Information-Energy Capacity Region for SWIPT Systems with Semantic Communication

In this paper, we study the fundamental limits of simultaneous semantic information and power transfer. In particular, a three-party communication system is considered, where an information transmitter aims to simultaneously convey semantic information to an information receiver (IR) and deliver energy to an energy harvesting receiver. An achievable and a converse region in terms of information and energy rates (in bits per channel use and energy-units per channel use, respectively) are presented for the discrete memoryless (DM) channel. The achievable region is obtained by using the asymptotic equipartition property (AEP) and a converse region is obtained by using outer bounds on the semantic information rates. In addition, we characterize an achievable region for the Gaussian case by using a power splitting technique between the information and the semantic context parts. A converse region is also obtained that provides an estimate of the information-energy capacity while taking into account semantics. Numerical results show a higher performance in terms of information and energy rates by considering a low semantic ambiguity code in comparison to the classical coding scheme (without semantic).


I. INTRODUCTION
When Shannon laid the theoretical foundation of communications engineering back in 1948, he deliberately excluded semantic aspects from the system design [1].However, based on Weaver [2], communication could be classified into three levels: i) the transmission of symbols (Shannon's classical Information Theory); ii) the precision in which the transmitted symbols convey the desired meaning; iii) the effects of semantic information exchange.The first level mainly concerns the accuracy at which the symbols of communication are transmitted.The second level deals with the semantic information which is sent from the transmitter and the meaning interpreted at the receiver, named as semantic communication.The third level mainly focuses on how the received meaning affect the communication and the coding schemes.
In the last few decades, wireless communication systems have experienced significant development from the first generation (1G) to the fifth generation (5G) with the system capacity gradually approaching the Shannon limit [3].Nevertheless, the meaning behind the transmitted data in the classic Shannon's This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 819819).
framework is expected to play an important role in 6G communications.Therefore, semantic communication techniques can be used to enable wireless devices to extract and transmit the meaning of original data to reduce the heavy congestion of current wireless networks, thus improving communication efficiency [4].However, using semantic communication techniques for data transmission faces several challenges such as semantic information modeling and extraction, original data recovery, and the definition of appropriate semantic metrics that can capture the effects of wireless factors (e.g., transmit power, packet errors) on the semantic communications [5].The authors in [6] use a novel approach to model the semantics through a Bayesian game, where the semantic similarity is used as a semantic error metric.The work in [7] proposes a signal shaping method by minimizing the semantic loss, which is measured by the pretrained bidirectional encoder representation from transformers (BERT).In [8], the authors propose a deep learning based multi-user semantic communication system that can extract the semantic information of image and text from different users.The authors in [9] propose a novel framework that enables users to communicate with a base station using a semantic communication and energy harvesting (EH) technique.
Simultaneous wireless information and power transfer (SWIPT) is a technology that exploits the duality of the radio frequency (RF) signals, which can carry both information and energy [10] through appropriate information-energy codesign and co-engineering.Semantic data can be compressed to a proper size for transmission by using a lossless method [11], which utilizes the semantic relationship between different messages.Specifically, in a SWIPT communication network with semantic, each user can use the energy savings associated with semantic encoder's compression to enhance the energy harvested at the EH receiver (ER).The combination of semantic information and EH has the potential to be applied in various fields such as wireless sensor networks, Internet of Things (IoT), and smart grid systems.Therefore, intelligent energy management systems can be developed to optimize the use of energy in various applications by using a semantic communication protocol.In a smart grid system, for instance, semantic information can be employed to identify the most energy-efficient route for energy transmission.By utilizing this approach, energy consumption can be minimized while ensuring optimal performance.The use of semantic communication protocol enables these systems to make more informed  decisions based on the context and meaning of the data being processed.This leads to more efficient and effective energy management, reducing waste and cutting costs.Overall, the development of intelligent energy management systems using semantic communication protocols holds great potential for improving energy efficiency in various domains [12].However, none of the previous works considered the optimization of semantic communication over SWIPT networks.
The main contribution of this work is a novel framework that enables the study of the fundamental limits of SWIPT with semantic communication for a discrete memoryless (DM) channel and a Gaussian channel.In particular, we consider a basic semantic SWIPT communication system, where a transmitter simultaneously sends data to an information receiver and power to an ER.Specifically, we propose an information-theoretic framework to characterize an achievable information-energy region as well its converse for both the DM channel and the Gaussian channel also by including the semantic context into the communication.The achievable region is obtained by using the asymptotic equipartition property (AEP) and the converse region is obtained that provides an estimate of the informationenergy capacity while taking into account semantics.Numerical results shows that by considering a low semantic ambiguity code, a higher performance is observed in comparison to conventional communication approaches (i.e., without semantics).
II. SYSTEM MODEL Consider a three-party semantic communication setup, in which a transmitter aims to simultaneously convey information to an information receiver (IR) and power to an ER.In particular, we consider two cases: a DM channel (see Fig. 1) and a Guassian channel (see Fig. 2).The study of the DM channel and the Gaussian channel is crucial as it provides the foundation for characterizing the fundamental limits of SWIPT systems with semantic communication.Specifically, gaining insights into these channels can enhance our understanding of how SWIPT systems operate and how they can be optimized for maximum efficiency.By studying the DM channel and Gaussian channel, we can gain a deeper understanding of the impact of noise, interference, and other factors on the performance of SWIPT systems.This knowledge can be leveraged to design efficent SWIPT systems that are more resilient and reliable, with improved information and energy capabilities.

