Power-Efficient Optimization for Coexisting Semantic and Bit-Based Users in NOMA Networks

Shannon and Weaver first introduced the concept of semantic communication in [12], 1949. After that, research on semantic communication has continued to progress steadily, such as the concept of semantic web [13] and the novel framework of semantic communication [14]. With the rapid development in artificial intelligence and machine learning in recent years, semantic communication has entered a new era. Many studies focus on improving performance, particularly in terms of semantic similarity or semantic accuracy, across various data types, including text, speech, images, and video. The authors of [15] proposed DeepSC for text transmission, which outperforms than conventional schemes. The authors of [8] extended DeepSC to a multi-user scenario for text data transmission. Then, the research shifts focus to various data types. For example, the authors of [9] presented a deep learning-enabled semantic communication system that converts speech transmission to text-related semantic features, significantly reducing data requirements while maintaining high performance, the authors of [10] proposed an end-to-end semantic communication system for efficient image transmission by implementing a deep learning-based classifier at the sender and a diffusion model at the receiver and the authors of [11] extended the method in [10] to transmit videos by converting videos into frames. Besides the above works that primarily focus on enhancing transmission performance, some works investigate semantic communication from a task-oriented perspective. For example, the authors of [16] established a multi-user semantic communication system called MU-DeepSC, which leverages correlated image and text data for the visual question answering (VQA) task. MU-DeepSC effectively processes and combines semantic information from images and text to accurately predict answers. The authors in [17] developed a semantic communication system based on deep learning that simultaneously performs image recovery and classification tasks by integrating JSCC. The system employs a novel loss function to enhance robustness and reduce communication overhead under varying channel conditions.

The aforementioned studies focus primarily on optimizing semantic communication at the structural level. Specifically, they emphasize designing innovative encoders and decoders or developing new system architectures by integrating novel components. However, since most semantic devices function within wireless communication networks, some research shifts the focus to network-level optimization. This includes adopting NOMA techniques and designing efficient resource allocation schemes to enhance overall performance. For example, recent work [18] studied the downlink scenario where the base station (BS) serves multiple semantic users by adopting NOMA. However, conventional bit communication still dominates at current stage, it is unrealistic to completely replace all bit-based devices with semantic devices. Hence, it is more worthy to investigate the practical scenario where semantic users and bit-based users co-exist. Recently, work [19] studied the simplest case where one semantic user and one traditional user simultaneously transmit data to a single antenna BS by adopting NOMA. Based on [19], the authors also proposed a semi-NOMA scheme for this case in [20].

Given the significant performance benefits of both NOMA and semantic communication, extending the semantic-bit NOMA network to a multi-user scenario is a critical research direction. Additionally, with the growing emphasis on green communication, many applications demand systems that deliver high performance while minimizing energy consumption. Therefore, it is essential to investigate resource allocation strategies in multi-user semantic-bit NOMA networks to achieve minimal transmission power. To the best of our knowledge, no existing work has addressed resource allocation optimization in multi-user NOMA systems where semantic and bit-based users coexist. In this paper, we focus on this challenging scenario where one BS serves multiple semantic users and multiple conventional bit users simultaneously by adopting NOMA. To introduce spatial diversity, the BS is assumed to be equipped with multiple antennas, which consequently introduces more challenges. We propose a multi-cluster NOMA network to accommodate each two users into one cluster. Two MA transmission schemes are proposed: the P-CNOMA scheme and the H-CNOMA scheme. For the P-CNOMA scheme, an iteration-free algorithm is developed to efficiently solve the optimization problem. The optimization problem for the H-CNOMA scheme is reformulated as a constraint-free problem, which is effectively solved using a deep neural network.

The main contributions of this paper are summarized as follows:

• •

  We propose a multi-cluster NOMA network that groups semantic and conventional bit users into clusters, with each cluster comprising two users of the same type to take care of two different hardware architectures. SIC is applied only within each cluster, treating signals from other clusters as interference, thus significantly reducing the complexity of the SIC process. The BS, equipped with multiple antennas, simultaneously serves multiple clusters by generating a unique beam for each cluster. Within the same cluster, two users share the same beam, performing power allocation accordingly.

