Performance Analysis of NOMA Over η − µ Fading Channels with imperfect SIC

In this paper, a downlink non-orthogonal multiple access (NOMA) network with two users is considered. In particular, the performance of NOMA is evaluated by assuming perfect and imperfect channel state information (CSI). We derive the closed-form expressions for the outage probability over η − µ fading channels in the special case of two users. Moreover, the proposed system model-based NOMA always achieves better performance than that with perfect CSI in the medium SNR region. Monte Carlo simulations are then performed to confirm a good match with the analytical results.


Introduction
There are many challenges in the fifth generation (5G) wireless communication networks and beyond when performing the massive number of connected devices with low delay time, high spectral efficiency, and data rates [1]. To solve these problems, in recent years, non-orthogonal multiple access (NOMA) technology, which many researchers have gained much attention to design communication systems [2]- [4]. NOMA is considered a potential candidate to significantly improve the efficiency spectrum compared to conventional orthogonal multiple access (OMA) [5]. For example, compared to conventional without NOMA, power allocation in NOMA to users which have worse channel conditions is more powerful than users with better channel conditions to guarantee user fairness. NOMA uses superposition coding (SC) signals on the transmitter side, and the receiver side performs successive interference cancellation (SIC). To separate signals, different power levels can be assigned to users at the same time and frequency. In SIC, the signal of a stronger user is decoded and removed first, and then the weaker user decodes its own message [6]. It comes from the benefits of NOMA to improve system performance. For instance, * Corresponding author. Email: tranquyhuu@iuh.edu.vn the authors in [7] discussed the advantages and challenges of NOMA for 5G networks. In [8], NOMA for wireless downlink, called multi-tier heterogeneous superposition transmission, is being investigated. In [9], NOMA and OMA, the results showed that NOMA achieves superior energy efficiency in comparison with OMA. Furthermore, [3] also studied the cooperative NOMA, in which BS communicates with the users with weaker channel conditions based on helping relays to decode and forward the signals to others. Thus, establishing reliable communication between the base station and users far away. In order to enhance energy efficiency in NOMA, simultaneous wireless information and power transfer (SWIPT) has been investigated [4], as using stochastic geometry to consider positions of users and radio energy harvesting (EH) at relay to help forwarding the signals to far user. In [2], the authors have considered performance systems to achieve the capacity of the broadcasting channel for the single-antenna downlink NOMA strategy. In [10], the authors have demonstrated MIMO NOMA systems can further provide significant gains compared to OMA schemes. In addition, the uplink and downlink singlecell NOMA communication system wherein multiple users communicate with a single BS with generalized fading channels based on the outage probability (OP) was proposed in [11]. Moreover, the optimal power allocation issue based on average channel state information (CSI) and MIMO-NOMA systems with CSI at the transmitter using ergodic capacity, and OP also was considered in [12]- [13]. However, the works by [12] and [13], are only discussed in cases where the distance between users is predefined. Therefore, most of the existing literature about NOMA requires the knowledge of perfect CSI. In practice, the assumption of perfect CSI in transmitters might be hard to achieve. Furthermore, towards 5G and beyond, there are mobile devices, a number of large users at high-speed, rapidly changing channels, perfect CSI at the transmitter is a challenge. As evident from different aspects, it is necessary to study NOMA users in different 5G-fading realities environments. Therefore, there are many generalized fading models which are used to describe these changing characteristics of the environment, in which the η − µ model is the general model of fading which encompasses propagating in a heterogeneous environment. The OP and sum rate performance of a NOMA downlink system considering η − µ and κ − µ fading channels were studied in [14]. The η − µ fading model inspects a general non-line-of-side propagation strategy by considering two form parameters, η and µ. In [15], the authors have analyzed the OP of the wireless system affected by the η − µ fading and co-channel interference, in an AF relay channel without diversity techniques at the receiver. The outage performance for downlink C-NOMA networks over Nakagami-m fading channels with the imperfect CSI taken into account was investigated in [16]. Moreover, in [17], the authors analyzed the performance of generalized frequency division multiplexing in Rayleigh, Nakagami-m, and Nakagami-q fading channels. In addition, the connection between Rayleigh and Hoyt distributions that facilitate the analysis of wireless communication systems is considered in [18]. Motivated by these constraint conditions, we investigate the use of partial CSI, which reduces the complexity of the system and enhances the spectral efficiency of NOMA in practice. In this paper, we consider a downlink NOMA network, where the base station and users are equipped with a single antenna. We also assume the channel estimation error model, where the BS and the users perform the channel estimation and predict the variance of the estimation error. For performance evaluation, we also carried out the expressions of OP over η − µ fading channels. The rest of the paper is organized as follows. Section 2 presents the system model and channel characteristics. Section 3 analysis of the outage performance and the system throughput of NOMA schemes with perfect and imperfect CSI. In Section 4, numerical results are shown and Monte Carlo simulations are performed to verify the theoretical analysis. Finally, Section 5 concludes the paper.

