Spectrum Allocation and Power Control Based on Newton’s Method for Weighted Sum Power Minimization in Overlay Spectrum Sharing

Abstract

As the popularity of smartphones, wearable devices, intelligent vehicles, and countless other devices continues to rise, the surging demand for mobile data traffic has resulted in an increasingly crowded electromagnetic spectrum. Spectrum sharing serves as a solution to optimize the utilization of wireless communication channels, allowing various types of users to share the same frequency band securely. This paper investigates spectrum allocation and power control problems in overlay spectrum sharing, with a focus on promoting green communication. Maximizing weighted sum energy efficiency (WSEE) requires solving complex multiple-ratio fractional programming (FP) problems. In contrast, weighted sum power (WSP) minimization offers a more straightforward approach. Moreover, because WSP is directly related to users’ power consumption, we can dynamically adjust their weights to balance their residual energy. We prioritize WSP minimization over the more common WSEE maximization. This choice not only simplifies computation but also maintains users’ quality of service (QoS) requirements. The joint optimization for multiple primary users (PUs) and secondary users (SUs) can be decomposed into two components: a weighted bipartite matching problem and a series of convex resource allocation problems. Utilizing Newton’s method, our system-level simulation results show that the proposed scheme achieves optimal performance with minimal computational time. We explore strategies to accelerate the proposed scheme by refining the selection of initial values for Newton’s method.

1. Introduction

1.1. Background

Over the last few decades, multifarious devices, including smartphones, wearable devices, and intelligent vehicles, have steadily gained popularity. All these devices rely on the same wireless frequency band to share data, voice, and images. The accompanying surge in demand for mobile data traffic is crowding the electromagnetic spectrum [1].

Spectrum sharing can effectively address this issue and improve overall system efficiency and accessibility. It also maintains the quality of service (QoS) for primary users (PUs) [2]. Current wireless communication systems have widely deployed this technology, e.g., device-to-device (D2D) communications in cellular networks and 5G New Radio (NR) over Long-Term Evolution (LTE) carriers. It can be foreseen that spectrum sharing will continue to be a vital research frontier in next-generation wireless networks [3].

Cognitive radio (CR) [4] has played a significant role in motivating the development of spectrum sharing. It enables a secondary user (SU) to access a PU’s licensed band, typically through either underlay, overlay, or interweave mode, thereby enhancing system spectrum efficiency (SE). Specifically, in the underlay mode, SUs can operate simultaneously on PUs’ licensed bands under the interference temperature limits. As a result, the SUs must carefully manage spectrum access and power control. In the overlay mode, SUs opportunistically access idle licensed bands through spectrum sensing. This approach guarantees no interference to PUs. The interweave mode is similar to the overlay mode. This mode also allows users to access the licensed spectrum bands opportunistically. Differently, it dilutes the concepts of PUs and SUs. Ongoing users are considered PUs, while incoming users are considered SUs [5].

Beyond the three typical methods, cooperative spectrum sharing—a form of overlay sharing that uses relaying techniques—provides a critical supplement [6]. It essentially belongs to overlay spectrum sharing. In this mode, SUs act as relays, helping PUs with poor channel conditions communicate. After the PUs complete their data transmission, the SUs can access the corresponding licensed bands. By dedicating part of their energy budget to relay PU transmissions, SUs gain temporary access to licensed bands. This creates a win–win scenario where both user types benefit [7].

In the abovementioned modes, spectrum allocation and power control play a significant role in enhancing overall system performance while ensuring the QoS of different priority levels for heterogeneous users. On the one hand, spectrum allocation involves strategically assigning frequency bands to different users to optimize bandwidth usage and minimize interference [8]; on the other hand, power control refers to managing transmission power levels across different devices, which can enhance the corresponding received signal-to-noise ratio (SNR) levels while reducing overall energy consumption [9]. Effective spectrum allocation and precise power control facilitate seamless communication and ensure the required QoS for all users. These combined effects contribute to robust and efficient communication systems.

Considering sustainable development in areas like the economy, the environment, and health, achieving green communication has become a key focus in designing and operating current and future wireless communication networks [10]. Non-Orthogonal Multiple Access (NOMA) is widely adopted in 5G applications like massive machine-type communication (mMTC). However, Enhanced Mobile Broadband (eMBB)—which relies on orthogonal frequency division multiple access (OFDMA)—will remain a dominant use case in the near future. Therefore, research on overlay spectrum sharing is still of substantial significance. This paper investigates spectrum allocation and power control in overlay spectrum sharing to align with green communication.

1.2. Related Work

Effective spectrum allocation and power control enable both primary links (PLs) and secondary links (SLs) to operate at a high-performance level. Various mathematical frameworks and performance metrics have been applied to develop network-centric and user-centric schemes [11]. A network-centric scheme requires a central controller and typically adopts mathematical frameworks such as convex optimization [12,13] and fractional programming (FP) [14,15,16]. In contrast, a user-centric scheme is implemented in a distributed manner and mainly relies on frameworks such as contract theory [17,18] and game theory [19,20,21].

For an extended period, beginning with the early days of CR, the primary emphasis in spectrum allocation and power control has been optimizing overall system throughput or achievable user rates. For example, the researchers of [12] introduced a cooperative D2D communication framework where cellular and D2D users can switch between the underlay, overlay, or cooperative spectrum-sharing mode. Based on this framework, they proposed an adaptive mode selection and spectrum allocation scheme to maximize the D2D users’ overall throughput. The simulation results showed that the proposed framework and scheme are effective in various scenarios. The study in [13] proposed an overlay spectrum-sharing framework using three technologies: cooperative relay, full-duplex (FD), and NOMA. Additionally, it integrated simultaneous wireless information and power transfer (SWIPT) techniques. Under this framework, they derived the throughput in tight closed-form approximations for the primary and secondary networks. After that, a rapid convergent iterative algorithm was proposed to optimize the system throughput.

In [17], the researchers examined cooperative spectrum sharing with incomplete information, i.e., SUs’ wireless channel characteristics are unknown to a PU. Using contract theory, they modeled the interaction between the PU and SUs as a labor market. Then, they studied the optimal contract design and proposed an algorithm in which the PU aims to maximize the average transmission rate. At the same time, each SU aims to choose relay power and time allocation to maximize its normalized payoff. The authors of [18] further extended the research of [17] to the scenario of OFDMA-assisted cognitive Internet of Things (IoT) networks.

Based on matching game theory, the authors in [19] developed a two-timescale wireless resource allocation scheme for cooperative overlay D2D communication systems. The scheme operates at two timescales: (1) long-term matching between cellular and D2D links, and (2) short-term transmission time allocation for matched pairs. After that, they proposed a distributed pairing algorithm to optimize the expected data rate of the D2D link while guaranteeing the QoS of the cellular link. In [20], the researchers studied the performance of overlay D2D communication links based on carrier-sense multiple access (CSMA) protocols. Each D2D user aims to achieve its data rate requirement while maximizing its payoff. A Stackelberg game was established, where the base station (BS) acts as a leader in regulating the D2D users’ payoff. In [21], the researchers designed a group-based framework without channel quality indicators (CQIs) that realizes a pricing-based Stackelberg game for mode selection and spectrum allocation feedback. They proposed an incentive-compatible pricing strategy to optimize the achievable rate of potential D2D links while effectively preventing selfish D2D users from negatively impacting overall system performance.