A. DM channel with SWIPT and semantics
The transmitter sends a message w from a set W with a probability distribution P (W = w) to an encoder.The message w is encoded into a channel input x = (x 1 , x 2 , . . ., x n ) ∈ X n by using an encoding function φ : W → X n , where n denotes the number of channel uses.The output at the decoder is given by y = (y 1 , y 2 , . . ., y n ) from the set Y n and is observed at the IR with probability where P (y i |x i ) is the transition probability distribution.
In contrast to conventional communication systems, we assume the existence of side information provided via a genieaided channel to the IR.This side information is likely to be useful in helping the IR to better understand and interpret the information being transmitted.As a result, the communication system becomes more intelligent and effective.Specifically, the communication between the transmitter and the receiver is taking place within a specific context, which means that the information being transmitted is related to a particular topic or area of interest [6].Therefore, semantic context can influence the IR on how it decodes the received signals, depending on the tasks and/or actions to be executed.The semantic context [6] is characterized by a random variable Q with respect to a probability distribution P (Q|W ), which satisfies We define the semantic distance between the words w and ŵ as, where 0 ≤ sim(w, ŵ) ≤ 1 denotes the semantic similarity between w and ŵ.By following similar steps as in [6], the average semantic error denoted by P SE is given by where ψ(y, q) : Y n × Q → W denotes the decoding function.
The output at the ER is given by s = (s 1 , s 2 , . . ., s n ) from the set S. Let E : S → R + be the function that determines the average energy harvested, which is given by where g : S → R + is a positive real-valued function that determines the energy harvested from the output symbols.The probability of energy-shortage when transmitting the message w is given by where b denotes the achieved energy rate at the ER.The system is said to be operating at the semantic information-energy rate (R, b) ∈ R 2 + when both transmitter-receiver pairs use a transmit-receive configuration such that: (i) reliable semantic communication at rate R is ensured; and (ii) reliable energy transmission at energy rate b is ensured.A formal definition is given below.
Definition 1. From a semantic communication standpoint, a semantic information-energy rate R(b) is achievable, if the probability of miss-interpretation of the message w, given the context Q, satisfies the limit P SE → 0, and the energy shortage probability, P ES (b), satisfies P ES (b) → 0, for n → ∞.
By using Definition 1, the fundamental limits of semantic information and energy transfer in a DM channel can be described by the semantic-energy capacity region, defined as follows.
Definition 2. The information-energy capacity region C(b) is defined as the maximum rate over all the achievable rates, i.e.,

B. Gaussian channel with SWIPT and semantics
The message w is mapped in this case into a vector x ∈ R n , where n denotes the number of channel uses.The channel gains for the IR and ER are considered constant and denoted by h 1 ∈ R and h 2 ∈ R, respectively.The received signal at the IR is given by Y and the ER observes where Z ∼ N (0, 1) is the Gaussian noise with unit variance and is assumed to be independent of the signal X.We define the conditional probability as The semantic encoder is subject to an average power constraint on the form where P req is the transmit power constraint.By following similar steps as in [7], an upper bound of the semantic loss for the Guassian case when maximum likelihood (ML) detector is used, it is given by where Q(•) is the tail distribution function of the standard normal distribution and where B Φ (•), represents the pretrained BERT model [13], which provides an efficient way to quantify the semantic similarity between two different messages.For simplicity of the analysis, we adopt a linear model for energy harvesting 1 .Hence, let E : R → R + , where E(y) determines the average harvested energy function, which is given by Given an energy constraint b, the energy shortage probability is given by

III. SWIPT WITH SEMANTICS
Analogous to the conventional communication system, a semantic channel has a capacity limit such that the information rate R and energy rate b can be achieved when n → ∞.First, we explain some notations to be used in the theorem.
• I(X; Y ) = H(X) − H(X|Y ) is the mutual information between X and Y , where H(X) and H(X|Y ) are the corresponding entropy of X and the conditional entropy of X given Y .• H(X|W ) is the equivocation of the semantic encoder.
Specifically, a higher H(X|W ) means higher semantic redundancy in semantic coding.• H(Q) measures the semantic context source, or the local information available at the transmitter and the IR.Specifically, a higher H(Q) means strong ability of the IR to interpret received messages.The main results of this paper consist of the description of the semantic information-energy region.Such a description is presented in the form of an approximation in the sense of an achievable and a converse region.

