# A Novel Expectation-Maximization-Based Blind Receiver for Low-Complexity Uplink STLC-NOMA Systems
## 3 Performance Analysis  
We mathematically analyze the performance of the proposed novel low-complexity uplink STLC-NOMA system employing \(M\)-ASK modulation. In particular, we derive the bit-error-rate (BER) performance and spatial diversity order of each STA. Since the signals of both STAs are aligned orthogonally to each other on the I-Q plane, the BER performance of each STA can be derived straightforwardly from the BER expression for the \(M\)-ary pulse amplitude modulation (PAM). Specifically, the BER of the \(n\)th STA, denoted by \(P_{b,n}\), can be derived from Ref. [27] as follows:  
\[\begin{split} P_{b,n}&=\mathbb{E}_{\gamma_{n}}\Bigg{[} \frac{1}{M\log_{2}M}\sum_{k=1}^{\log_{2}M}\Bigg{[}\sum_{i=0}^{(1-2^{-k})M-1} \Bigg{\{}\Phi(i,k,m)\cdot\text{erfc}\Bigg{(}(2i+1)\sqrt{\frac{3\cdot\rho_{n} \cdot\gamma_{n}}{2(M^{2}-1)}}\Bigg{)}\Bigg{\}}\Bigg{]}\Bigg{]}\\ &=\frac{1}{M\log_{2}M}\sum_{k=1}^{\log_{2}M}\Bigg{[}\sum_{i=0}^{(1 -2^{-k})M-1}\Bigg{\{}\Phi(i,k,m)\cdot\mathbb{E}_{\gamma_{n}}\Bigg{[}\text{erfc }\Bigg{(}(2i+1)\sqrt{\frac{3\cdot\rho_{n}\cdot\gamma_{n}}{2(M^{2}-1)}}\Bigg{)} \Bigg{]}\Bigg{\}}\Bigg{]},\end{split} \tag{6}\]  
where \(\rho_{n}\)\((=P_{n}/N_{0})\) denotes the transmit SNR of the \(n\)th STA and  
\[\Phi(i,k,M)=(-1)^{\lfloor\frac{2^{k-1}}{M}\rfloor}\Bigg{(}2^{k-1}-\Bigg{\lfloor} \frac{i\cdot 2^{k-1}}{M}+\frac{1}{2}\Bigg{\rfloor}\Bigg{)},\]  
where \(\lfloor\cdot\rfloor\) denotes the floor function that gives the greatest integer number less than or equal to the input real number. Furthermore, we define a random variable for the effective channel gain of the STA as \(X_{n}:=\gamma_{n}\). The probability density function of \(X_{n}\) is given by \(f_{X_{n}}(x_{n})=(x_{n}/\beta_{n}^{2})\exp(-x_{n}/\beta_{n})\) from the definition of \(\gamma_{n}\); hence the expectation term in (6) can be derived as follows:  
\[\begin{split}\mathbb{E}_{\gamma_{n}}&\Bigg{[}\text{ erfc}\Bigg{(}(2i+1)\sqrt{\frac{3\cdot\rho_{n}\cdot\gamma_{n}}{2(M^{2}-1)}} \Bigg{)}\Bigg{]}=\int_{0}^{\infty}\text{erfc}\Bigg{(}(2i+1)\sqrt{\frac{3\cdot \rho_{n}\cdot x_{n}}{2(M^{2}-1)}}\Bigg{)}f_{X_{n}}(x_{n})dx_{n}\\ &=\int_{0}^{\infty}\text{erfc}\Bigg{(}(2i+1)\sqrt{\frac{3\cdot \rho_{n}\cdot x_{n}}{2(M^{2}-1)}}\Bigg{)}\frac{x_{n}}{\beta_{n}^{2}}\exp(- \frac{x_{n}}{\beta_{n}})dx_{n}\\ &=1-\frac{\sqrt{\frac{3(2i+1)^{2}\cdot\beta_{n}\cdot p_{n}}{2(M^ {2}-1)}}}{\sqrt{\frac{3(2i+1)^{2}\cdot\beta_{n}\cdot p_{n}}{2(M^{2}-1)}}+1} \Bigg{[}1+\frac{1/2}{\frac{3(2i+1)^{2}\cdot\beta_{n}\cdot p_{n}}{2(M^{2}-1)} }+1\Bigg{]}.\end{split} \tag{7}\]  
By substituting (7) into (6), the analytical BER performance closed-form for the \(n\)th STA in the proposed low-complexity uplink STLC-NOMA network is obtained. It is worth noting that each STA achieves the same BER performance as using orthogonal resources.  
Moreover, the expectation term can be approximated by using Taylor series expansion under the high SNR regime as follows:  
\[\mathbb{E}_{\gamma_{n}}\Bigg{[}\text{erfc}\Bigg{(}(2i+1)\sqrt{\frac{3\cdot \rho_{n}\cdot\gamma_{n}}{2(M^{2}-1)}}\Bigg{)}\Bigg{]}\approx\frac{3}{8\Big{(} \frac{3(2i+1)^{2}\cdot\beta_{n}\cdot p_{n}}{2(M^{2}-1)}\Big{)}^{2}}=\frac{1}{6 }\bigg{(}\frac{M^{2}-1}{(2i+1)^{2}\cdot\beta_{n}}\bigg{)}^{2}\rho_{n}^{-2}. \tag{8}\]  
By substituting (8) into (6), the asymptotic BER performance of the \(n\)th STA is obtained, which represents the BER behavior in the high SNR regime.  
Finally, from the asymptotic BER expression, we can derive the spatial diversity order of each STA in the proposed STLC-NOMA system as follows:  
\[\eta_{n}\triangleq-\lim_{\rho_{n}\rightarrow\infty}\frac{\log P_{b,n}}{\log \rho_{n}}=2. \tag{9}\]  
We can clearly state that the proposed uplink STLC-NOMA system achieves the optimal spatial diversity order of two for both STAs.