Green communication is now a cornerstone of wireless system design, driven by a growing global emphasis on sustainable development. Consequently, energy efficiency (EE)—the ratio between global achievable throughput and power consumption—has become a primary metric for evaluating system cost-effectiveness [22]. For instance, the authors in [14] addressed the resource allocation problem for one PL and SL pair in the overlay or underlay mode. The optimization goal is to maximize the SL’s EE while guaranteeing the PL’s minimum required rate. For the overlay mode, two algorithms were proposed: one solves a sequence of more accessible FP to fulfill the first-order optimality conditions of the original problem; the other enjoys a weaker optimality claim but with even lower computational complexity. In [15], researchers addressed spectrum scarcity and energy limitations in cognitive radio networks. They proposed a joint optimization of transmission time allocation and power control for radio frequency energy harvesting systems. They converted the non-convex EE maximization to a convex problem via the proposed approximate convex policy for the co-frequency interference (CO-ACP) scheme. Then, they used Frank–Wolfe (FW) with one-dimensional linear programming to achieve the optimal solution. The numerical results indicated that the CO-ACP can achieve a tight lower-bound optimum solution.

Since EE focuses on the entire system performance, taking EE as the objective function as resource allocation problems will not indicate each user’s priority. Weighted sum energy efficiency (WSEE), defined as the weighted sum of users’ EE, considers individual users’ EE and overall system EE. It is suitable for heterogeneous networks with diverse EE requirements for different users. For this reason, more and more studies have adopted WSEE rather than EE as the objective function for systems such as D2D communications and cognitive radio networks. For example, the authors in [16] investigated D2D communications overlaying cellular networks with spectrum–power trading. In their system, D2D users act as cellular users’ (CUs’) relays, exchanging their transmit power for the opportunities to access the licensed bands. The proposed spectrum–power trading schemes analyze two scenarios: public-interested and self-interested D2D users. The optimization objectives include maximizing WSEE via relay selection, spectrum allocation, and power control.

Nevertheless, EE and WSEE maximization problems are challenging due to their non-convex form. Specifically, EE maximization results in a single-ratio FP problem, while the latter involves multiple-ratio FP problems [23]. Conventional approaches like Charnes–Cooper transformation (CCT) and Dinkelbach’s algorithm effectively solve single-ratio FP problems. However, they struggle with multiple-ratio FP cases. Quadratic transform is able to handle multiple-ratio FP problems but a series of prerequisites must be satisfied [24]. Overall, these methods have already established a very mature solution framework. However, significant computational complexity is required within this framework to obtain a near-optimal solution. In addition, when jointly optimizing spectrum allocation and power control for EE or WSEE maximization, the resulting mixed-integer fractional programming (MIFP) problems will be more challenging.

In recent years, cellular networks have rapidly evolved toward being ultra-dense and heterogeneous, raising increasingly higher requirements for wireless resource management [25]. With this trend, artificial intelligence (AI) technology deeply integrates with wireless communication networks and plays a crucial role in spectrum allocation and power control [26]. With the aid of neural networks (NNs), machine learning (ML) methods such as supervised learning, unsupervised learning, and reinforcement learning (RL) have overcome the difficulties mentioned above to a certain extent when dealing with EE or WSEE maximization problems. For example, the authors in [27] proposed a resource allocation strategy based on deep learning (DL) for underlay cognitive radio networks with multiple-input–single-output (MISO) interference channels. The proposed scheme optimizes SUs’ transmit power and beamforming vector to maximize their sum EE while ensuring the QoS requirements for both PUs and SUs. The authors designed a deep neural network (DNN) architecture with three interconnected units. This DNN model effectively approximates the optimal resource allocation strategy. The researchers in [28] investigated transmit power control in D2D communication networks via a deep reinforcement learning (DRL) method. To maximize overall system EE, a dynamic power optimization scheme with two parallel deep Q networks (DQNs) was proposed. The numerical results showed that the proposed scheme can achieve higher EE while guaranteeing system throughput requirements. However, as is well-known, ML tasks require a large amount of training data while also bringing enormous computational burdens and energy costs [26].

1.3. Motivation, Contribution, and Organization

The 3GPP committed to developing 6G specifications from December 2023 onward. Heterogeneous network paradigms such as D2D communications are expected to expand significantly in the 6G era [25]. However, introducing new technologies like terahertz communication will substantially increase network infrastructures and connected devices, which will, in turn, drive up energy costs [29]. Moreover, there is no doubt that 6G will have increasingly diversified and stringent requirements for the QoS of smartphones, whose standby time is limited by the capacity of lithium-ion batteries. Thus, it is crucial to reduce and balance users’ energy consumption while ensuring their QoS requirements in energy-limited communication systems.

Firstly, contrary to the concept of green communication, some studies have placed a one-sided emphasis on maximizing SE, whether in terms of system throughput or user sum rates, e.g., [12,13,17,18,19,20,21].

Secondly, schemes maximizing EE or WSEE under the FP framework require stringent prerequisites and result in high computational complexity, e.g., [14,15,16].

Thirdly, although complex FP problems such as MIFP can achieve near-optimal solutions by leveraging DL, ML tasks depend on extensive training data and considerable computing resources. These demands can result in increased energy costs, e.g., [27,28].

Finally, a considerable portion of the research effort primarily focuses on maximizing the performance of the secondary system while guaranteeing PUs’ minimum QoS requirements, e.g., [12,14,16,19,27].

In the upcoming 6G era, the primary focus will be reducing and balancing terminal energy consumption while satisfying users’ QoS requirements, thereby extending network lifetime as much as possible [1,10]. This paper proposes weighted sum power (WSP) as a novel objective function for overlay spectrum sharing. WSP is defined as the sum of weighted users’ power consumption, addressing resource allocation challenges while aligning with 6G’s EE goals. Unlike WSEE in multiple-ratio FP form, WSP minimization can typically be converted into a convex form. Additionally, because WSP is directly related to users’ power consumption, it allows for effectively managing their power budgets by dynamically adjusting the corresponding weights.

To sum up, the trade-offs between WSP minimization, EE maximization, WSEE maximization, and other energy-related objectives are indeed critical in 6G green communication systems. WSP minimization focuses on reducing and balancing absolute power consumption across users. This aligns with 6G’s emphasis on extending terminal battery life and managing energy costs in dense networks. Its linear formulation enables low-complexity convex optimization, making it suitable for dynamic resource allocation in heterogeneous networks. EE maximization focuses on system EE, which optimizes the ratio of throughput to power consumption. WSEE maximization prioritizes EE per user, which ensures fairness in energy utilization across diverse user requirements (e.g., high-rate vs. low-power devices). Their FP nature introduces non-convexity, leading to higher computational complexity and stringent prerequisites for convergence. In scenarios with strict latency requirements (e.g., industrial IoT), WSP minimization ensures timely transmissions by prioritizing power budget adherence, whereas EE or WSEE maximization might sacrifice latency to maintain high EE ratios. Therefore, WSP minimization is particularly advantageous for 6G’s energy-limited, latency-sensitive, and heterogeneous environments. While EE or WSEE maximization remains popular for most use cases, the simplicity, scalability, and alignment with 6G’s green communication goals make WSP minimization a pragmatic choice.

