# 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  
In the two previous subsections, we proposed two deep learning-based channel estimation methods: FDNN-based and CNN-based channel estimation approaches. However, those two methods have no ability to exploit the long-term correlation of the channels, and thus they could not reach the optimal performance in general. To address this issue, one good choice is to apply a neural network that has the ability to study the behaviors of the channel correlations, such as a recurrent neural network (RNN). The simple structure of a one-layer RNN is given in Figure 7. As we can see from this figure, the input of the RNN cell in the current time step is the output of the RNN cell in the previous time step. Working in this way, the RNN can remember the past information of the input. The basic RNN cell is the computation unit, which performs the following calculation [41]:  
\[\mathbf{h}_{t} =f(\mathbf{W}_{ih}\mathbf{x}_{t}+\mathbf{b}_{ih}+\mathbf{W}_{hi} \mathbf{h}_{t-1}+\mathbf{b}_{hh}), \tag{38}\] \[\mathbf{Y}_{t} =f(\mathbf{W}_{ho}\mathbf{h}_{t}+\mathbf{b}_{ho}), \tag{39}\]  
where \(f(\cdot)\) is the activation function; \(\mathbf{h}_{t}\) and \(\mathbf{h}_{t-1}\) are the hidden states at the time step \(t\) and \(t-1\), respectively; \(\mathbf{x}_{t}\) and \(\mathbf{Y}_{t}\) are the input and the output at the time step \(t\); \(\mathbf{W}_{ih}\mathbf{W}_{hh}\), and \(\mathbf{W}_{ho}\) are the weights for the input layer to the hidden layer, the hidden layer to the next hidden layer, and the hidden layer to the output layer, respectively; and \(\mathbf{b}_{ih}\), \(\mathbf{b}_{hh}\), and \(\mathbf{b}_{ho}\) are the corresponding biases. However, the simple RNN cell has several weaknesses: first, it has no ability to exploit the future information of the data, while the channel at the time step \(t\) has a relation not only with the past but also the future. Thus, the bidirectional network should be used in this case to obtain better performance. Second, another problem with using a  
Figure 7: The illustration of the proposed RNN model. simple RNN cell is that it cannot capture long-term information. One solution for this problem is to use LSTM instead. Consequently, in this paper, we propose a bidirectional-long short-term memory (bi-LSTM) network for 5G channel estimation to overcome the above-mentioned weaknesses. The structure of the proposed bi-LSTM network for the channel estimation is illustrated in Figure 8. In the bi-LSTM structure, the simple RNN cell is replaced by the corresponding LSTM cell, which has the structure shown in the top of Figure 8. The computation of the LSTM cell will give the result as shown in the following equations [41]:  
\[\mathbf{f}_{t} =f(\mathbf{W}_{f}\mathbf{h}_{t-1}+\mathbf{U}_{f}\mathbf{X}_{t}+ \mathbf{b}_{f}), \tag{40}\] \[\mathbf{i}_{t} =f(\mathbf{W}_{i}\mathbf{h}_{t-1}+\mathbf{U}_{i}\mathbf{X}_{t}+ \mathbf{b}_{i}),\] (41) \[\mathbf{c}_{t}^{\prime} =\tanh(\mathbf{W}_{c}\mathbf{h}_{t-1}+\mathbf{U}_{c}\mathbf{X}_{t} +\mathbf{b}_{c}),\] (42) \[\mathbf{c}_{t} =\mathbf{f}_{t}\odot\mathbf{c}_{t-1}+\mathbf{i}_{t}\odot \mathbf{c}_{t}^{\prime},\] (43) \[\mathbf{h}_{t} =f(\mathbf{W}_{ih}\mathbf{x}_{t}+\mathbf{b}_{ih}+\mathbf{W}_{hi} \mathbf{h}_{t-1}+\mathbf{b}_{hh}), \tag{38}\]0 (44) \[\mathbf{h}_{t} =f(\mathbf{W}_{ih}\mathbf{x}_{t}+\mathbf{b}_{ih}+\mathbf{W}_{hi} \mathbf{h}_{t-1}+\mathbf{b}_{hh}), \tag{38}\]1  
where \(\tanh\) is the hyperbolic tangent function, and \(\mathbf{W}_{f}\), \(\mathbf{W}_{i}\), \(\mathbf{W}_{c}\), \(\mathbf{W}_{o}\), \(\mathbf{U}_{f}\), \(\mathbf{U}_{i}\), \(\mathbf{U}_{c}\), \(\mathbf{U}_{o}\), \(\mathbf{b}_{f}\), \(\mathbf{b}_{i}\), \(\mathbf{b}_{c}\), and \(\mathbf{b}_{o}\) are correspondingly the weights of matrices and biases.