• •

  We develop the P-CNOMA scheme to minimize the total transmission power. The formulated problem is non-convex due to the coupling of the beamforming vector and the power allocation coefficient. Moreover, since the objective function is not directly related to the power allocation coefficient, the alternating algorithm may exhibit poor convergence performance. We propose an iteration-free algorithm to solve the problem. The formulated problem is first transformed to a problem without composite constraints ¹¹1The composite constraint in this paper refers to a constraint that can be equivalently decomposed into two or more individual constraints. For example, $\min(A,B)\geq C$ is equivalent to $A\geq C$ and $B\geq C$, hence, $\min(A,B)\geq C$ is a composite constraint. By determining the upper and lower bounds of the power allocation coefficient, the primal problem is approximately transformed into a single-variable optimization problem. Subsequently, semi-definite relaxation (SDR) is employed to efficiently solve this problem.

• •

  We develop the H-CNOMA scheme to compensate for the shortcomings of P-CNOMA. Although the aforementioned P-CNOMA scheme can achieve the lowest transmission power, it faces feasibility issues, particularly when conventional bit users demand high data rates or when semantic users require high semantic similarity. In the H-CNOMA scheme, the primal problem can be solved on a per-cluster basis due to the absence of inter-cluster interference. It is reformulated into a constraint-free problem, and a deep neural network is introduced to efficiently find the solution. This scheme can meet the high data rate demands of bit users and the high semantic similarity requirements of semantic users with a slight increase in transmission power. However, the maximum achievable semantic rate is constrained by the use of multiple frequency sub-channels. Consequently, these two schemes complement each other, and the choice between them should be made based on specific system requirements.

• •

  The simulation results introduce the concepts of effective semantic similarity, the semantic similarity domination region, and the semantic rate domination region. We analyze the impact of these two regions on the effective semantic similarity from the simulation results. Generally, the effective semantic similarity matches the target semantic similarity when the target semantic rate falls within the similarity domination region. However, when the target semantic rate falls within the semantic rate domination region, the effective semantic similarity depends solely on the target semantic rate. Moreover, reveal how the two proposed MA schemes effectively complement one another.

The rest of the paper is organized as follows. In Section II, the system model is introduced and the total transmission power minimization problem is formulated. In Section III, two schemes are proposed along with the optimization algorithms. In Section IV, simulation results are provided. Finally, a conclusion is summarized in Section V.

Notations: $\mathbf{X}$, $\mathbf{x}$ and $x$ represent matrix, vector and scalar, respectively. $\mathbf{x}^{H}$ represents the conjugate transpose of vector $\mathbf{x}$. $\mathbb{C}^{N\times 1}$ represents the space of a $N\times 1$ complex vector and $\mathbb{C}^{N\times M}$ represents the space of a $N\times M$ complex matrix. $||.||_{2}$ represents $l_{2}$ norm.

It is critical to define the semantic rate for network-level performance optimization. In this paper, we assume the typical DeepSC text transmission system described in [24] is deployed. Let $K$ denote the average number of semantic symbols per word and $\gamma$ denote the SNR. According to [24], semantic similarity, which describes the similarity between the recovered sentence and the original sentence in a DeepSC text transmission system, is highly related to $K$ and $\gamma$. The resultant expression of the semantic similarity can be a function of $K$ and $\gamma$, i.e., $\varepsilon\left(K,\gamma\right)$. [25] evaluated the semantic rate based on the semantic similarity, which is given by

where $W$ denotes the transmission bandwidth as well as the symbol rate and $I$ denotes average semantic information per sentence measured by semantic unit per second (suts). To obtain a traceable closed-form expression of the semantic similarity, [20] utilized a data regression method to approximate $\varepsilon\left(K,\gamma\right)$ by a generalized logistic function. For any given $K$, the semantic similarity can be expressed as follows:

where $A_{K,L}$ and $A_{K,R}$ are both positive and respectively denote the left asymptote and the right asymptote. $l_{k}$ is the logistic growth rate, $\gamma_{k,0}$ affects the logistic mid-point and $v_{k}$ affects near which asymptote maximum growth occurs. Note that $\varepsilon_{K}\left(\gamma\right)$ is monotonically increasing with $\gamma$, hence, we have $A_{K,R}>A_{K,L}$.

