# Low-Complexity Convolutional Neural Network for Channel Estimation
## 5 Computational Cost
The first table includes the computation process for LS and MMSE techniques in order to provide a benchmark against the traditional channel associated limitations: LS lacks in accuracy, while MMSE has high complexity. The second table provides all the information per stage to compute the complexity of the analyzed DL-based channel estimation methods. This helps to highlight the possible reduction in complexity avoiding some stages. Additionally, in the prospect of massive MIMO application, a useful prediction of the complexity growth is given by the increased number of antennas at one end. Figures 7 and 8 show the total number of operations required to calculate the full channel matrix from the pilot signals for different numbers of antennas at the receiver for each solution. The MMSE and LS techniques represent the two benchmarks for evaluating the feasibility on hardware of the rest of the techniques. Being the MMSE traditionally discarded for implementation, clearly, the CNN network presented in [8] is not suitable, as the complexity is 2 orders of magnitude higher than the MMSE. This highlights that CNN-based structure, where the number of neurons in the network is not dependent on the number of antennas, have a completely different order of complexity, indicating that the required computational cost varies with design choices. Indeed, the implemented CNNs with pre-interpolation are comparable to the MMSE complexity. Although there is a big difference in number of layers, the computational cost varies by half an order of magnitude. In this range, the FCDNN and LSTM networks presented in [3; 8] are included. The growth of the FCDNN complexity is faster than the LSTM, having the number of neurons per layer dependent on the number of antennas in the system. Figure 7: Computational cost analysis among several DL channel estimations varying the number of antennas at the receiver. The transmitter (user terminal, UT) uses two antennas. The calculation is obtained from the above table considering the implemented designs and those reported in the literature [3; 8; 25]. As expected, the LS channel estimation with interpolation in frequency and time domains is traditionally the least expensive in terms of computational cost. However, our proposal provides the next lowest complexity, outperforming all other DL-based techniques. The improvement results in terms of complexity are due to the full channel being obtained as the output of the neural network, completely avoiding the interpolation stage and a hardware-oriented CNN design. The other three important factors in the choice of the hardware-oriented neural network when it comes to their implementation are the memory usage, the opportunities offered for parallelization, and the type of required operations. Because of the use of locality, the CNN requires less memory for storing parameters, and each local section can be parallelized. The other advantage is related to the type of involved operations since convolution is predominant. In the FCDNNs, each neuron is connected to the neuron in the subsequent layer. Therefore, the generated information has to be stored. This is true also for the LSTM because it captures a recurrent property over time. As a consequence, data have to be stored and propagated to the next cell to obtain the final result, therefore each result is dependent on the previous one. Additionally, the latter uses sigmoids and hyperbolic tangents in each cell, which are not hardware-friendly operations. In conclusion, the proposed technique outperforms other DL alternatives when it comes to implementation. In the next section, we evaluate the performance in terms of accuracy in the channel estimation. Since FCDNN and LSTM present unfavorable characteristics when it comes to their hardware implementation, the focus will be oriented to the CNN structures. The LSTM is excluded, while the comparison with the FCDNN structure is purely based on the neural network presented in [3].