## 4 Proposed Scheme  
A block diagram of the proposed scheme is shown in Figure 2. Compared with the conventional DMT (C-DMT), two additional modules, extra MB-Hadamard precoding and clipping, are implemented at the transmitter. Hadamard is a class of Jacket matrix [22; 23; 24] which are simple to calculate and easily element-wise inverse. Assuming \(\mathbf{H}_{M}=\left(h_{m,l}\right)_{M}\) signifies a unit \(M\times M\) Hadamard matrix, the inverse Hadamard matrix \(\mathbf{H}_{M}^{-1}\) can be calculated directly using  
\[\mathbf{H}_{M}^{-1}=\mathbf{H}_{M}^{T}. \tag{17}\]  
Figure 1: Distribution of the effective signal-to-noise ratio (SNR) for different clipping ratio and bits resolution. Figure 2: Diagram of the proposed scheme at transmitter. For example,  
\[\mathbf{H}_{[1]}=\frac{1}{\sqrt{2}}\left[ \begin{array}{cc}1&1\\ 1&-1\end{array} \right], \tag{18}\]  
where \([\kappa\;]=2^{\kappa}\) denotes the matrix order. Note that the Hadamard matrix \(\mathbf{H}_{[\kappa]}\) with an order of the power-of-2 is adopted in this paper. Assuming that there are \(N=M\cdot L\) sub-carriers, \(X_{m}^{l}(l=0,\cdots,\;L-1,\;m=0,\cdots,\;M-1)\) is the \((m+1)\)th data symbol in the \((l+1)\)th band. The original symbol \(X_{m}^{l}\) is spread by an \(M\times M\) Hadamard matrix firstly, and then a new set symbols is generated with  
\[S_{k^{\prime}}^{l}=\sum_{m=0}^{M-1}X_{m}^{l}h_{m,k^{\prime}},k^{\prime}=0, \cdots,M-1, \tag{19}\]  
where \(h_{m,k^{\prime}}\) is an element of \(\mathbf{H}_{M}\) and \(S_{k^{\prime}}^{l}\) are sub-band mapped onto the \(N\)-point symbol vector, expressed by  
\[\mathbf{C}_{MB}=\left\{\underbrace{C^{0}(0),\cdots}_{1-th}\underbrace{C^{1}( M),\cdots}_{2-th},\cdots,\underbrace{C^{L-1}((L-1)M),\cdots}_{L-th}\right\}, \tag{20}\]  
where \(C^{l}(k)\in\mathbf{C}_{MB}\) are the modulated symbols according to the sub-band constellation mapping, expressed as  
\[C^{l}(k)=\left\{ \begin{array}{ll}S_{k^{\prime}}^{l},&k=k^{\prime}+Ml\\ 0,&\text{others}\end{array} \right.,k=0,\cdots,N-1. \tag{21}\]  
In addition, the vector \(\mathbf{C}_{MB-n}=\left[\left(\mathbf{C}_{MB}\right)^{T},\left(\mathcal{Z}( \mathbf{C}_{MB})\right)^{T}\right]^{T}\) is obtained by performing Hermitian symmetry, and then fed into to the IFFT block to generate the time domain signals \(\mathbf{x}_{MB-n}\). By introducing both the ACF expression from Equation (6) and the ACF sum from Equation (7), we can arrive at  
\[\mathbb{E}\{\xi_{\mathbf{C}_{MB-n}}\}<\mathbb{E}\{\xi_{\mathbf{C}}\}. \tag{22}\]  
Note that Equation (22) will be verified by the numerical result which is demonstrated in the Section 5.1. Furthermore, \(\xi_{\mathbf{C}_{MB-n}}\) approaches an approximate Gaussian distribution as the value of \(N\) is large enough. The reduction in \(\mathbb{E}\{\xi_{\mathbf{C}_{MB-n}}\}\) implies a lower value of \(\xi_{\mathbf{C}_{MB-n}}\). Thus, we obtain  
\[\xi_{\mathbf{C}_{MB-n}}<\xi_{\mathbf{C}} \tag{23}\]  
According to Equation (7), there is a close relationship between \(\mathbf{Y}_{d}\) and the correlations of the input frequency-domain data-symbols. Therefore, the more \(\xi_{\mathbf{C}_{MB-n}}\) can be reduced, the lower the PAPR value and the higher the \(SNR_{eff}\) can be obtained by MB-Hadamard precoding. After that, the precoded DMT is added with cyclic prefix (CP) and then fed into the clipping module to further reduce the peaks. In addition, we can also perform the filtering procedure iteratively to both remove the out-of-band interference and suppress the regrowth of the peak power. The received optical signal is transformed into an electrical signal \(\mathbf{y}_{MB-n}\) by the PD detector. Note that the quantization noise is negligibly small when the bit resolution is greater than seven.