## 3 Proposed Recognition Algorithm
We have demonstrated, depending on (5), that \[\mathrm{E}\Bigl{[}s_{\widehat{\alpha}}^{(f)}(n)\Big{|}\mathbf{r}, \widehat{\alpha}(i),\widehat{\overline{\mathbf{h}}}(i)\Big{]}=\frac{1}{\sqrt{ N+\
u}}\sum_{k=0}^{N-1}\] (27) \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \mathrm{E}\Bigl{[}u_{\alpha}^{(f)}(k)\Big{|}\mathbf{r},\widehat{\alpha}(i), \widehat{\overline{\mathbf{h}}}(i)\Big{]}e^{\beta 2\pi nk/N},\] where \[\Lambda(\alpha)=\int_{\overline{\mathbf{S}}(\alpha)}\overline{\mathbf{S}}^{H} (\alpha)\overline{\mathbf{S}}(\alpha)\Pr\Bigl{(}\overline{\mathbf{S}}(\alpha )\Big{|}\mathbf{r},\widehat{\alpha}(i),\widehat{\overline{\mathbf{h}}}(i)\, \Bigr{)}d\overline{\mathbf{S}}(\alpha), \tag{21}\]0 (28) \[\Lambda(\alpha)=\int_{\overline{\mathbf{S}}(\alpha)}\overline{\mathbf{S}}^{H} (\alpha)\overline{\mathbf{S}}(\alpha)\Pr\Bigl{(}\overline{\mathbf{S}}(\alpha )\Big{|}\mathbf{r},\widehat{\alpha}(i),\widehat{\overline{\mathbf{h}}}(i)\, \Bigr{)}d\overline{\mathbf{S}}(\alpha), \tag{21}\]1 Here, \(\varpi\) denotes each possible point in the modulation constellation \(\Omega\). In reality, rather than relying on the mysterious \(s_{\alpha}^{(f)}(n)\) supplied by the transmitter, the recipient utilizes \(\mathrm{E}\Bigl{[}s_{\widehat{\alpha}}^{(f)}(n)\Big{|}\mathbf{r},\widehat{ \alpha}(i),\widehat{\overline{\mathbf{h}}}(i)\Big{]}\) to generate the matrix of \(\Phi(\alpha)\). Moreover, because an interleaver is present, \(\Lambda(\alpha)\) is efficiently calculated as \(\Phi^{H}(\alpha)\Phi(\alpha)\). 2. It has been noted that determining \(\mathrm{Pr}\Bigl{[}u_{\alpha}^{(f)}(k)=\alpha\Big{|}\mathbf{r},\widehat{ \alpha}(i),\widehat{\overline{\mathbf{h}}}(i)\Big{]}\) is essential for the suggested recognizer. This likelihood is determined from the decoder findings of any error control coding scheme that can establish soft representations. More information about this concern is provided in [39]. 3. Sending pilot symbols from the broadcaster to the destination provides the proposed estimator and recognizer with a starting point. With more known symbols, the original assessment becomes more accurate. Nevertheless, more known symbols means less power is accessible for information and implies more bandwidth is needed. Consequently, it is essential that there be as few known symbols as possible in comparison to data symbols. The offered expectation maximization method is effective for this purpose because it treats the results provided by the channel decoders as if they were already known symbols. This incrementally enhances the first estimate without adding more training symbols. Using a small number of training symbols while initializing the unknown data symbols at zero, we obtain first-order predictions of the uncertain variables. The suggested classifier is derived via a series of rounds that use the soft outputs given by the error control decoder to produce \(\Pr\left[u_{\alpha}^{(f)}(k)=\sigma\middle|\mathbf{r},\widehat{\alpha}(i), \widehat{\mathbf{h}}(i)\right]\). It is well known that all contemporary decoding procedures produce a posteriori probabilities of the transmitted symbols for use in their iterative frameworks. The proposed recognition and estimation approach exploits these probabilities to produce expected values of the transmitted symbols, which are employed as if they were known symbols. 4. By adopting a computational complexity analysis as stated in [26; 33], we demonstrate that the count of floating point operations (flps) requested for a single iteration is represented by \[\Lambda(\alpha)=\int_{\overline{\mathbf{S}}(\alpha)}\overline{\mathbf{S}}^{H} (\alpha)\overline{\mathbf{S}}(\alpha)\Pr\Bigl{(}\overline{\mathbf{S}}(\alpha )\Big{|}\mathbf{r},\widehat{\alpha}(i),\widehat{\overline{\mathbf{h}}}(i)\, \Bigr{)}d\overline{\mathbf{S}}(\alpha), \tag{21}\]2 (29) Consider \(N=512\), \(L=7\), \(\
u=6\), and a processor that runs at 10 Teraflps per second, then \(\rho=20.5\times(10)^{6}\) flps. This necessitates a computation time of 2.05 \(\upmu\)s, which seems to be insignificant for real situations.