To reduce the complexity associated with SIC and hardware requirements, it is assumed that each cluster consists of only two users of the same type. In each cluster, which consists of either two S-users or two B-users ²²2It is assumed that $M$ is even, which means each type of users can be allocated into $\frac{M}{2}$ clusters., the user with the higher channel gain is referred as the strong user, while the one with the lower channel gain is referred as the weak user. In the $i$-th S-user cluster, the strong and weak users are denoted as ${\rm U}_{s,i}^{\rm S}$ and ${\rm U}_{s,i}^{\rm W}$, respectively. Likewise, in the $i$-th B-user cluster, they are represented as ${\rm U}_{b,i}^{\rm S}$ and ${\rm U}_{b,i}^{\rm W}$. $x_{s,i}^{\rm S}$ and $x_{s,i}^{\rm W}$ are the corresponding signals of ${\rm U}_{s,i}^{\rm S}$ and ${\rm U}_{s,i}^{\rm W}$, while $x_{b,i}^{\rm S}$ and $x_{b,i}^{\rm W}$ are corresponding signals to ${\rm U}_{b,i}^{\rm S}$ and ${\rm U}_{b,i}^{\rm W}$. The superimposed signal transmitted by the BS can be expressed as follows:

$\mathbf{w}_{s,i}\in\mathbb{C}^{N\times 1}$ and $\mathbf{w}_{b,i}\in\mathbb{C}^{N\times 1}$ represent beamforming vectors for the $i$-th S-user cluster and $i$-th B-user cluster, respectively. $\alpha_{s,i}$ and $\alpha_{b,i}$ respectively denote power allocation coefficients in the $i$-th S-user cluster and B-user cluster. Thus, the received signal by ${\rm U}_{s,i}^{\rm S}$ and ${\rm U}_{s,i}^{\rm W}$ can be expressed as:

where $\mathbf{h}_{s,i}^{\rm S}$ and $\mathbf{h}_{s,i}^{\rm W}$ denote the channel vectors on the BS-${\rm U}_{s,i}^{\rm S}$ link and the BS-${\rm U}_{s,i}^{\rm W}$ link, respectively, and $n_{s,i}^{\rm S}$ and $n_{s,i}^{\rm W}$ denote the zero-mean additive white Gaussian noise (AWGN) with variance $\sigma^{2}$. Similarly, the received signal by ${\rm U}_{b,i}^{\rm S}$ and ${\rm U}_{b,i}^{\rm W}$ in the $i$-th B-user cluster can be expressed as:

Let first consider the SIC procedure in S-user clusters. As SIC is only applied in each cluster, the decoding order is assumed to be that the strong user first decodes the weak user’s signal and then its own signal and the weak user directly decodes its own signal. During SIC, signals from users in other clusters are considered as interference. We assume the perfect channel state information (CSI) is available to all users and all signals have the unit power, satisfying $\mathbb{E}\left(|x_{s,i}^{\rm S}|^{2}\right)=\mathbb{E}\left(|x_{s,i}^{\rm W}|^{2}\right)=\mathbb{E}\left(|x_{b,i}^{\rm S}|^{2}\right)=\mathbb{E}\left(|x_{b,i}^{\rm W}|^{2}\right)=1,\forall i$ with the expectation operation $\mathbb{E}(\cdot)$. For the simple notation, we define a function to express inter-cluster interference to the $i-$th S-user cluster through channel fading $\mathbf{x}$, which is expressed as

Similarly, the inter-cluster interference function of the $i$-th B-user cluster is expressed as

Therefore, the signal-to-interference-plus-noise ratio (SINR) of ${\rm U}_{s,i}^{\rm W}$’s signal decoded by ${\rm U}_{s,i}^{\rm S}$ is given by

After ${\rm U}_{s,i}^{\rm W}$’s signal was successfully decoded, ${\rm U}_{s,i}^{\rm S}$ removes intra-cluster interference. Hence, the SNR of ${\rm U}_{s,i}^{\rm S}$’s signal is given by

The SNR of ${\rm U}_{s,i}^{\rm W}$’s signal decoded by itself is given by

Then, the achievable semantic rate of ${\rm U}_{s,i}^{\rm S}$ and ${\rm U}_{s,i}^{\rm W}$ can be expressed as

and

respectively.

