Outage Probability Analysis of Relay Communication Systems for Semantic Transmission

Abstract

This paper conducts an in-depth study on the outage probability performance of relay-based semantic communication systems and proposes a multi-mode intelligent relay design framework to address complex scenarios such as background knowledge differences, channel quality fluctuations, and computational limitations at the destination node. Based on a three-node two-hop communication model (source node–relay node–destination node) and integrating the DeepSC model, the study achieves cross-layer collaboration between semantic encoding/decoding and channel encoding/decoding. The proposed relay node operates in four innovative modes: semantic cooperative decode-and-forward, semantic adaptive forwarding, semantic-enhanced forwarding, and semantic-bit hybrid forwarding, each tailored to different levels of background knowledge matching, channel conditions, and computational constraints at the destination node. Through theoretical derivations, this paper presents the first closed-form expressions for the outage probability of the four relay modes, systematically quantifying the coupling effects of semantic symbol redundancy, background knowledge differences, and computational conversion efficiency on system reliability. The results show that semantic adaptive forwarding significantly reduces outage probability when background knowledge differences are minimal. When the destination node has limited computational power, the semantic-bit hybrid mode enhances communication reliability by flexibly adjusting the transmission strategy. Moreover, proper configuration of semantic symbol redundancy plays a crucial role in maintaining semantic information integrity and resisting channel interference. Monte Carlo simulations validate the theoretical analysis, demonstrating that the dynamic switching mechanism of the multi-mode relay outperforms single-mode strategies. This research provides theoretical support for reliable transmission and resource optimization in 6G semantic communication systems, uncovering the potential of joint optimization between semantic parameters and dynamic channel conditions. It holds significant implications for advancing future intelligent communication systems.

1. Introduction

In recent years, the application of semantic communication (SC) in wireless relay networks has received extensive attention. Numerous studies have been conducted on semantic information transmission, forwarding strategies, and optimized architectures to enhance the efficiency and reliability of communication.

Firstly, in the aspect of deep learning-driven semantic relay communication, Study Wang et al. [1] proposed a two-way relay network based on deep joint source-channel coding (DeepJSCC-TWRN), focusing on optimizing the image transmission quality. In addition, Erkantarcı et al. [2] investigated integrating advanced deep learning models such as ResNet-18, GoogLeNet, and enhanced GoogLeNet into the semantic forwarding (SF) framework in 6G networks to improve the performance of collaborative communication. Zhang et al. [3] further proposed a task-oriented relay strategy, performing re-encoding and forwarding (RF) based on a deep neural network (DNN) to optimize the performance of classification tasks in device-edge collaborative inference systems.

In terms of semantic forwarding optimization, Luo et al. [4] proposed a semantic communication scheme for wireless relay channels based on autoencoders (AESC) and designed a new semantic forwarding (SF) mode, which is particularly suitable for scenarios where there is a lack of common knowledge between the source node and the destination node. Tang et al. [5] further proposed an on-demand semantic forwarding framework to enhance the flexibility and adaptability of semantic information transmission. Arda et al. [6] studied collaborative semantic text communication and proposed two strategies, namely semantic lossy forwarding (SLF) and semantic predictive forwarding (SPF), to optimize the transmission performance under different channel conditions.

In terms of semantic relay architecture, Ma et al. [7] designed an intelligent relay-assisted semantic communication system, combining two modes of amplify-and-forward (AF) and decode-and-forward (DF). By using an intelligent semantic relay (Sem-Relay) to continuously share the knowledge background with the transmitter and receiver, the transmission efficiency is improved. Mu et al. [8] explored a new type of heterogeneous communication system that enables access points (APs) to simultaneously transmit semantic streams and bit streams to semantic users (S-users) and bit users (B-users). Yin et al. [9] studied semantic communication (SC) in multi-hop relay networks for long-distance or high path loss attenuation conditions to enhance the reliability of long-distance information transmission. In addition, Yan et al. [10] proposed the concept of semantic spectral efficiency (S-SE), providing a new measure for the transmission efficiency of semantic information.

In the development of semantic relay (SemRelay), Liu et al. [11] proposed a SemRelay architecture equipped with a semantic receiver, specifically designed to assist in text transmission from a resource-rich base station (BS) to resource-constrained mobile devices. Hu et al. [12] further expanded this architecture to support text transmission for multiple resource-constrained mobile devices. Zhao et al. [13] proposed an uplink SemRelay-assisted wireless communication system, using the probability graph shared between the SemRelay and the base station (BS) to provide efficient semantic communication services for multiple users. Hu et al. [14] regarded the SemRelay as an edge server, providing deep learning-based semantic communication (DeepSC) services for semantic users (SemUsers) with rich computing resources and traditional users (ConUsers) with limited resources.

In summary, these studies have promoted the application and development of semantic communication in wireless relay networks from multiple aspects such as deep learning methods, forwarding optimization, architecture design, and the evolution of SemRelay, laying the foundation for future efficient and intelligent semantic communication systems.

