# 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
Note that \(L_{n}^{left},L_{n}^{right}>0\forall n\); otherwise, the power weight ratio \(p_{1}/p_{2}\) is unallowable, and the PD-NOMA signal is formed wrong. The reference constellation point \(x_{ins}+y_{n}\) is between \(L_{n}^{left}\) and \(L_{n}^{right}\); thus, it is decoded correctly when \(\Re_{x_{ins}+y_{n}}\) under noise falls between \(L_{n}^{left}\) and \(L_{n}^{right}\), respectively. We obtain these probabilities by using this Gauss error function for the left side  
\[\rho\Big{(}\Re_{x_{ins}+y_{n}}<L_{n}^{left}\Big{)}=\frac{1}{2}erf\left(\frac{ L_{n}^{left}}{\sqrt{N_{0}}}\right), \tag{17}\]  
and this one for the right side  
\[\rho\Big{(}\Re_{x_{ins}+y_{n}}<L_{n}^{right}\Big{)}=\frac{1}{2}erf\left(\frac{ L_{n}^{right}}{\sqrt{N_{0}}}\right),\]where \(erf(x)\) represents the error function which is defined as \(erf(x)=\frac{2}{\sqrt{\pi}}\frac{x}{0}e^{-x^{2}}dt\). Then, \(\rho\big{(}\Re_{x_{ins}+y_{n}}\in O_{ins}\big{)}\) is given by  
\[\rho(\Re_{x_{ins}+y_{n}}\in O_{ins})=\frac{1}{2}erf\Bigg{(}\frac{L_{n}^{left}}{ \sqrt{N_{0}}}\Bigg{)}+\frac{1}{2}erf\Bigg{(}\frac{L_{n}^{right}}{\sqrt{N_{0}}} \Bigg{)}.\]  
By substituting the top expression to (14), we can write  
\[\rho(x|s)=\rho(\Re_{s}\in O_{x})\times\rho(\Im_{s}\in O_{x}), \tag{13}\]0  
Let us denote the new variable as follows  
\[\rho(x|s)=\rho(\Re_{s}\in O_{x})\times\rho(\Im_{s}\in O_{x}), \tag{13}\]1  
Then, \(\rho(\Re_{x_{ins}}\in O_{ins})=Z_{1}\), and the probability of the error decoding \(x_{ins}\) is obtained by  
\[\rho(x|s)=\rho(\Re_{s}\in O_{x})\times\rho(\Im_{s}\in O_{x}), \tag{13}\]2  
Next, let us find the probability of the error decoding for the corner symbols \(\rho(\overline{x}_{cor}|s_{cor})\) in the same way as for the inside symbols. The decoding area of the corner constellation point is limited only by one side in the real and imaginary regions equally (the corner symbol in Figure 1). To be precise, it is the same as for the inside symbol \(\rho(x_{cor}|s_{cor})=\rho(\Re_{s_{cor}}\in O_{cor})^{2}\). The symbols in the right and left corners are symmetric around zero; therefore, we obtain the equation for the symbols in the right corner and apply it to all the corner symbols. Thus, \(\Re_{x_{cor}+y_{n}}\in O_{cor}\) is performed when \(\Re_{x_{cor}+y_{n}}\in(L_{n}^{left};+\infty)\), where \(L_{n}^{left}\) is defined by (15). We obtain \(\rho\left(\Re_{x_{cor}+y_{n}}<L_{n}^{left}\right)\) by (17), and \(\rho(\Re_{x_{cor}+y_{n}}<+\infty)=\frac{1}{2}\) always. Then, \(\rho(\Re_{x_{cor}}\in O_{cor})\) is given by  
\[\rho(x|s)=\rho(\Re_{s}\in O_{x})\times\rho(\Im_{s}\in O_{x}), \tag{13}\]3  
Simplifying the last equation, we denote  
\[\rho(x|s)=\rho(\Re_{s}\in O_{x})\times\rho(\Im_{s}\in O_{x}), \tag{13}\]4  
and transform \(\rho(\Re_{x_{cor}}\in O_{cor})=Z_{2}\). Consequently, \(\rho(x_{cor}|s_{cor})=Z_{2}^{2}\) and the probability of the error decoding for \(x_{cor}\) can be obtained by  
\[\rho(x|s)=\rho(\Re_{s}\in O_{x})\times\rho(\Im_{s}\in O_{x}), \tag{13}\]5  
Finally, let us obtain \(\rho(\overline{x}_{out}|s_{out})\). For the first half of the outside constellation points, the decoding area is limited by one side in the real region and by two sides in the imaginary region and vice versa for the remaining points (the outside symbol in Figure 1). Based on the discussion above, the error decoding probability for the outside symbols is obtained by  
\[\rho(x|s)=\rho(\Re_{s}\in O_{x})\times\rho(\Im_{s}\in O_{x}), \tag{13}\]6After obtaining the error decoding probabilities for each type of a symbol position, we can derive the common expression for \(\rho(\overline{x})\) by substituting (11), (19), (21), and (22) into (12) as follows  
\[\rho(x|s)=\rho(\Re_{s}\in O_{x})\times\rho(\Im_{s}\in O_{x}), \tag{13}\]7  
which can be simplified to  
\[\rho(x|s)=\rho(\Re_{s}\in O_{x})\times\rho(\Im_{s}\in O_{x}), \tag{13}\]8  
where \(Z_{1},~{}Z_{2}\) are obtained by (17), (19).