## 3 Proposed SS-ELM Equalizer
Obviously, it begins when the FFT module is performed and ends with the constellation symbols to be demapped. As can be seen, a pre-equalization with the LS method is done right after the Fourier-based block. The reason behind this stage is because the channel estimation which resulted from the LS scheme for the time domain OFDM symbols is fairly accurate, and results in a good initial mitigation of the channel effects with relatively low computational impact [13]. In addition, an extra design parameter (\(\delta\)) is introduced to strengthen the learning phase in the time domain. This determines how many pilots in the time domain are used for each training step of each slot. This parameter is important as it uses the time dependency of the channel to improve the estimation/equalization processes by considering that the vehicular environment changes rapidly over the time domain. According to our observations, and as is expected in this scenario, a high value of \(\delta\) might have a negative impact on the system performance. Evidently, the proposed scheme (Figure 7) replaces the channel estimation and equalization blocks seen in Figure 2, by combining them into a single operation after the FFT module. To depict the general operation of the novel SS-ELM equalizer step by step, the SS-ELM training and testing is presented in Algorithm 3. It is important to consider that the actual value of unlabeled data used for the training of each data frame is the multiplication of \(\mu\) with the number OFDM symbols \(N_{symbols}\) and the number of slots. Even though the values that are shown in the next Section are small, these values do not directly represent the total amount of data used for the unsupervised learning step of the SS-ELM scheme. Finally, each slot can be processed parallel to each other, by allowing the reduction in the execution time needed to process a full data frame. In fact, this procedure is done in Section 4.4 to estimate the complexity of the proposal. Figure 7: Block diagram of the equalization process. ```
\(N_{symbol}\) is the same in Equation (1) and depicts the total number of OFDM symbols Input: Training stage: Labeled data, \(U_{l}=\{Y_{T1},Y_{T2},P^{R}\}\), \(T_{l}=\{X_{T1},X_{T2},P^{T}\}\) where \(X_{T1},Y_{T1}\) and \(X_{T2},Y_{T2}\) are the two long training symbols and \(P^{T},P^{R}\), are the pilot vector transmitted and received respectively constructed as: \(\{P^{T},P^{R}\}=\{Y_{i}(p),X_{i}(p)\}_{t=i-\delta/2}^{i+\delta/2}\) Unlabeled data, \(U_{\mu}=\{Y_{i}^{R}(k)\}_{k=1}^{P}\) Testing stage: Corrupted data, \(Y_{i}(k)\) Output: Training stage: \(\beta_{ELM}\) Testing stage: \(Y_{i}(k)\) for\(i=1\)to\(N_{symbol}\)do Training stage: train the SS-ELM computing \(\beta_{ELM}\) based on Algorithm 2: 1: Construct the graph Laplacian \(L\) from both \(U_{l}\) and \(U_{\mu}\) 2: Initiate an ELM network of \(n_{h}\) hidden neurons with random input weights and biases, calculate the output matrix of the hidden neurons \(F\) 3: Select the hyper-parameters \(C\) and \(\lambda\) 4: Find the output weights if\(n_{h}<=N_{m}\)then \(\beta_{ELM}=(I_{n_{h}}+F^{T}CF+\lambda F^{T}LF)^{-1}F^{T}C\widetilde{T}\) else \(\beta_{ELM}=F^{T}(I_{l+\mu}+CFF^{T}+\lambda LFF^{T})^{-1}C\widetilde{T}\)  end for Testing: The equalization step for the \(i^{th}\) OFDM symbol is perform by feeding the slot signal \(Y_{i}^{p}\) to the model. 1: Compute the output of the hidden layer \(F_{signal}\) 2: Compute the output of the ELM with \(F_{signal}\beta_{ELM}\)  end
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**Algorithm 3**SS-ELM training and equalization.