With WSP as the objective function, we jointly optimize the performance of both PUs and SUs. The joint power control, spectrum allocation, and link matching is formulated as an integer programming problem, which can be decomposed into two parts: (1) a series of spectrum allocation and power control sub-problems, and (2) a weighted bipartite graph matching between potentially matched primary and secondary link pairs. Given the linear nature of WSP, these sub-problems can be easily proved convex and effectively solved using Newton’s method. For the weighted bipartite matching problem, we employ the well-established Kuhn–Munkres (KM) algorithm to find the optimal solution.

In our prior work [30], we have proposed a suboptimal resource allocation scheme with low computation time based on Newton’s method and graphical method for the cooperative spectrum sharing in cognitive radio networks (i.e., the scenario where SUs serve as relays in exchange for the opportunity to access PUs’ authorized spectrum). This work was also dedicated to minimizing system WSP. However, unlike the prior work, this paper focuses on a more common spectrum-sharing mode, i.e., overlay spectrum sharing. We derive an optimal spectrum allocation and power control scheme using Newton’s method and the KM algorithm. The differences between [30] and this paper are listed in

Table 1. Comparison between [30] and the current work.

Comparison Dimension	Prior Work [30]	Current Work
Scenario	Cooperative spectrum sharing	Overlay spectrum sharing
Methodology	Newton’s method + graphical method	Newton’s method + KM algorithm
Optimization goal	WSP minimization	WSP minimization
Solution type	Low-complexity, suboptimal	Optimal

.

The main contributions of our paper are summarized below:

1. To the best of our knowledge, it is the first time that WSP has been taken as the objective function to handle resource allocation problems for overlay spectrum sharing. We also present the system model and frame structure associated with the proposed centralized scheme;

2. Utilizing Newton’s method, the optimal solutions for the spectrum allocation and power control sub-problems can be found with very short computational time;

3. We also propose corresponding heuristic approaches to accelerate the convergence of Newton’s method, which makes our scheme more suitable for practical real-time applications. Through simulation, the effectiveness and efficiency of our proposed centralized scheme and heuristic approaches are validated.

The rest of this paper is organized as follows: In Section 2, we present the system model and formulate a joint problem of spectrum allocation, power control, and link-matching to minimize system WSP. Section 3 proposes a scheme based on the KM algorithm and Newton’s method. In this section, we also discuss the initial value selection in Newton’s method and weight adjustment. In Section 4, we present simulation results to evaluate the performance of the proposed scheme. Finally, we conclude this paper in Section 5.

2. System Model and Problem Formulation

We define a primary transmitter (PT) and its corresponding primary receiver (PR) as a PL; a secondary transmitter (ST) and its corresponding secondary receiver (SR) are referred to as an SL. This paper considers a scenario with multiple PLs and SLs. Figure 1 illustrates an example of our system model and its corresponding frame structure as an application for D2D communications in cellular networks based on OFDMA. In this example, a CU and the BS serve as a PL, while a D2D pair—consisting of a D2D transmitter (DT) and a D2D receiver (DR)—acts as an SL. A D2D pair can opportunistically access licensed bands in the overlay mode. Since cellular downlink transmission operates at relatively high power levels and has a sufficient energy budget, our focus will be on uplink transmission. For simplicity, the transmission directions of all the SLs remain unchanged. We denote $\mathcal{P}=\left\{{\mathcal{p}}_1,\dots ,{\mathcal{p}}_M\right\}$ and $\mathcal{S}=\left\{{\mathcal{s}}_1,\dots ,{\mathcal{s}}_N\right\}$ as the sets of the PLs and SLs, where $M=\left|\mathcal{P}\right|$ and $N=\left|\mathcal{S}\right|$. A list of the main symbols in this paper can be found in

Table 2. List of main nomenclature.

Symbol	Definition
$\mathcal{P}$	PL set
$\mathcal{S}$	SL set
$M$	Number of PLs
$N$	Number of SLs
${\mathcal{p}}_m$	PL $m$
${\mathcal{s}}_n$	SL $n$
$∆p$	Granularity of power levels
$∆\theta$	Granularity of spectrum-sharing factor levels
$n_0$	Additive white Gaussian noise power density
$g_{\mathcal{p},m}$	Channel gain between PT $m$ and PR $m$(the BS)
$g_{\mathcal{s},n}$	Channel gain between ST $n$ and SR $n$
$p_{\mathcal{p},mn}$	Transmit power of PT $m$(when ${\mathcal{p}}_m$ matches with ${\mathcal{s}}_n$)
$p_{\mathcal{s},mn}$	Transmit power of ST $n$
${\theta }_{mn}$	Spectrum-sharing factor for ${\mathcal{p}}_m$ and ${\mathcal{s}}_n$
$S_{\mathcal{p},mn}$	Achievable SE of ${\mathcal{p}}_m$ within one subframe
$S_{\mathcal{s},mn}$	Achievable SE of ${\mathcal{s}}_n$ within one subframe
$P_{\mathcal{p},mn}$	Power consumption of ${\mathcal{p}}_m$ within one subframe
$P_{\mathcal{s},mn}$	Power consumption of ${\mathcal{s}}_n$ within one subframe
${\alpha }_{mn}$	Matching indicator for ${\mathcal{p}}_m$ and ${\mathcal{s}}_n$
${\mu }_m$	Weight of ${\mathcal{p}}_m$
${\nu }_n$	Weight of ${\mathcal{s}}_n$
$Q_{\mathcal{p},m}$	QoS requirement of ${\mathcal{p}}_m$
$Q_{\mathcal{s},n}$	QoS requirement of ${\mathcal{s}}_n$
$p_{min}$	Minimum transmit power
$p_{max}$	Maximum transmit power
$w_{mn}$	WSP of link pair ${\mathcal{p}}_m$ and ${\mathcal{s}}_n$

.

In LTE and 5G NR, resource blocks (RBs) are the basic units of wireless time-frequency resources allocated to user equipment (UE). Generally, the PR, i.e., the BS, allocates a single RB or contiguous RBs to a PT, i.e., a CU [31]. In this paper, referring to [12,19], the number of RBs is assumed to equal $M$, and the BS only assigns one RB to one PL. Furthermore, each SL can access only one RB, while the PL allows only one SL to occupy its RB, so the orthogonality of spectrum resources can be preserved.

Our frame structure is compatible with LTE and 5G NR. Each frame starts with a preprocessing stage that includes pilot transmission, channel state information (CSI) reporting, and wireless resource allocation, followed by ten spectrum-sharing subframes. The PTs and the STs first send pilots. Then, the PR and the SRs estimate the channel gains and report CSI. With the CSI, the resource allocation can be determined. The details of spectrum sharing between one matched PL and SL pair are described as follows.

