## 3 Simulation Results
### Simulation Results  
In this experiment, the first method is shown in Figure 2, which uses an equalizer instead of the inner detector. The channel output \(y[j,k]\) was inputted to 2D equalizer. This equalizer has a size of \(5\times 5\). At the same time, the coefficients of the equalizer and the GPR target were determined by calculating the error values \(e[j,k]\) and minimizing this error with the MMSE algorithm [13]. Then, the output of the equalizer \(z[j,k]\) was passed to the horizontal equalizer, which had a size of \(5\times 5\), and the coefficients were updated by finding the error values \(e_{i}[j,k]\) and using the LMS algorithm to reduce ISI. Finally, the outputs of the horizontal equalizer, \(s[j,k]\), were transferred to the vertical detector to reduce ITI and restore the original input data \(\tilde{a}[j,k]\). The channel signal-to-noise ratio (SNR) is defined as 10 log\({}_{10}\)(1/\(\sigma^{2}\)), where \(\sigma^{2}\) is AWGN power. To simulate the second method, we built a scheme as shown in Figure 3. The system with the six-level output signal for serial detection was built in the same manner as Figure 1. In our experiment, we refer to the system in Figure 3 as soft-output horizontal (inner) detection, the system in Figure 2 as horizontal (inner) equalizer, and system in Figure 1 as conventional serial detection. First, we compare the bit error rate (BER) performance of three systems as shown in Figure 4. We simulated 10 pages with a size of 1200 \(\times\) 1200 bits and an AD of 3 Tb/in\({}^{2}\) (\(T_{x}=T_{z}=14.5\) nm) [16]. To simplify the problem, the coefficients of the channel without TMR effect and media noise used in the simulation were given as  
\[\mathbf{H}=\left[ \begin{array}{ccc}0.0824&0.3876&0.0824\\ 0.2125&1&0.2125\\ 0.0824&0.3876&0.0824\end{array} \right]. \tag{16}\]  
At SNR = 20 dB, the PR target was calculated as  
\[\mathbf{G}=\left[ \begin{array}{ccc}0.0503&0.4348&0.0503\\ 0.1158&1&0.1158\\ 0.0503&0.4348&0.0503\end{array} \right]. \tag{17}\]  
The results showed that the system with soft output of the inner detector achieves better BER performance than the system with hard output of the inner detector. At a BER of 10\({}^{-6}\), the gain of the soft output horizontal detector is ~1.6 dB higher than that of the conventional serial detection and 0.5 dB higher than that of the horizontal equalizer. Because soft output preserves the interference information, the outer detector decides more accurately. In Figure 3, the soft-output inner detector has two functions. One is to reduce errors compared to a six-level signal, which improves the accuracy  
Figure 4: Bit error rate (BER) performance of the proposed models without track mis-registration (TMR). of the inner detector. The other is the preservation of interference information. The resulting noise information is then provided to the outer detector, which is the second detector in the serial detector, thus improving the performance of the entire serial detector. For GPR target to reach the optimal coefficient, which is the same as the channel coefficients, we choose the matrix \(\mathbf{G}=\mathbf{H}\). The result of this case is represented by the dashed line in Figure 4, and the proposed scheme is close to the optimal performance. In the next simulation, we include the effect of TMR and the channel matrix \(\mathbf{H}\) is changed according to the level of TMR (%). We show the examples of 10 and 20% TMRs. The system is simulated when AD is 3 Tb/in\({}^{2}\). The BER performance of this simulation with 10% and 20% TMR is shown in Figures 5 and 6, respectively. The coefficients of channels with 10% and 20% TMR were shown as  
\[\mathbf{H}_{10\%}=\left[ \begin{array}{ccc}0.0675&0.3176&0.0675\\ 0.2105&0.9906&0.2105\\ 0.0986&0.4641&0.0986\end{array} \right] \tag{18}\]  
and  
\[\mathbf{H}_{20\%}=\left[ \begin{array}{ccc}0.0543&0.2554&0.0543\\ 0.2046&0.9628&0.2046\\ 0.1159&0.5452&0.1159\end{array} \right], \tag{19}\]  
Meanwhile, the targets estimated for the channels with 10% and 20% TMR were  
\[\mathbf{G}_{10\%}=\left[