The DeepSC system Xie et al. [15] proposed a deep learning-based semantic communication framework, but it did not analyze the outage probability in relay cooperation and multi-user scenarios. Although L-DeepSC Xie and Qin [16] optimized lightweight semantic transmission, it is only applicable to direct connection links and does not consider the semantic conversion function of relay nodes. The SemRelay framework Hu et al. [14] introduced semantic relay for the first time, but it did not quantify the impact of background knowledge differences on the outage probability.

There are two key limitations in the current research in the field of semantic relay communication: Firstly, most of the existing work focuses on the optimization of semantic rate, but ignores the modeling of the outage probability, a key reliability indicator. In particular, it fails to establish a theoretical correlation between the outage probability and semantic parameters (such as the matching degree of background knowledge, the symbol compression rate K, the computing power of the destination node, etc.), which leads to a blind spot in the performance evaluation of the system in actual complex channel environments. Existing models usually assume that the destination node (D) has sufficient computing power to support complex semantic decoding. However, in practical scenarios, the limited computing resources of node D may lead to the failure of semantic information parsing, and the dynamic coupling relationship between computing power and semantic decoding ability has not been quantitatively analyzed. Secondly, when traditional relay strategies (such as AF/DF) are directly applied to semantic transmission scenarios, their design does not fully consider the impact of semantic decoding failure on system reliability. For example, when there is a mismatch in background knowledge or semantic symbols are damaged, the simple amplification or forwarding of traditional relays will significantly reduce the fidelity of semantic information, thereby increasing the risk of system outage. These limitations make it difficult for existing methods to meet the requirements of 6G communication for highly reliable semantic transmission, and there is an urgent need to establish an outage probability analysis framework that integrates semantic features and channel dynamics.

This paper focuses on the relay semantic communication system and conducts an in-depth study of its outage probability performance under different conditions. It innovatively proposes a multi-mode intelligent relay design framework. Through four working modes of a single relay node (semantic collaborative decoding-forwarding, semantic adaptation forwarding, semantic enhancement forwarding, semantic-bit hybrid forwarding), this framework achieves the collaborative optimization of the matching degree of background knowledge, channel quality, and the computing power of the destination node. Based on the three-point two-line communication model (source node-relay node-destination node), the study combines the DeepSC model to achieve cross-layer collaboration of semantic encoding and decoding and channel encoding and decoding at the relay node. For complex scenarios such as limited computing power of the destination node, mismatch of background knowledge, and channel degradation, the closed-form expressions of the outage probability of the four relay modes are derived for the first time, and the coupling influence mechanism of semantic symbol redundancy, background knowledge differences, and computing power conversion efficiency on system reliability is systematically quantified. Theoretical analysis shows that:

In scenarios with small background knowledge differences, the semantic adaptation forwarding mode can significantly reduce the system outage probability;

When the computing power of the destination node is limited, the semantic-bit hybrid mode can effectively improve communication reliability by flexibly switching transmission strategies;

The reasonable configuration of semantic symbol redundancy plays a key role in maintaining the integrity of semantic information and resisting channel interference.

Verified by Monte Carlo simulation, in dynamic channel conditions and heterogeneous background knowledge scenarios, the dynamic switching mechanism of the multi-mode relay shows significant performance advantages compared with a single strategy, and the outage probability in different scenarios is greatly reduced. This study provides theoretical support for the reliable transmission and resource optimization of 6G semantic communication systems, reveals the potential value of the collaborative optimization of semantic parameters and channel dynamics, and lays an important theoretical foundation for the design and performance evaluation of future intelligent communication systems.

The organization of this paper is as follows: The second part establishes the system model. The third part deals with the outage probability performance of the proposed relay selection strategy and gives the exact and high signal-to-noise ratio approximation. The fourth part presents the simulation and analysis results, and the fifth part summarizes the full text and gives our conclusions.

Notation: ${\mathbb{C}}^{m\times n}$ and ${\mathbb{R}}^{m\times n}$ denote the sets of complex matrices and real matrices of size $m\times n$, respectively. Boldface variables represent matrices or vectors. $x\sim \mathcal{CN}(\mu ,\mathrm{\Sigma })$ indicates that the variable x follows a circularly symmetric complex Gaussian distribution with mean $\mu$ and covariance $\mathrm{\Sigma }$.

2. System Model

We consider a three-point two-line communication transmission model, as shown in Figure 1, which includes the source node (S), the relay node (R), and the destination node (D), with no direct link. The relay operates in half-duplex mode and uses DeepSC as the communication transmission model. Semantic and channel encoders and decoders are deployed at both S and R, where the semantics of the text can be effectively extracted through the Transformer model. It is assumed that the DeepSC transceivers are trained at a base station or cloud platform.

In this model, S inputs a sentence $s=[w_1,w_2,\dots ,w_L]$, where $w_L$ represents the L-th word in the sentence. The DeepSC transmitter consists of two parts: the semantic encoder and the channel encoder, which are responsible for extracting semantic information from s and ensuring its successful transmission over the physical channel. The sentence is input into the DeepSC transmitter and mapped into a semantic symbol vector $X_n=[x_1,x_2,\dots ,x_{KL}]$, where $X_n\in {\mathbb{R}}^{KL\times 1}$, and $KL$ is the length of the semantic symbol vector after transforming the sentence. We note that the length of $X_n$ changes with L, allowing more effective extraction of semantic information from sentences of varying lengths. In this model, K represents the average number of semantic symbols used for each word, and each semantic symbol can be directly transmitted over the communication medium.

