. 7, which shows an excellent match between the practical and theoretical results of the proposed coded NOMA system for three users and for different power allocation coefficients.

C. Complexity Analysis
This section investigates the complexity of the proposed coding scheme by evaluating the number of floating point operations (FLOPS). Low latency is one of the key goals of the 5G and 6G communications systems. Hence, determining the number of the required FLOPS for each proposed algorithm and/or scheme is of great importance, as less number of FLOPS means less processing time and less latency. This fact has motivated us to introduce a low-complexity coding scheme to improve the performance of the conventional uncoded NOMA system.
Without the loss of generality, each mathematical operation between any two real-valued numbers are treated as a single FLOP in this work [26], [27], [28]. A closer look at (7) and (8) show that the decoding of the received signal at user 1 and user 2 is done as follows.
1) First, we divide the received sequence into odd and even subsequences. Consequently, we will have two subsequences with a size of N/2, one for the odd signals and one for the even signals. 2) Then, we preform addition between these sub-sequences at user 1 and subtraction between them at user 2 to obtain the decision sequence of size N/2, which is used to decode the received signals. 3) Finally, we compare each element of the decision frame with zero to decide whether the received signal is demodulated as "1" or "0." So, the demodulation of the received signal frame will require performing N/2 additions or subtractions, followed by N/2 comparison with zero. Hence, the complexity order of the proposed algorithm can be expressed as Extending our NOMA system to three users will require extra arrangements for demodulating and decoding the third user's data. First, the proposed decoding algorithm is used to eliminate the effect of user 2 by simple addition process, as illustrated by (43), and then user 1 data can be obtained, which is then remodulated and multiplied by √ a 1 to mimic user 1 signal. Next, this signal is removed from the total received signal by subtracting it from that signal. After that, the proposed scheme is performed again to obtain the data of the third user. Hence, the complexity order of the proposed approach will increase to be On the other hand, for the uncoded NOMA system and for the first user, i.e., the user that has been assigned the higher power allocation factor, the demodulation of the received frame is done by comparing each signal with zero to decide whether this signal is demodulated as "0" or "1." Thus, the complexity order of obtaining the data of user 1 can be given as For user 2 , we need to detect the data of user 1 ; then, we remodulate the data to BPSK symbols. After that, these BPSK symbols will be multiplied by the power allocation factor assigned for Fig. 8. Complexity of the proposed scheme and that of the SIC versus frame size N . user 1 and subtracted the resulting signal from the received signal to obtain the signal of user 2 . The obtained signal is demodulated using the comparison operation with zero to obtain the data of user 2 . Hence, the complexity order of demodulating the data of user 2 can be presented as Analogous to user 2 , the third user performs SIC to obtain its data. First, the data intended for the first user are detected and demodulated and then used to generate a copy of the first user's signal. Next, we subtract this signal from the received signal and use the subtraction output to detect and obtain user two data. Then, these data will be used to mimic the signal of user two, which will be subtracted from the total received signal. Finally, the remainder signal after the two stages of SIC will be utilized to obtain the data of the third user. Consequently, the complexity order of detecting the data of user 3 can be expressed as (49) Fig. 8 shows the complexity order of the proposed scheme and that of the conventional uncoded NOMA. Moreover, a closer look at Fig. 8 reveals that the complexity order of the proposed scheme at user 2 is 5 N less than that of the uncoded NOMA with SIC. Furthermore, this gap is increased at user 3 to 6 N . Consequently, it is fair to say that our scheme provides higher efficiency, less latency, and better BER performance, which were the main motivations for our work.

