## 3 Proposed Recognition Algorithm  
Using (6) and (9) in (10), one writes  
\[\mathbf{r}=\sum_{f=0}^{F-1}\mathbf{s}^{(f)}_{\alpha}\star\mathbf{h}^{(f)}_{1} +\mathbf{s}^{(f)*}_{\alpha}\star\mathbf{h}^{(f)}_{2}+\overline{\mathbf{n}}, \tag{11}\]  
where \(\overline{\mathbf{n}}\) is the noise contribution and  
\[\mathbf{h}^{(f)}_{1}=\eta_{R}\eta^{(f)}\mathbf{h}^{(f)}+\mu_{R}\mu^{(f)*} \mathbf{h}^{(f)*}, \tag{12}\]  
\[\mathbf{h}^{(f)}_{2}=\eta_{R}\mu^{(f)}\mathbf{h}^{(f)}+\mu_{R}\eta^{(f)*} \mathbf{h}^{(f)*}. \tag{13}\]  
Matrix representation is used to rewrite (11) for ease of calculation as  
\[\mathbf{r}=\sum_{f=0}^{F-1}\mathbf{S}^{(f)}(\alpha)\mathbf{h}^{(f)}_{1}+ \mathbf{S}^{(f)*}(\alpha)\mathbf{h}^{(f)}_{2}+\overline{\mathbf{n}}, \tag{14}\]  
where \(\mathbf{S}^{(f)}(\alpha)\) is an \((N+L-1)\times L\) matrix with its item at row \(x_{1}\) and column \(x_{2}\) being provided as  
\[S^{(f)}_{x_{1},x_{2}}(\alpha)=\left\{ \begin{array}{ll}s^{(f)}_{\alpha}(x_{1} \!-\!x_{2})&\text{ for }x_{1}=0,\cdots,N+L\!-\!1\\ 0&\text{ for }x_{2}=0,\cdots,L\!-\!1,x_{1}\!\geq\!x_{2},\end{array} \right. \tag{15}\]  
and \(s^{(f)}_{\alpha}(x_{1}-x_{2})\) is the \((x_{1}-x_{2})\)th item of sequence \(\mathbf{s}^{(f)}_{\alpha}\). The concise format of (14) is represented as  
\[\mathbf{r}=\overline{\mathbf{S}}(\alpha)\overline{\mathbf{h}}\!+\!\overline{ \mathbf{n}}, \tag{16}\]where \(\overline{\mathbf{S}}(\alpha)=\left[\mathbf{S}^{(0)}(\alpha),\mathbf{S}^{(0)*}( \alpha),\cdots,\mathbf{S}^{(F-1)}(\alpha),\mathbf{S}^{(F-1)*}(\alpha)\right]\) and \(\overline{\mathbf{h}}=\left[\mathbf{h}_{1}^{(0)T},\mathbf{h}_{2}^{(0)T},\cdots,\mathbf{h}_{1}^{(F-1)T},\mathbf{h}_{2}^{(F-1)T}\right]^{T}\). The maximum likelihood prediction of \(\alpha\) is stated as  
\[\hat{\alpha}=\arg\max_{\alpha}\,\log\Pr\Bigl{(}\mathbf{r}|\overline{\mathbf{S} }(\alpha),\overline{\mathbf{h}}\Bigr{)}, \tag{17}\]  
with \(\Pr(\circ|\circ)\) being the probability density function of \(\circ\) given \(\circ\), and  
\[\Pr\Bigl{(}\mathbf{r}|\overline{\mathbf{S}}(\alpha),\overline{\mathbf{h}} \Bigr{)}\propto\exp\left(-\left\|\mathbf{r}-\overline{\mathbf{S}}(\alpha) \overline{\mathbf{h}}\right\|^{2}/\sigma_{n}^{2}\right), \tag{18}\]  
where \(\left\|\cdot\right\|\) is the vector norm action. A straightforward development of a maximum likelihood strategy is not possible since the recipient has no previous knowledge of the broadcast matrix \(\overline{\mathbf{S}}(\alpha)\), the IQD parameters \(\eta^{(f)}\), \(\mu^{(f)}\), \(\eta_{R}\), and \(\mu_{R}\), or the channel coefficients \(\overline{\mathbf{h}}\). Expectation maximization approaches are used to determine the regional maximum likelihood parameters of a mathematical framework when the calculations cannot be solved directly. These models often include hidden variables in addition to uncertain variables and existing data inputs. Typically, obtaining a maximum likelihood strategy entails determining the derivatives of the likelihood distribution with respect to all of the unknown values, variables, and latent factors, and concurrently solving the resultant equations. Here, latent factors are the variables that can only be inferred indirectly through a mathematical model from other observable variables that can be directly observed or measured. Instead, the outcome is generally a series of interconnected formulas in which the solution to the variables necessitates the quantities of the latent factors, but inserting one series of formulas into the other yields an unsolvable problem. Convergence of an expectation maximization procedure is guaranteed because the likelihood of the predictions does not decrease. In other words, it is guaranteed to converge to a point with zero gradient with respect to the estimated parameters. An expectation maximization process consists of two stages: the expectation (E-stage) and the maximization (M-stage).