When ${\mathcal{p}}_m$ matches with ${\mathcal{s}}_n$, where $m\in \left\{1,\dots ,M\right\}$ and $n\in \left\{1,\dots ,N\right\}$, each corresponding spectrum-sharing subframe is split into two phases. In phase one, PT $m$ transmits its data to the PR using power $p_{\mathcal{p},mn}$ while ST $n$ remains silent. In phase two, when PT $m$ has completed its uplink transmission, ST $n$ can fulfill its data transmission to its corresponding SR using power $p_{\mathcal{s},mn}$ on the RB of PT $m$. If we denote the duration of phase one as ${\theta }_{mn}$ of an RB in the time domain, phase two will last $1-{\theta }_{mn}$ of an RB, where ${\theta }_{mn}$ denotes the spectrum-sharing factor and ${\theta }_{mn}\in \left(\mathrm{0,1}\right)$.

Discrete power levels are widely used in current wireless communication standards, such as LTE-A and 5G [32]. This approach is particularly beneficial for transmitters with limited capabilities due to hardware constraints, as seen in machine-type communications. Thus, this paper chooses the values of $p_{\mathcal{p},mn}$ and $p_{\mathcal{s},mn}$ from $\left\{p_{min},p_{min}∆p,p_{min}{\left(∆p\right)}^2,\dots ,p_{max}\right\}$, where $∆p$ is the granularity of the power levels. Furthermore, we assume that the spectrum-sharing factor levels are also discrete. ${\theta }_{mn}$ chooses its value from $\left\{∆\theta ,2∆\theta ,\dots ,1-∆\theta \right\}$.

The achievable SEs of ${\mathcal{p}}_m$ and ${\mathcal{s}}_n$ within one subframe are given as follows:

$$S_{\mathcal{p},mn}={\theta }_{mn}{\mathit{log}}_2\left(1+p_{\mathcal{p},mn}{g}_{\mathcal{p},m}/n_0\right),$$

(1)

$$S_{\mathcal{s},mn}=\left(1-{\theta }_{mn}\right){\mathit{log}}_2\left(1+p_{\mathcal{s},mn}{g}_{\mathcal{s},n}/n_0\right),$$

(2)

where $n_0$ is the additive white Gaussian noise power density; $g_{\mathcal{p},m}$ and $g_{\mathcal{s},n}$ denote the channel gains between PT $m$ and PR $m$ or ST $n$ and SR $n$, respectively. The channel characteristics include both path loss and multipath fading. We use a block fading model, where all channel gains change independently from one frame to the next. This means that within a single frame, all channel gains are considered to be constant.

The transmit power consumption of ${\mathcal{p}}_m$ and ${\mathcal{s}}_n$ within the corresponding subframe can be expressed as outlined below:

$$P_{\mathcal{p},mn}={\theta }_{mn}{p}_{\mathcal{p},mn},$$

(3)

$$P_{\mathcal{s},mn}=\left(1-{\theta }_{mn}\right)p_{\mathcal{s},mn}.$$

(4)

Green communication focuses on minimizing network energy consumption while meeting the QoS requirements. This strategy also supports sustainability goals, including carbon footprint reduction and environmental health preservation. As mentioned in the introduction, EE is the most critical performance metric in line with green communication. It focuses more on system performance but neglects users’ priority. WSEE balances system-level EE and user fairness. This metric is particularly effective for networks with diverse user priorities. Network operators can leverage predefined weights to tailor their performance more effectively. Nevertheless, a WSEE maximization problem is often challenging since it belongs to multiple-ratio FP. If any ratio of the sum does not follow a concave–convex form, or its feasible region is non-convex, its optimization will be extremely complex. In this instance, standard global optimization algorithms will face limitations in practice, often exhibiting exponential complexity. When it comes to joint spectrum allocation and power control, the resulting MIFP will be even more difficult to solve.

In the upcoming 6G era, the energy consumption of network infrastructure and connected terminals will further increase with the activation of the high-frequency spectrum. In order to alleviate the contradiction between stagnant battery energy storage technology and the increasing energy consumption of mobile terminals, some researchers have regarded energy consumption as the objective function in resource allocation optimization for a long time [33]. Energy consumption minimization, like EE optimization, ignores user priorities. This oversight risks depleting specific users’ energy budgets prematurely, which shortens the overall network lifetime. Inspired by the above analysis, we introduce WSP as the objective function for our optimization problem.

Omitting the circuit power consumption, we express the WSP of a matched link pair ${\mathcal{p}}_m$ and ${\mathcal{s}}_n$ as follows:

$$w_{mn}={{\mu }_{m}P}_{\mathcal{p},mn}+{\nu }_n{P}_{\mathcal{s},mn},$$

(5)

where ${\mu }_m$ and ${\nu }_n$ are the positive weights of ${\mathcal{p}}_m$ and ${\mathcal{s}}_n$.

A value of ${\alpha }_{mn}=1$ indicates that ${\mathcal{p}}_m$ and ${\mathcal{s}}_n$ are matched, otherwise ${\alpha }_{mn}=0$. Denoting ${\mathbf{p}}_{\mathcal{p}}=\left\{p_{\mathcal{p},\mathrm{m}\mathrm{n}}\right\}$, ${\mathbf{p}}_{\mathcal{s}}=\left\{p_{\mathcal{s},mn}\right\}$, $\mathsf{\alpha }=\left\{{\alpha }_{mn}|{\alpha }_{mn}=\right.\left.\mathrm{0,1}\right\}$, $\mathsf{\theta }=\left\{{\theta }_{mn}|{\theta }_{mn}\in \left(\mathrm{0,1}\right)\right\}$, the optimization problem for the overall system WSP minimization is expressed as stated below:

$$\underset{\left\{{\mathit{p}}_{\mathcal{p}},{\mathit{p}}_{\mathcal{s}},\mathit{\alpha },\mathit{\theta }\right\}}{\mathit{min}}{\sum }_{m=1}^M{\sum }_{n=1}^N{\alpha }_{mn}{w}_{mn},$$

(6)

$$\mathrm{s}.\mathrm{t}. \mathrm{C}1: S_{\mathcal{p},mn}\ge Q_{\mathcal{p},m},$$

$$\hspace{1em}\hspace{1em}\mathrm{C}2: S_{\mathcal{s},mn}\ge Q_{\mathcal{s},n},$$

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{C}3: p_{min}\le p_{\mathcal{p},mn}\le p_{max},$$

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{C}4: p_{min}\le p_{\mathcal{s},mn}\le p_{max},$$

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{C}5: {\sum }_{m=1}^M{\sum }_{n=1}^N{\alpha }_{mn}\in \left\{\mathrm{0,1}\right\},$$

$$\hspace{1em}\hspace{1em}\mathrm{C}6: 0<{\theta }_{mn}<1,$$

where C1 and C2 ensure the minimum SE requirements for all the links; C3 and C4 constrain their minimum transmit power ($p_{min}$) and maximum transmit power ($p_{max}$); C5 refers to the one-to-one correspondence matching between $\mathcal{P}$ and $\mathcal{S}$; C6 constrains the value of the spectrum-sharing factor.

Equation (6) is a mixed integer nonlinear programming (MINLP) problem. Due to C5 and the structure of its objective function, we can decompose Equation (6) into a series of spectrum allocation and power control sub-problems at a lower level, along with a minimum weighted bipartite matching at a higher level. This decoupling naturally leads to a centralized solution.

