## 2 BPMR Channel Model
To utilize the mutual information from the upper and lower tracks, H1 and H3 were moved toward the desired track by half the track pitch. This helped increase the correlation of this mutual information, thus enabling us to improve the BER performance of the desired track using our proposed methods. It is important to highlight that track misregistration is a concern during the reading process; however, prediction and correction techniques using the array reader may be adopted to cope with this unavoidable situation, as proposed in [31; 32; 33]. Figure 3: Configuration of the staggered BPMR array at an AD of 3.0 Tb/in\({}^{2}\), showing the extent of the read head sensitivity response. To obtain the readback signal, \(s(x,y)\), as a function of \(x\) and \(y\) coordinates (in nanometers), the magnetization of each bit island in the recording medium, \(m(x,y)\)\(\in\)\(\left[\pm 1\right]\), was convolved with the reader sensitivity function, \(h(x,y)\), [34,35]. Therefore, the readback signal \(s(x,y)\) can be expressed as  
\[s(x,y)=\iint m(\xi,\eta)h(x-\xi,y-\eta)d\xi d\eta. \tag{3}\]  
Considering the reading process in the time domain, the continuous-time readback signal, \(r(t)\), can be obtained by setting \(s(t)=s(t,i)\), where \(i\) is the center of the reader sensitivity function in the cross-track direction. Thus, the continuous-time readback signal, \(r(t)\), can be obtained by  
\[r(t)=s(t)+n(t), \tag{4}\]  
where \(n(t)\) is additive white Gaussian noise, with the power spectral density being flat across the entire sampling bandwidth. Here, \(n(t)\) can be calculated from  
Figure 1:0  
where  
Figure 1:1  
and \(Z\sim N\)(0,1) is a random variable that follows the standard normal distribution with a zero mean and a standard deviation/variance of 1, \(A=1\) is the peak amplitude of the normalized readback signal for an isolated transition, and SNR is the signal-to-noise ratio in decibels (dBs). Assuming perfect synchronization, the discrete-time readback sequences for the \((l-1)\)-th and \((l+1)\)-th tracks, \(r_{k,l-1}\) and \(r_{k,l+1}\), were obtained by over-sampling \(r_{l-1}(t)\) and \(r_{l+1}(t)\) at time \(t=0.5 kT_{x}\). However, the data sequence of the \(l\)-th track, \(r_{k,l}\), was obtained by sampling the readback signal \(r_{l}(t)\) at time \(t=kT_{x}\). Therefore, three data sequences or three mutual data sequences were produced. Next, the three readback signal sequences \(\left\{r_{k,l-1},r_{k,l},r_{k,l+1}\right\}\) were fed to the equalizers to output the equalized data sequences \(\left\{z_{k,l-1},z_{k,l},z_{k,l+1}\right\}\) according to their target responses. The equalized data were then sent to the SOVA detectors to estimate the most likely recording bit sequences in the form of soft-output information, or LLR, of each track \(\left\{\lambda_{k,l-1},\lambda_{k,l},\lambda_{k,l+1}\right\}\). Then, to improve the reliability of LLR sequences produced from such detectors before making a final decision, four proposed LLR improvement techniques based on mutual data were employed, namely, HIF, MSI, BSD, and MLP. Finally, the estimated user bits were compared with the actual user bits to calculate the BER. Figure 1:2 Block diagram of the three-reader one-track detection BPMR channel with the proposed improvement techniques.