# A Novel Expectation-Maximization-Based Blind Receiver for Low-Complexity Uplink STLC-NOMA Systems
## 2 System Model
\begin{split}\widetilde{y}_{2l^{\prime}-1}&=y_{1}^{ (2l^{\prime}-1)}+\left(y_{2}^{(2l^{\prime})}\right)^{*}=\sum_{n=1}^{2}\sqrt{P_ {n}}\sqrt{\beta_{n}}\|\mathbf{h}_{n}\|\tilde{y}^{n-1}x_{n}^{(2l^{\prime}-1)}+w _{1}^{(2l^{\prime}-1)}+\left(w_{2}^{(2l^{\prime})}\right)^{*},\\ \widetilde{y}_{2l^{\prime}}&=\left(y_{2}^{(2l^{ \prime}-1)}\right)^{*}-y_{1}^{(2l^{\prime})}=\sum_{n=1}^{2}\sqrt{P_{n}}\sqrt{ \beta_{n}}\|\mathbf{h}_{n}\|\tilde{y}^{n-1}x_{n}^{(2l^{\prime})}+\left(w_{2}^ {(2l^{\prime}-1)}\right)^{*}-w_{1}^{(2l^{\prime})},\end{split}  \tag{3}\]  
where \(\widetilde{y}_{l}\) denotes the \(l\)th STLC decoding signal and \(\mathbf{h}_{n}\triangleq[h_{n,1},h_{n,2}]^{\mathrm{T}}\). We can observe that the phase distortion of the transmitted signals caused by the wireless channels is completely compensated thanks to the fundamental nature of STLC, hence each STA's symbols \(x_{n}^{(l)}\) are still aligned for each axis in the I-Q plane. From (3), the STLC decoding frame at the AP, denoted by \(\widetilde{\mathbf{y}}\ (=[\widetilde{y}_{1},\ldots,\widetilde{y}_{l},\ldots, \widetilde{y}_{L}]^{\mathrm{T}}\in\mathbb{C}^{L\times 1})\), can be stated as  
\[\widetilde{\mathbf{y}}=\sqrt{P_{1}}\sqrt{\beta_{1}}\|\mathbf{h}_{1}\|\mathbf{ x}_{1}+j\sqrt{P_{2}}\sqrt{\beta_{2}}\|\mathbf{h}_{2}\|\mathbf{x}_{2}+\widetilde{ \mathbf{w}}, \tag{4}\]  
where \(\mathbf{x}_{n}\ (=[x_{n}^{(1)},\ldots,x_{n}^{(L)}]^{\mathrm{T}}\in\mathbb{C}^{L \times 1})\) denotes the length-\(L\) frame transmitted from the \(n\)th STA, each consisting of \(L\) M-ASK symbols; and \(\widetilde{\mathbf{w}}\) represents the equivalent AWGN vector at the AP following an i.i.d. \(\mathcal{CN}(\mathbf{0},2N_{0}\cdot\mathbf{I}_{L})\). Note that the noise variance is doubled due to the nature of STLC. Let \(\gamma_{n}\) be the effective channel gain of the \(n\)th STA defined as \(\gamma_{n}\triangleq\|\mathbf{g}_{n}\|^{2}\), where \(\mathbf{g}_{n}\ (\triangleq\sqrt{\beta_{n}}[h_{n,1},h_{n,2}]^{\mathrm{T}}\in \mathbb{C}^{2\times 1})\) is the wireless channel vector including the large-scale fading component between the \(n\)th STA and the AP, i.e., \(\mathbf{g}_{n}\sim\mathcal{CN}(\mathbf{0},\beta_{n}\cdot\mathbf{I}_{2})\). We will design a blind energy estimation (BEE) algorithm for the proposed uplink STLC-NOMA system in Section 4. This allows the AP to blindly estimate the transmit power and effective channel gain of each STA. The AP then detects the transmitted \(M\)-ASK symbols for all \(n\) and \(l\) through a simple maximum-likelihood (ML) detector from (4) as follows:  
\[ \begin{split}\widehat{x}_{1}^{(l)}&=\arg\min_{x\in \mathcal{X}}\Bigl{|}\Re[\widetilde{y}_{l}]-\sqrt{P_{1}\gamma_{1}x}\Bigr{|}^{2},\quad\text{for the first STA},\\ \widehat{x}_{2}^{(l)}&=\arg\min_{x\in\mathcal{X}} \Bigl{|}\Im[\widetilde{y}_{l}]-\sqrt{P_{2}\gamma_{2}x}\Bigr{|}^{2},\quad\text {for the second STA},\end{split}  \tag{5}\]  
where \(\Re[\cdot]\) and \(\Im[\cdot]\) denote the real and imaginary components of a complex number, respectively; \(\widehat{x}_{n}^{(l)}\) is the estimate of the \(l\)th symbol of the \(n\)th STA, and \(\mathcal{X}\) is a set of the normalized \(M\)-ASK symbols defined as  
\[\mathcal{X}=\left\{x\bigg{|}x=\frac{2m-1-M}{\sqrt{\sum_{m=1}^{M}(2m-1-M)^{2}/M} },m=1,2,\ldots,M\right\}.\]  
It is worth noting that the proposed novel low-complexity STLC-NOMA system has \(\mathcal{O}(L\cdot M)\) computational complexity for the detector, whereas the conventional STLC-NOMA employing a joint ML detector has \(\mathcal{O}(L\cdot M^{2})\) complexity in the two-user uplink IoT network [23].