System model
It can be seen from Fig. 1, the system includes a single antenna source (S) that serves two NOMA users at the downlink. We have h 1 and h 2 , respectively, characterize link BS-user D 1 and link BS-user D 2 . Two users, D 1 and D 2 , are facilitated with half-duplex (HD) mode. During EH time αT , in which T is the block time and α is the fraction of the block time allocated for EH, and its condition is 0 ≤ α < 1. The harvested energy, E D i is given by where P S depicts the normalized transmission power at the source (S), υ is the energy conversion efficiency.
Depending on the quality of EH electric circuitry 0 < υ < 1. From obtained E D i in (1), the transmitted power from node S, P D i is given by The received signal at D 1 and D 2 , respectively, are given by where ω 1 and ω 2 depicts the additive white Gaussian noise (AWGN) with zero mean and variance of N 0 at the D 1 , and D 2 , respectively, and x j is assumed to be normalized the unity power signal for the j − th user, i.e., E x 2 j = 1 where E {} is the expectation operator. The j − th user ′ s power allocation factor v j is for the fairness of users. In the first phase, the signal-to-interference-plus-noise ratio after treating x 1 as interference can be computed by where γ i ∆ = |h i | 2 , i ∈ {1, 2} and the transmit signal-tonoise ratio (SNR) is determined at the S as ρ S = P S N 0 . Note that γ 1 and γ 2 are independent random variables (RVs). It should be noted that imperfect SIC (ipSIC) occurs, and the SINR to detect x 2 is given as  Similarly, the instantaneous SINR at D 2 for detecting x 2 is given by

Channel Characteristics
The probability density function (PDF) of γ = γ 1 = γ 2 is given by [15,Eq.(1)] According to [16,Eq.(2)] the cumulative distribution functions (CDF) of γ can be determined by Next, we have PDF and CDF of |h I | 2 are given by and

Outage probability
First, the OP of D 1 is calculated as where Then, P 1 can be calculated by 3 EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 12 2022 -05 2023 | Volume 10 | Issue 1 | e5 Case 1: when µ ∈ N, ∀µ ≥ 0 then we use PDF of (11) and CDF of (10), (14) is given by Case 2: when µ ∈ I, ∀µ ≥ 0 then we use PDF of (10) and CDF of (9), (14) can be given by Specifically, we set q = x λ I with the help of Gauss-Laguerre integration in [21,Eq.(25.4.45). The closedform approximation of the OP 1 at D 1 is given by where Θ (q n ) = (ρ S λ I q + 1) , χ n and q n are the weight and abscissas for the Gauss-Laguerre integration, respectively. More specifically, q n is the n − th zero of Laguerre polynomial L N (q n ) and the corresponding the n-th weight is given by χ n = (N !) 2 q n [L N +1 (q n )] 2 . The parameter N is to ensure a complexity-accuracy trade off. Finally, the OP at D 2 is calculated as Case 1: when µ ∈ N, ∀µ ≥ 0 then we use CDF of (10), (19) is given by Case 2: when µ ∈ I, ∀µ ≥ 0 then we use CDF of (9), (19) is given by

System Throughput
With a given constant R, the transmitted information of the source node depends on the OP performance due to wireless fading channels. Therefore, the system throughput is determined by Where OP 1 is given in (14)- (16).

