## 5 Performance Analyses  
This section numerically and comparatively analyzes the performance of HS-OFDM, OCDM, SCCP, and OTFDM under the presence of the CP length violation and the STO when a narrowband PLC channel, which is frequency selective, is disturbed by additive noise. The PLC channel is generated following the channel model in [29] and considering the parameters found in [30] (Annex D) with a frequency bandwidth of \(B=500\) kHz starting in \(f=0\) Hz, \(L_{h}=30\), and \(N=256\). Based on the values of \(L_{h}\) and \(N\), we obtain \(I=1\). Consequently, the interference may come only from the \((l-1)\)th and \((l+1)\)th \(\widetilde{N}\)-length symbol blocks. Since our attention is on the frequency selectivity trait of PLC channels, we consider only the background aspect of the PLC noise. Therefore, the additive random process is modeled as a zero-mean and white Gaussian random process, which will be named additive white Gaussian noise (AWGN), or a zero-mean and colored Gaussian random process, which will be named additive colored Gaussian noise (ACGN). The discrete PSD of the AWGN model is equal to \(P_{\text{n}_{\text{W}}}[k,k]=N_{0}/2\), \(k=0,1,\ldots,N-1\). For the ACGN model, the discrete PSD is expressed as \(P_{\text{nc}}[k,k]=\frac{\eta}{2}e^{(-v|\Delta f|k)}+P_{\text{n}_{\text{W}}}[k,k]\), \(k=0,1,\ldots,N-1\), where the constants \(v,\eta\in\mathbb{R}^{+}\) are equal to \(1.2\times 10^{-5}\) and \(1.0\times 10^{-15}\), respectively [31], and \(\Delta f=B/N\). The parameters used for comparing these data communication schemes are the data rate and BEP. The data rate analysis is based on the UA and OA, through the water-filling algorithm [8], while the BEP analysis relies on the use of UA and adaptive modulation. A discussion about the numerical results obtained with the deduced closed-form expressions and the Monte Carlo simulation, which allows us to check the accuracy of the deduced closed-form expressions, are also detailed. To perform the data rate analysis based on OA and UA, the assumptions made concerning the channel model (i.e., filtered PLC channel disturbed by the presence of the additive and Gaussian random process) results in the following expression for the achievable data rate [32]:  
\[R=\frac{1}{2(2N+L_{\text{CP}})T_{s}}\sum_{k=0}^{2N-1}\text{log}_{2}\Bigg{(}1+ \frac{\gamma_{\text{SINR}}[k]}{\text{Y}}\Bigg{)}, \tag{39}\]  
Figure 6: Computational complexity of the schemes when \(N=256\) and \(\mathcal{E}_{\times}=\mathcal{E}_{+}=0\). The abscissa is \(\mathcal{B}\)-geometry, which shows that we can have different OTFDM schemes with different computational complexities for a value of \(N\). where Y is the gap factor from [PERSON]'s capacity curve, \(T_{s}\) is the sampling time, and \(\gamma_{\text{SINR}}[k]\) is the \(k\)th diagonal element of (19) for a given data communication scheme (HS-OFDM, OCDM, SCCP, or OTFDM). Moreover, applying (20) in (39) produces the upper bound for the achievable data rate, which will be denoted by \(R^{\text{UB}}\). The additive noise and interference are considered Gaussian random processes in the frequency domain since the central limit theorem applies in the transformation to obtain the frequency or Fresnel domains. Consequently, the average BEP is expressed as  
\[\mathbb{P}_{\text{e}}=\frac{1}{2N}\sum_{k=0}^{2N-1}P_{\text{e}}[k]. \tag{40}\]  
with \(P_{\text{e}}[k]\) being the BEP associated with the \(k^{\text{th}}\) subcarrier, subchirp, subslot, or subtle, and given by  
\[P_{\text{e}}[k]=\frac{4}{\log_{2}(M)}\left[\left(1-\frac{1}{\sqrt{M}}\right) \mathcal{Q}\left(\sqrt{\frac{3\gamma_{\text{SINR}}[k]}{M-1}}\right)\left(1- \frac{1}{\sqrt{M}}\right)^{2}\mathcal{Q}\left(\sqrt{\frac{3\gamma_{\text{SINR }}[k]}{M-1}}\right)^{2}\right], \tag{41}\]  
where \(M\) is the constellation order and \(\mathcal{Q}(\cdot)\) is the Q-function. Furthermore, we can apply (20) in (40) to obtain the lower-bound bit error probability denoted as \(\mathbb{P}_{\text{e}}^{\text{LB}}\).