# Low-Complexity Convolutional Neural Network for Channel Estimation
## 5 Computational Cost  
Towards an efficient hardware implementation, the computational cost is a fundamental criterion to analyze. With offline training, the complexity depends strongly on the neural network dimension and topology. Because the design choices largely affect the number of operations involved in the computation, an analysis among the state-of-the-art DL-channel estimation techniques is presented. When considering the computational cost, the primary focus lies in identifying the dominant factors that contribute the most to the overall growth rate of the algorithm's complexity. This helps to simplify the calculation, focusing on the most significant operations. The steps of this analysis are summarized in Tables 2 and 3, where each column highlights an additional stage before reaching the total computational cost. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Channel Estimation Techniques** & **Complexity per Reference** & **Complexity per Reference Symbols** & **Complexity of the Interpolation** & **Complexity per Tx and Rx Antennas** \\ \hline MMSE & \(\begin{array}{c}C_{rElem}=N_{RefElim}^{3}+\\ 3\;N_{RefElim}^{2}+3\;N_{RefElim}\end{array}\)[25] & \(\begin{array}{c}C_{rs}=C_{Flem}\;N_{RefSymbol}\end{array}\) & \(\begin{array}{c}C_{interp}=C_{rs}+3\;N_{noRefEISlat}\end{array}\) & \(\begin{array}{c}C_{MMSE}=C_{interp}\\ N_{TxArts}N_{RxArts}\end{array}\) \\ \hline LS & \(\begin{array}{c}C_{rElem}=4\;N_{RefElim}\end{array}\)[25] & \(\begin{array}{c}C_{rs}=C_{rElem}\;N_{RefSymbol}\end{array}\) & \(\begin{array}{c}C_{interp}=C_{rs}+3\;N_{noRefEISlat}\end{array}\) & \(\begin{array}{c}C_{LS}=C_{interp}\\ N_{TxArts}N_{RxArts}\end{array}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Complexity derivation for traditional methods per stage. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Channel Estimation Techniques** & **Complexity of the Input Computation** & **Complexity of the Method per Layer** & \begin{tabular}[]{c} **Complexity of the Post-Interpolation (\(C_{intpost}\))** \\ \end{tabular}  & **Total Complexity** \\ \hline CNN-L3 & \( \begin{array}{c}C_{interp}=\\ C_{rs-LS}+3\;N_{noRefEISlat}\end{array} \) & \( \begin{array}{c}C_{CNN}=\\ N_{C}\;M_{C}K_{H}K_{W}N_{H}N_{W}\end{array} \) & \( \begin{array}{c}N/A\end{array} \) & \( \begin{array}{c}C_{tot}=(C_{interp}+3\;C_{CNN})\\ N_{TxArts}N_{RxArts}\end{array} \) \\ \hline CNN-L6 & \( \begin{array}{c}C_{interp}=\\ C_{rs-LS}+3\;N_{noRefEISlat}\end{array} \) & \( \begin{array}{c}C_{CNN}=\\ N_{C}\;M_{C}K_{H}K_{W}N_{H}N_{W}\end{array} \) & \( \begin{array}{c}N/A\end{array} \) & \( \begin{array}{c}C_{tot}=(C_{interp}+6\;C_{CNN})\\ N_{TxArts}N_{RxArts}\end{array} \) \\ \hline FCDNN [3] & \( \begin{array}{c}C_{LS-interpInFreq}=\\ (C_{rs-LS}+3\;N_{noRefEISympt})\end{array} \) & \( \begin{array}{c}C_{FCDNN}=\\ N_{neur}\;M_{input}\;+N_{neur}\end{array} \) & \( \begin{array}{c}C_{intpost}=\\ N_{nx}\;M_{xParts}\end{array} \) & \( \begin{array}{c}C_{tot}=C_{LS-interpInFreq}+\\ N_{RxFISlat}\end{array} \) \\ \hline CNN [8] & \( \begin{array}{c}C_{LS-interpInFreq}=\\ (C_{rs-LS}+3\;N_{noRefEISympt})\end{array} \) & \( \begin{array}{c}C_{CNN}=\\ N_{C}\;M_{C}K_{H}K_{W}N_{H}N_{W}\end{array} \) and \( \begin{array}{c}C_{intpost}=\\ N_{RxFISlat}\end{array} \) & \( \begin{array}{c}C_{tot}=C_{LS-interpInFreq}+\\ N_{RxFISplit}\end{array} \) & \( \begin{array}{c}C_{tot}=C_{LS-interpInFreq}+\\ N_{RxFISplit}\end{array} \) \\ \hline LSTM [8] & \( \begin{array}{c}C_{LS-interpInFreq}=\\ (C_{rs-LS}+3\;N_{noRefEISympt})\end{array} \) & \( \begin{array}{c}C_{LSTM}=\\ 4N_{s}\;D_{HL}(2N_{feat}+2D_{HL}+1)\end{array} \) & \( \begin{array}{c}C_{tot}=\\ C_{intpost}=\\ N_{RxFISpub}\end{array} \) & \( \begin{array}{c}C_{tot}=C_{LS-interpInFreq}+\\ N_{RxFISpub}\end{array} \) \\ \( \begin{array}{c}C_{TxArts}N_{RxArts}\end{array} \) & \(\begin{array}{c}C_{FCDNN}=\\ N_{RxFISlat  
The considered total computation is the complexity needed to estimate the full channel (\(\text{N}_{\text{sc}}\times\text{N}_{\text{symb}}\times K\times M\)) from the reference elements in the grid.