## 3 Proposed SS-ELM Equalizer  
Initially, we used the original ELM with supervised learning (as a reference case). To this end, we performed the training step by employing the two long training symbols as input training data and by following the steps of the Algorithm 1, then the frame was equalized by feeding each data subcarrier to the ELM. As any equalizer, this procedure is done frame by frame. Based on the poor BER resulting from this type of ELM (as will be depicted in the next Section), for this manuscript, we explored the SS learning paradigm for the ELM algorithm along with localized mapping. In the following paragraphs, we will define and explain the proposed SS-ELM algorithm. The proposed SS-ELM algorithm is based on Algorithm 2 and, as a common machine learning approach, has two stages: training and testing. The first one consists of the training phase, in which we use the pilots and the two long training symbols to calculate the internal parameters of the ELM (\(\beta_{ELM}\)). Notice that the weights and biases between the input and hidden layers are arbitrarily generated (a main characteristic of the ELM algorithm as mentioned in Section 2.3) based on the uniform distribution on [\(-\)1,1] [21; 22]. Using these parameters, we then perform the evaluation phase. Here, we equalize the constellation symbols and, afterwards, calculate the BER metric by using the standard demodulation. The time-frequency fluctuations in the channel can be seen in Equations (2) and (3) where \(\tau(t)\) represents the general time delay variations. Instead, the several parameters on Equations (4) and (5) show the frequency oscillations associated with the Doppler effect. Consequently, if we feed all the data directly to the model, we will not properly address this problem (as observed in the next Section). Consequently, we perform the training and equalization by considering the frequency and time domains for the SS-ELM algorithm. The basic ELM architecture consists of two real inputs \(\mathbb{R}\{Y_{i}(k)\}\) and \(\mathbb{I}\{Y_{i}(k)\}\), one for the real part of the constellation symbol and other for its imaginary part, \(n_{h}\) hidden neurons, and two neurons on the output, one for each part of the constellation symbol. For the sake of simplicity, we will refer to the inputs as \(Y_{i}(k)\) and the outputs as \(\hat{Y}_{i}(k)\). First of all, by considering that the channel changes rapidly in the frequency domain, the information from the pilot carriers is used in a localized manner, as shown in Figure 6.  
Therefore, four localized equalization processes following the IEEE 802.11p standard are done at the same time, i.e., one for each pilot subcarrier. We define the slot signal as follows  
\[X_{i}^{p}(k),Y_{i}^{p}(k)=X_{i}(k),Y_{i}(k)_{k=p}^{p+q}, \tag{16}\]  
where \(i\) is the number of OFDM symbol, \(k\) denotes the subcarrier number, \(p\) represents the pilot corresponding to the slot being process, and \(q\) comes to be the length of the frequency slot. With the localized mapping, there are 12 data carriers for each pilot. This number of data carriers was adopted because it corresponds to the relationship \(N_{SD}/N_{SP}\), where \(N_{SD}\) and \(N_{SP}\) are given by the parameters exposed in Table 1. Consequently, we take  
Figure 6: Subcarrier distribution slots. advantage of the frequency dependency of the channel in order to diminish the impact of the frequency variations in the channel in the system performance, since the term \(\lambda\) in Equations (4) and (5) is the closest to the pilot subcarrier term. Then, for each slot, the process is performed symbol by symbol in the time domain by considering the definition given in Equation (1). This is done in order to track the time fluctuations in the channel. For illustration purposes, Figure 6 also represents how the subcarriers are distributed for each slot. In addition, the block diagram of the equalization process is presented in Figure 7 for clarification purposes.