# A Novel Expectation-Maximization-Based Blind Receiver for Low-Complexity Uplink STLC-NOMA Systems
## 4 Blind Energy Estimation (BEE)  
We can observe from (4) that the STLC decoded signal at the AP \(\widetilde{y}_{l}\) conforms to an i.i.d. mixture of Gaussian distributions  
\[\widetilde{y}_{l}\sim\frac{1}{|\mathcal{X}|^{2}}\sum_{x_{\mathbf{i}}^{(l)}\in \mathcal{X}}\mathcal{CN}\Big{(}\sqrt{P_{1}\gamma_{1}}x_{1}^{(l)}+j\sqrt{P_{2} \gamma_{2}}x_{2}^{(l)},2N_{0}\Big{)},\]  
because this is affected by AWGN as illustrated in the first column in Figure 1. Furthermore, since the signals of both STAs are aligned on the in-phase and quadrature axes, respectively, they can be divided into two components as follows:  
\[ \begin{split}\Re[\widetilde{y}_{l}]&\sim\frac{1}{| \mathcal{X}|}\sum_{x_{1}^{(l)}\in\mathcal{X}}\mathcal{N}\Big{(}\sqrt{P_{1} \gamma_{1}}x_{1}^{(l)},N_{0}\Big{)},\quad\text{ for the first STA},\\ \Im[\widetilde{y}_{l}]&\sim\frac{1}{|\mathcal{X}|} \sum_{x_{2}^{(l)}\in\mathcal{X}}\mathcal{N}\Big{(}\sqrt{P_{2}\gamma_{2}}x_{2} ^{(l)},N_{0}\Big{)},\quad\text{ for the second STA},\end{split}  \tag{10}\]  
as shown in the second and third columns in Figure 1. It is worth noting that each of (10) still follows a mixture of Gaussians and can be modeled as a one-dimensional (1D) Gaussian mixture model (GMM). We now design an expectation-maximization (EM)-based blind energy estimation (BEE) scheme for blindly estimating the transmit power and effective channel gain, \(P_{n}\gamma_{n}\), of each STA at the AP. Briefly, we exploit the EM algorithm [24], also known as soft \(K\)-means clustering, which infers the parameters of the GMM given the number of STAs and modulation type of each STA [25; 26]. Recall that although the constellation consisting of the superimposed \(M\)-ASK symbols from the two STAs configures \(M^{2}\) clusters as shown in the first column in Figure 1, the signals of each STA are aligned on each axis. Hence, we consider 1D GMMs with \(M\) parameters for the real and imaginary axes in (10). Algorithm 1 represents the pseudo-code of the proposed BEE scheme for the two-user uplink STLC-NOMA system employing \(M\)-ASK modulation, where \(\mathcal{N}(z|\mu,\sigma^{2})\) denotes a Gaussian probability density function with mean \(\mu\) and variance \(\sigma^{2}\) expressed as  
\[\mathcal{N}(z|\mu,\sigma^{2})=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\!\left(- \frac{(z-\mu)^{2}}{2\sigma^{2}}\right)\!. \tag{11}\]  
The AP executes this algorithm by taking the STLC decoding signals \(\widetilde{\mathbf{y}}\) in (4) as observed data to infer the parameters of a GMM compromising \(M\) means, variances, and weights for each axis. It can be stated that the objective of Algorithm 1 is to maximize the likelihood function with respect to the GMM parameters for all observations,  
\[\Pr(\mathbf{z}|\mu_{1},\mu_{2},\cdots,\mu_{M},\sigma^{2})=\prod_{l=1}^{L}\sum _{k=1}^{M}\pi_{k}\mathcal{N}(z_{l}|\mu_{k},\sigma^{2}), \tag{12}\]  
where \(\mathbf{z}\ (=[z_{1},\ldots,z_{L}]^{\mathrm{T}}\in\mathbb{R}^{L\times 1})\) represents the observations, \(\mu_{k}\) and \(\pi_{k}\) denote the mean and weight of the \(k\)th Gaussian distribution, respectively, and \(\sum_{k=1}^{M}\pi_{k}=1\). Although the canonical EM algorithm for a GMM infers the mean \(\mu_{k}\), variance \(\sigma_{k}^{2}\), and weight \(\pi_{k}\) of each Gaussian distribution [24], we exploit a shared common variance \(\sigma^{2}\) for all Gaussian distributions in a GMM. This is because all symbols in a frame experience AWGN with the same variance, so the clusters formed at the AP may have almost the same density. More specifically, Algorithm 1 is as follows. Figure 1: The constellation diagrams of \(\tilde{\mathbf{y}}\) (first column), in-phase axis energy histograms \(\Re[\tilde{\mathbf{y}}]\) (second column), and quadrature axis energy histograms \(\Im[\tilde{\mathbf{y}}]\) (third column), for \(M=2,\ 4,\ 8,\ 16\), respectively, where \(P_{1}=P_{2}=1\), \(\gamma_{1}\approx 1.1032\), and \(\gamma_{2}\approx 1.0303\). For each case, the system SNR \(\rho_{n}\) and frame length \(L\) were set appropriately to clearly present the characteristics.