The encoded symbol stream can be expressed as:

(1)$x=C_{\alpha }\left(S_{\beta }\left(s\right)\right)$

where $x\in {\mathbb{C}}^{M\times 1}$, $S_{\beta }$ is the semantic encoder network with parameter set $\beta$, and $C_{\alpha }(·)$ is the channel encoder with parameter set $\alpha$.

At R, the received signal is:

(2)$y_R=h_{SR}{x}_S+n_{SR}$

where $y_R\in {\mathbb{C}}^{M\times 1}$, h represents the Rayleigh fading channel, which follows a $\mathcal{CN}(0,1)$ distribution, and $n\sim \mathcal{CN}(0,{\sigma }_n^2)$. The decoded signal can be expressed as:

(3)$\widehat{s}=S_{\chi }^{-1}\left(C_{\delta }^{-1}\left(y\right)\right)$

where $\widehat{s}$ is the recovered sentence, $C_{\delta }^{-1}(·)$ is the channel decoder with parameter set $\delta$, and $S_{\chi }^{-1}(·)$ is the semantic decoder network with parameter set $\chi$.

At D, depending on D’s computational power, either bit transmission or semantic transmission is employed. Specifically, if D has sufficient computational power to decode the semantic stream, semantic transmission is used over the $RD$ link. If D lacks sufficient computational power, traditional bit transmission is used.

Both the $SR$ (Source-Relay) link and $RD$ (Relay-Destination) link employ Rayleigh fading channel transmission models. In a Rayleigh fading channel, $|h_{SR}{|}^2$ follows an exponential distribution with parameter ${\displaystyle \frac{1}{{\sigma }^2}}$, that is:

(4)$f_{{\left|h\right|}^2}\left(z\right)={\displaystyle \frac{1}{{\sigma }^2}}{e}^{-{\displaystyle \frac{z}{{\sigma }^2}}},z\ge 0.$

For The signal-to-noise ratio of the Source-Relay Link(${\gamma }_{SR}$), its expression is:

(5)${\gamma }_{SR}={\displaystyle \frac{P_S{\left|h_{SR}\right|}^2}{W{N}_S}},$

where $P_S$ is the transmit power, and $W{N}_S$ is the noise power. ${\gamma }_{SR}$ is a linear transformation of $|h_{SR}{|}^2$, which is:

(6)${\gamma }_{SR}={\displaystyle \frac{P_S}{W{N}_S}}{\left|h_{SR}\right|}^2.$

It can be derived that ${\gamma }_{SR}$ follows an exponential distribution with parameter ${\lambda }_{SR}$, whose probability density function is:

(7)$f_{{\gamma }_{SR}}\left(y\right)={\lambda }_{SR}{e}^{-{\lambda }_{SR}y},y\ge 0,$

where:

(8)${\lambda }_{SR}={\displaystyle \frac{W{N}_S}{P_S{\sigma }^2}}.$

Similarly, we can derive the following for the $RD$ link: In a Rayleigh fading channel, $|h_{RD}{|}^2$ also follows an exponential distribution with parameter ${\displaystyle \frac{1}{{\sigma }^2}}$, that is:

(9)$f_{|h_{RD}{|}^2}\left(z\right)={\displaystyle \frac{1}{{\sigma }^2}}{e}^{-{\displaystyle \frac{z}{{\sigma }^2}}},z\ge 0.$

For The signal-to-noise ratio of the Relay-Destination Link(${\gamma }_{RD}$), its expression is:

(10)${\gamma }_{RD}={\displaystyle \frac{P_R{\left|h_{RD}\right|}^2}{W{N}_R}},$

where $P_R$ is the transmit power and $W{N}_R$ is the noise power. ${\gamma }_{RD}$ is a linear transformation of $|h_{RD}{|}^2$, which is:

(11)${\gamma }_{RD}={\displaystyle \frac{P_R}{W{N}_R}}{\left|h_{RD}\right|}^2.$

It can be derived that ${\gamma }_{RD}$ follows an exponential distribution with parameter ${\lambda }_{RD}$, whose probability density function is:

(12)$f_{{\gamma }_{RD}}\left(y\right)={\lambda }_{RD}{e}^{-{\lambda }_{RD}y},y\ge 0,$

where:

(13)${\lambda }_{RD}={\displaystyle \frac{W{N}_R}{P_R{\sigma }^2}}.$

Based on the computational power of D and the semantic-transmission-suitability of channel conditions, the channel transmission strategy falls into four distinct scenarios:

Both SR and RD links engage in semantic transmission, and S and D share identical background knowledge ($BK$): Here, D has ample storage and computational resources to decode the semantic information being transmitted. Moreover, its semantic encoding logic aligns with that of S, indicating shared $BK$. R, functioning much like a traditional decode-and-forward (DF) relay, decodes and re-encodes data semantically. This ensures the integrity and efficient conveyance of semantic information.