The SIC procedure in B-user clusters is similar to the above. Therefore, the SINR of ${\rm U}_{b,i}^{\rm W}$’s signal decoded by ${\rm U}_{b,i}^{\rm S}$ is given by

After removing intra-cluster interference, the SNR when ${\rm U}_{b,i}^{\rm S}$ decodes its own signal is given by

As ${\rm U}_{b,i}^{\rm W}$ directly decodes its own signal, the SNR can be expressed as follows:

Then, the achievable data rate of ${\rm U}_{b,i}^{\rm S}$ and ${\rm U}_{b,i}^{\rm W}$ can be expressed as

and

respectively.

It is assumed that all S-users have the same target semantic rate $S_{0}$ and target semantic similarity $\varepsilon_{0}$, and all B-users have the same target data rate $R_{0}$. The transmission power minimization problem can be formulated as follows:

where $\mathbf{W}\in\mathbb{C}^{N\times M}$ is a beamforming matrix collecting all beamforming vectors and $\bm{\alpha}\in\mathbf{R}^{M}$ is a vector collecting all power allocation coefficients. Constraint (18b) guarantees each S-user to achieve the target semantic rate. Constraint (18c) is introduced to guarantee the minimal requirement of semantic similarity. The reason to introduce constraint (18c) is that the semantic rate can be large if the transmission bandwidth $W$ is sufficiently large even the semantic similarity is small according to (1). Therefore, the semantic similarity is another critical metric to evaluate the performance of semantic communication. Constraint (18d) guarantees each B-user to achieve the target data rate and constraints (18b), (18c) and (18d) jointly guarantee a successful SIC procedure.

It is noted that two optimization variables, the beamforming vector and the power allocation coefficient, are coupled together in ${\rm P_{0}}$. One common method to solve a multi-variable optimization problem is the alternating algorithm, where one variable is fixed and only another is optimized. However, the power allocation coefficient will not affect the objective function of ${\rm P_{0}}$ directly. As a result, the alternating algorithm may have the difficulty on convergence when solving this problem. The idea is to transfer ${\rm P_{0}}$ to a problem only related to beamforming and then solve it by convex optimization.

According to the definition of semantic rate, constraint (18b) can be rewritten as

where $\tilde{\varepsilon}_{0}=\frac{KLS_{0}}{WI}$. It is noted that constraint (18c) and constraint (19) can be combined as one constraint, which is given by

where $\bar{\varepsilon}_{0}=\max\{\varepsilon_{0},\tilde{\varepsilon}_{0}\}$ denotes the effective semantic similarity. Therefore, ${\rm P_{0}}$ can be rewritten as

${\rm P_{1}}$ is a non-convex problem because of constraint (18d) and constraint (20). According to optimization theory, constraint (20) can be equivalently split into three sub-constraints $\varepsilon_{K}\left(\gamma_{s,i}^{\rm S}\right)\geq\bar{\varepsilon}_{0}$, $\varepsilon_{K}\left(\gamma_{s,i}^{\rm W}\right)\geq\bar{\varepsilon}_{0}$ and $\varepsilon_{K}\left(\gamma_{s,i}^{\rm W\rightarrow S}\right)\geq\bar{\varepsilon}_{0}$, for any given $i$. These three sub-constraints can be further recast into $\gamma_{s,i}^{\rm S}\geq\varepsilon_{K}^{-1}(\bar{\varepsilon}_{0})$, $\gamma_{s,i}^{\rm W}\geq\varepsilon_{K}^{-1}(\bar{\varepsilon}_{0})$ and $\gamma_{s,i}^{\rm W\rightarrow S}\geq\varepsilon_{K}^{-1}(\bar{\varepsilon}_{0})$, where $\varepsilon_{K}^{-1}$ denotes the inverse function of (2). Similarly, constraint (18d) can be equivalently split into two sub-constraints $R_{i}^{\rm S}\geq R_{0}$ and $R_{i}^{\rm W}\geq R_{0}$, which can be further rewritten as $\gamma_{b,i}^{\rm S}\geq\gamma_{0}$, $\gamma_{b,i}^{\rm W\rightarrow S}\geq\gamma_{0}$ and $\gamma_{b,i}^{\rm W}\geq\gamma_{0}$ with $\gamma_{0}=2^{\frac{R_{0}}{W}}-1$. As a result, ${\rm P_{1}}$ can be recast into

