# Low-Complexity Convolutional Neural Network for Channel Estimation
## 4 DL-Channel Estimation
### Proposed Convolutional Neural Network  
Following the previous considerations among the pertinent neural network models, the need for lower computational cost and simpler operations suggests focusing on the convolutional neural network. However, the configuration of a CNN structure can make a substantial difference, as there are several degrees of freedom that can be explored before disqualifying a specific structure, since the complexity and the accuracy of a neural network varies with the parameters of a specific application. For example, a neural network can have a simple structure, but this could be highly dependent on the dimension of the input, and to achieve a certain level of accuracy, the input size has to increase. Therefore, a more complex neural network, in which computational cost varies less with the input, could be the preferred choice. In light of the above, the choice of the CNN structure is not enough to guarantee favorable characteristics, but all the parameters involved in the system play a significant role. In the literature, the DL-channel estimation is used to improve the outcomes of the LS technique. In [3; 8], this resulted in performing the LS calculations only on the reference signal and applying the neural network on the outcome. This increases the accuracy of the estimated channel in a few subcarriers generating a partial outcome. The latter is, then, interpolated across the subcarriers to obtain the full channel matrix. In other designs (e.g., [3; 8; 11]), the interpolation is a preliminary or middle stage of the channel estimation process using the neural network. Figure 4: Example of a convolutional layer having \(M_{C}=3\), \(K_{H}=K_{W}=3\), \(N_{C}=6\). The yellow volume refers to the input neurons, while the light green volume refers to the output neurons. Accordingly, the orange block is the input sliding window, while the green block indicates the output neurons interested by that window. Unlike previous art, our proposal exploits the CNN to: (a) assist the channel estimation of a preliminary LS stage on the reference symbols only; (b) avoid the interpolation stage; (c) completely leverage only one CNN for the estimation of the full channel matrix. The proposed architecture is illustrated in Figure 5.  
The estimation of the channel matrix, \(\mathbf{H}\), is arranged as a 2-D image having \(\mathrm{N_{SC}}\) rows and \(\mathrm{N_{OFDM}}\) columns, which are the number of subcarriers and symbols, respectively. Since each value is a complex number, it is split into its real and imaginary parts, thus composing two different images, which are provided to the CNN in a parallel manner. In this application, the full channel grid is denoted with \(N_{\mathrm{SC}}=288\) and \(N_{OFDM}=14\), but only the DMRS resource elements are provided as input to the network (\(N_{\mathrm{SC}}=288\), \(N_{OFDM}=3\)). In the first layer, the symmetric padding is used to increase the horizontal size from 3 to 7. The same process is applied in the second and fifth layers, while the third and the fourth layers are classic convolutional layers, and the output has the same dimension of the input. An activation function follows each layer: the hyperbolic tangent (tanh) is applied after the first three layers, while the Rectified Linear Unit (ReLU) is used after the fourth and fifth layers. At the end, this neural network structure provides the full channel matrix as an output. To the best knowledge of the authors, this is the first time where the CNN is used to increase the resolution of the input without additional processing. Another novelty in the structure is related to the number of channels per layer. Considering the simplicity of an image which characterizes the estimated channel, the enormous amount of channels per layer proposed in the literature is not needed. This can show signs of overfitting in the networks, especially when the amount of training data is limited. Additionally, fewer  
Figure 5: Diagram of the proposed neural network.