Both SR and RD links carry out semantic transmission, yet S and D possess different $BK$: In this case, D still has enough computational resources for semantic-information decoding. However, its background knowledge diverges from that of S. This disparity may cause discrepancies in semantic-information expression and comprehension. Therefore, R must undertake additional tasks, such as semantic-information conversion and adaptation, to enable D to correctly interpret the received semantic information.

Both SR and RD links perform semantic transmission. S and D share the same $BK$, but signal degradation impedes D’s decoding: Although D has sufficient computational resources and shares background knowledge with S, long-distance propagation and poor channel quality result in significant signal degradation. This prevents D from accurately decoding the semantic information. Consequently, R operates as a traditional amplify-and-forward (AF) relay, amplifying and forwarding the signal. In this situation, the system’s channel capacity is equivalent to that of a two-hop AF relay.

The SR link uses semantic transmission while the RD link employs bit transmission: In this scenario, S and R communicate using semantic transmission, whereas R and D rely solely on bit transmission. This model is suitable when D has limited computational resources. R must decode the semantic information and re-encode it as a bit stream for transmission to D. Here, the transmission performance of the RD link is determined by the bit-transmission channel capacity.

In summary, these four cases demonstrate the optimization strategies for transmission under different system conditions. By combining semantic transmission with traditional transmission methods, these strategies address various limitations, such as background knowledge matching, channel quality, and computational resources, providing a theoretical basis and practical guidance for efficient semantic transmission in relay communication systems.

Based on the above four transmission strategies, four types of working modes assumed by the relay node in different scenarios can be further summarized, namely semantic collaborative decoding-forwarding, semantic adaptation forwarding, semantic enhancement forwarding, and semantic-bit hybrid forwarding. Under different modes, the relay node has different functional focuses to adapt to various constraints such as the matching of background knowledge, channel quality, and computing resources, providing flexible solutions for achieving efficient semantic communication.

3. Performance Analysis

To evaluate the performance of semantic communication in text transmission, this paper uses semantic similarity as the primary performance metric. Specifically, the semantic similarity $\xi$ is defined as:

(14)$\xi ={\displaystyle \frac{B\left(s\right)B{\left(s\right)}^T}{\parallel B\left(s\right)\parallel \parallel B\left(s\right)\parallel }}$

where $B(·)$ represents the bidirectional encoder representations from Transformers (BERT) model, which has made significant improvements in state-of-the-art sentence embedding methods. The pre-trained Sentence-BERT model is used. Unlike other semantic evaluation metrics like BLEU, the BERT similarity can more accurately measure the semantic distance between two sentences. Notably, the range of semantic similarity values is $0\le \xi \le 1$, where $\xi =1$ indicates that the two sentences are identical, and $\xi =0$ means they have no similarity.

Based on semantic similarity, a new performance metric called semantic rate is introduced in literature [10] to measure the semantic information transmission rate achieved by DeepSC. Let I represent the semantic units (suts), i.e., the average amount of semantic information contained in a sentence s. Therefore, the semantic information for each semantic symbol can be expressed as ${\displaystyle \frac{I}{KL}}$ (units: suts/symbol). Recall that the symbol rate is equal to the transmission bandwidth W. Thus, the effective semantic rate (units: suts/s) can be expressed as:

(15)$S=W·{\displaystyle \frac{I}{KL}}·\xi (K,\gamma )$

where $\xi$ is a function dependent on the specific semantic communication system and physical channel conditions. The value of $\xi (K,\gamma )$ can be obtained by running the DeepSC tool, with the mapping relationship shown in Figure 2.

Semantic similarity has been proven to be a function of received signal-to-noise ratio (SNR) $\gamma$ and K. For any given K, the semantic similarity function $ϵ\in [0,1]$ typically follows an “S” curve as SNR changes. Therefore, a generalized logistic regression method can be used to approximate it as a sigmoid function, expressed as:

(16)$\xi =a_1+{\displaystyle \frac{a_2}{1+e^{-(c_1\gamma +c_2)}}}$

where $a_1$, $a_2$, $c_1$, and $c_2$ are constant coefficients dependent on K.

Based on this, the channel capacity formulas for four scenarios can be derived:

3.1. Semantic Collaborative Decoding-Forwarding

(17)$R_{SR}^{\mathrm{sem}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{W}{KL}}\xi (K,{\gamma }_{SR})$

where W is the channel bandwidth, I is the average amount of semantic information per sentence, measured in semantic units (suts), K is the average number of semantic symbols per word in the original sentence, L is the number of words in the sentence, and $ϵ(K,{\gamma }_{SR})$ is the semantic similarity.

The bit rate for transmission is:

(18)$R_{SR}^{\mathrm{bit}}=\mu {\displaystyle \frac{R_{SR}^{\mathrm{sem}}}{I/L}}={\displaystyle \frac{1}{2}}\mu {\displaystyle \frac{W}{K}}\xi (K,{\gamma }_{SR})$

where $\mu$ is the conversion factor for encoding the source information into bits, indicating the average number of bits per word.