IV. RESULTS AND DISCUSSION
In this section, the results of the proposed system will be presented in different scenarios and compared with the uncoded and convolutional-coded NOMA system on AWGN channel. Before starting the comparison, the performance of the comparative systems needs to be clarified in the conventional communication system in order to have a fair comparison to determine their efficiency in detecting and correcting errors when employed in NOMA. Fig. 9. BER performance comparison between the proposed code, uncoded scheme, and the convolutional code in conventional communication system. Fig. 10. BER performance comparison between uncoded, convolutional, and the proposed coded two users' NOMA system. Fig. 9 illustrates the BER performance comparison between the proposed code, uncoded, and convolutional code in the case of the conventional communication system, and clearly, the convolutional codes achieve very low BER at 6 dB while the proposed one performs just like the uncoded system in this scenario, and the gain in power will be about 4 dB in the advantage of convolutional code over the proposed code. For more information about the performance of the convolutional code used in this article can be found in the following sources [29], [30]. Although the potential and ability of convolutional code to detect and correct errors in the OMA communication system are much more than the proposed code, the proposed system will outperform the NOMA convolutional system as it is specifically designed for this type of communication, and this is clearly shown in the following figures. Fig. 10 shows the BER performance of the proposed coded NOMA, the uncoded NOMA system, and convolutional coded NOMA versus SNR when the power allocation coefficients a 1 Fig. 11. BER performance comparison between uncoded, convolutional, and the proposed coded three users' NOMA system. and a 2 are 0.75 and 0.25, respectively. In this figure, the gain in BER for both users of the proposed system over the uncoded users is about 7.5 and 2.3 dB for user 1 and user 2 , respectively. Since a 1 a 2 , in other words, when the difference in user power allocation coefficients is huge, the proposed system is performing better than the convolutional coded system for the far user, and the gain in performance over the convolutional NOMA is about 2 dB. In contrast, the convolutional coded NOMA system with such values will perform better for the second user by 2 dB as well.
Adding more users to the system will clearly show the difference in the performance of the proposed coded system from the rest of the conventional coded and uncoded NOMA systems. Fig. 11 shows the BER curves for three users' coded NOMA and uncoded system. Obviously, the proposed system has a huge gain in performance over the uncoded one, by roughly 17.5, 15.5, and 13 dB for users 1, 2, and 3, respectively. Even though the BER of the proposed system of user 2 in Fig. 10 is worse than the convolutional coded NOMA system, the difference in overall performance will be in the interest of the suggested system when the number of users increased and also when the values of the power allocation coefficients are very close. Fig. 11 carries the idea of the comparison between the proposed coded NOMA and convolutional coded NOMA for three users. The proposed system outperforms the convolutional coded NOMA system in error detection and correction, so the gain in favor of the proposed system is about 10, 7.5, and 5 dB for users 1, 2, and 3, respectively.
Furthermore, to show the impact of the power allocation factors on the performance of the coded NOMA systems, Fig. 12 will carry out the effect of different values of the power allocation coefficients on the overall system performance. It is absolutely clear that if the a 1 = a 2 , the convolutional coded NOMA systems are unable to detect and/or correct errors, whereas the proposed system can provide excellent performance in detecting and correcting errors even when a 1 is close or equal to a 2 .
To find a relationship between the performance of the NOMA system and the power allocation factors in order to choose the  optimum parameters, Fig. 13 presents the relationship between the BER performance of two users coded and uncoded NOMA systems on fixed SNR (10 dB) versus a 1 . Since a 2 = 1 − a 1 and a 1 > a 2 , this figure is considering the values of a 1 ∈ [0.5, 1] and a 2 ∈ [0.5, 0].
Finally, Fig. 13 also shows that the lowest BER (3.4 × 10 −2 ) for both users of the uncoded NOMA system can be achieved when the a 1 = 0.8 (i.e., a 2 = 0.2 and a 1 >> a 2 ), whereas the lowest BER of the proposed system was (7.8 × 10 −4 ) when a 1 = a 2 = 0.5. This clearly demonstrates the ability of the proposed system to recover the transmitted data more efficiently and does not require a large difference in power allocation factor between users, and this gives more freedom to increase the number of users or assign close values of power allocation coefficients, unlike the conventional system that requires a large difference in power allocation factor between users so that the system can function.
From the simulation results in Figs. 9-13 and the complexity analysis, it can be seen that our proposed coding scheme has achieved a large performance gain over the uncoded and convolutional coded NOMA systems with low complexity. However, our proposed scheme and analysis are limited to two-user and three-user cases so far, and it is worth generalizing our scheme to more users. Another challenge of our scheme is that our work is restricted to AWGN channel and BPSK modulation. In the future, fading channels and higher-order modulation schemes can be further explored.

V. CONCLUSION
Taking into account the various characteristics of the NOMA system to think of coding and decoding methods that suit these features, it is possible to gain better performance with lower complexity. This is what was considered and implemented in this research article, and the results were worthwhile to some extent compared with the simplicity of the proposed system. The proposed system outperformed the traditional systems that were compared, such as uncoded NOMA and convolutional coded NOMA systems in many cases and scenarios that were practically implemented and mathematically proven. Comparison with the classical NOMA system and the proposed system when using the same power allocation parameters showed a significant performance superiority of the proposed system by 7.5 and 2.3 dB for user 1 and user 2 , respectively, in the case of a two-user system, whereas in the case of a three-user system, the gain was 17.5, 15.5, and 13 dB in favor of the proposed system and for the first, second, and third users, respectively. In addition, this superior performance was achieved without adding complexity to the system; on the contrary, the proposed scheme needs the same number of FLOPs at user 1 and achieves a reduction in the number of FLOPs by 5 N and 6 N at user 2 and user 3 , respectively. Moreover, the research has proven the validity and conformity of the obtained results for the proposed system mathematically, and accurate mathematical equations were derived to estimate the performance of the system. Finally, the theoretical and practical results in this article were congruent. In the future, we will investigate the generalization of our work to more practical scenarios, i.e., more users, high-order modulations, and fading channels.