3. The Proposed Scheme for WSP Minimization

3.1. Spectrum and Power Allocation Based on Newton’s Method for a Matched PL-SL PAIR

In this subsection, we first discuss the spectrum allocation and power control for one matched PL-SL pair, i.e., ${\mathcal{p}}_m$ and ${\mathcal{s}}_n$, during one spectrum-sharing subframe, expressed as follows:

$$\underset{\left\{p_{\mathcal{p},\mathit{mn}},p_{\mathcal{s},\mathit{mn}},{\theta }_{\mathit{mn}}\right\}}{\mathit{min}}{w}_{mn},$$

(7)

$$\mathrm{s.t.} \mathrm{C}1, \mathrm{C}2, \mathrm{C}3, \mathrm{C}4, \mathrm{and} \mathrm{C}6.$$

When introducing the proposed scheme in this subsection, we omit the subscripts $\mathrm{m}$, $\mathrm{n}$, and $\mathrm{m}\mathrm{n}$ for clarity and simplicity.

From C1 and C3, we can obtain a lower bound on $p_{\mathcal{p}}$ as outlined below:

$$p_{\mathcal{p}}^{low}=max\left\{p_{min},\left(2^{Q_{\mathcal{p}}/\theta }-1\right)n_0/g_{\mathcal{p}}\right\}.$$

(8)

Similarly, a lower bound on $p_{\mathcal{s}}$ can be obtained from C2 and C4 as follows:

$$p_{\mathcal{s}}^{low}=max\left\{p_{min},\left[2^{Q_{\mathcal{s}}/\left(1-\theta \right)}-1\right]n_0/g_{\mathcal{s}}\right\}.$$

(9)

It is easy to validate that the objective function in Equation (7) monotonically increases with $p_{\mathcal{p}}$ and $p_{\mathcal{s}}$. Their solutions can be expressed as functions of $\theta$, i.e., $p_{\mathcal{p}}^{\mathrm{*}}=p_{\mathcal{p}}^{low}$ and $p_{\mathcal{s}}^{\mathrm{*}}=p_{\mathcal{s}}^{low}$. Therefore, Equation (7) can be rewritten as presented below:

$$\underset{\theta }{\mathit{min}}\mu \theta p_{\mathcal{p}}^{low}+\nu \left(1-\theta \right)p_{\mathcal{s}}^{low}.$$

(10)

$$\mathrm{s}.\mathrm{t}. 0<\theta <1.$$

Next, we will discuss the optimal solution for $\theta$ based on four possible combinations of $p_{\mathcal{p}}^{low}$ and $p_{\mathcal{s}}^{low}$.

• Case 1

If $p_{\mathcal{p}}^{low}=\left(2^{Q_{\mathcal{p}}/\theta }-1\right)n_0/g_{\mathcal{p}}$ and $p_{\mathcal{s}}^{low}=\left[2^{Q_{\mathcal{s}}/\left(1-\theta \right)}\right.$ $\left.-1\right]n_0/g_{\mathcal{s}}$, defining the following auxiliary functions

$$f_1\left(x\right)={\displaystyle \frac{\mu n_0}{g_{\mathcal{p}}}}x\left(2^{Q_{\mathcal{p}}/x}-1\right),$$

(11)

$$f_2\left(x\right)={\displaystyle \frac{\nu n_0}{g_{\mathcal{s}}}}\left(1-x\right)\left[2^{Q_{\mathcal{s}}/\left(1-x\right)}-1\right],$$

(12)

the objective function in Equation (10) can be rewritten as outlined below:

$$w\left(x\right)=f_1\left(x\right)+f_2\left(x\right).$$

(13)

In this case, we can prove that Equation (10) is convex as follows.

Proposition: optimization problem Equation (10) in case 1 is convex.

Proof: All that is needed is to check the convexity of Equation (13). The first and second derivatives of $f_1\left(x\right)$ and $f_2\left(x\right)$ are calculated as follows:

$$f_1^{\prime }\left(x\right)={\displaystyle \frac{\mu n_0}{g_{\mathcal{p}}}}\left[2^{Q_{\mathcal{p}}/x}\left(1-{\displaystyle \frac{Q_{\mathcal{p}}\mathit{ln}2}{x}}\right)-1\right],$$

(14)

$$f_2^{\prime }\left(x\right)={\displaystyle \frac{\nu n_0}{g_{\mathcal{s}}}}\left[2^{Q_{\mathcal{s}}/\left(1-x\right)}\left({\displaystyle \frac{Q_{\mathcal{s}}\mathit{ln}2}{1-x}}-1\right)+1\right],$$

(15)

$$f_1^{\prime \prime }\left(x\right)={\displaystyle \frac{\mu n_0}{g_{\mathcal{p}}}}{\displaystyle \frac{{\left(Q_{\mathcal{p}}\mathit{ln}2\right)}^2}{x^3}}{2}^{Q_{\mathcal{p}}/x},$$

(16)

$$f_2^{\prime \prime }\left(x\right)={\displaystyle \frac{\nu n_0}{g_{\mathcal{s}}}}{\displaystyle \frac{{\left(Q_{\mathcal{s}}\mathit{ln}2\right)}^2}{{\left(1-x\right)}^3}}{2}^{Q_{\mathcal{s}}/\left(1-x\right)}.$$

(17)

Since $\theta \in \left(\mathrm{0,1}\right)$, obviously, $f_1^{\prime \prime }\left(x\right)>0$ and $f_2^{\prime \prime }\left(x\right)>0$. $w^{\prime \prime }\left(x\right)>0$ holds. Therefore, $w\left(x\right)$ is a convex function [34].

It is clear that $f_1^{\prime }\left(0^{+}\right)=-\mathrm{\infty }$ and $f_2^{\prime }\left(1^{-}\right)=+\mathrm{\infty }$, while $f_2^{\prime }\left(0^{+}\right)$ and $f_1^{\prime }\left(1^{-}\right)$ are not infinite. Thus, $w^{\prime }\left(0^{+}\right)=-\mathrm{\infty }$ and $w^{\prime }\left(1^{-}\right)=+\mathrm{\infty }$. According to the convexity of (13), $w^{\prime }\left(x\right)$ is monotonical so there must exist an $x^{\mathrm{*}}$ satisfying $w^{\prime }\left(x^{\mathrm{*}}\right)=0$, and the solution of $\theta$ equals $x^{\mathrm{*}}$.

Although the analytical solution of $x^{\mathrm{*}}$ cannot be obtained, we can access its numerical solution through iterative methods such as the Newton–Raphson method [35]. Specifically, if we initialized an $x_0\in \left(\mathrm{0,1}\right)$ and obtained $x_k$ after $k$ iterations, a more approximate $x_{k+1}$ is given by using Newton’s method as outlined below:

$$x_{k+1}=x_k-w^{\prime }\left(x_k\right)/w^{\prime \prime }\left(x_k\right).$$

(18)

When the stopping criterion holds, i.e., $\left|x_{k+1}-x_k\right|\le \epsilon$, $x_{k+1}$ is obtained as an approximation for $x^{\mathrm{*}}$, where $\epsilon$ is a predefined step tolerance.

