# Extreme Learning Machines to Combat Phase Noise in RoF-OFDM Schemes
## 4 Proposed Phase Noise Suppression Methods through Extreme Learning Machines  
In this section, we introduce the real and complex ELMs to diminish phase and amplitude noises in OFDM-based RoF systems following the OFDM frame used by the default pilot-assisted channel correction (see Figure 1). In other words, both novel methods are distinguished by no additional overhead and/or no modification of the transmitter and receiver of an overall OFDM scheme. For the proposed methods, pilots must be modulated by the desired constellation points in order to get successful learning in the artificial neural network and, thus, track the phase noise variations in each constellation symbol. The minimum number of pilots is therefore given by the M-ary subcarrier modulation format; 64 pilot subcarriers for a 64 QAM-OFDM signal, for instance. If the subcarriers are modulated with high spectral efficiency formats, the required pilots for these novel phase error correction methods would become the main disadvantage owing to transmission efficiency reasons. In this scenario, the pilot-based equalization technique could be adopted. The fully-complex ELM possesses a single input and a single output to reinforce the training stage, where the training and the input testing parts are pilot and data subcarriers, respectively, during each OFDM symbol period. Remembering the previous section, the learning stage is hence formed by the following steps: (i) arbitrarily assign the input weights \(w_{k}\) and the biases of the hidden nodes \(b_{k}\); (ii) given the activation function \(g(\cdot)\) and the hidden neuron number \(\widetilde{L}\), calculate the Moore-Penrose generalized inverse of the hidden layer output matrix \(H\); and (iii) according to Formula (7), solve the output weight \(\beta\). In other words, the learning phase by using the information given by the pilot subcarriers can be summarized as:  
\[\beta_{\widetilde{L}\times 1}=H_{\widetilde{L}\times L}^{\dagger}P_{L\times 1}, \tag{8}\]  
where the samples \(L\) matches the number of pilots \(N_{p}\) and \(H^{\dagger}\) and \(P\) depend on the received \(\overline{p_{i}}\) and transmitted \(p_{i}\) reference symbols, respectively. In order to diminish the phase noise effects in the received OFDM signal, the test performance of the ELM algorithm is finally carried out:  
\[D_{L\times 1}=H_{L\times\widetilde{L}}\beta_{\widetilde{L}\times 1}, \tag{9}\]  
where the output matrix \(D\) contains the corrected data symbols \(\overline{d_{i}^{x}}\), the number of samples is equal to the number of data subcarriers (\(L=N_{d}\)), and the hidden layer output matrix \(H\) is a function of the received data symbols without equalization \(\overline{d_{i}}\). As a pilot-assisted equalization method, all  
Figure 2: The ELM architecture. this procedure is done OFDM symbol-by-OFDM symbol in order to follow the rapid variations in the time and frequency domains, which come from the optical and/or wireless channels. Evidently, as the training samples (the number of pilots) increases, the success rate in the testing stage increases, i.e., the BER improves. However, the relationship between the number of data subcarriers and pilot subcarriers is about 10:1 for spectral efficiency purposes [20]. For the real ELM, the complex signal is divided into two inputs (one for the real number and one for the imaginary part), fed into different real-valued ELMs, and joined in the Cartesian form for recovering the corrected data subcarrier. The ELM phase noise suppression pseudo-codes can be summarized as:  
```
Input: The training test (the pilots subcarriers) \(\Pi=\{(\overline{p_{i}},p_{i})\mid\overline{p_{i}}\in\mathbb{C},p_{i}\in \mathbb{C},i=1,...,N_{p}\}\). The complex activation function \(g(\cdot)\). The number of hidden nodes \(\widetilde{L}\). Output: The corrected data symbols \(\Delta=\{\overline{d_{i}^{c}}\in\mathbb{C},i=1,...,N_{d}\}\). Assign arbitrarily the complex input weights \(w_{i}\) and the complex biases \(b_{i}\).