## 5 Offered Detector  
The base station segments the received signals into chunks of \(N+v\) samples for processing. The base station suppresses the multiple access interference caused by the transmissions of other users in order to detect the data of user \(k\). Next, we get rid of the \(v\) samples that matched the cyclic prefix, keeping \(N\) samples for subsequent inspection. After performing \(N\)-point FFT, frequency demapping, and \(M\)-point IFFT, the \(m\)th outputs of the IFFT unit of the \(g\)th vector for the first and second-time slots are, respectively, expressed as  
\[R_{g,1}^{(k)}(m)=h_{1}^{(k)}(m)d_{g}^{(k)}(m)+h_{2}^{(k)\dagger}(-m)d_{g}^{(k) \dagger}(-m)+W_{g,1}^{(k)}(m), \tag{36a}\] \[R_{g,2}^{(k)}(m)=h_{3}^{(k)}(m)d_{g}^{(k)}(m)+h_{4}^{(k)\dagger}(-m)d_{g}^{(k) \dagger}(-m)+W_{g,2}^{(k)}(m), \tag{36b}\]  
where \(W_{g,1}^{(k)}(m)\) and \(W_{g,2}^{(k)}(m)\) are the noise elements and \(h_{x}^{(k)}(m)\) is the frequency domain channel coefficient that includes the channel response along with the IQM parameters found at user \(k\), relay, and base station, \(x=1,\cdots,4\). We rewrite (36a) and (36b) in a compact vector format as  
\[\mathbf{R}_{g}^{(k)}(m)=\overline{\mathbf{h}}^{(k)}(m)\mathbf{d}_{g}^{(k)}(m )+\mathbf{W}_{g}^{(k)}(m), \tag{37}\]  
where \(\mathbf{R}_{g}^{(k)}(m)=\left[R_{g,1}^{(k)}(m),R_{g,2}^{(k)}(m)\right]^{ \dagger}\), \(\mathbf{d}_{g}^{(k)}(m)=\left[d_{g}^{(k)}(m),d_{g}^{(k)\dagger}(-m)\right]^{\dagger}\), and  
\[\overline{\mathbf{h}}^{(k)}(m)=\left[ \begin{array}{cc}h_{1}^{(k)}(m)&h_{2} ^{(k)\dagger}(-m)\\ h_{3}^{(k)}(m)&h_{4}^{(k)\dagger}(-m)\end{array} \right]. \tag{38}\]  
We write the a posteriori probability of \(\mathbf{d}_{g}^{(k)}(m)\) as  
\[ \begin{array}{c}\Pr\Bigl{(}\mathbf{d}_{g}^{(k)}(m)\Big{|}\mathbf{R}_{g}^{( k)}(m),\overline{\mathbf{h}}^{(k)}(m)\Bigr{)}\propto\\ \\ \exp\Bigl{(}\frac{-1}{\sigma_{\pi}^{2}}\Big{\|}\mathbf{R}_{g}^{(k)}(m)- \overline{\mathbf{h}}^{(k)}(m)\mathbf{d}_{g}^{(k)}(m)\Big{\|}^{2}\Bigr{)} \Pr\Bigl{(}\mathbf{d}_{g}^{(k)}(m)\Bigr{)}.\end{array}  \tag{39}\]  
We presumed that \(Z^{(k)}=\log_{2}\Bigl{(}\left\langle\Omega^{(k)}\right\rangle\Bigr{)}\) bits identify each information symbol, \(d_{g}^{(k)}(m)=\mathcal{L}\Bigl{[}b_{g}^{(k)}(m,0),\cdots,b_{g}^{(k)}(m,Z-1) \Bigr{]}\), where \(\mathcal{L}[\cdot]\) is the label function and \(b_{g}^{(k)}(k,z)\) is the \(z\)th bit of the information symbol \(d_{g}^{(k)}(m)\). Therefore, the bit metric \(\lambda\Bigl{(}b_{g}^{(k)}(m,z)=o\Bigr{)}\), for \(o=0,1\), is computed as  
\[ \begin{array}{c}\lambda\Bigl{(}b_{g}^{(k)}(m,z)=o\Bigr{)}=\sum\limits_{\theta =F(o,z)}\sum\limits_{d_{g}^{(k)}(-m)\in\left(\left\langle\Omega^{(k)}\right\rangle \right)}\\ \Pr\Bigl{(}\mathbf{d}_{g}^{(k)}(m)\Big{|}\mathbf{R}_{g}^{(k)}(m),\overline{ \mathbf{h}}^{(k)},d_{g}^{(k)}(m)=\vartheta\Bigr{)},\end{array}  \tag{40}\]where  
\[F(b,z)=\mathcal{L}\bigg{\{}\Big{[}b_{g}^{(k)}(m,0),\cdots,b_{g}^{(k)}(m,Z-1)\Big{]} \bigg{|}b_{g}^{(k)}(m,z)=o\bigg{\}}. \tag{41}\]  
After de-interleaving, the bit metrics are sent to the channel error-control decoder to derive the a posteriori probability of coded bits. Next, the probabilities are interleaved and transferred to the a posteriori computation unit. The offered estimator and detector algorithms use these probabilities as a priori input, as stated in (31a), (31b), and (39), respectively. The decoder makes hard choices based on the a posteriori probability of the transmitted bits in the last iteration. The conceptual block diagram of the recommended detection and estimation technique is shown in Figure 3. The proposed approach is summarized in Algorithm 1. 1. Use a few training symbols to create the matrix \(\mathcal{S}^{(k)}\), for \(k=0,1,\cdots,K-1\), with setting unknown data symbols to zeros. Then, employ (31a) and (31b) to obtain preliminary assessments of the unknown parameters \(\widehat{\mathrm{H}}_{1}^{(k)}(0)\) and \(\widehat{\mathrm{H}}_{2}^{(k)}(0)\). 2. For \(\zeta=0:\rho-1\), where \(\rho\) is the total number of iterations for each user \(k\), where \(k=0,\cdots,K-1\).