# 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
### Second Layer Decoding  
In this subsection, we derive SER and BER for the second non-orthogonal layer. According to the SIC, the decoding of the second layer is performed after decoding, regeneration, and cancellation of the first layer. If it is cancelled perfectly (without error), the updated received symbol is the superposition of the power weighted \(Q_{2}\)-QAM and noise realization. In this case, the probability of the error decoding is simply obtained in a similar way to (8), considering the symbol energy \(p_{2}\), and given by  
\[\rho(\overline{y}|correct~{}x)=2\bigg{(}1-\frac{1}{\sqrt{Q_{2}}}\bigg{)}erfc \bigg{(}\sqrt{\frac{p_{2}}{M_{2}N_{0}}}\bigg{)}-\bigg{(}1-\frac{2}{\sqrt{Q_{2}} }+\frac{1}{Q_{2}}\bigg{)}erfc^{2}\bigg{(}\sqrt{\frac{p_{2}}{M_{2}N_{0}}} \bigg{)}.\]  
However, the error cancelling of \(x\) causes additional corruption of the second layer and increases the probability of its error decoding. In general, the probability of error decoding in the second non-orthogonal layer is obtained by  
\[\rho(\overline{y})=1-\rho\big{(}\Re_{s}\in O_{y}\big{)}\rho\big{(}\Im_{s}\in O _{y}\big{)},\]  
where \(\rho\big{(}\Re_{s}\in O_{y}\big{)}\) and \(\rho\big{(}\Im_{s}\in O_{y}\big{)}\) are the probabilities that the real and imaginary components of the received symbol \(s\) fall into the decoding area of \(y\). Note that \(\rho\big{(}\Re_{s}\in O_{y}\big{)}\) and \(\rho\big{(}\Im_{s}\in O_{y}\big{)}\) are identical and can be obtained by the same expression. Thus, we can modify the equation for \(\rho(\overline{y})\) as  
\[\rho(\overline{y})=1-\rho\big{(}\Re_{s}\in O_{y}\big{)}^{2}. \tag{25}\]  
Now, it is enough to derive the expression for the probability of the correct detection in the real region, and further, we focus only on it. The overall decoding area \(O_{y}\) for each \(y\) is divided into several local areas. Let us look at the PD-NOMA constellation that is shown in Figure 1. The symbol \(y\) is decoded correctly when \(s\) falls in one of the sixteen (\(Q_{1}=16\)) areas near \(\sqrt{p_{1}}x+\sqrt{p_{2}y}\), \(\forall x\in\mathbf{A}_{1}\). In general, the number of the detecting sections in the real region is \(\sqrt{Q_{1}}\), and we obtain \(\rho\big{(}\Re_{s}\in O_{y}\big{)}\) as follows  
\[\rho(\Re_{s}\in O_{y})=\sum_{i=1}^{\sqrt{Q_{1}}}\rho(x_{i}+y)\rho\big{(}\Re_ {x_{i}+y}\in O_{y}\big{)},\]  
where \(\rho(x_{i}+y)\) is the occurrence probability of the combination \(x_{i}+y\) in the received symbol, and \(\rho\big{(}\Re_{x_{i}+y}\in O_{y}\big{)}\) is the probability that the real component of the received symbol falls into its own detecting area. The main problem is that \(\rho\big{(}\Re_{x_{i}+y}\in O_{y}\big{)}\) are not identical for different \(y\) because the detection sections are different. For example, Figure 2 shows the upper part of the PD-NOMA constellation from Figure 1. It is observed that \(y\) symbols on the left side of \(x_{i}\) (blue dots) have detection segment boundaries, which are different from the symbols on the right side (green dots) as well as from the inside symbols (red dots).