## 3 Proposed Recognition Algorithm
Based on the predictions of the unknown variables obtained in the previous iteration \(i\), the E-stage calculates the conditional mean of the log likelihood function of \(\overline{\mathbf{S}}(\alpha)\) with respect to \(\overline{\mathbf{h}}\). In mathematical terms, we write  
\[ \begin{split} U\Bigl{(}\alpha,\overline{\mathbf{h}}\Bigr{|} \widehat{\alpha}(i),\widehat{\mathbf{h}}(i)\,\Bigr{)}=&\,\mathrm{ E}\Bigl{[}\log\Pr\Bigl{(}\mathbf{r},\overline{\mathbf{S}}(\alpha)\Big{|} \mathbf{r},\widehat{\alpha}(i),\widehat{\mathbf{h}}(i)\Bigr{)}\Bigr{]}\\ =&\,\widehat{\mathbf{S}}_{(\alpha)}\log\Pr\Bigl{(} \mathbf{r},\overline{\mathbf{S}}(\alpha)\Big{|}\mathbf{r},\widehat{\alpha}(i ),\widehat{\mathbf{h}}(i)\Bigr{)}\\ &\times\Pr\Bigl{(}\overline{\mathbf{S}}(\alpha)\Big{|}\mathbf{r},\widehat{\alpha}(i),\widehat{\overline{\mathbf{h}}}(i)\Bigr{)}d\overline{ \mathbf{S}}(\alpha)\end{split}  \tag{19}\]  
Substituting (18) into (19) while disregarding the extraneous components, the E-step is represented as  
\[U\Bigl{(}\alpha,\overline{\mathbf{h}}\Bigr{|}\widehat{\alpha}(i),\widehat{ \overline{\mathbf{h}}}(i)\,\Bigr{)}\propto-\overline{\mathbf{h}}^{H}\Lambda( \alpha)\overline{\mathbf{h}}+2\Re\Bigl{\{}\mathbf{r}^{H}\Phi(\alpha) \overline{\mathbf{h}}\Bigr{\}}, \tag{20}\]  
where \((\cdot)^{H}\) is the matrix Hermitian transpose action, \(\Re\{\cdot\}\) denotes the real-part of a complex-number,  
\[\Lambda(\alpha)=\int_{\overline{\mathbf{S}}(\alpha)}\overline{\mathbf{S}}^{H} (\alpha)\overline{\mathbf{S}}(\alpha)\Pr\Bigl{(}\overline{\mathbf{S}}(\alpha )\Big{|}\mathbf{r},\widehat{\alpha}(i),\widehat{\overline{\mathbf{h}}}(i)\, \Bigr{)}d\overline{\mathbf{S}}(\alpha), \tag{21}\]  
and  
\[\Phi(\alpha)=\int_{\overline{\mathbf{S}}(\alpha)}\overline{\mathbf{S}}( \alpha)\Pr\Bigl{(}\overline{\mathbf{S}}(\alpha)\Big{|}\mathbf{r},\widehat{ \alpha}(i),\widehat{\overline{\mathbf{h}}}(i)\,\Bigr{)}d\overline{\mathbf{S}}( \alpha). \tag{22}\]M-stage revises the predictions as  
\[\left[\widehat{\alpha}(i+1),\widehat{\overline{\mathbf{h}}}(i+1)\right]=\arg\max_ {\alpha,\overline{\mathbf{h}}}\;U\Big{(}\alpha,\overline{\mathbf{h}}\Big{|} \widehat{\overline{a}}(i),\widehat{\overline{\mathbf{h}}}(i)\Big{)}. \tag{23}\]  
Here, we obtain the revised estimates by maximizing the expectation function with respect to \(\alpha\) and \(\overline{\mathbf{h}}\), using the known baseline values for those parameters. We design the following tactic to ease the execution of the two-dimensional optimization process shown in (23). We revise \(\overline{\mathbf{h}}\) by optimizing (20) for each possible element of \(\alpha\) as  
\[\widehat{\overline{\mathbf{h}}}_{\alpha}(i+1)=\left(\Lambda(\alpha)\right)^{- 1}(\Phi(\alpha))^{H}\mathbf{r}. \tag{24}\]  
Inserting (20) and (24) into (23), the revised value of \(\alpha\) is represented as  
\[\widehat{\alpha}(i+1) = \arg\max_{\alpha}\biggl{\{}-\widehat{\overline{\mathbf{h}}}_{ \alpha}^{H}(i+1)\Lambda(\alpha)\widehat{\overline{\mathbf{h}}}_{\alpha}(i+1)\] \[\qquad+2\Re\Bigl{\{}\mathbf{r}^{H}\Phi(\alpha)\widehat{\overline {\mathbf{h}}}_{\alpha}(i+1)\Bigr{\}}\biggr{\}}.\]  
The last channel adjustment is acquired as  
\[\widehat{\overline{\mathbf{h}}}(i+1)=\widehat{\overline{\mathbf{h}}}_{ \widehat{\alpha}}(i+1). \tag{26}\]  
Figure 2 shows the basic structure of the offered solution. It should be noted that the proposed approach necessitates the participation of a few pilots in order to provide preliminary estimates of the unknown parameters. The following concerns pertaining to reality are of significance:  
1.