NUMERICAL RESULTS
The Monte-Carlo simulation results from an average of over 10 7 independent channel realizations. Target data rates at D 1 and D 2 are R 1 = 1 (BPCU) and R 2 = 0.5 (BPCU), respectively. Mean values of the interference signal channel power gains λ I = −30 (dB) . The power allocation coefficients v 1 = 0.2 and v 2 = 0.8. The Gauss Laguerre quadratures have several points, N = 40. Figure 2 plots the OP versus SNR according to different fair values of η and µ. The figures show that the curves of the OP for User 2 decrease quickly while that for User 1 decrease gradually. In SNR range of from 40 to 60 dB, the OP for User 1 is almost constant. The OP of User 1 is always higher than that for User 2. For effects of η on the OP, the figure shows that the lower the η, the higher the OP. On contrary, the higher the µ, the lower the OP. The OP for ipSIC only decreases in a certain range of SNR, e.g., herein from 0 to 40 dB, then tends to keep a constant value. Figure 3 plots the OP versus SNR according to perfect SIC and ipSIC. One can see from the figure that the OP curves for both users in case of perfect SIC decrease linearly in the SNR range of from 20 to 60 dB. The OP for user 2 is always lower than that for user 1, regardless 4 EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 12 2022 -05 2023 | Volume 10 | Issue 1 | e5 of the variation of SNR. When the ipSIC occurs, the OP curves of user 1 generate inflection points and tend to decrease gradually in the SNR range from 0 to 40 dB. However, these OP curves are almost constant when SNR increases from 40 to 60 dB. The higher the λ, the higher the OP for ipSIC. We can conclude that the ipSIC considerably impacts on the OP. Figure 4 plots the OP versus power allocation coefficient v 1 of from 0 to 0.5 in cases of perfect and ipSIC. The figure shows that the OP for user 1 decreases in the v 1 range of from 0 to 0.35 and then increases in the v 1 range, from 0.35 to 0.5, while that for user 2 is always an increase in the overall v 1 range. Therefore, as an obvious results, the OP for two users exists some intersection points. On the left side of the intersection, the OP of the user 2 is significantly lower than that of user 1. On the contrary, this OP at the right side of the intersection points tends to coincide with each other. Besides, the OP for ipSIC is always the highest. It can be concluded that an ipSIC can cause negative impacts on the OP. In addition, the simulations also consider the different transmitting power, such as 10, 15 and 20 dB, respectively. The OP for both users increases when the transmitting power P S decreases. This is due to the fact that the stronger the transmitting signal, the better the quality of the signal. We can observe from figure 5 that the OP for user 2 is lower than that for user 1. The OP for ipSIC is always higher than that for perfect SIC. Besides, when the transmitting power decreases, the OP increases in the overall target rate range. Moreover, the OP for both users increases highly and is asymptotic to 1 when the target rate increases. Specifically, in this work, the OP for user 1 reaches 1 when the target rate is equal to 2. The OP for user 2 reaches 1 when the target rate is equal to 2.25. Fig. 6 plots the system throughput according to different values of power allocation factor v 2 . We can see from this figure that the system throughput of user 1 in the case of perfect CSI is higher than imperfect CSI and both cases decrease when increasing v 2 from 0.55 to 0.95. The throughput of user 1 is larger than that of user 2 with the values of v 2 (0.5, 0.75) in the case perfect CSI and v 2 (0.5, 0.8) in the case imperfect CSI. However, the system throughput of user 2 reaches saturation when increasing the values of v 2 . It can be explained that the SNR at user 2 used to detect x 1 and x 2 is higher than the minimum value of SNR at user 1 used to detect x 2 . Figure 7 describes the throughput versus transmitting SNR ρ (dB) with different values λ I . From the figure, it is shown that the throughput of user 1 with perfect SIC is considerably higher than that with ipSIC when increasing ρ (dB) about (10 ÷ 30) dB. Additionally, when decreasing the residual interference level caused by ipSIC (λ I ), the throughput of user 1 is increasing. Moreover, at medium transmit SNR about (0 ÷ 20) dB, the throughput of user 2 is higher than that for user 1.

Conclusion
In this paper, we have studied the OP and the throughput of NOMA schemes for the two users. The numerical derivations of NOMA protocols in two cases with perfect and imperfect CSI. The analytical evaluations of NOMA protocols based on perfect CSI at medium SNR always outperforms NOMA with imperfect CSI. In addition, the outage performance of user 1 is always higher than that for user 2. The results shows that the η lower, the outage performance is higher. On the contrary, the higher µ, OP is lower. Moreover, the analytical expressions showed that the ipSIC considerably impacts on the OP and the system throughput.

Data Availability
The data used to support the findings of this study are included in the paper.

Conflicts o f Interest
The authors declare there is no conflict of interest in this manuscript.

Acknowledgement
This study was self-funded by the authors. 5 EAI Endorsed Transactions on Industrial Networks and Intelligent Systems 12 2022 -05 2023 | Volume 10 | Issue 1 | e5 Huu Q. Tran and Ong Mau Dung