# Symbol Error-Rate Analytical Expressions for a Two-User PD-NOMA System with Square QAM
## 1 Introduction  
The power domain non-orthogonal multiple access (PD-NOMA) is a perspective technology for user multiplexing in future networks [1; 2; 3; 4; 5; 6; 7]. It is based on non-orthogonal channel multiplexing by user signal superposition in the power domain at the same time-frequency resource segment (for example, a subcarrier in orthogonal frequency division multiplexing). Thus, there is a controlled co-channel interference which is cancelled at the receiver side by the successive interference cancellation (SIC) method [8] in the SIC detector. In comparison to orthogonal multiple access (OMA), such as orthogonal frequency division multiple access (OFDMA), the PD-NOMA exploits the channel state differences between users. Thus, the available power of a resource segment is divided between multiplexed users, according to the required quality of the service and channel state information of each user. Therefore, resource allocation occurs in power, frequency, and time domains simultaneously. Due to the non-orthogonal power allocation techniques, the spectral and energy efficiency advantages of multiuser communication systems can be obtained.  
There is a significant number of PD-NOMA articles; most of them are aimed at the design of new analytical resource scheduling [9; 10; 11; 12; 13; 14; 15] and research of system gains in comparison to OMA systems [16; 17]. Mostly, these results are based on the Shannon-Hartley theorem [18] and do not take into account bit error rate (BER) and symbol error rate (SER) performances, which are different from OMA systems. However, it is necessary to consider BER and SER to obtain a more reliable result because the BER and SER are the main performance parameters of real signals in an additive white Gaussian noise (AWGN) channel. Usually, the signal construction based on quadrature amplitude modulation (QAM) is used in both OMA and PD-NOMA. However, there is an issue that the analyticalexpressions for calculating the error probability are not equal. Those expressions are given in [19] for OMA. For the PD-NOMA, this issue has not been well-studied, and only a few works provide a solution for these problems.  
The SER and BER analysis using SIC detectors has received attention in recent works. In [20], the BER is explored for PD-NOMA under Rayleigh fading channel where exact and approximate expressions are obtained. However, the presented expressions are applicable only for two-user PD-NOMA with binary phase shift keying (BPSK) and quadrature phase shift keying (QPSK). The BER expressions over Nakagami-\(m\) channel for two and three-user scenarios using QPSK are derived in [21]. In [22], the authors present BER expressions for PD-NOMA systems while using only QPSK in both user channels. Work [23] also considers the study of SER and BER in PD-NOMA systems, and the authors present a closed-form expression within the conditions of Gray coding and QPSK. In [24], the researchers study the analytical SER in visible light communication systems with the NOMA; however, QAM is not considered. SER expression using QAM is presented in [25] which has non-ideal accuracy for the second non-orthogonal layer. As can be seen from the discussion above, there is no paper that considers the exact SER expressions of toe first and second non-orthogonal user in PD-NOMA with square QAM.  
The purpose of our work is to derive the analytical SER and BER expressions for a two-layer PD-NOMA system under the conditions of square QAM multiplexing. First of all, we describe a two-layer PD-NOMA system model and known expressions for conventional square QAM in the OMA system. Secondly, we give SER and BER expressions for a QAM signal in the first and second non-orthogonal layers consistently. Finally, the simulation and analytical calculation results of BER and SER are compared.