The capacity of the RD link is:

(19)$C_{RD}^{\mathrm{bit}}={\displaystyle \frac{1}{2}}W{log}_2\left(1+{\displaystyle \frac{P_{R}g{\left|h_{RD}\right|}^2}{N_D}}\right)$

where the bit stream includes encoded semantic symbols, and $r_c$ represents the average number of bits per semantic symbol using the traditional joint encoder. Thus, the effective semantic symbol rate on the RD link is:

(20)$R_{RD}^{\mathrm{sem}}={\displaystyle \frac{C_{RD}^{\mathrm{bit}}I}{r_{c}KL}}$

The effective semantic bit rate on the SR link is:

(21)$R_{RD}^{\mathrm{bit}}=\mu {\displaystyle \frac{R_{RD}^{\mathrm{sem}}}{I/L}}=\mu {\displaystyle \frac{C_{RD}^{\mathrm{bit}}}{r_{c}K}}={\displaystyle \frac{1}{2}}\mu {\displaystyle \frac{W}{r_{c}K}}{log}_2\left(1+{\displaystyle \frac{P_{R}g{\left|h_{RD}\right|}^2}{N_D}}\right)$

Assuming the predefined channel capacity is C, the signal-to-noise ratio (SNR) threshold is ${\gamma }_{\mathrm{th}}$, and the outage probability is:

(22)$P_{\mathrm{out}}\left(R\right)=P_r(R_{SD}^{\mathrm{bit}}<C)=1-P_r(R_{SD}^{\mathrm{bit}}>C)=1-P_r(min\{R_{SR}^{\mathrm{bit}},R_{RD}^{\mathrm{bit}}\}>C)$

Based on $R_{SR}^{\mathrm{bit}}>C$, we can derive:

(23)${\displaystyle \frac{1}{2}}\mu {\displaystyle \frac{W}{K}}\xi (K,{\gamma }_{SR})>C\Rightarrow \xi >{\displaystyle \frac{2CK}{\mu W}}$

Substituting:

(24)$\xi (K,{\gamma }_{SR})=a_1+{\displaystyle \frac{a_2}{1+e^{-(c_1{\gamma }_{SR}+c_2)}}}$

where $\xi \in [0,1]$, we can obtain:

(25)${\gamma }_{SR}>{\displaystyle \frac{-ln\left(\frac{a_2}{\frac{2CK}{\mu W}-a_1}-1\right)-c_2}{c_1}}$

The SR link follows a Rayleigh channel model, so the probability $P_r(R_{SR}^{\mathrm{bit}}>C)$ is:

(26)$P_r(R_{SR}^{\mathrm{bit}}>C)=1-F_{{\gamma }_{SR}}\left({\displaystyle \frac{-ln\left(\frac{a_2}{\frac{2CK}{\mu W}-a_1}-1\right)-c_2}{c_1}}\right)$

Using the CDF for the Rayleigh distribution, this becomes:

(27)$P_r(R_{SR}^{\mathrm{bit}}>C)=e^{-{\lambda }_{SR}\left({\displaystyle \frac{-ln\left(\frac{a_2}{\frac{2CK}{\mu W}-a_1}-1\right)-c_2}{c_1}}\right)}$

Similarly, for $R_{RD}^{\mathrm{bit}}>C$, we get:

(28)${\gamma }_{RD}>2^{{\displaystyle \frac{2r_{c}KC}{\mu W}}}-1$

and the probability:

(29)$P_r({\gamma }_{RD}>2^{{\displaystyle \frac{2r_{c}KC}{\mu W}}}-1)=e^{-{\lambda }_{RD}(2^{{\displaystyle \frac{2r_{c}KC}{\mu W}}}-1)}$

Hence, the outage probability is:

(30)$P_r(R_{SD}^{\mathrm{bit}}<C)=1-e^{{\displaystyle \frac{ln\left(\frac{a_2}{\frac{2CK}{\mu W}-1}-1\right)+c_2}{c_1}}}\times e^{-{\lambda }_{RD}(2^{{\displaystyle \frac{2r_{c}KC}{\mu W}}}-1)}$

3.2. Semantic Adaptation Forwarding

For the SR link, the effective semantic rate is:

(31)$R_{SR}^{\mathrm{sem}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{W}{KL}}ϵ(K,{\gamma }_{SR})$

which can be converted to the bit transmission rate:

(32)$R_{SR}^{\mathrm{bit}}=\mu {\displaystyle \frac{R_{SR}^{\mathrm{sem}}}{I/L}}={\displaystyle \frac{1}{2}}\mu {\displaystyle \frac{W}{K}}\xi (K,{\gamma }_{SR})$

For the RD link, the effective semantic rate is:

(33)$R_{RD}^{\mathrm{sem}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{W}{KL}}ϵ(K,{\gamma }_{RD})$

and the bit transmission rate is:

(34)$R_{RD}^{\mathrm{bit}}=\mu {\displaystyle \frac{R_{RD}^{\mathrm{sem}}}{I/L}}={\displaystyle \frac{1}{2}}\mu {\displaystyle \frac{W}{K}}\xi (K,{\gamma }_{RD})$

