## 4 Data Communication Scheme Derivation
### General Comments  
The four data communication schemes behave differently. Moreover, a valuable illustration is to show the time-frequency domain divisions provided by these schemes when a \(2N\)-length symbol block is considered. In this sense, Figure 5 shows the geometric figures for the time-frequency domain occupation for subcarriers (HS-OFDM), subchirps (OCDM), subslots (SCCP), and subtiles (OTFDM) when \(2N=8\). We can see that these geometric figures are different and, consequently, the four data communication schemes attain different performances in general. Moreover, the previous subsections confirm that the generic transmitter and receiver let us see the data communication schemes mentioned above from a unique formulation. Considering interference-free condition (i.e., \(L_{\text{CP}}\geq L_{h}-1\), \(\Delta=0\)), Table 1 shows that nSNR expressions (26), (32), and (38) are very similar to each other since all of them are harmonic means of (27). Moreover, the interference-free condition combined with the flatness of the CFR magnitude and a constant PSD of the additive noise results in the same value as the SNR for the four schemes. We can also see that the SNR for the OTFDM scheme (i.e., (38)) is the harmonic mean of consecutive \(\beta_{k}\) subbands, meaning that (38) will be the same land subtiles that have the same central frequency \(\
u[k]\), or that are in the same voice. Moreover, an OTFDM scheme using \(\mathcal{B}\)-geometry was exploited in [21] to reduce the computational complexity of the OA techniques applied to the OTFDM scheme. Furthermore, a comparison between the nSNR of the four data communication schemes, see Table 1, shows that each subcarrier occupies only a single subband with an nSNR that can be different from other subbands since the CFR of the PLC channel and the PSD of the additive noise may not be flat. Meanwhile, each subchirp of the OCDM scheme occupies a constant bandwidth that linearly shifts in frequency as time evolves. Consequently, the nSNR associated with the OCDM scheme can be interpreted as the harmonic mean of the nSNRs of all subbands. The nSNR for the SCCP scheme is also a harmonic mean of the nSNRs in all subbands since a subslot occupies the whole frequency band. Regarding the OTFDM scheme, we can see that each subtile can occupy pieces of subbands within a given group of subslots. Therefore, the nSNR in a subtile is the harmonic mean of the nSNRs of subbands that constitute a subtile. Figure 5: Spectrogram of the four data communication schemes. (**a**) HS-OFDM; (**b**) OCDM; (**c**) SCCP; (**d**) OTFDM. Finally, in regard to the computational complexity of each multiplexing modulation scheme, Table 2 summarizes the numbers of basic operations (multiplication and sums) for all transmitters and receivers considered in this paper while considering the fast version of each discrete transform. The HS-OFDM, OCDM, and SCCP computational complexities are found in [16] while OTFDM's was derived based on [23]. In this context, the frequency domain equalization number of operators is the only distinction between the transmitter and receiver of HS-OFDM, OCDM, and OTFDM, while for the SCCP scheme, they are completely different. Since we are using three different types of equalizers, we define \(\mathcal{E}_{\times}\) and \(\mathcal{E}_{+}\) as the multiplication and sum numbers required by the equalizer, respectively, for a generic perspective between the modulation techniques. HS-OFDM and SCCP have the same computational complexity when considering the transmitter and receiver; however, all operations of the SCCP scheme are on the receiver side. Next, the OCDM scheme has more computational complexity than the other two mentioned above. Finally, the OTFDM scheme's complexity will vary with its \(\mathcal{B}\)-geometry but will always be lower than the OCDM scheme and higher than the HS-OFDM scheme, as shown in Figure 6.