Although there is no $\min\{\cdot\}$ constraint in ${\rm P_{2}}$, constraints (22b)-(22g) are non-convex. Hence, ${\rm P_{2}}$ is still a NP hard problem, which is difficult to be solved in polynomial time. After the algebraic transformation, constraint (22b) can be rewritten as follows

From (23), a relationship between $\alpha_{s,i}$ and $\mathbf{w}_{s,i}$ is obtained, which is

Similarly, another two relationships between $\alpha_{s,i}$ and $\mathbf{w}_{s,i}$ can be obtained from (22c) and (22d), which are

and

Considering $0\leq\alpha_{s,i}\leq 1$ and $\frac{\varepsilon_{K}^{-1}(\bar{\varepsilon}_{0})\left(\mathcal{I}_{s,i}\left(\mathbf{h}_{s,i}^{{\rm S}H}\right)+\sigma^{2}\right)}{|\mathbf{h}_{s,i}^{{\rm S}H}\mathbf{w}_{s,i}|^{2}}\geq 0$, the lower bound of $\alpha_{s,i}$ is given by

and the upper bound of $\alpha_{s,i}$ is given by

where $f_{i}(\mathbf{x})\triangleq\frac{|\mathbf{x}^{H}\mathbf{w}_{s,i}|^{2}-\varepsilon_{K}^{-1}(\bar{\varepsilon}_{0})\left(\mathcal{I}_{s,i}\left(\mathbf{x}^{H}\right)+\sigma^{2}\right)}{\left(\varepsilon_{K}^{-1}(\bar{\varepsilon}_{0})+1\right)|\mathbf{x}^{H}\mathbf{w}_{s,i}|^{2}}$.

According to Lemma 1, the upper bound of $\alpha_{s,i}$ is

A feasible $\alpha_{s,i}$ always exists as long as $L_{s,i}\leq U_{s,i}$ holds.

The next step is to find the lower bound and upper bound of $\alpha_{b,i}$. According to (22e)-(22g), we have

and

Similarly, we have the lower bound of $\alpha_{b,i}$

and the upper bound of $\alpha_{b,i}$

where $g_{i}(\mathbf{x})\triangleq\frac{|\mathbf{x}^{H}\mathbf{w}_{b,i}|^{2}-\gamma_{0}\left(\mathcal{I}_{b,i}\left(\mathbf{x}^{H}\right)+\sigma^{2}\right)}{\left(\gamma_{0}+1\right)|\mathbf{x}^{H}\mathbf{w}_{b,i}|^{2}}$. $g_{i}(\mathbf{x})$ can be proved less than 1 by following the method in Lemma 1. Then, the upper bound of $\alpha_{b,i}$ is

A feasible $\alpha_{b,i}$ always exists as long as $L_{b,i}\leq U_{b,i}$ holds.

Consequently, ${\rm P_{2}}$ can be recast into the following form:

Only beamforming vectors in ${\rm P_{3}}$ need to be optimized. The next step is to deal with constraint (36b). Constraint (36b) is equivalent to the following two constraints

and

It is noted that $L_{s,i}$ and $f_{i}\left(\mathbf{h}_{s,i}^{{\rm S}}\right)$ have the same denominator, hence, (37) can be rewritten into

which becomes a quadratic form. As for (38), it represents a quadratic fractional form, which has significant challenges when attempting to transform it into a convex form.

