# Machine Learning-Based 5G-and-Beyond Channel Estimation for MIMO-OFDM Communication Systems
## 3 Deep Learning-Based Channel Estimation
### Long Short-Term Memory-Based Channel Estimation
The forget function \(\mathbf{f}_{t}\) defines which information will be forgotten by the LSTM cell, \(\mathbf{c}_{t}\) is the cell state that contains the important information from the past, and \(\mathbf{c}_{t}^{\prime}\) is a new candidate value that defines which information will be updated to the cell state \(\mathbf{c}_{t}\) and \(\mathbf{h}_{t}\) is the hidden state function of the LSTM cell. By working in this way, the LSTM cell can capture the important information from the past and avoid the redundant information, thus providing a greater ability to capture the information compared to the simple RNN cell. The bottom of Figure 8 shows the structure of the bi-LSTM network. As we can see, the bi-LSTM approach is the combination of two LSTM networks with two different directions. The output of the bi-LSTM takes the outputs of the two LSTM cells into consideration via the linear layer as  
\[\mathbf{h}_{t} =f(\mathbf{W}_{ih}\mathbf{x}_{t}+\mathbf{b}_{ih}+\mathbf{W}_{hi} \mathbf{h}_{t-1}+\mathbf{b}_{hh}), \tag{38}\]2  
where \(\mathbf{H}_{t}\) is the hidden state concatenated from the forward hidden state \(\mathbf{h}_{t}\) and the backward hidden state \(\mathbf{h}_{t}^{\prime}\), and \(\mathbf{W}\) and \(\mathbf{b}\) are the weights and biases of the linear layer, respectively. Therefore, the bi-LSTM approach can exploit the relation of both history and the future with the data in the current time step. To apply the bi-LSTM model for our system, we first gather the LS-estimated channels from all antennas and then define a realization of the input for the training process as  
\[\mathbf{h}_{t} =f(\mathbf{W}_{ih}\mathbf{x}_{t}+\mathbf{b}_{ih}+\mathbf{W}_{hi} \mathbf{h}_{t-1}+\mathbf{b}_{hh}), \tag{38}\]3  
where \(L\) is the sequence length considered for bi-LSTM network. Note that the input of bi-LSTM \(\hat{\mathbf{h}}_{LS}\) is the LS-estimated channel for all \(N_{T}\times N_{R}\) channel streams, so the number of features for the input is \(2N_{T}N_{R}\). The output of the bi-LSTM network is the corresponding true channel as  
\[\mathbf{h}_{t} =f(\mathbf{W}_{ih}\mathbf{x}_{t}+\mathbf{b}_{ih}+\mathbf{W}_{hi} \mathbf{h}_{t-1}+\mathbf{b}_{hh}), \tag{38}\]4The purpose of using a bi-LSTM network is to minimize the MSE between the predicted channel and the true channel; thus, the MSE loss function is considered. The objective function of bi-LSTM network is expressed as  
\[\mathbf{h}_{t} =f(\mathbf{W}_{ih}\mathbf{x}_{t}+\mathbf{b}_{ih}+\mathbf{W}_{hi} \mathbf{h}_{t-1}+\mathbf{b}_{hh}), \tag{38}\]5  
where \(\mathbf{h}^{n}(i)\) is the true channel corresponding to \(\hat{\mathbf{h}}^{n}(i)\); \(\mathcal{W}\) and \(\mathcal{B}\) are all the weights and biases of bi-LSTM; \(N\) is the total number of training samples; and the superscript \(n\) denotes the \(n\)-th training sample. The loss function can be minimized by updating \(\mathcal{W}\) and \(\mathcal{B}\) using gradient descent algorithms. We note that this paper considers the perfect instantaneous channels to be available for the training stage, and therefore we emphasize the imperfect channel state information as a potential extension of our work in the future. **Remark 1**. : _The deep learning-based channel estimation framework studied in this paper is based on the assumption that the perfect CSI is available during the training stage. Such information can be very accurately estimated by the orthogonal pilot signals with a sufficiently large power budget. Even though these conditions for the pilot signals increase the cost for the training stage, the neural networks can learn the channel profile properly. The effects of imperfect channels on the training of neural networks along with the performance reduction in the testing stage as a consequence are of practical interest, which will lead to solid works in the future._