• Case 2

If $p_{\mathcal{p}}^{low}=p_{min}$ and $p_{\mathcal{s}}^{low}=\left[2^{Q_{\mathcal{s}}/\left(1-\theta \right)}-1\right]n_0/g_{\mathcal{s}}$, defining

$$f_3\left(x\right)=\mu p_{min}x,$$

(19)

we can rewrite the objective function in Equation (10) as follows:

$$w\left(x\right)=f_2\left(x\right)+f_3\left(x\right).$$

(20)

The first and second derivatives of $f_3\left(x\right)$ are $f_3^{\prime }\left(x\right)=\mu p_{min}$ and $f_3^{\prime \prime }\left(x\right)=0$. Similarly, $w^{\prime \prime }\left(x\right)>0$. $w\left(x\right)$ is still convex. In this case, $w^{\prime }\left(0^{+}\right)=\mu p_{min}+{\displaystyle \frac{\nu n_0}{g_{\mathcal{s}}}}\left[2^{Q_{\mathcal{s}}}\left(Q_{\mathcal{s}}\ln 2-1\right)+1\right]$ and $w^{\prime }\left(1^{-}\right)=+\mathrm{\infty }$.

If $w^{\prime }\left(0^{+}\right)<0$, the numerical solution of $x^{\mathrm{*}}$ can be found using Newton’s method described in case 1.

If $w^{\prime }\left(0^{+}\right)\ge 0$, $w^{\prime }\left(x\right)>0$ holds within $\left(\mathrm{0,1}\right)$. In order to minimize (20), $\theta$ should take its minimum. Thus, ${\theta }^{\mathrm{*}}=∆\theta$.

• Case 3

If $p_{\mathcal{p}}^{low}=\left(2^{Q_{\mathcal{p}}/\theta }-1\right)n_0/g_{\mathcal{p}}$ and $p_{\mathcal{s}}^{low}=p_{min}$, defining

$$f_4\left(x\right)=\nu \left(1-x\right)p_{min},$$

(21)

the objective function in Equation (10) can be rewritten as follows:

$$w\left(x\right)=f_1\left(x\right)+f_4\left(x\right).$$

(22)

The first and second derivatives of $f_4\left(x\right)$ are given by $f_4^{\prime }\left(x\right)$ $=-\nu p_{min}$ and $f_4^{\prime \prime }\left(x\right)=0$. $w^{\prime \prime }\left(x\right)>0$ also holds. In this case, $w^{\prime }\left(0^{+}\right)=-\mathrm{\infty }$ and $w^{\prime }\left(1^{-}\right)=-\nu p_{min}+{\displaystyle \frac{\mu n_0}{g_{\mathcal{p}}}}\left[2^{Q_{\mathcal{p}}}\left(1-Q_{\mathcal{p}}\ln 2\right)-1\right]$.

If $w^{\prime }\left(1^{-}\right)<0$, $w^{\prime }\left(x\right)<0$ holds and $\mathrm{w}\left(x\right)$ monotonically decreases within $\left(\mathrm{0,1}\right)$. In order to minimize (22), $\theta$ should take its maximum. Thus, ${\theta }^{\mathrm{*}}=1-∆\theta$.

If $w^{\prime }\left(1^{-}\right)\ge 0$, we can use Newton’s method to find the numerical solution of $x^{\mathrm{*}}$ similarly.

• Case 4

When $p_{\mathcal{p}}^{low}=p_{\mathcal{s}}^{low}=p_{min}$, the objective function in Equation (10) is rewritten as presented below:

$$w\left(x\right)=\left[\mu x+\nu \left(1-x\right)\right]p_{min}.$$

(23)

There are three possible optimal solutions for $\theta$ as outlined below:

If $\mu >\nu$, ${\theta }^{\mathrm{*}}=∆\theta$;

If $\mu <\nu$, ${\theta }^{\mathrm{*}}=1-∆\theta$;

If $\mu =\nu$, then ${\theta }^{\mathrm{*}}$ can take any value within the spectrum-sharing factor levels.

In summary, we discussed all possible optimal solutions regarding $\theta$. Then, the corresponding optimal solutions for $p_{\mathcal{p}}$ and $p_{\mathcal{s}}$ can be obtained through Equations (8) and (9).

3.2. Link Matching for Multiple PLs and SLs

Based on the above derivation, the spectrum allocation and power control scheme via Newton’s method for one matched PL-SL pair is taking shape. It should be noted that this scheme is centralized and requires a network central controller. In this subsection, we discuss the link matching to extend this scheme to general scenarios with multiple PLs and SLs.

In our system—shown in Figure 1—the public PR, i.e., the BS, performs as a central controller and determines the matching between $\mathcal{P}$ and $\mathcal{S}$ through the KM algorithm [36]. We assume complete CSI is available at the BS. According to the matching rule C5, we can formulate the link matching, which aims to minimize the system WSP as a minimum cost bipartite matching with the weights $\left\{w_{mn}\right\}$ obtained from Equation (7). Through the proposed spectrum allocation and power control strategy described in the previous subsection, a weight matrix can be formed at the PR as follows:

$$\mathit{W}=\left[\begin{array}{cccc}{w}_{11}& w_{12}& \cdots & w_{1N}\\ w_{21}& w_{22}& \cdots & w_{2N}\\ ⋮& ⋮& \ddots & ⋮\\ w_{M1}& w_{M2}& \cdots & w_{MN}\end{array}\right].$$

(24)

To allocate wireless resources for $M$ PLs and $N$ SLs, the central controller, namely, the BS, should collect the instantaneous CSI via control channels at the beginning of every frame. Suppose a CSI report contains $A$ bits. The CSI of the SLs is required but the CSI of the PLs is not; therefore, the overhead for the CSI reporting is $MNA$ bits. As $M$ or $N$ increases, the CSI overhead linearly increases. One of our future research directions will be how to reduce CSI feedback and implement our scheme in partial CSI situations.

The time complexity of the KM algorithm is $O\left({\mathrm{m}\mathrm{a}\mathrm{x}\left\{M,N\right\}}^3\right)$. In typical 5G eMBB scenarios, the user capacity of a single cell usually ranges from hundreds to thousands. As the number of links increases, the execution time of the KM algorithm will also increase rapidly. Thus, the scalability of the proposed method in large-scale network scenarios is an important topic that concerns its practicality. In our prior work [37], we introduced cooperative link sets to accelerate the proposed deep reinforcement learning (DRL)-based hybrid centralized–distributed scheme and reduce signaling exchange. Unfortunately, similar mechanisms are challenging to apply to the proposed scheme in this paper.

Distance-constrained spanning trees (DCSTs) are vital for optimizing spectrum allocation and power control. They ensure efficient connectivity while minimizing interference and power consumption. In overlay spectrum sharing, structuring users within specific distance constraints enhances signal quality, upholds the QoS requirements, and reduces unnecessary power expenditure. This approach aligns with our goal of minimizing WSP. In [38], the authors address fault tolerance and robust communication networks by constructing multiple disjoint spanning trees that adhere to strict distance constraints.