The overall communication capacity is determined by the minimum value of the two links:

(35)$R_{SD}^{\mathrm{bit}}=min\{R_{SR}^{\mathrm{bit}},R_{RD}^{\mathrm{bit}}\}$

Thus, the outage probability can be expressed as:

(36)$\begin{array}{c}\hfill P_{\mathrm{out}}\left(R\right)=P_r(R_{SD}^{\mathrm{bit}}<C)=1-P_r(R_{SD}^{\mathrm{bit}}>C)=1-P_r\left(min\{R_{SR}^{\mathrm{bit}},R_{RD}^{\mathrm{bit}}\}>C\right)\end{array}$

Using the probability properties, we get:

(37)$P_{\mathrm{out}}\left(R\right)=1-P_r(R_{SR}^{\mathrm{bit}}>C)·P_r(R_{RD}^{\mathrm{bit}}>C)$

Substituting the expressions for $R_{SR}^{\mathrm{bit}}$ and $R_{RD}^{\mathrm{bit}}$, we have:

(38)$\begin{array}{c}{P}_{\mathrm{out}}\left(R\right)=1-P_r\left({\gamma }_{SR}>{\displaystyle \frac{-ln\left(\frac{a_2}{\frac{2CK}{\mu W}-1}-1\right)-c_2}{c_1}}\right)\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}·P_r\left({\gamma }_{RD}>{\displaystyle \frac{-ln\left(\frac{a_2}{\frac{2CK}{\mu W}-1}-1\right)-c_2}{c_1}}\right)\end{array}$

3.3. Semantic Enhancement Forwarding

In a two-hop relay communication system, the channel capacity is:

(39)$C_{SD}^{\mathrm{bit}}={\displaystyle \frac{1}{2}}W{log}_2\left(1+{\displaystyle \frac{{\gamma }_{SR}·{\gamma }_{RD}}{1+{\gamma }_{SR}+{\gamma }_{RD}}}\right)$

The semantic channel capacity is derived as:

(40)$R_{SD}^{\mathrm{sem}}={\displaystyle \frac{C_{SD}^{\mathrm{bit}}}{r_c}}·{\displaystyle \frac{I}{KL}}$

The bit rate for transmission is:

(41)$R_{SD}^{\mathrm{bit}}=\mu {\displaystyle \frac{R_{SD}^{\mathrm{sem}}}{I/L}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{W\mu }{K{r}_c}}{log}_2\left(1+{\displaystyle \frac{{\gamma }_{SR}·{\gamma }_{RD}}{1+{\gamma }_{SR}+{\gamma }_{RD}}}\right)$

Based on this, the outage probability is:

(42)$P_{\mathrm{out}}\left(R\right)=P_r(R_{SD}^{\mathrm{bit}}<C)=1-P_r(R_{SD}^{\mathrm{bit}}>C)$

Substituting the expression for $R_{SD}^{\mathrm{bit}}$, we get:

(43)$P_{\mathrm{out}}\left(R\right)=1-P_r\left({\displaystyle \frac{1}{2}}{\displaystyle \frac{W\mu }{K{r}_c}}{log}_2\left(1+{\displaystyle \frac{{\gamma }_{SR}·{\gamma }_{RD}}{1+{\gamma }_{SR}+{\gamma }_{RD}}}\right)>C\right)$

Let $M=2^{{\displaystyle \frac{2K{r}_{c}C}{W\mu }}}-1$, and simplify:

(44)$P_{\mathrm{out}}\left(R\right)=1-P_r\left({\displaystyle \frac{{\gamma }_{SR}·{\gamma }_{RD}}{1+{\gamma }_{SR}+{\gamma }_{RD}}}>M\right)$

Under high SNR conditions, where ${\gamma }_{SR}\gg 1$ and ${\gamma }_{RD}\gg 1$, we can approximate:

(45)${\displaystyle \frac{{\gamma }_{SR}·{\gamma }_{RD}}{1+{\gamma }_{SR}+{\gamma }_{RD}}}\approx {\displaystyle \frac{{\gamma }_{SR}·{\gamma }_{RD}}{{\gamma }_{SR}+{\gamma }_{RD}}}$

and the final expression for the outage probability involves integrating the joint probability of two independent exponential random variables:

(46)$P_{\mathrm{out}}\left(R\right)=1-{\int }_0^{\infty }{\int }_0^{{\displaystyle \frac{1}{M-u}}}{\lambda }_{SR}{e}^{-{\displaystyle \frac{{\lambda }_{SR}}{u}}}·{\lambda }_{RD}{e}^{-{\displaystyle \frac{{\lambda }_{RD}}{v}}}{\displaystyle \frac{1}{u^2}}{\displaystyle \frac{1}{v^2}}dvdu$

3.4. Semantic-Bit Hybrid Forwarding

In this case, the effective semantic rate for the SR link is:

(47)$R_{SR}^{\mathrm{sem}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{W}{KL}}\xi (K,{\gamma }_{SR})$

and the bit rate is:

(48)$R_{SR}^{\mathrm{bit}}=\mu {\displaystyle \frac{R_{SR}^{\mathrm{sem}}}{I/L}}={\displaystyle \frac{1}{2}}\mu {\displaystyle \frac{W}{K}}\xi (K,{\gamma }_{SR})$

For the RD link, the bit transmission rate is:

(49)$R_{RD}^{\mathrm{bit}}={\displaystyle \frac{1}{2}}W{log}_2\left(1+{\displaystyle \frac{P_R{\left|h_{RD}\right|}^2}{W{N}_R}}\right)$

The overall communication capacity is:

(50)$R_{SD}^{\mathrm{bit}}=min\{R_{SR}^{\mathrm{bit}},R_{RD}^{\mathrm{bit}}\}$

The outage probability is:

(51)$P_{\mathrm{out}}\left(R\right)=P_r(R_{SD}^{\mathrm{bit}}<C)$

Simplifying:

(52)$P_{\mathrm{out}}\left(R\right)=1-P_r(R_{SD}^{\mathrm{bit}}>C)=1-P_r(min\{R_{SR}^{\mathrm{bit}},R_{RD}^{\mathrm{bit}}\}>C)$

The probabilities for the SR and RD links are derived as:

(53)$P_r(R_{SR}^{\mathrm{bit}}>C)=e^{{\displaystyle \frac{ln\left(\frac{a_2}{\frac{2CK}{\mu W}-1}-1\right)+c_2}{c_1}}}$

and:

(54)$P_r(R_{RD}^{\mathrm{bit}}>C)=e^{-{\lambda }_{RD}(2^{{\displaystyle \frac{2C}{W}}}-1)}$

The final outage probability is:

(55)$P_{\mathrm{out}}\left(R\right)=1-e^{{\displaystyle \frac{ln\left(\frac{a_2}{\frac{2CK}{\mu W}-1}-1\right)+c_2}{c_1}}-{\lambda }_{RD}(2^{{\displaystyle \frac{2C}{W}}}-1)}$

After completing the theoretical derivations of the outage probabilities in various scenarios and systematically summarizing the key performance indicators and their mathematical expressions of different transmission strategies, this section will summarize and compare the analysis results of the outage probabilities in the four situations. Table 1 details the expressions of the channel capacities of the SR link and the RD link in each scenario, the overall channel capacity model of the SD link, as well as the corresponding closed-form formulas for the outage probabilities.

4. Simulation and Results

This section presents the theoretical analysis of outage probability for the four proposed semantic communication systems and verifies it through Monte Carlo simulations. The experimental parameters are shown in Table 2. In this simulation, the outage probability for four relay-based semantic communication models is tested under different values of K and transmit power of the source.

This simulation is implemented on the MATLAB R2022a platform, making full use of its numerical calculation and stochastic process simulation capabilities. The core modules of the simulator include the channel model, semantic similarity mapping, and outage event determination.

The channel model adopts the Rayleigh fading channel, in which the channel gains $h_{SR}$ and $h_{RD}$ follow an exponential distribution, and the instantaneous signal-to-noise ratios (SNRs) ${\gamma }_{SR}$ and ${\gamma }_{RD}$ are calculated. Semantic similarity mapping calculates the semantic similarity $\xi$ based on the pre-trained Sentence-BERT model, and then approximates it using the Sigmoid function. The outage event determination checks, according to the outage probability formula derived in the paper, whether each transmission meets the capacity threshold C in sequence.

The Monte Carlo simulation runs 2000 independent iterations. Each iteration includes the generation of a complete channel state, semantic encoding and decoding, calculation of the transmission rate, and outage determination. This number of iterations balances computational efficiency while ensuring the stability of the statistical results. The final outage probability is obtained by calculating the ratio of the frequency of outage events to the total number of iterations.

4.1. First Case

In the BD channel, bit transmission is performed, so the SD link’s channel capacity is actually determined by the SR link. Since the signal-to-noise ratio (SNR) ${\gamma }_{SR}$ is much greater than 1, no matter how the transmit powers of S and R are changed, $\epsilon (K,{\gamma }_{RD})$ is approximately equal to 1, meaning the outage probability can be approximated to 0. The SD link’s channel capacity decreases as K increases. As shown in Figure 3.

4.2. Second Case

By adjusting the transmit power P of S and R, the semantic similarity $\xi$ between the SR and RD links can be modified, thereby altering the channel capacity of the SD and RD links. As the value of P increases, the SNR of the link increases. At this point, the SD link’s channel capacity is determined by the link with the lower SNR. When K exceeds 6, the outage probability decreases and approaches 0.As shown in Figure 4.

4.3. Third Case

In this case, the outage probability decreases as K increases. Since the relay performs an amplify-and-forward (AF) function, this case is consistent with a traditional three-node communication model. The outage probability decreases as the SNR increases. As shown in Figure 5.

4.4. Fourth Case

In this case, the outage probability decreases as K increases. As shown in Figure 6.

