提出首个语义通信性能与信噪比的理论公式，可指导系统优化。

Due to the black-box characteristics of deep learning based semantic encoders and decoders, finding a tractable method for the performance analysis of semantic communications is a challenging problem. In this paper, we propose an Alpha-Beta-Gamma (ABG) formula to model the relationship between the end-to-end measurement and SNR, which can be applied for both image reconstruction tasks and inference tasks. Specifically, for image reconstruction tasks, the proposed ABG formula can well fit the commonly used DL networks, such as SCUNet, and Vision Transformer, for semantic encoding with the multi scale-structural similarity index measure (MS-SSIM) measurement. Furthermore, we find that the upper bound of the MS-SSIM depends on the number of quantized output bits of semantic encoders, and we also propose a closed-form expression to fit the relationship between the MS-SSIM and quantized output bits. To the best of our knowledge, this is the first theoretical expression between end-to-end performance metrics and SNR for semantic communications. Based on the proposed ABG formula, we investigate an adaptive power control scheme for semantic communications over random fading channels, which can effectively guarantee quality of service (QoS) for semantic communications, and then design the optimal power allocation scheme to maximize the energy efficiency of the semantic communication system. Furthermore, by exploiting the bisection algorithm, we develop the power allocation scheme to maximize the minimum QoS of multiple users for OFDMA downlink semantic communication Extensive simulations verify the effectiveness and superiority of the proposed ABG formula and power allocation schemes.