A. Achievable/Converse region for the DM channel
Define b 1 = min s∈S g(s) and b 0 = max s∈S g(s), where b 1 and b 0 denotes the minimum and the maximum harvested energy per symbol, respectively.By using this definition, the following theorem introduces an achievable information-energy region for semantic communication over a DM channel.Proof: The proof is presented in Appendix A.
Remark 1. From the above expression, the achievable semantic energy region could be lower or higher than the conventional Shannon capacity region sup{I(X; Y )} depending on the term −H(X|W ) + H(Q).Specifically, we may achieve a higher semantic information-energy region by considering a semantic encoder with lower ambiguity codebook in respect of the semantic context Q.
The following theorem introduces an upper bound for the SWIPT semantic information-energy region.
Proof: The proof is presented in Appendix B. Thus, Theorem 2 provides a fundamental limit on the semantic rate at which information can be transmitted over a noisy channel, while ensuring a certain level of energy reliability.It establishes a lower bound on the error probability of any reliable communication scheme and provides a guideline for designing efficient coding schemes that approach the semantic capacity.

B. Achievable/Converse regions for the Gaussian channel
In the memoryless Gaussian channel, the alphabets are continuous.Nonetheless, information and energy transfer can be described similarly to the DM channel (where the finite input and output alphabets are replaced by R).The following Theorem introduces a lower bound of the information-energy capacity region for the Gaussian channel.
Theorem 3. The information-energy capacity C : R + → R + is lower bounded as follows where λ 1 ≥ 0 and λ 2 ≥ 0 denote the fraction of the power dedicated to the input X and the semantic context Q, respectively and h(X|W ) denotes the conditional differential entropy of the input X given the message W .
Proof: The proof is presented in Appendix C. The following theorem introduces an upper bound for the semantic information-energy region Theorem 4. The information-energy capacity region for the SWIPT semantic communication is upper bounded by the function C : R + → R + , i.e., where µ 1 ≥ 0 and µ 2 ≥ 0 denote the fraction of the power dedicated to the input X and the semantic context Q, respectively.
Proof: The proof is presented in Appendix D.

IV. NUMERICAL RESULTS
For the DM channel, without loss of generality, we consider the special case of a binary symmetric channel (BSC) with crossover probability ρ, i.e., where l(y, x) is the Hamming distance between y and x.As an example, we consider a binary set of context Q = {q 1 , q 2 }, which satisfies the following distribution [6] P The average energy harvested at the ER for the BSC case is given by [15] with b 0 = g(0) and b 1 = g(1).A benchmark word set from semantic similarity literature is used with P (W = w) = 1 |W| .Fig. 3 plots the achievable semantic information-energy capacity and its corresponding converse region.A trade-off between the information rate and energy rate is observed and becomes evident as b increases.As shown in Fig. 4, by considering a low semantic ambiguity code, an upper bound for the conventional information-energy capacity region is obtained by the semantic code.
Fig. 5 shows the impact of the semantic context on the information-energy capacity region over the Gaussian channel.By setting λ 2 = 0, i.e., where λ 2 denotes the fraction of the power dedicated to the semantic component by adopting a power splitting technique, we obtain the same region as [16].By slightly increasing λ 2 , i.e., (λ 2 = 0.2) we observe an enlargement of the semantic information-energy region, due to the fact that semantic boost the performance of the information transfer task as well as EH task.However, by setting λ 2 = 0.8 corresponding to a higher ambiguity code, we observe a lower performance in comparison with the conventional Shannon region.

APPENDIX A PROOF OF THEOREM 1
The achievability scheme used to obtain the lower bound on the semantic energy capacity region relies on the AEP [17] and random coding arguments.By assuming that the semantic context Q is independent from W , the joint probability distribution is simplified as follows  We generate 2 nR n-length codewords i.i.d, according to the distribution According to the AEP, the set of all possible sequences is divided into typical sets, where the sample entropy is close to the entropy of individual variables with high probability, i.e., P | − 1 n log p(x n ) − H(X)| < > 1 − η and other nontypical sets with low probability.In the following, we discuss the typical sets, and their properties hold with high probability for all sequences.A semantic error appears, if a received message is not decoded by the receiver using the context Q.Let n be a sufficiently large number.Assume that Q 1 , Q 2 , . . ., Q n is the sequence of the observed context, X 1 , X 2 , . . ., X n is the sequence of the transmitted signals, and Y 1 , X 2 , . . ., Y n is the sequence of the received signals.According to the AEP, there are 2 nH(Q) typical sequences of context.By using the channel coding theorem, there are 2 (I(X;Y )−R)n typical input sequences.For a typical sequence X, there are 2 −nH(X|W ) typical sequences of X, given the context.Hence, there are 2 (I(X;Y )−H(X|W )+H(Q))n typical sequences of input, given the context.Specifically, if which follows from the definition of I(X; Y ) in [17].By using the data procession inequality [17], i.

Theorem 2 .
The information-energy capacity region for semantic communication with SWIPT C : [b 1 , b 0 ] → R + is upper bounded as follows C(b) ≤ max

Fig. 5 :P
Fig.5: Impact of semantic context on the information-energy capacity region over the Gaussian channel.