## 2 Principles  
The block diagram of the probabilistic shaping bit loading scheme is shown in Figure 1. First of all, channel estimation is done by sending a quadrature phase-shift keying (QPSK) DMT signal as the training sequence. DMT is a common form of multicarrier modulation. The principle of DMT modulation can refer to [19]. In the experiment, we assume the transmission link between transmitter and receiver is relatively stable and the change of distance will only influence the signal-to-noise ratio (SNR) of the received signal. So, the impact of the distance link will not be considered in detail. SNR for each DMT subcarrier can be calculated through the error vector magnitude (EVM) of the received signal [20] as  
\[SNR=EVM_{\text{RMS}}^{2}, \tag{1}\]  
the \(EVM_{\text{RMS}}\) is defined as  
\[EVM_{\text{RMS}}=\left[\frac{\frac{1}{T}\sum_{t=1}^{T}\left|I_{t}-I_{0,t} \right|^{2}+\left|Q_{t}-Q_{0,t}\right|^{2}}{\frac{1}{N}\sum_{n=1}^{N}\left[ \left(I_{0,n}\right)^{2}+\left(Q_{0,n}\right)^{2}\right]}\right]^{\frac{1}{2}}, \tag{2}\]  
where \(I_{t}\), \(Q_{t}\) denote the in-phase (I) and quadrature (Q) components for the \(t\)-th received symbol, \(I_{0,t}\), \(Q_{0,t}\) denote the I and Q components for the \(t\)-th transmitted symbol, \(T\) is the total number of the received symbols, and \(N\) is the number of the unique symbols. Then, the PS bit loading signal is generated according to the SNR. The source entropy of each subcarrier is set to the Shannon capacity as  
\[H=\log_{2}(1+SNR), \tag{3}\]  
For quadrature amplitude modulation (QAM) constellation, the Gaussian-like constellation satisfies the Maxwell-Boltzmann (MB) distribution [17] and the MB distribution for PS M-QAM is expressed as  
\[P_{X}(x_{i})=e^{-v|x_{i}|^{2}}/\sum\
olimits_{j=0}^{M-1}e^{-v|x_{j}|^{2}}, \tag{4}\]  
where \(v\) is adjusted to match the source entropy and the set \(\{x_{0},x_{1},\ldots,x_{M-1}\}\) contains the complex value of the M-QAM constellation. In this paper, the Constant Composition Distribution Matching (CCDM) algorithm [21] is utilized to realize probabilistic constellation shaping, through which the random input bits can be mapped to a probabilistically-shaped output constellation with specific source entropy. And the famous probabilistic amplitude shaping (PAS) scheme with MB distribution is employed to realize the PS system. The PAS scheme can be referred to [22]. Then the FEC encoder [23] is followed by the CCDM and it should be noted that the FEC encoder will not change the probability of the symbol. Theoretically, the PS M-QAM constellation of a subcarrier is generated by employing a CCDM separately. So, the complexity of the proposed modulation format is relatively high considering the implementation of a large amount of CCDM for the overall subcarriers. However, in practical application, the complexity can be reduced by sharing the CCDM [24]. Through adaptively partitioning and precoding the subcarriers, the SNRs of several subcarriers are averaged and the source entropy loaded on these subcarriers can be the same, which means that the use of a CCDM is able to finish the data generation for several subcarriers. Besides, with the rapid development of chip integration and computing power, the complexity of the proposed modulation format will not restrict its realization. In the PS system, GMI and NGMI are often used to measure the system performance for convenience [25]. And NGMI has been proved to be suitable for the prediction of the bit error rate of the forward error correction (FEC) after soft decision bit-metric decoding [26]. The GMI can be calculated as  
\[GMI\approx\frac{1}{N}{\sum\
olimits_{i=1}^{N}{[-P_{X}(x_{i})\log_{2}P_{X}(x_{i} )]}}-\frac{1}{N}{\sum\
olimits_{i=1}^{N}{\sum\
olimits_{j=1}^{m}{\log_{2}(1+ e^{(-1)^{h_{ij}}\Lambda_{ij})}}}}, \tag{5}\]  
Figure 1: Block diagram of probabilistic shaping bit loading scheme. where \(b_{i,j}\) are the transmitting bits of \(i\)-th symbol and \(m=\log_{2}M\)[25].