## 5 Results and Discussion
### PAPR Reduction  
To assess the performance of the MB-Hadamard precoding in reducing the PAPR, the \(N=\)512 value is adopted from Table 1, and the signal bandwidth is calculated to be approximately 250 MHz. In our simulations, 4000 DMT symbols are generated randomly. Figure 5 shows the signal amplitudes of our proposed scheme for various sub-band numbers \(L\) as well as the signal amplitude of C-DMT. As shown in the figure, the signal amplitude is reduced significantly by the proposed MB-Hadamard precoding for the cases of \(L=1\), \(L=4\) and \(L=8\). In addition, the energy distribution becomes more uniform, and most of the large amplitudes are compressed below the clipping threshold, which indicates that the signal distortion in the following clipping could be mitigated to some extent.  
Figure 6 illustrates the CCDF performance of the proposed scheme. As shown in the figure, the C-DMT scheme achieves the highest PAPR; however, the smaller the band split (e.g., \(L~{}=~{}1\)), the lower the PAPR obtained by the proposed. When the CCDF = \(10^{-3}\), the PAPR of the C-DMT is 14.1 dB, whereas those of the proposed are 10.15 dB and 11.53 dB for \(L=1\) and \(L=4\), respectively,  
\begin{table}
\begin{tabular}{c c} \hline \hline
**DMT Parameter** & **Value** \\ \hline Sampling frequency (GHz) & 1.0 \\ IFFT/FFT length & 2048 \\ Subcarrier spacing (MHz) & 0.488 \\ Subcarrier number N & 1–1024 \\ Cyclic prefix ratio & 1/64 \\ Sub-band number L & 1, 4, 8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Modulation parameters.  
Figure 5: Waveforms of proposed scheme with various numbers of split bands: (**a**) \(L=1\) (**b**) \(L=4\) (**c**) \(L=8\).  
resulting in respective 3.95 dB and 2.57 dB performance improvements. In addition, the figure also indicates that the CCDF performance of the proposed scheme with \(L=8\) improves little because an insufficient volume of reliable data exists in the split band, which affects the correlations of the input symbols. The CCDF of a well-known PTS in which the parameters \(U\), \(V\) and \(W\) denote the numbers of corresponding modified candidate sequences, disjoint sub-vectors and phase angles, respectively, is also demonstrated. The PAPR performance of the PTS exceeds that of C-DMT but is inferior to those of the proposed scheme for both \(L=1\) and \(L=4\).  
The calculated \(\mathbb{E}\Big{\{}\left|\rho_{\mathbf{C}}(q)\right|^{2}\Big{\}}\) values with respect to both the C-DMT and the proposed scheme under various values of \(L\) are shown in Figure 7a, which illustrates that these curves move gradually away from the C-DMT as \(L\) decreases. As seen in the figure, the corresponding relationship \(\mathbb{E}\big{\{}\xi_{\mathbf{C}_{MB-n}}\big{\}}<\mathbb{E}\big{\{}\xi_{ \mathbf{C}}\big{\}}\) can easily be obtained based on the definition of \(\rho_{\mathbf{C}}(q)\) give in Equation (6). Taking \(N=512\) as an example, the calculated probability density function (PDF), in terms of both \(\xi_{\mathbf{C}}\) and \(\xi_{\mathbf{C}_{MB-n}}\), is shown in Figure 7b. The figure reveals that the curves have similar shapes and are approximately Gaussian distributed. Compared to \(\xi_{\mathbf{C}}\), \(\xi_{\mathbf{C}_{MB-n}}\) and \(\mathbb{E}\big{\{}\xi_{\mathbf{C}_{MB-n}}\big{\}}\) are decreased, which implies that the MB-Hadamard precoding reduces the PAPR effectively. In addition, as discussed in Section 4, the establishment of inequalities Equations (22) and (23) are also verified by the results shown in Figure 7.