# Estimating Interference with a Two-Dimensional Viterbi Algorithm for Bit-Patterned Media Recording
## 4 Simulation and Results  
For simulation, we generated random data for 10 pages. Each page includes a 1 \(\times\) 1,440,000-bits block \(u[k]\). First, \(u[k]\) is converted into \(a[j,k]\) with a size of 1200 \(\times\) 1200 bits. We used the first page to estimate parameters of the GPR target, \(\mathbf{G}\), and the equalizer, \(\mathbf{F}\), using the model shown in Figure 1. The remaining pages were applied to the proposed detection model, as shown in Figure 2, to evaluate the bit error rate (BER) performance. The BPMR channel was set up with an AD of 3 Tb/in\({}^{2}\) (0.465 Tb/cm\({}^{2}\)) (\(T_{x}=T_{q}=14.5\) nm) [30]. In this article, the signal-to-noise ratio (SNR) is definite as SNR = 10 log\({}_{10}(1/\sigma^{2})\). First, we experimented with the proposed model on the BPMR channel without the TMR effect (0% TMR). The coefficient of the channel, \(\mathbf{H}\), is given in [23]. As shown in Figure 2, the ISI estimator (\(IS[j,k]\) is the output of the estimator) or the \(\Pi\Pi\) estimator (\(ITl[j,k]\) is the output of the estimator) can be used. For the second 1D Viterbi detection, either horizontal or vertical detection (depending on the interference target). If the \(\Pi\Pi\) target is applied, horizontal detection is used. If the ISI target is applied, vertical detection is used. Therefore, t there are four cases, including the combinations of \(\Pi\Pi\) estimator and vertical direction detector (\(\Pi\Pi\)-VD), ITI estimator and horizontal direction detection (ITI-HD), ISI estimator and vertical direction detector (ISI-VD), and ISI estimator and horizontal direction detector (ISI-HD). In the first experiment, we determined the best estimator-detector combination for the proposed model. From Figure 3 shows that the \(\Pi\Pi\)-VD structure has the most favorable BER. \(\Pi\Pi\)-VD includes the \(\Pi\Pi\) estimator and ISI target. Then second 1D Viterbi detection was applied to the vertical direction. Since the \(\Pi\Pi\) component is larger than the ISI in channel, \(\mathbf{H}\), of the simulation for the BPMR, the BER performance of this combination is outstanding compared with the that of other cases. In addition, based on this experiment, we can see how to choose the estimator between \(ITl[j,k]\) and \(IS[j,k]\). When the \(\Pi\Pi\) component is larger than the ISI component in the channel, \(ITl[j,k]\) is chosen, and when the ISI component is larger than the \(\Pi\Pi\) component in the channel, \(IS[j,k]\) is chosen. We then compared the proposed model with the previous studies in the channel without TMR effect (\(\Delta_{off}=0\)) and media noise (\(\Delta_{x}=\Delta_{q}=0\)). The results are presented in Figure 4.  
Figure 4 shows that the proposed model with the optimal structure can improve the gain from 0.5 to 2.5 dB at a BER of \(10^{-4}\). For the three-way GPR target with feedback [18], the model estimates either the upper or lower interference. Our proposed model estimates both the upper and lower interference. Thus, our proposed model can achieve higher performance compared to the model in [18]. Comparing the serial soft output [24], our proposed model removes the interference from the equalized signal. Thus, it preserves the noise information better than the serial soft output, which estimates noise information by using the feedback from the hard output. Finally, with the ITI subtraction technique [25] and estimation with a neural network [28], a general filter to estimate the ITI. For ITI  
Figure 4: BER performance of proposed model [18; 24; 25; 28]. Figure 3: BER performance of the four combinations of the estimator and detection structures. subtraction technique, the authors used one-layered filter to predict the ITI. For neural networks [28], a multi-layered filter can be used to predict the ITI. In our proposed model, we used the 2D VA, which gives a more specific rule to predict the ITI (or ISI). Thus, the proposed model achieves better optimization compared to the models in [25; 28].