# Probabilistically-Shaped DMT for IM-DD Systems with Low-Complexity Fast WHT-Based PDSP
## 2 Operation Principle
### Fast WHT-Based PDSP Scheme  
Ideally, only one probability distribution is required for the PS-DMT transmission system with a small range of SC SNR variations. However, the SC SNR may be largely fluctuated due to the imperfect frequency response of the optical/electrical devices, as illustrated in Figure 2a. In the conventional FDSP case, all of the data-carrying SCs are  
Figure 1: Schematic diagram of the PS-enabled DMT system.  
regarded as one group and used for performing precoding (see Figure 2b,c). As mentioned above, the number of data-carrying SCs (\(F\)) is usually not an integer power of two due to the HS constraint. Thus, the hardware implementation of the FDSP is a big challenge when a large FFT size is applied for DMT modulation/demodulation. For the proposed PDSP scheme, we divide \(F\) data-carrying SCs into two groups: \(P\) data-carrying SCs in group 1, where \(P\) is a number of an integer power of two, on both sides are selected for precoding, while no precoding is performed for the left \(F\)-\(P\) data-carrying SCs in group 2, as shown in Figure 2d,e. Note that the \(P\) data-carrying SCs selected from the successive data-carrying SCs in high/low-frequency whose SNRs are lower than average SNR and successive low-frequency data-carrying SCs with highest SNRs. In this case, the implementation complexity can be further reduced with the PDSP scheme in the PS-DMT transmission system.  
Unlike other PDSP schemes with a complex-valued precoded matrix (e.g., DFT and OCT), the real-valued WHT-based PDSP technique has lower computational complexity. According to [16], the normalized Hadamard matrix of order \(P\) can be given by  
\[D_{P}=\frac{1}{\sqrt{p}}\begin{bmatrix}D_{P/2}&D_{P/2}\\ D_{P/2}&-D_{P/2}\end{bmatrix},D_{2}=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\ 1&-1\end{bmatrix} \tag{4}\]  
where \(D_{2}\) is the normalized Hadamard matrix of order 2, and \(P\) must be either 1, 2, or an integer multiple of 4. \(D_{P}\) can be constructed from \(D_{2}\). In the PDSP scheme, \(P\) is an integer number of two, the inverse matrix \(D_{P}{}^{-1}\) equals to \(D_{P}\).  
Once the PS is done, the shaped QAM symbols can be expressed as \(\mathbf{X}_{F}=[X_{1},\,X_{2},\,\ldots,\,X_{F}]^{T}\) and \(\mathbf{X}_{P}=[X_{1},\,X_{2},\,\ldots,\,X_{P}]^{T}\), where \(F>P\) and \(T\) denotes the transpose operation. Thus, by multiplying the fast WHT-based precoding matrix \(D_{P}\), the PDSP precoded symbols \(Y_{P}=[Y_{1},\,Y_{2},\,\ldots,\,Y_{P}]^{T}\) can be given by  
\[\mathbf{Y}_{P}=D_{P}\mathbf{X}_{P} \tag{5}\]  
After fiber channel transmission, without considering nonlinear distortions, the recovered PDSP symbols \(\mathbf{R}_{P}=[R_{1},\,R_{2},\,\ldots,\,R_{P}]\) after FFT operation can be given by  
\[\mathbf{R}_{P}=HY_{P}+W=HD_{P}\mathbf{X}_{P}+W \tag{6}\]  
Figure 2: Schematic diagram of the PDSP scheme in the PS-enabled DMT system. (**a**) original SCs SNR, (**b**) SCs selected for FDSP, (**c**) SNR balance for FDSP, (**d**) SCs selected for PDSP and (**e**) SNR balance for PDSP.  
The channel transfer matrix, \(H\), is a diagonal matrix that can be expressed as diag (\(h_{1},h_{2},\ldots,h_{P}\)), and the frequency response of the \(k\)-th SC is \(h_{k}\). Similarly, the noise on the \(k\)-th SC is denoted by \(W_{k}\), which obeys Gaussian distribution with variance and zero mean. The corresponding noise vector in the frequency domain is expressed as \(W=[W_{1},W_{2},\ldots,W_{P}]\). Assuming that the accurate channel equalization is performed, the PDSP-decoded QAM symbols \(\mathbf{Z}_{P}=[Z_{1},Z_{2},\ldots,Z_{P}]\) can be written as  
\[\mathbf{Z}_{P}=D_{p}^{-1}H^{-1}\mathbf{R}_{P}=\mathbf{X}_{P}+D_{p}^{-1}H^{-1}W=\mathbf{X}_{P}+D _{p}H^{-1}W \tag{7}\]  
After performing PDSP decoding, these data-carrying SCs can be treated equally and only one kind of CCDM and one symbol modulation scheme are required for the implementation of the PS-DMT transmission system.