## 4 Simulation Results and Discussions
In the subsequent experiment, we used feedback to create an iterative algorithm. The model is similar to the model in the first experiment. However, in vertical detection, the signal \(b[j,k]\) was exploited to calculate \(P_{e}(R_{i})\) in (6). Then, \(P_{e}(R_{i})\) was supplied to the horizontal detection, as in (3). To optimize the number of iterations, we implemented the experiments and show the results in Figure 7. We fixed the blur \(\sigma_{b}\) at 1.85 and 0% misalignment (\(m_{q}=m_{t}=0\)) during the second experiment. In Figure 7, we assume that the inter-track interference (ITI) is known when we can completely remove the ITI and remind the ISI. We can detect the 1D Viterbi with the ISI to achieve the ideal case with the best performance. By applying feedback, the proposed model can be improved to the ideal case and optimized after two iterations. In addition, when applying the feedback line, a gain of 1 dB can be achieved at a BER of \(10^{-3}\) compared with the proposed model without feedback. In the serial structure, the 2D interference can be analyzed into two 1D interferences. Therefore, the horizontal detector (the first detector) lacks vertical interference information and vice versa. When the symbol information from the vertical detector (the second detector) is feedbacked to the first detector, the vertical interference is supplied to the first detector. This helps the first detector have information about both the horizontal and vertical interferences. Therefore, the feedback line can improve the BER performance of the system. However, in the second iteration, because the feedback information did not change, the performance was not improved compared with the first iteration. In this study, we investigated the effects of blurring. When the level of blur increases, the desired pixel gets a significant 2D interference from neighboring pixels. We varied the blur \(\sigma_{b}\) from 1.8 to 3 with SNR = 13 dB. The results are shown in Figure 8. When the blur increases to three, our proposed model achieves the same serial detection performance as Viterbi. Therefore, the proposed model can resist the blur effect of less than 3. Figure 7: BER performance of the proposed model with the iterative algorithm. We set a 0% misalignment for the HDS channel in the above experiments. In the next experiment, we added a 10% misalignment (\(m_{q}=m_{t}=0.1\)) to the HDS channel to close the reality. This effect caused detection to miss the position of the data. Therefore, the misalignment effect decreased the detection ability. We set SNR = 13 dB, added a 10% misalignment, and varied the blur from 1.8 to 3, as shown in Figure 9. When misalignment appears, the GPR target can estimate the level of misalignment and compensate for that difference with the equalizer. The output of the equalizer was approximately the desired signal and only left the 2D interference, which is suitable for our proposed model to detect the signal. However, when the level of misalignment was too large, the GPR target could not estimate the misalignment which led to the proposed model not working. In addition, we set blur \(\sigma_{b}=1.85\), 10% misalignment, and varied the SNR from 10 to 16, as shown in Figure 10. Our proposed model achieved the best performance in blurred and misaligned environments. As shown in Figure 10, the proposed model achieved a gain of 3 dB at a BER of \(10^{-2}\). Figure 8: BER performance of the proposed model according to blur. Figure 9: BER performance of the proposed model according to blur with 10% misalignment. Finally, we show the complexity of our proposed model and that of previous studies in Table 2. The operators per the detected bit are counted from the equalizer to the detection in each method. In Table 2, the operators of our proposed model are counted without iteration. We can see the trade-off between the BER performance and the complexity of the models.