# A Novel Expectation-Maximization-Based Blind Receiver for Low-Complexity Uplink STLC-NOMA Systems
## 2 System Model  
We consider an uplink STLC-NOMA network consisting of two single-antenna IoT devices (stations: STAs) and a dual-antenna access point (AP), where each STA has its local CSI and the AP does not have any CSI (CSIT) as in Ref. [23]. Although we consider a simple system model for the convenience of explanation, the framework proposed in this paper can be easily extended to the MIMO system model by employing the STLC encoder for multiple transmit antennas and low-rate STLC combiner for multiple receive antennas [17]. Unlike Ref. [23], both STAs employ \(M\)-ary amplitude-shift keying (ASK) modulators on the in-phase and quadrature axes, respectively. Without loss of generality, we assume that the first STA occupies the in-phase axis; thus, the second STA uses the quadrature axis. One frame of the STA consists of \(L\) symbols, where \(L\) is defined as the frame length, and block fading channels are assumed while at least one frame is transmitted, i.e., the wireless channels remain constant during a frame time. It is also assumed that two STAs in the network are synchronized by observing the common pilot signals periodically broadcast from the AP. Both STAs simultaneously transmit their frame to the AP exploiting STLC over the same radio resource blocks. The STLC encoded signal for the \(l\) (\(\in\{1,2,\ldots,L\}\))th symbol of the \(n\) (\(\in\{1,2\}\))th STA, denoted by \(s_{n}^{(l)}\), can be expressed equivalently as follows:  
\[ \begin{split} s_{n}^{(2l^{\prime}-1)}&=\sqrt{P_{n}} \frac{h_{n,1}^{*}\Big{(}\
icefrac{{p^{n-1}x_{n}^{(2l^{\prime}-1)}}}{{n}}\Big{)} +h_{n,2}^{*}\Big{(}\
icefrac{{p^{n-1}x_{n}^{(2l^{\prime})}}}{{n}}\Big{)}^{*} }{\sqrt{|h_{n,1}|^{2}+|h_{n,2}|^{2}}},\\ s_{n}^{(2l^{\prime})}&=\sqrt{P_{n}}\frac{h_{n,2}^{* }\Big{(}\
icefrac{{p^{n-1}x_{n}^{(2l^{\prime}-1)}}}{{n}}\Big{)}^{*}-h_{n,1}^{* }\Big{(}\
icefrac{{p^{n-1}x_{n}^{(2l^{\prime})}}}{{n}}\Big{)}}{\sqrt{|h_{n,1}| ^{2}+|h_{n,2}|^{2}}},\end{split}  \tag{1}\]  
where \(j\) (\(\triangleq\sqrt{-1}\)) represents the imaginary unit and \(l^{\prime}\) (\(\in\{1,2,\ldots,L/2\}\)) is defined to indicate the odd (\(2l^{\prime}-1\)) and even (\(2l^{\prime}\)) \(M\)-ASK symbols in the frame; \(x_{n}^{(l)}\) denotes the \(l\)th normalized \(M\)-ASK symbol of the \(n\)th STA, such that \(\mathbb{E}[x_{n}^{(l)}]=1\), and \(h_{n,m}\) denotes the wireless channel between the \(n\)th STA and the \(m\) (\(\in\{1,2\}\))th antenna of the AP. All channels are assumed to follow an independent and identically distributed (i.i.d.) complex Gaussian distribution with zero mean and unit variance, i.e., \(h_{n,m}\sim\mathcal{CN}(0,1),\ \forall n,m\). As aforementioned, static channels are assumed to one frame, so \(h_{n,m}\) remains constant for each frame of length \(L\) symbols. Furthermore, \(P_{n}\) is the transmission power of the \(n\)th STA, such that \(\mathbb{E}[\|[s_{n}^{(2l^{\prime}-1)},s_{n}^{(2l^{\prime})}]^{\mathrm{T}}\|^ {2}/2]=P_{n}\). Equation (1) shows that two consecutive \(M\)-ASK symbols are encoded into two STLC signals within the frame. The received frame at the \(m\)th antenna of the AP, denoted by \(\mathbf{y}_{m}\) (\(=[y_{m}^{(1)},\ldots,y_{m}^{(L)}]^{\mathrm{T}}\in\mathbb{C}^{L\times 1}\)), is then written as follows:  
\[\mathbf{y}_{m}=\sum_{n=1}^{2}\sqrt{\beta_{n}}\cdot h_{n,m}\cdot\mathbf{s}_{n }+\mathbf{w}_{m}, \tag{2}\]  
where \(\beta_{n}\) is the large-scale fading component, \(\mathbf{s}_{n}\) (\(=[s_{n}^{(1)},\ldots,s_{n}^{(L)}]^{\mathrm{T}}\in\mathbb{C}^{L\times 1}\)) represents the transmit frame of the \(n\)th STA, and \(\mathbf{w}_{m}\) (\(=[w_{m}^{(1)},\ldots,w_{m}^{(L)}]\in\mathbb{C}^{L\times 1}\)) denotes the additive white Gaussian noise (AWGN) at the \(m\)th antenna of the AP that follows an i.i.d. \(\mathcal{CN}(\mathbf{0},N_{0}\cdot\mathbf{I}_{L})\), where \(N_{0}\) and \(\mathbf{I}_{L}\) represent the noise power and the \(L\times L\) identity matrix, respectively. The AP combines the received signals in (2) according to the linear combining procedure of the STLC as follows:  
\[