## 4 Simulation Results and Discussions  
In the simulation, we used hardware with CPU i5-11400 2.6 GHz, 16 GB RAM, and a GTX 1650 graphic card. For the software, we created the program with MATLAB 2022b. To estimate the parameters of the serial target \(\mathbf{G}\) and equalizer \(\mathbf{F}\), we input a data page with a size of 1200 \(\times\) 1200 into the system and collected the output. The coefficients of the target were calculated and applied to detection using the method described by [PERSON] and [PERSON] [26]. We input ten data pages to the model in Figure 5 to estimate the BER performance of the system. To be simpler and shorter, we did not draw modulation and demodulation in Figure 5. In modulation, we arranged the 1D user data \(u[k]\in\{0,\,1\}\) into the 2D modulated signal \(a[j,k]\). We can see that the signal \(a[j,k]\) represents the light and dark pixels in SLM. Here, to fit the MAP algorithm, we use \(-1\) and \(1\) to present light and dark pixels, respectively. Then, the signal \(a[j,k]\) was inputted into the HDS channel. In the HDS channel, the coefficients of \(h[j,k]\) was created, as in Section 3. The output of HDS channel \(y[j,k]\) was obtained, as in (7). The signal \(z[j,k]\), which is close to the desired signal \(d[j,k]\), was created by the convolution of \(y[j,k]\) and the equalizer matrix \(\mathbf{F}\) as below. \[z[j,k]=y[j,k]\otimes\mathbf{F}, \tag{11}\]  
The signal \(z[j,k]\) was inputted to the serial detection in Section 2 to restore the signal \(a[j,k]\), which is the estimation of \(a[j,k]\). Finally, we recovered the user data \(a[k]\) by arranging the signal \(a[j,k]\). We collected and compared the signal \(a[k]\) with \(u[k]\) for calculating the BER performance. In Table 1, we present the results of the estimation of the matrix target \(\mathbf{G}\) and equalizer \(\mathbf{F}\). The matrix \(\mathbf{F}\) helps convert the received signal \(y[j,k]\) into the signal \(z[j,k]\), which is similar to the desired signal \(d[j,k]\). Therefore, we varied SNR from 10 to 16 dB and compared the signals \(z[j,k]\) and \(d[j,k]\). We used mean square error as below. \[MSE=E[(z[j,k]-d[j,k])^{2}], \tag{12}\]  
where \(E\) denotes the expectation. From Table 1, we can see that when the SNR increases, the MSE decreases, and the accuracy of the estimation is improved. The estimator was designed to predict the 2D interference in the channel. However, the AWGN disturbs the estimator too much to exactly predict the 2D interference. Therefore, the equalizer \(\mathbf{F}\) just converts the signal \(z[j,k]\) into an approximation of the desired signal \(d[j,k]\). In the first experiment, the signal \(z[j,k]\) was inputted to the horizontal detection to extract the probability of the symbols \(b[j,k]\). Then, the signal \(b[j,k]\) is supplied to the vertical detection to estimate the probability of bits 0 and 1 in \(d[j,k]\). We did not use the feedback in this experiment, and the signal \(d[j,k]\) is restored into the user data \(u[k]\). We fixed the blur \(\sigma_{b}\) at 1.85 and 0% misalignment (\(m_{q}=m_{t}=0\)) during the first experiment. As shown in Figure 6, our proposed model can improve the BER performance compared with previous studies. We implemented the GPR target and 1D detection for the 1D GPR target reported by [PERSON] and [PERSON] [15]. Serial detection using Viterbi is a serial target detection method [26]. Serial MAP detection achieved a gain of 3.5 dB at a BER of \(10^{-3}\). Because the soft output signal was created in the inner part of the serial structure when applying the MAP algorithm, the proposed model can improve the BER performance compared with the VA. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**SNR** & **10** & **11** & **12** & **13** & **14** & **15** & **16** \\ \hline MSE & 0.4036 & 0.39 & 0.3776 & 0.3677 & 0.3580 & 0.3497 & 0.3425 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: MSE of the estimation according to SNR (dB). Figure 6: BER performance of the proposed model without feedback.