# Symbol Error-Rate Analytical Expressions for a Two-User PD-NOMA System with Square QAM
## 4 BER and SER for PD-NOMA Square QAM Signal
### First Layer Decoding  
The received PD-NOMA symbol \(s\) is expressed in (3), where the information symbol \(x\) of UE\({}_{1}\) is transmitted in the first non-orthogonal layer. Denote \(\rho(\overline{x})\) and \(\rho_{b}(\overline{x})\) as symbol and bit error probabilities while decoding \(x\) from \(s\). Similar to conventional QAM, we define three types of the received symbols: the corner symbol \(s_{cor}\) contains corner symbol \(x_{cor}\); \(s_{ins}\) contains \(x_{cor}\), and \(s_{out}\) contains \(x_{out}\). The occurrence probabilities are the same as for QAM, and they can be obtained from (7) by  
\[ \begin{split}\rho(s_{cor})&=\frac{4}{Q_{1}}\\ \rho(s_{ins})&=\frac{Q_{1}-4(\sqrt{Q_{1}}-1)}{Q_{1}} \\ \rho(s_{out})&=\frac{4(\sqrt{Q_{1}}-2)}{Q_{1}} \end{split}  \tag{11}\]  
Then, \(\rho(\overline{x})\) is obtained by using the same ideas as in (6) and given by  
\[\rho(\overline{x})=\rho(s_{cor})\rho(\overline{x}_{cor}|s_{cor})+\rho(s_{ins}) \rho(\overline{x}_{ins}|s_{ins})+\rho(s_{out})\rho(\overline{x}_{out}|s_{out}), \tag{12}\]where \(\rho(\mathfrak{F}_{cor}|s_{cor})\), \(\rho(\mathfrak{F}_{ins}|s_{ins})\), \(\rho(\mathfrak{F}_{out}|s_{out})\) are the probabilities of the error decoding \(x_{cor}\), \(x_{ins}\), \(x_{out}\) from \(s_{cor}\), \(s_{ins}\), \(s_{out}\). In turn, the probability of the error decoding for \(\forall x\in\mathbf{A}_{1}\) is given by \(\rho(\mathfrak{F}|s)=1-\rho(x|s)\), where \(\rho(x|s)\) is the probability of the correct decoding \(x\). Further, the symbol \(x\) is decoded correctly when the real and imaginary components of \(s\) fall into its own decoding area. Let \(\Re_{s}\) and \(\Im_{s}\) be the real and imaginary components of \(s\), then \(\rho(x|s)\) is calculated by  
\[\rho(x|s)=\rho(\Re_{s}\in O_{x})\times\rho(\Im_{s}\in O_{x}), \tag{13}\]  
where \(O_{x}\) is the decoding area of \(x\). The symbol \(x\) can be simultaneously combined with \(\forall y\in\mathbf{A}_{2}\); thus, we denote \(x+y_{n}\) as the combination presented in one region (real or imaginary) for \(n=1\ldots\sqrt{Q_{2}}\). For example, \(x+y_{n}\) in the real region are shown on the group constellation in Figure 1. Further, we focus on the real region and \(\Re_{s}\). Thus, let us note the equation for \(\rho(\Re_{s}\in O_{x})\) as follows  
\[\rho(\Re_{s}\in O_{x})=\sum_{n=1}^{\sqrt{Q_{2}}}\rho(x+y_{n})\times\rho\big{(} \Re_{x+y_{n}}\in O_{x}\big{)},\]  
where \(\rho(x+y_{n})\) is the occurrence probability of the \(x+y_{n}\) combination, and \(\rho(\Re_{x+y_{n}}\in O_{x})\) is the probability that the real component of \(x+y_{n}\) falls into \(O_{x}\). The occurrence probabilities are equal for all the combinations; therefore, \(\rho(x+y_{n})=1/\sqrt{Q_{2}}\). Then, the expression for \(\rho(\Re_{s}\in O_{x})\) can be rewritten by  
\[\rho(\Re_{s}\in O_{x})=\frac{1}{\sqrt{Q_{2}}}\sum_{n=1}^{\sqrt{Q_{2}}}\rho \big{(}\Re_{x+y_{n}}\in O_{x}\big{)}. \tag{14}\]  
Now, we need to obtain the probabilities for the corner, inside, and outside symbols, using (14), and put them in (12). Firstly, we obtain the expression for finding the correct decoding probability for \(x_{ins}\). The decoding area \(O_{ins}\) is limited by two sides in the real and imaginary regions (Figure 1) equally, so, using (13), we have \(\rho(x_{ins}\big{|}s_{ins})=\rho(R_{s_{ins}}\in O_{ins})^{2}\). In turn, \(\rho(\Re_{s_{ins}}\in O_{ins})\) is obtained by (14). As we can see, it is enough to find \(\rho\big{(}\Re_{x_{ins}+y_{n}}\in O_{ins}\big{)}\) and substitute it to (14). The Euclidian distance from the reference constellation point \(\sqrt{p_{1}}x_{ins}+\sqrt{p_{2}}y_{n}\) to the left bound of \(O_{ins}\) is  
\[L_{n}^{left}=\frac{1}{2}\Big{[}D_{1}-\Big{(}\sqrt{Q_{2}}-2n+1\Big{)}D_{2} \Big{]}, \tag{15}\]  
and to the right bound is  
\[L_{n}^{right}=\frac{1}{2}\Big{[}D_{1}+\Big{(}\sqrt{Q_{2}}-2n+1\Big{)}D_{2} \Big{]}. \tag{16}\]  
where \(D_{1}\) and \(D_{2}\) are obtained by (2).