# Large-Language-Model-Enabled Text Semantic Communication Systems
## 3 System Model and Mechanism
### Problem Description
For a specific received sequence, \(P(o_{1},o_{2},\cdots,o_{T})\) is a fixed value and cannot be optimized. Therefore, the final optimization objective is set by Equation (8). \[\underset{(s_{i}\in\mathbb{S})}{\arg\max}\,P(s_{1},s_{2},\cdots,s_{t}\mid o_{1 },o_{2},\cdots o_{t})=\quad\underset{(s_{i}\in\mathbb{S})}{\arg\max}\,P(s_{1},s_{2},\cdots,s_{t})\prod_{i=1}^{t}P(o_{i}\mid s_{i}) \tag{8}\]  
The problem now revolves around identifying a transmission system where the receiver can access both \(P(s_{1},s_{2},\ldots,s_{t})\) and \(P(o_{i}\mid s_{i})\). The length of \(t\) determines the coding rate. If such a system exists, it would be optimal in view of the current coding rate. The two components of this requirement will be analyzed separately in the subsequent sections. First, \(P(o_{i}\mid s_{i})\) represents the probability distribution of the received signal at the receiver through the channel. For an AWGN channel, this distribution is given by Equation (9):  
\[P(o_{i}\mid s_{i})=\frac{1}{\sqrt{2\pi}\sigma}e^{\left(-\frac{(s_{i}-o_{i})^{ 2}}{2\sigma^{2}}\right)}, \tag{9}\]  
where \(\sigma^{2}\) denotes the noise power. Considering fading channels beyond AWGN channels, if \(h_{i}\) can be accurately estimated at time \(i\), then based on Equation (3), \(P(o_{i}\mid h_{i}\otimes s_{i})\) follows a normal distribution, as expressed in Equation (10):  
\[P(o_{i}\mid h_{i}\otimes s_{i})=\frac{1}{\sqrt{2\pi}\sigma}e^{\left(-\frac{( s_{i}-s_{i}\otimes\mathbb{h})^{2}}{2\sigma^{2}}\right)}, \tag{10}\]  
When \(h_{i}\) remains constant, the fading channel effectively behaves like an AWGN channel. The '\(\otimes\)' in Equation (10) corresponds to channel equalization in classical technical communication systems, where \(s_{i}\) represents the constellation sequences corresponding to the transmitted semantic symbol. In digital communication, the estimated \(h_{i}\) is utilized to equalize \(o_{i}\) by multiplying the conjugate transposed matrix of \(o_{i}\) and \(h_{i}\). However, we use \(h_{i}\) to convolve \(s_{i}\) to the advantage of the design of our subsequent system. Thus, our objective transforms into identifying a function \(\varphi\) that maps the sequence \(X\) to the sequence \(S\), while accurately modeling the probability of \(S\). This probability model should be consistent and computable at both transmitter and receiver ends, where the length of \(S\) reflects the coding rate at the transmitter. In technical communication systems, determining the probability distribution of \(S\) is challenging, often assuming an equiprobable transmission of information bits. LLMs process text into numerical token sequences and predict the probabilities of subsequent tokens in natural language, closely resembling our problem. These models can effectively model \(P(S)\). In subsequent sections, we detail the use of LLMs for the probabilistic modeling of text. Assuming that \(P(S)\) is accessible, we continue discussing the function \(\varphi\). The goal of \(\varphi\) is to minimize the length \(t\) of \(S\), thereby maximizing data transmission rates, ensuring that \(P(s_{1},s_{2},\cdots,s_{t})\) reaches its maximum:  
\[\hat{X}=\varphi^{-1}(\hat{S}), \tag{2}\]0  
\[\hat{X}=\varphi^{-1}(\hat{S}), \tag{2}\]1  
When transmitted symbols are independent and equiprobable, the mapping process from \(X\) to \(S\) approaches the entropy limit in compression. This scenario mirrors source coding achieving the entropy limit without channel coding in technical communication [2]. The optimal decoding strategy involves comparing the Euclidean distance between the received signal and the transmission constellation. Bit errors are irreparable without prior information; once an error occurs at the receiver, the information becomes unrecoverable. However, technical communication systems typically exhibit some correlations among elements within \(S\), enabling the receiver to correct erroneous symbols using prior knowledge. This correlation is analogous to the use of parity bits in channel coding within technical communication systems.