# Machine Learning-Based 5G-and-Beyond Channel Estimation for MIMO-OFDM Communication Systems
## 3 Deep Learning-Based Channel Estimation
### Fully Connected Deep Neural Network-Based Channel Estimation  
To overcome the aforementioned drawbacks of LS and LMMSE estimation approaches, we propose a FDNN-aided estimation that minimizes the MSE between the channel estimate obtained by LS estimation and the actual channel. The structure of the proposed FDNN-based channel estimation is depicted in Figure 5. As shown in this figure, the proposed FDNN structure is organized as layers including the input layer, hidden layers, and output layer. Notice that an FDNN may have many hidden layers. However, for the considered MIMO-OFDM system, the proposed FDNN structure is designed with 3 hidden layers that include multiple neurons. In particular, a neuron is a computational unit that performs the following calculation:  
Figure 5: The illustration of the FDNN-based channel estimation. \[o=f(z)=f\Bigg{(}\sum_{i=1}^{M}w_{i}x_{i}+b\Bigg{)}, \tag{26}\]  
where \(M\) is the number of inputs to the neuron for which \(x_{i}\) is the \(i\)-th input (\(i=1,\ldots,M\)); \(w_{i}\) is the \(i\)-th weight corresponding to the \(i\)-th input; \(b\) is a bias; and \(o\) is the output of this neuron. In Equation (26), \(f(. )\) is an activation function that is used to characterize the non-linearity of the channel data. In our proposed FDNN-based channel estimation, we borrow the \(\tanh\) function as the activation function, which is defined as  
\[f(z)=\frac{e^{z}-e^{-z}}{e^{z}+e^{-z}}, \tag{27}\]  
where \(e\) is Euler's number. To minimize the mean square error, the FDNN-based channel estimation is used to learn the actual channel information provided by the channel estimates obtained from the LS estimation as the input. In more detail, we define a realization of the input for the training process as  
\[\mathcal{M}_{\rm n-FDNN}=\Big{\{}\text{Re}\Big{\{}\big{[}\hat{\mathbf{h}}_{ \rm LS}^{n}(t)\big{]}_{0}\Big{\}},\text{Im}\Big{\{}\big{[}\hat{\mathbf{h}}_{ \rm LS}^{n}(t)\big{]}_{0}\Big{\}},\ldots,\text{Re}\Big{\{}\big{[}\hat{\mathbf{ h}}_{\rm LS}^{n}(t)\big{]}_{K}\Big{\}},\text{Im}\Big{\{}\big{[}\hat{\mathbf{h}}_{ \rm LS}^{n}(t)\big{]}_{K}\Big{\}}\Big{\}}, \tag{28}\]  
where \(\hat{\mathbf{h}}_{\rm LS}^{n}(t)\) is LS-estimated channel gathered from all received antennas, where the superscript \(n\) denotes the \(n\)-th realization; K is the number of channel samples that FDNN can handle; and the \(\text{Re}\{\cdot\}\) and \(\text{Im}\{\cdot\}\) operators give the real and imaginary part of a complex number, respectively. The output of the neural network is formulated as  
(29)  
where \(\hat{\mathbf{h}}^{n}(t)\) is the output of the neural network at the \(n\)-th realization. In Equations (28) and (29), we separate the channel estimates into the real and imaginary parts to handle the complex numbers for the use of the FDNN neural network. The learning process handles the one-by-one mapping as  
\[\Big{(}\text{Re}\Big{\{}\big{[}\hat{\mathbf{h}}_{\rm LS}^{n}(t)\big{]}_{s} \Big{\}},\text{Im}\Big{\{}\big{[}\hat{\mathbf{h}}_{\rm LS}^{n}(t)\big{]}_{s} \Big{\}}\Big{\}}\Big{)}\rightarrow\Big{(}\text{Re}\Big{\{}\big{[}\hat{ \mathbf{h}}^{n}(t)\big{]}_{s}\Big{\}},\text{Im}\Big{\{}\big{[}\hat{\mathbf{h}} ^{n}(t)\big{]}_{s}\Big{\}}\Big{)},s=0,\ldots,K. \tag{30}\]  
As desired, the output of the neural network should be identical to the actual channels. Alternatively, the purpose of the FDNN-aided estimation is to minimize the MSE between the prediction and actual channels on average; thus, the loss function utilized for the training phase is defined as  
\[\mathcal{L}_{\rm FDNN}(\mathcal{W},\mathcal{B})=\frac{1}{NK}\sum_{n=1}^{N} \sum_{t=1}^{T}\big{\|}\hat{\mathbf{h}}^{n}(t)-\mathbf{h}^{n}(t)\big{\|}_{2}^{2}, \tag{31}\]  
where \(N\) is the number of realizations used for training, and \(\mathbf{h}^{n}(t)\) is the actual channel corresponding to \(\hat{\mathbf{h}}^{n}(t)\). \(\mathcal{W}\) and \(\mathcal{B}\) include all the weights and biases, respectively.