# Symbol Error-Rate Analytical Expressions for a Two-User PD-NOMA System with Square QAM
## 3 BER and SER for Conventional Square QAM Signal  
First, we shortly describe a conventional square QAM decoding process and show the known analytical SER and BER over the \(E_{s}/N_{0}\) expressions for it. Let the received square \(Q\)-QAM symbol be \(u=u+w\), where \(u\) is the transmitted symbol using alphabet \(\mathbf{A}\) (\(u\prime\in\mathbf{A}\)), and \(w\) is the AWGN realization. The receiver decodes \(u\) by the minimum Euclidean distance criterion as follows  
\[\hat{u}=arg\underset{u\in\mathbf{A}}{min}|u-u\prime|^{2}.\]  
There are \(Q\) possible states on the \(Q\)-QAM constellation, which can be divided into three groups: corner, inside, and outside symbols. It is observed on the 16-QAM constellation of UE\({}_{1}\) shown in Figure 1. Thus, let \(\rho(u_{cor})\), \(\rho(u_{ins})\), \(\rho(u_{out})\) be occurrence probabilities, and \(\rho(\overline{u}|u_{cor})\), \(\rho(\overline{u}|u_{ins})\), \(\rho(\overline{u}|u_{out})\) be error decoding probabilities of the corner, inside, and outside symbols. Next, we obtain the symbol error probability \(\rho(\overline{u})\) as follows  
\[\rho(\overline{u})=\rho(u_{cor})\rho(\overline{u}|u_{cor})+\rho(u_{ins})\rho( \overline{u}|u_{ins})+\rho(u_{out})\rho(\overline{u}|u_{out}), \tag{6}\]  
where occurrence probabilities are known from the ratio number of each type of symbol position to the constellation order \(Q\) and given by  
\[\begin{array}{l}\rho(u_{cor})=\frac{4}{Q}\\ \rho(u_{ins})=\frac{Q-4(\sqrt{Q}-1)}{Q}\\ \rho(u_{out})=\frac{4(\sqrt{Q}-2)}{Q}\end{array} \tag{7}\]  
Figure 1: Signal constellations.  
The corner, inside, and outside symbols have different correct decoding areas. The calculation of the decoding probabilities is based on obtaining the detecting areas and the probability \(u\) that falls outside due to AWGN with the \(E_{s}/N_{0}\) ratio. The derivation of the analytical expressions for the error decoding probabilities of the square QAM is well-described in [26]. We introduce these expressions below  
\[\begin{split}\rho(\overline{u}|u_{cor})&=erfc\Big{(} \sqrt{\frac{E_{s}}{\text{M}\text{N}_{0}}}\Big{)}-\frac{1}{4}erfc^{2}\Big{(} \sqrt{\frac{E_{s}}{\text{M}\text{N}_{0}}}\Big{)}\\ \rho(\overline{u}|u_{ins})&=2 erfc\Big{(}\sqrt{\frac{E _{s}}{\text{M}\text{N}_{0}}}\Big{)}-erfc^{2}\Big{(}\sqrt{\frac{E_{s}}{\text{M} \text{N}_{0}}}\Big{)}\\ \rho(\overline{u}|u_{out})&=\frac{3}{2}erfc\Big{(} \sqrt{\frac{E_{s}}{\text{M}\text{N}_{0}}}\Big{)}-\frac{1}{2}erfc^{2}\Big{(} \sqrt{\frac{E_{s}}{\text{M}\text{N}_{0}}}\Big{)}\end{split} \tag{8}\]  
where \(erfc(x)\) represents the complementary error function, which is defined as \(erfc(x)=\frac{2}{\sqrt{\pi}}\int\limits_{x}^{\infty}e^{-x^{2}}dx\).  
The common probability of error decoding \(\rho(\overline{u})\) in the AWGN channel with the \(E_{s}/N_{0}\) ratio is obtained by putting (7) and (8) into (6) and simplifying the resulting expression as follows  
\[\rho(\overline{u})=2\Big{(}1-\frac{1}{\sqrt{Q}}\Big{)}erfc\Bigg{(}\sqrt{\frac {E_{s}}{\text{M}\text{N}_{0}}}\Big{)}-\Big{(}1-\frac{2}{\sqrt{Q}}+\frac{1}{Q} \Big{)}erfc^{2}\Bigg{(}\sqrt{\frac{E_{s}}{\text{M}\text{N}_{0}}}\Bigg{)}, \tag{9}\]  
Finally, the BER value \(\rho_{b}(\overline{u})\) can be obtained under the same conditions by averaging the result of (9) with help of \(log_{2}Q\) by  
\[\rho_{b}(\overline{u})\approx\frac{\rho(\overline{u})}{log_{2}Q}. \tag{10}\]