5. Conclusions

This paper provides an in-depth study of the outage probability performance in relay-based semantic communication systems and proposes four transmission strategies under different background knowledge, computational capabilities, and channel conditions. By establishing a three-node communication model, the channel capacity and outage probability expressions for each scenario were derived and verified through Monte Carlo simulations. The research findings indicate that:

The effectiveness of semantic transmission is influenced by the degree of background knowledge matching. When the background knowledge (BK) of the source node (S) and the destination node (D) is consistent, the semantic transmission performance of the communication system is optimal.

Relay nodes play a crucial role in semantic conversion and adaptation, particularly when background knowledge does not match. Relay nodes can optimize semantic information transmission, improving the accuracy of semantic decoding.

Channel conditions have a significant impact on the success rate of semantic transmission. When channel quality is poor, the relay node needs to use an amplify-and-forward (AF) strategy to enhance the signal, thereby reducing the outage probability.

Hybrid strategies of semantic and bit transmission are suitable for scenarios with limited computational resources. When the computational power of the destination node is insufficient, performing semantic decoding at the relay node and converting it to bit streams can enhance the overall transmission reliability.

The theoretical analysis and simulation results in this study provide new insights for optimizing relay strategies in semantic communication systems and are of significant importance for improving the stability and reliability of future intelligent communication systems. Future research can further consider full-duplex relay modes and joint optimization of resource allocation to further enhance the performance of semantic communication systems.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The author declares no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1 System Model.

Figure 2 The semantic similarity for DeepSC.

Figure 3 The first case: the variation of SD channel capacity with respect to the K value.

Figure 4 The second case: the variation curve of outage probability with respect to the K value.

Figure 5 The third case: the variation curve of outage probability with respect to the K value.

Figure 6 The fourth case: the variation curve of outage probability with respect to the K value.

Table 1

Comparison of different working modes.

Working Modes	Channel Capacity of SR Link ${\mathit{R}}_{\mathbf{SR}}^{\mathbf{bit}}$	Channel Capacity of RD Link ${\mathit{R}}_{\mathbf{RD}}^{\mathbf{bit}}$	Channel Capacity of SD Link ${\mathit{R}}_{\mathbf{SD}}$	Outage Probability ${\mathit{P}}_{\mathbf{out}}$
semantic collaborative decoding-forwarding	${\displaystyle \frac{\mu W}{2K}}\xi (K,Y_{SR})$	${\displaystyle \frac{\mu W}{2r_{c}K}}{log}_2(1+Y_{RD})$	$min\{R_{SR}^{bit},R_{RD}^{bit}\}$	$1-e^{{\displaystyle \frac{ln(\frac{a_2}{2CK/\mu W^{-1}})+c_2}{c_1}}-{\lambda }_{RD}(2^{{\displaystyle \frac{2r_{c}c}{W\mu }}}-1)}$
semantic adaptation forwarding	${\displaystyle \frac{\mu W}{2K}}\xi (K,Y_{SR})$	${\displaystyle \frac{\mu W}{2K}}\xi (K,Y_{RD})$	$min\{R_{SR}^{bit},R_{RD}^{bit}\}$	$1-e^{{\displaystyle \frac{ln(\frac{a_2}{2CK/\mu W^{-1}})+c_2}{c_1}}}$
semantic enhancement forwarding	–	–	${\displaystyle \frac{1}{2}}·{\displaystyle \frac{W\mu }{K{r}_c}}{log}_2\left(1+{\displaystyle \frac{Y_{SR}·Y_{RD}}{1+Y_{SR}+Y_{RD}}}\right)$	$$\begin{gathered} 1-{\int }_{u=0}^{\infty }{\int }_{v=0}^{1/M-u}{\lambda }_{SR}{e}^{-{\lambda }_{SR}/u}·{\displaystyle \frac{1}{u^2}}· \\ {\lambda }_{RD}{e}^{-{\lambda }_{RD}/v}·{\displaystyle \frac{1}{v^2}}dvdu, \end{gathered}$$
				where $M=2^{{\displaystyle \frac{2r_{c}c}{W\mu }}}-1$
semantic-bit hybrid forwarding	${\displaystyle \frac{\mu W}{2K}}\xi (K,Y_{SR})$	${\displaystyle \frac{1}{2}}W{log}_2\left(1+{\displaystyle \frac{P_R{\left|h_{RD}\right|}^2}{W{N}_R}}\right)$	$min\{R_{SR}^{bit},R_{RD}^{bit}\}$	$1-e^{{\displaystyle \frac{ln(\frac{a_2}{2CK/\mu W^{-1}})+c_2}{c_1}}-{\lambda }_{RD}(2^{{\displaystyle \frac{2c}{W}}}-1)}$

Table 2

Parameter values.

Parameter Name	Value
S—Transmission Power ($P_S$)	100 W
Relay Transmission Power ($P_R$)	100 W
Noise Power—Spectral Density ($N_0$)	$4\times {10}^{-24}$ W/Hz
Capacity Threshold (C)	$8\times {10}^8$ bit/s
Source—Coding Factor ($\mu$)	48
Average Number of Semantic Symbols per Word (K)	4 symbols/word
Average Number of Bits ($r_c$)	8
System Bandwidth (W)	$1.8\times {10}^8$ Hz
Standard Deviation of Channel Gain ($\sigma$)	1

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