On the other hand, DCSTs help reduce computational complexity by limiting the number of potential connections. This limitation accelerates the KM algorithm for link matching and enhances the convergence speed of Newton’s method. To address scalability challenges in ultra-dense networks, meta-heuristic approaches (e.g., genetic algorithms, particle swarm optimization, or simulated annealing) can efficiently find near-optimal solutions. This integration ensures fair energy distribution and improved spectrum reuse without excessive computational overhead. By structuring the network into clusters or subgraphs, DCSTs combined with meta-heuristic methods provide a more scalable and computationally feasible approach to real-time spectrum allocation and power control in next-generation wireless networks. In our future research, we will aim to further address the scalability issues of the proposed scheme through the use of DCSTs.

3.3. Discussion of the Initial Value Selection of Newton’s Method and Weight Adjustment

The rate of convergence indicates how quickly a real-valued sequence $\left\{x_k\right\}$ can approach its local optimum $x^{\mathrm{*}}$, which is essential for evaluating the performance of an iterative method. Newton’s method has a quadratic rate of convergence with order of convergence (time complexity) $O\left(\log \log \left(1/\epsilon \right)\right)$ and is extremely fast. However, Newton’s method does not necessarily converge. Depending on where it starts, it may “quadratically quickly” either converge or diverge. In practice, $x_0$ must be sufficiently close to $x^{\mathrm{*}}$. Otherwise, it may result in a large number of iterations or even fail to converge. To avoid this situation, we should carefully select its initial value $x_0$. Although there are sufficient conditions to ensure the convergence of Newton’s method, selecting initial values based on them is often quite complex. Therefore, this paper adopts a simplified method as follows.

First, when the spectrum sharing begins, we select an $x_0$, which fulfills

$$w^{\prime \prime \prime }\left(x_0\right)\ne 0.$$

(25)

and

$${\left|w^{\prime \prime }\left(x_0\right)\right|}^2>\left|{w\left(x_0\right)w}^{\prime \prime }\left(x_0\right)\right|/2.$$

(26)

This operation ensures Newton’s method converges in most cases [35].

Then, if the wireless environment changes slowly, we can use the optimal value of $\theta$ in the previous frame as the initial value of $\left\{x_k\right\}$.

Through the proposed joint spectrum allocation, power control, and link matching described above, the system WSP can be effectively minimized within each frame. Then, we will discuss how to extend the lifetime of the entire network, including all the PLs and SLs, as much as possible.

The minimum QoS requirement ensures the priority level of a link in terms of SE. On the other hand, it is essential to clarify the role of link weights in WSP. In WSEE maximization, the weights assigned to links determine their priority in terms of EE. Specifically, when we choose WSEE as the optimization objective, a link with a higher weight indicates a higher expected EE after resource allocation. Similarly, once the proposed WSP minimization scheme is implemented, a link with a higher weight will consume less energy. By reasonably setting their weights within each transmission frame, not only will the system power consumption be reduced but the energy consumption among different links will also be balanced.

As analyzed earlier, in WSP minimization, the weight of a link reflects the willingness to use its energy budget. Many studies are dedicated to extending network lifetime in the fields of wireless sensor networks and unmanned aerial vehicle (UAV)-assisted wireless networks. Most solutions, however, tend to have high computational complexity, and some require ML approaches [39]. However, there are some heuristic weight dynamic adjustment strategies. For example, we can set the weight of each link as inversely proportional to its residual energy at the beginning of the current transmission frame. By performing it this way, different links’ energy consumption rates can be balanced. In future research, we will jointly optimize weights and wireless resources to maximize the network lifetime.

Table 3. Proposed algorithm.

Begin

for $m=1:M$

for  $n=1:N$

select $x_0$ to fulfill $w^{\prime \prime \prime }\left(x_0\right)\ne 0$, ${\left|w^{\prime \prime }\left(x_0\right)\right|}^2>{\displaystyle \frac{\left|{w\left(x_0\right)w}^{\prime \prime }\left(x_0\right)\right|}{2}}$.

repeat

calculate $x_{k+1}$ using (18)

until $\left|x_{k+1}-x_k\right|\le \epsilon$

set $x^{\mathrm{*}}\approx x_{k+1}$, obtain $\left(p_{\mathcal{p},mn}^{*},p_{\mathcal{s},mn}^{*},{\theta }_{mn}^{*}\right)$

end for

end for

obtain ${\mathsf{\alpha }}^{*}$ through the KM algorithm with matrix $\mathit{W}$

End

shows the proposed algorithm with the heuristic initial value selection strategy.

3.4. Practical Deployment Limitations

While the proposed scheme demonstrates theoretical advantages in WSP, its real-world implementation faces three critical challenges that warrant further investigation:

Reliance on perfect CSI assumptions: Full CSI assumption is adopted in foundational works like [12,28], where algorithms achieve theoretical performance bounds but require high-overhead feedback. Partial CSI assumption reduces feedback overhead at the cost of performance degradation. These align better with dynamic environments where the CSI changes rapidly (e.g., >100 Hz Doppler shifts). While partial CSI assumptions may degrade absolute performance metrics (e.g., 2–3 dB loss in SINR estimation), they preserve the fundamental performance trends demonstrated in most existing works. The proposed scheme’s performance is contingent upon perfect CSI availability, which is challenging in dynamic wireless environments such as high-mobility scenarios. In practical deployment, the proposed scheme should integrate robust CSI estimation techniques like compressed sensing to address this challenge.

Scalability of centralized computation: As mentioned in Section 3.2, the $O\left({\mathrm{m}\mathrm{a}\mathrm{x}\left\{M,N\right\}}^3\right)$ complexity of the KM algorithm poses significant latency challenges in ultra-dense networks. A promising alternative lies in meta-heuristic optimization, which can help reduce computational complexity by limiting the number of potential connections, thereby accelerating the Kuhn–Munkres (KM) algorithm for link matching and enhancing the convergence speed of Newton’s method.

Power consumption imbalance: In battery-constrained IoT devices, the circuit power dominates total consumption, diminishing the benefits of transmission power optimization. Although our optimization focuses on transmission power, future work on holistic power models is required. Considering constant baseband processing overhead, implementing adaptive sleep scheduling could better align transmission–circuit power tradeoffs.

4. Simulation and Discussion

In this section, we validate the performance of the proposed scheme through simulation.

4.1. Simulation Setup and Baseline Schemes

The PTs (CUs), STs (DTs), and SRs (DRs) are randomly located in a single-cell cellular network. Unless otherwise specified, all numerical results derive from statistically rigorous Monte Carlo simulations based on 1000 independent trials conducted under varying channel conditions. We conduct the simulation evaluations on a laptop with a Ryzen 9 7945HX, an NVIDIA GeForce RTX 4080, and 64 GB memory. Some common simulation parameters are listed as follows and can also be found in

Table 4. Simulation parameters.

