## 3 Proposed Iterative Signal Detection Scheme  
A block diagram of the proposed iterative detection scheme is shown in Figure 1. The partial response maximum likelihood (PRML) detector, which comprises a PR equalizer and a maximum likelihood sequence detector, is used to manage the ISI and ITI. In this study, a 2D equalizer and a 1D PR target based on the MMSE criterion and Viterbi detectors is used for equalization and detection, respectively. An MLP is utilized to improve the system performance using the MLP output as the a priori information for equalization and the extrinsic information for detection. Figure 2 shows the structure of the MLP that is composed of three types of layers, i.e., input layer, hidden layer, and output layer. In all neurons in the hidden layers in the MLP, a rectified linear unit (ReLU), expressed as \(f(y)=\max(0,y)\) is used as the activation function. For the neuron in the output layer of the MLP, we use a sigmoid function, \(s(v)=1/(1+e^{-v})\) as the activation function. For training, the MLP is trained by [PERSON] in Tensorflow 2.0, which is a high-level neural network application programming interface, written in Python and capable of operating on top of TensorFlow [12]. To obtain the weights that minimize the loss function, adaptive moment estimation (Adam) is used as the optimizer [13]. After user data are passed through BPMR channel, the proposed detection scheme is performed as follows: (Step 1) First,the readback signal \(r_{j,k}=\left[ \begin{array}{cc}r_{j+M_{m},k+N_{m}}&r_{j+M_{m},k+N_{m}-1}&\cdots& r_{j,k}&\cdots&r_{j-M_{m},k-(N_{m}-1)}&r_{j-M_{m},k-N_{m}}\end{array} \right]\) is used as the input sequence of the MLP, and then the MLP is implemented. (Step 2) To exploit the MLP output \(s(v)\) as reliable information, the information \(p_{j,k}\), which is used as a priori information in the equalizer and extrinsic information in the detector, is modified as follows:  
\[p_{j,k}=\alpha\cdot(2s(v)-1) \tag{4}\]  
where \(\alpha\) is the scaling factor of the information. (3) Using \((2M_{w}+1)\times(2N_{w}+1)\) 2D equalizer with a \((2N_{g}+1)\) 1D PR target, the equalizer output \(s_{j,k}\) and desired output \(d_{j,k}\) can be calculated as follows:  
\[s_{j,k}=\sum_{m=-M_{w}}^{M_{w}}\sum_{n=-N_{w}}^{N_{w}}w_{m,n}\left(r_{j-m,k-n} +p_{j-m,k-n}\right)=\sum_{m=-M_{w}}^{M_{w}}\sum_{n=-N_{w}}^{N_{w}}w_{m,n}q_{j- m,k-n}=\mathbf{w}^{T}\mathbf{q}_{j,k} \tag{5}\]  
\[d_{j,k}=\sum_{n=-N_{g}}^{N_{g}}g_{n}\cdot q_{j,k-n}=\mathbf{g}^{T}\mathbf{a}_ {j,k} \tag{6}\]  
where \(\mathbf{w}^{T}=\left[ \begin{array}{cc}w_{-M_{w},-N_{w}}&w_{-M_{w},-N_{w}+1} \cdots&w_{0,0}\cdots w_{M_{w},N_{w}-1}&w_{M_{w},N_{w}}\end{array} \right]\) is the equalizer coefficient, \(\mathbf{q}_{j,k}=\left[ \begin{array}{cc}q_{j+M_{w},k+N_{w}}&q_{j+M_{w},k+N_{w }-1}&\cdots&q_{j+M_{w},k-(N_{w}-1)}&q_{j-M_{w},k-N_{w}}\end{array} \right]\) is the equalizer input that combines the readback signal \(\mathbf{r}_{j,k}\) and the a priori information \(p_{j,k}\), \(\mathbf{g}^{T}=\left[ \begin{array}{cc}g_{-N_{g}}&\cdots&g_{N_{g}}\end{array} \right]\) is the target polynomial coefficients, and \(\mathbf{a}_{j,k}=\left[ \begin{array}{cc}a_{k+N_{g}}&\cdots&a_{k-N_{g}}\end{array} \right]\) is the user data sequences. To design the equalizer and target, we utilize the MMSE criterion [14,15].