## 5 Numerical Results  
This section presents numerical results for the fair comparison of the eOFDM scheme with the CP-OFDM and TDS-OFDM schemes and describes the development of a simulation model. The spectral efficiency, energy efficiency, BER and capacities of the systems were compared. The symbol energy, block size \((N)\), length of the guard sequence PN/CP, and multipath channel impulse response (CIR) for one block remained the same for all schemes. The numerical results for \(\eta_{s}\) and \(\eta_{e}\) are shown in Table 2. It is worth mentioning that eOFDM is more spectrally efficient than CP-OFDM for \((\mathbf{x}_{i}=\mathbf{x}_{j})\) and equals TDS-OFDM for \((\mathbf{x}_{i}\
eq\mathbf{x}_{j})\). Based on the \(\eta_{e}\) comparison, it is concluded that eOFDM is as energy efficient as TDS-OFDM for \((\mathbf{x}_{i}=\mathbf{x}_{j})\), while it outperforms both TDS-OFDM and CP-OFDM for \((\mathbf{x}_{i}\
eq\mathbf{x}_{j})\). Table 3 provides the simulation parameters for the fair comparison of eOFDM. It is worth mentioning here that the proposed system model did not impose any restrictions on the simulation parameters. Generally, any higher values of these parameters, such as \(N\), are supported. These parameter values were merely selected to avoid any unnecessary complexity and brevity of the proposed model. The simulation model was developed according to the proposed system model, as shown in Figure 1. The uniformly distributed binary source represents the data source, which output binary 1s and 0s. This sequence was then modulated using the BPSK modulation scheme. A vector of modulated symbols was formed for the IFFT operation. The PN sequence was selected for insertion as the guard interval. This selection was based on a vector of every \(J\)-th symbol from \(B_{g}\). PN sequence generation and selection is a unique block that was included in the eOFDM system model. A set of PN sequences was formed using a conventional PN sequence generator. Then, a subset, which fulfilled the condition stated in (6), was selected. The PN sequences in this subset were used for insertion in OFDM symbols. Figure 5 compares the BER results obtained for BPSK modulation in an additive white Gaussian noise (AWGN) channel. It can be observed that eOFDM performs better than CP/TDS OFDM for the given range of SNR. This SNR gain is the result of the high probability of detection of \(\mathbf{x}_{d}\) or \(G\) data symbols in the block, as mentioned in [36]. The SNR gain for the given BER is more prominent in Figure 6. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **CP-OFDM** & **TDS-OFDM** & **eOFDM** & **eOFDM** \\  & & & \((\mathbf{x}_{i}\
eq\mathbf{x}_{j})\) & \((\mathbf{x}_{i}=\mathbf{x}_{j})\) \\ \hline \(\eta_{s}\) & 60\% & 80\% & 80\% & 66.67\% \\ \(\eta_{e}\) & 65.23\% & 66.67\% & 80\% & 66.67\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of \(\eta_{s}\) and \(\eta_{e}\) for \(m=N/4\). These results were obtained for the ITU pedestrian B channel model [38]. The variations of channel realization for each OFDM block were considered according to the average path delays (seconds) and path gain (dB) vectors, as mentioned in the Table 3. The channel impulse responses with \(L=6\) were generated for a 50 Hz Doppler frequency (\(f_{D}\)). Notably, the eOFDM scheme shows an approximately 3 dB SNR gain for the BER \(10^{-1}\). The SNR gain of eOFDM can be observed for any value of BER, which is high for a high BER and has a decreasing trend for lower BER values. These results validate that, based on a fair comparison, the eOFDM scheme outperforms the CP-OFDM and TDS-OFDM schemes for the given range of SNR in a frequency-selective Rayleigh fading channel. The system capacity of the eOFDM scheme was compared with CP-OFDM and TDS-OFDM, as depicted in Figure 7. These graphs are plotted according to Equation (10). It is worth mentioning that the hard decision decoding and lower modulation order made it feasible to represent system capacity in bits/seconds/Hz (b/s/Hz) for eOFDM, CP-OFDM and TDS-OFDM.