## 3 Proposed Model  
In the proposed model, we used an RNN instead of the Viterbi algorithm in [20] to estimate the LRI and ULI. The model is illustrated in Figure 2.  
We used the RNN structure shown in Figure 3. It includes an input layer, two hidden layers, and an output layer. Because the received signal \(b[j,k]\) is used as the vector \(\mathbf{y}\), we created 25 neurons in the input layer. We designed m and n neurons in hidden layers 1 and 2, respectively, and used a neuron in the output layer. To select \(r\) and \(t\), we implemented a simulation by varying their values. In this experiment, we simulated the horizontal detection branch of the model, as shown in Figure 2. The results are summarized in Table 1. Based on the performance shown in Table 1, we selected \(r=30\) and \(t=20\) to obtain the best BER performance. To create a feedback line for each neuron, we used three delay terms: \(z^{-1}\), \(z^{-2}\), and \(z^{-3}\). With feedback lines in the neurons, an RNN is an advantageous technique for processing sequential data. Similar to the 2D data received from the output channel, the RNN has memory to store the previous state information. Therefore, it can provide better predictions than the previous model. In addition, we used the hyperbolic tangent function for the activation function \(f(x)=2/(1+\exp(-2x))-1\). Figure 3: RNN model. Figure 2: Proposed model. In Figure 2, to create the signal \(a[j,k]\), the user signal \(u[k]\) is modulated. Subsequently, \(a[j,k]\) is stored in the HDS channel. The readback signal from HDS is \(b[j,k]\), which is distorted by the 2D interference and Gaussian noise \(w[j,k]\). The received signal \(b[j,k]\) was supplied in three parts. The first is the equalizer part, which converts \(b[j,k]\) into \(z[j,k]\). Signal \(z[j,k]\) is approximate to the desired signal \(d[j,k]\), which is interfered with by the 2D target \(\mathbf{G}\). The second is the RNN with horizontal detection creating the signal \(z_{v2 naw}[j,k]\). Signal \(z_{v2 naw}[j,k]\) is similar to the desired signal \(d_{v2 naw}[j,k]\), which is interfered with by the 1D target \(\mathbf{G}_{v2 naw}\). The third is the RNN with vertical detection. Signal \(z_{h2 naw}[j,k]\) is approximate to the desired signal \(d_{h2 naw}[j,k]\), which is interfered with by the 1D target \(\mathbf{G}_{h2 naw}\). After the RNN converts the received \(y[j,k]\) into signals \(z_{v2 naw}[j,k]\) and \(z_{h2 naw}[j,k]\), which include 1D interference, we can use the conventional 1D VA to detect \(z_{v2 naw}[j,k]\) and \(z_{h2 naw}[j,k]\) in \(\overline{a}_{v}[j,k]\) and \(\overline{a}_{h}[j,k]\), respectively. Thereafter, the signals \(\overline{a}_{v}[j,k]\) and \(\overline{a}_{h}[j,k]\) pass through the VI and HI targets to achieve the estimated signals of the ULI and LRI, respectively, as follows:  
\[ \begin{array}{ll}ULI[j,k]&=\overline{a}[j+1,k+1]\beta+\overline{a}[j+1,k]p+ \overline{a}[j+1,k-1]\beta\\ &+\overline{a}[j-1,k+1]\beta+\overline{a}[j-1,k]p+\overline{a}[j-1,k-1]\beta \end{array} , \tag{20}\]  
\[ \begin{array}{ll}LRI[j,k]&=\overline{a}[j+1,k+1]\beta&+\overline{a}[j+1,k-1 ]\beta\\ &+\overline{a}[j,k+1]q&+\overline{a}[j,k-1]q\\ &+\overline{a}[j-1,k+1]\beta&+\overline{a}[j-1,k-1]\beta\end{array} . \tag{21}\]  
After equalizing the received signal \(b[j,k]\) into \(z[j,k]\), we can present the signal \(z[j,k]\) as follows:  
\[ \begin{array}{ll}z[j,k]&\approx d[j,k]\\ &\approx a[j+1,k+1]\beta&+a[j+1,k]p&+a[j+1,k-1]\beta\\ &+a[j,k+1]q&+a[j,k]g_{0,0}&+a[j,k-1]q\\ &+a[j-1,k+1]\beta&+a[j-1,k]p&+a[j-1,k-1]\beta\end{array} . \tag{22}\]  
Next, we can remove \(ULI[j,k]\) or \(LRI[j,k]\) from the signal \(z[j,k]\) as follows:  
* Removing \(ULI[j,k]\): \[ \begin{array}{ll}z[j,k]-ULI[j,k]&\approx a[j+1,k+1]\beta&+a[j+1,k]p&+a[j+1,k-1 ]\beta\\ &+a[j,k+1]q&+a[j,k]g_{0,0}&+a[j,k-1]q\\ &+a[j-1,k+1]\beta&+a[j-1,k]p&+a[j-1,k-1]\beta\\ &-\overline{a}[j+1,k+1]\beta&-\overline{a}[j+1,k]p&-\overline{a}[j+1,k-1] \beta\\ &-\overline{a}[j-1,k+1]\beta&-\overline{a}[j-1,k]p&-\overline{a}[j-1,k-1] \beta\end{array}