## 2 Principle  
### PS Based on Single-Layer LUT  
We propose the LUT-PS scheme that employs a single-layer LUT for symbol mapping in both QAM modulation and demodulation. This scheme shapes the probability distribution into a quasi-Gaussian form. The schematic diagram of the proposed LUT-PS is shown in Figure 1. The operation principle of the LUT-PS involves mapping a uniform pseudo-random binary sequence (PRBS) tx_bits according to a specific regulation, using a fixed LUT and a flexible shaping proportion \(\mu\), which describes the degree of non-uniformity in the symbol probability distribution. A flag signal is generated to record the mapping state. After mapping, the data are modulated into 16 QAM symbols, and this step is followed by OFDM modulation. During PS decoding, the flag generated in the encoding process serves as a reference to allow accurate recovery of the original data using the inverse LUT mapping. This work is based entirely on a simulation framework and was conducted with the aim of evaluating the performance gain of the proposed LUT-PS scheme. In the simulation, the flag information is directly shared between the transmitter and receiver without being transmitted over the channel. This allows us to focus on assessing the performance of the probabilistic shaping itself without the results being affected by additional channel impairments. In practical communication systems, the flag can be transmitted by embedding it into the frame structure or through joint encoding with the data, which will be an important direction for our future work. Figure 1: Schematic diagram of LUT-PS. Taking the 16 QAM signal as an example, the contents of the LUT are shown in Table 1 and include the input, output, and symbol energy. Each set of 4 bits is treated as the input to the LUT, resulting in 16 possible combinations. The energy for each input is calculated and arranged in ascending order, while the corresponding outputs are arranged in descending order of energy. The mapping process is governed by a comparison between the input energy \(E[|X|^{2}]\) and the output energy \(E[|Y|^{2}]\). If \(E[|X|^{2}]{\leq}E[|Y|^{2}]\), the input symbol remains unchanged. If \(E[|X|^{2}]>E[|Y|^{2}]\), the input is mapped to the corresponding output value based on the lookup table (LUT). This approach effectively maps high-energy constellation points to low-energy ones, thereby reducing the average power of the signal. However, relying solely on the LUT causes all outer constellation points to be mapped onto the inner ones, resulting in a limited and inflexible shaping effect. To overcome this limitation, we introduce a shaping ratio parameter \(\mu(\mu=1/i)\), which controls the proportion of symbols subjected to shaping. Specifically, it indicates that one out of every \(i\) symbols will remain unshaped. A smaller value of \(\mu\) corresponds to a greater shaping depth, resulting in constellation points being more concentrated in the inner rings. This reduces the probability of symbol errors and enhances overall system performance. In this work, the value of the shaping parameter \(\mu\) is selected based on simulation results. Specifically, we manually adjust the value of \(\mu\) to observe its impact on system-performance metrics such as BER. As shown in Table 2, a smaller \(\mu\) generally leads to a lower BER, although the relationship is not strictly monotonic. A possible explanation is that when \(\mu\) becomes too small, the symbol probability distribution becomes highly skewed--some symbols are transmitted far more frequently, while others are rarely used. This imbalance makes the receiver more sensitive to noise affecting the rarely occurring symbols, thereby resulting in an increase in BER. However, the parameter \(\mu\) is essentially designed to flexibly adjust the shaping depth rather than serve as a direct performance-optimization variable. This flexibility offers the potential to adapt to different channel conditions or system requirements, which may be valuable in practical implementations.