# Improving Serial Detection Using MAP Algorithm for Bit-Patterned Media Recording
## 2 Proposed Model  
### Horizontal Detection and Vertical Detection  
Figure 1 shows the proposed detection model. The received signal \(y[j,k]\) passes through equalizer \(\mathbf{F}\), and the equalizer output signal \(z[j,k]\) is an approximation of the desired signal \(d[j,k]\). In other words, the horizontal detection receives the signal \(z[j,k]\), which is corrupted by the 2D channel interference and approximated to the desired signal by the GPR target \(\mathbf{G}\). In [10], the authors analyzed the 2D interference in target \(\mathbf{G}\) into two serial 1D interferences. Because the output of the inner detection (horizontal detection) is a 6-level signal \(\{-1-2p\), \(-1\), \(-1+2p\), \(1-2p\), \(1\), \(1+2p\)}, the trellis of the horizontal detection includes 36 states and six input branches in each state, as shown in Figure 2. We denote these six values as the \(R_{0}\), \(R_{1}\),..., \(R_{5}\) (\(R_{0}=-1-2p\), \(R_{1}=-1\), \(R_{2}=-1+2p\), \(R_{3}=1-2p\), \(R_{4}=1\), and \(R_{5}=1+2p\)) for convenience. Figure 1: Proposed detection model. In [13], the authors proposed \(P(u_{k}=+1\,|\,\mathbf{y})\) and \(P(u_{k}=-1\,|\,\mathbf{y})\), where \(u_{k}\) is the output of the MAP detection at time \(k\), and \(\mathbf{y}\) is the received symbol of MAP detection. However, in our proposed model, the output of the MAP has more than two states. Therefore, we simply calculate the conditional probability \(P(R_{i}\,|\,\mathbf{z})\) as follows:  
\[ \begin{array}{l}P(\,R_{i}\,|\,\mathbf{z})=\sum\limits_{R_{i}}P(s\prime,s| \,\mathbf{z})=\sum\limits_{R_{i}}\frac{P(s\prime,s,\mathbf{z})}{P(\mathbf{z})} \\ \qquad\qquad=\sum\limits_{R_{i}}\frac{P(s\prime,\mathbf{z}_{<k})P(\mathbf{z}_ {\oplus}s\prime)P(\mathbf{z}_{\oplus}s\mid s)}{P(\mathbf{z})}\\ \qquad\qquad=\sum\limits_{R_{i}}\frac{\alpha_{k-1}(s\prime)\gamma_{\mathrm{L} }(s\prime,s)\beta_{\mathrm{L}}(s)}{P(\mathbf{z})}\\ \qquad\qquad\text{with }i=0,1,\ldots,5\end{array}  \tag{1}\]  
where \(s\prime\) is the previous state, \(s\) is the current state, \(\mathbf{z}\) is the received symbol of the MAP detection in the proposed model, \(\alpha_{k-1}(s\prime)=P(s\prime,\mathbf{z}_{<k})\) is the joint probability of the previous state \(s\prime\) and the sequence received in the past received sequence \(\mathbf{z}_{<k}\), \(\gamma_{k}(s^{\prime},s)=P(\mathbf{z}_{k},s\mid s\prime)\) represents the conditional probability of the current state \(s\) and the current received symbol \(\mathbf{z}_{k}\) given the previous state \(s\prime\), and \(\beta_{k}(s)=P(\mathbf{z}_{\gtrsim k}\,|\,s)\) is the conditional probability of the future received sequence \(\mathbf{z}_{>k}\) under the current state \(s\). Figure 2: Proposed detection model. By using Equation (1), we can calculate \(P(R_{i}\,|\,\mathbf{z})\) for \(R_{0}\), \(R_{1}\),..., \(R_{5}\), Let \(r[j,k]=[P(R_{0}\,|\,\mathbf{z})\), \(P(R_{1}\,|\,\mathbf{z})\), \(P(R_{2}\,|\,\mathbf{z})\), \(P(R_{3}\,|\,\mathbf{z})\), \(P(R_{4}\,|\,\mathbf{z})\), and \(P(R_{5}\,|\,\mathbf{z})\)]. Then, we find the maximum \(P(R_{i}\,|\,\mathbf{z})\) in \(r[j,k]\). In other words, we have  
\[b[j,k]=\underset{R_{i}}{\operatorname{argmax}}(P(R_{i}|\,\mathbf{z})). \tag{2}\]  
With Equation (2), the signal \(b[j,k]\) is a hard decision value. To create a soft output for the signal \(b[j,k]\), we applied the feedback in [11]. The signal \(b[j,k]\) passes through the interference target \([r\,1\,r]\) to create the estimated signal \(\mathbb{z}[j,k]\). The difference of \(\mathbb{z}[j,k]\) and \(\mathbb{z}[j,k]\) is the estimated noise \(e_{b}[j,k]\). Finally, the soft output of signal \(b[j,k]\) is given as follows:  
\[t[j,k]=b[j,k]+e_{b}[j,k]. \tag{3}\]  
To detect the signal \(t[j,k]\) in Equation (3), we used vertical detection with the Bahl, Cocke, Jelinek, and Raviv (BCJR) algorithm in [13]. Thus, the outputs of the vertical detection \(s[j,k]\) are the conditional probabilities \(P(s_{k}=+1\,|\,\mathbf{t})\) and \(P(s_{k}=-1\,|\,\mathbf{t})\) for the signals \"\(1\)\" and \"\(-1\)\", respectively.