According to Proposition 1, constraint (36b) can be replaced by constraint (39). Follow the step above, constraint (36c) can be replaced by

when assuming $|\mathbf{h}_{b,i}^{{\rm S}H}\mathbf{w}_{b,i}|^{2}\leq|\mathbf{h}_{b,i}^{{\rm W}H}\mathbf{w}_{b,i}|^{2}$ and $\mathcal{I}_{b,i}\left(\mathbf{h}_{b,i}^{{\rm S}H}\right)\geq\mathcal{I}_{b,i}\left(\mathbf{h}_{b,i}^{{\rm W}H}\right)$. Then, ${\rm P_{3}}$ can be recast into

${\rm P_{4}}$ is still non-convex due to the quadratic term. One efficient way to deal with quadratic constraint is SDR [26]. Auxiliary matrices $\mathbf{W}_{s,i}=\mathbf{w}_{s,i}\mathbf{w}_{s,i}^{H}$, $\mathbf{H}_{s,i}^{\rm S}=\mathbf{h}_{s,i}^{\rm S}\mathbf{h}_{s,i}^{{\rm S}H}$, $\mathbf{H}_{s,i}^{\rm W}=\mathbf{h}_{s,i}^{\rm W}\mathbf{h}_{s,i}^{{\rm W}H}$, $\mathbf{W}_{b,i}=\mathbf{w}_{b,i}\mathbf{w}_{b,i}^{H}$, $\mathbf{H}_{b,i}^{\rm S}=\mathbf{h}_{b,i}^{\rm S}\mathbf{h}_{b,i}^{{\rm S}H}$ and $\mathbf{H}_{b,i}^{\rm W}=\mathbf{h}_{b,i}^{\rm W}\mathbf{h}_{b,i}^{{\rm W}H}$ are introduced. By applying SDR, the trace of matrix replaces the quadratic term. For example, $||\mathbf{w}_{s,k}||_{2}^{2}$ is replaced by ${\rm Tr}(\mathbf{W}_{s,i})$ and $|\mathbf{h}_{s,i}^{{\rm S}H}\mathbf{w}_{s,i}|^{2}$ is replaced by ${\rm Tr}(\mathbf{H}^{\rm S}_{s,i}\mathbf{W}_{s,i})$. The inter-cluster interference function after SDR can be expressed as

and

Hence, ${\rm P_{4}}$ can be recast as the following form

where $C_{1}=\varepsilon_{K}^{-1}(\bar{\varepsilon}_{0})(\varepsilon_{K}^{-1}(\bar{\varepsilon}_{0})+2)$ and $C_{2}=\gamma_{0}(\gamma_{0}+2)$. To ensure ${\rm P_{5}}$ is convex, we temporarily disregard the rank one constraints (44j) and (44k). As a result, we obtain a convex formulation that

Since the rank one constraint is disregarded in ${\rm P_{6}}$, the optimal value of ${\rm P_{6}}$ might not be the optimal value of ${\rm P_{4}}$. Let $\mathbf{W}_{s,i}^{*},\forall i$ and $\mathbf{W}_{b,i}^{*},\forall i$ denote the optimal solution of ${\rm P_{6}}$. If ${\rm rank}(\mathbf{W}_{s,i}^{*})=1,\forall i$ and ${\rm rank}(\mathbf{W}_{b,i}^{*})=1,\forall i$ hold, it suggests the optimal solution of ${\rm P_{5}}$ can be successfully recovered from $\mathbf{W}_{s,i}^{*},\forall i$ and $\mathbf{W}_{b,i}^{*},\forall i$ by eigen-decomposition. However, if the above equations do not hold, only the suboptimal solution of ${\rm P_{5}}$ can be obtained by Gaussian Randomization.

Although the aforementioned P-CNOMA scheme allows each user to fully utilize the whole spectrum and achieve the minimal transmission power, however, it has the feasibility issue that arises from the interference (6) and (7). As transmission power increases, interference power rises accordingly. Consequently, once the target QoS exceeds a certain threshold, further increases in transmission power cannot achieve the desired target. To address this issue, we propose the H-CNOMA scheme, where each cluster occupies one orthogonal frequency sub-channel, with two users within the same cluster sharing the sub-channel. In this scheme, OMA is adopted between clusters while NOMA is adopted between users within the same cluster. Due to orthogonality of sub-channels, the inter-cluster interference is eliminated, but at the cost of reduced bandwidth. Since each cluster is independent to other clusters, we can focus on a single cluster when formulating the optimization problem. Let take a S-user cluster as an example, the transmission power minimization problem can be expressed as