## 4 Proposed Scheme
After the FFT and CP removal, the digital received symbol in the frequency domain can be approximately expressed with a matrix form as  
\[\mathbf{Y}_{MB-n}=\mathcal{H}\mathbf{C}_{MB-n}+\mathcal{H}\mathbf{C}_{clip}+\mathbf{ W}, \tag{24}\]  
where \(\mathcal{H}\) is the frequency domain channel response, \(\mathbf{C}_{clip}\) is the \(c_{clip}(n)\) in the frequency domain and \(\mathbf{W}\) is channel noise. In what follows, the perfectly known channel response and the perfect synchronization in the demodulation process are assumed for every received frame. After zero-forcing equalization, the equalized symbol can be expressed as  
\[\mathbf{Y}_{eq-MB-n}=\mathbf{C}_{MB-n}+\mathbf{C}_{clip}(n)+\mathcal{H}^{-1}\mathbf{W}. \tag{25}\]  
With the Hermitian symmetric data of \(\mathbf{Y}_{eq-MB-n}\) removed, the new received symbol \(\mathbf{\hat{\mathbf{C}}}_{MB}\) is generated. Then, \(\mathbf{\hat{\mathbf{C}}}_{MB}\) is fed into the MB-Hadamard decoding module. A diagram of the proposed decoding operation is depicted in Figure 3, where the demodulation procedure is the reverse of those used in (19)-(21). After the sub-band demapping is completed, \(\mathbf{\hat{S}}_{k^{\prime}}^{l}\) is transformed by the inverse Hadamard matrix \(\mathbf{H}_{M}^{-1}\) and the final data symbol \(\mathbf{\hat{X}}_{m}^{l}\) is obtained. Then, a maximum likelihood (ML) estimator is adopted as  
\[\mathbf{\hat{\mathbf{X}}}_{F}=\arg\min_{ \begin{subarray}{c}\mathcal{X}_{i}\\ i=1,2,\cdots,\mathcal{I}\end{subarray} }\Big{|}\alpha^{-1}\mathbf{\hat{\mathbf{X}}}- \mathcal{X}_{i}\Big{|}, \tag{26}\]  
where \(\mathcal{X}_{i}\) is constellation point of QAM and \(\mathcal{I}\) is the constellation size. When bit-loading with the desired BER \(P_{\varepsilon}\) is employed, the SNR for each split band can be represented by  
\[SNR^{l}=M\Bigg{[}\sum_{m=0}^{M-1}\Bigg{(}\frac{\sigma^{2}\Gamma}{\left|\mathcal{ H}_{m}^{l}\right|^{2}P_{m}^{l}}\Bigg{)}\Bigg{]}^{-1}, \tag{27}\]  
where \(\sigma^{2}=\sigma_{total}^{2}/N\) is the average noise power per subcarrier, \(\Gamma\) is the modulation gap, \(\mathcal{H}_{m}^{l}\) and \(P_{m}^{l}\) denote the channel response and the allocated power at the \((m+1)\)th subcarrier in the \((l+1)\)th band, respectively. As we known, the clipping will introduce the offensive impulse noise among in-band and out-band. Fortunately, as seen in Equation (27), the system noise is spread averagely in each sub-band and the corresponding SNR gains will be achieved by the MB-Hadamard decoding, which is favorable for symbol estimation. When a uniform power allocation is adopted for each sub-band and the total used power is \(P_{t}\), then \(P_{m}^{l}\) can be calculated as \(P_{m}^{l}=P_{t}/M\). Therefore, for a frequency-selective channel such as a POF, the SNR of the entire transmission band can be written as  
\[\mathbf{SNR}_{MB}=\Big{[}SNR^{0},SNR^{1},\cdots,SNR^{L-1}\Big{]}^{T}. \tag{28}\]  
As shown in Equations (27) and (28), the \(SNR^{l}\) values within the same sub-band are equal, whereas those of adjacent sub-bands are different. Under these circumstances, it is appropriate to apply the bit-loading technique to each sub-band, which represents the primary difference between the proposed scheme and the DFT-S-DMT with respect to bit loading. Finally, the BER of the MB-Hadamard decoding in the \((l+1)\)th band can be approximately calculated as  
\[BER^{l}=\frac{2\Big{(}\sqrt{\mathcal{I}}-1\Big{)}}{\sqrt{\mathcal{I}}\log_{2}( \mathcal{I})}erfc\Bigg{(}\sqrt{\frac{3\log_{2}(\mathcal{I})M}{\mathcal{I}-1} \Bigg{[}\sum_{m=0}^{M-1}\Bigg{(}\frac{\sigma^{2}\Gamma}{\left|\mathcal{H}_{m}^ {l}\right|^{2}P_{m}^{l}}\Bigg{)}\Bigg{]}^{-1}}\Bigg{)}. \tag{29}\]