Symbol	Value
Cell radius, $R$	500 m
Number of PLs, $M$	10
Number of SLs, $N$	5~15
Granularity of power levels, $∆p$	1 dB
Granularity of spectrum-sharing factor levels, $∆\theta$	0.01
Gaussian noise spectral density, $n_0$	−174 dBm/Hz
Weight of ${\mathcal{p}}_m$, ${\mu }_m$	1
Weight of ${\mathcal{s}}_n$, ${\nu }_n$	1
QoS requirement of ${\mathcal{p}}_m$, $Q_{\mathcal{p},m}$	10 bps/Hz
QoS requirement of ${\mathcal{s}}_n$, $Q_{\mathcal{s},n}$	5 bps/Hz
Minimum transmit power, $p_{min}$	−40 dBm
Maximum transmit power, $p_{max}$	23 dBm
Step tolerance $\epsilon$	10−5

.

The cell radius is $R=500$ m. Referring to the 3GPP 38.811 S-band (2–4 GHz) urban scenario, we adopt a log-distance path-loss model for large-scale fading with a path-loss exponent 3.8 and path-loss coefficient ${10}^{3.453}$. For small-scale fading gains, we adopt a Rayleigh fading model and model them as independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian (CSCG) random variables, with a mean of zero and a variance of one.

The number of PLs is fixed as $M=10$, and the number of SLs is $N=5~15$. The Gaussian noise spectral density is $n_0=-174$ dBm/Hz. The weights of ${\mathcal{p}}_m$ and ${\mathcal{s}}_n$ are ${\mu }_m=1$ and ${\nu }_n=1$.

We set the granularities of power and spectrum-sharing factor levels as $∆p=1$ dB and $∆\theta =$0.01, respectively. The minimum and maximum transmit power are $p_{min}=-40$ dBm and $p_{max}=23$ dBm. The required minimum SEs are $Q_{\mathcal{p},m}=10$ bps/Hz and $Q_{\mathcal{s},n}=5$ bps/Hz. Moreover, the step tolerance is set as $\epsilon ={10}^{-5}$ for the stopping criterion.

The proposed scheme is compared with the following three baseline schemes.

1. Optimal scheme: This strategy found the optimal solution via the exhaustive search for all possible spectrum allocation and power control. Its link matching is still carried out through the KM algorithm because the KM algorithm is the optimal matching solution in our system;

2. DRL-based scheme: Referring to our prior work [37], we utilize DRL to handle spectrum allocation and power control problems within each link pair in a distributed manner. Then, this strategy uses the KM algorithm to solve the optimal link-matching problem between multiple link pairs. Unlike the original algorithm in work [37], we modify the objective function to minimize system WSP;

3. Random scheme: The spectrum allocation, power control, and link matching are randomly performed. It is utilized as a lower-performance benchmark in most related works;

4. Optimal WSEE maximization scheme: In this approach, we aim to maximize system WSEE. To be more specific, the spectrum allocation and power control are achieved via an exhaustive search while the link matching is implemented using the KM algorithm. The WSEE weight for each link is set to be equal.

4.2. Performance Analysis for Multiple Links

First, we focus on multiple PLs and SLs. In each snapshot of the Monte Carlo simulation, as shown in Figure 2, we only evaluate the system performance within the first transmission frame.

In Figure 3, we depict the overall WSP versus the number of SLs. The optimal scheme achieves the minimum overall WSP, while the random scheme performs the worst. It can be observed that the curves of the proposed scheme and the optimal scheme are overlapped. This phenomenon indicates that the proposed one is also optimal. The performance of the DRL-based scheme is very close to the optimal (proposed) scheme, with an average of 2–3 percentage points WSP higher than that of the latter two. In addition, the optimal WSEE maximization scheme performs between the optimal (proposed) and the random schemes. However, the average overall WSP of the optimal WSEE maximization scheme is far higher than that of the optimal (proposed) scheme, more than 400 times that of the latter. This result confirms that WSEE maximization may exacerbate energy consumption. Thus, the EE or WSEE are not very good optimization goals for energy-limited systems.

Figure 4 shows the SE of the PLs and SLs versus the number of SLs. Similarly, we can observe that the optimal and the proposed schemes have the same performance. This result further demonstrates that our proposed scheme is optimal. Although it is difficult to distinguish between the curve of the DRL-based scheme and the curve of the optimal (proposed) scheme, there are still subtle differences. The SE of the DRL-based scheme is approximately 1% higher than that of the latter two. It can be noticed that the link SEs of all the considered schemes linearly increase before $N=10$ and remain largely the same when $N>10$. The reason for this phenomenon is easy to understand: in our system model, the spectrum sharing among the PLs and SLs is one-to-one correspondence; thus, when the number of the SLs exceeds the number of the PLs, the link SLs almost no longer increase. Additionally, although the random scheme has the best link SEs in this configuration, it cannot ensure the QoS requirements of all the links. In the following subsection, we will prove it.

In Figure 5, we show the overall WSEE versus the number of SLs. The optimal WSEE maximization scheme achieved the best performance. We can estimate from the figure that the optimal (proposed or DRL-based) scheme can achieve an average of 80% WSEE of the former. The difference in WSEE between the two is not significant in the region where N is small. For example, when $N=7$, the WSEE of the optimal WSEE maximization scheme is about $1.903\times {10}^9$ bit/J and the WSEE of the optimal (proposed or DRL-based) scheme is about $1.684\times {10}^9$ bit/J.

From Figure 3, Figure 4 and Figure 5, we can verify that our scheme is optimal in terms of WSP while maintaining the performance gap with the optimal WSEE maximization scheme in terms of WSEE. Furthermore, although the above results are based on the configuration in Table 3, a similar trend will be observed when the configuration changes.

4.3. Performance Analysis for a Single Link Pair

This subsection analyzes the performance of our proposed scheme for a single link pair. Continuing with Figure 4, we will observe whether the proposed scheme can guarantee the minimum required QoS of either link. Then, we will verify the advantages of our scheme based on Newton’s method in terms of computational time.

We fix the distance between PT-PR and ST-SR pairs as 500 m each. The required QoS for the PL is fixed as 10 bps/Hz, and we set the QoS requirement for the SL to vary from 5 to 15 bps/Hz.

Figure 6 provides a description of the change in the link SE when the minimum QoS requirement of the SL increases. We still focus on the transmission within the first frame. All considered schemes (except the random scheme) ensure the minimum QoS requirements, with their link spectral efficiency (SE) precisely matching the required QoS. This demonstrates our solution’s capability to guarantee QoS for heterogeneous users.

In Figure 7, we show the computational time of all the schemes versus the QoS requirement of the SL. It can be noted that the optimal scheme and the optimal WSEE maximization scheme perform the same, and their average computational time is over 15 s. As expected, the random scheme gains the shortest computational time, i.e., $3.27\times {10}^{-5}$ s. The computational time of the proposed one is consistent with that of the random scheme in order of magnitude, i.e., $5.28\times {10}^{-5}$ s. The average computational time of the DRL-based scheme is approximately $3.17\times {10}^{-3}$ s. The above results indicate that our proposed scheme is not only optimal but also converges very quickly. Moreover, the proposed scheme does not need complex hardware conditions required by artificial intelligence methods, making it suitable for practical real-time deployment.

5. Conclusions

In this paper, we have studied the energy-efficient wireless resource allocation issue for overlay spectrum sharing. A centralized scheme based on the Newton–Raphson method and KM algorithm was developed to optimize spectrum allocation, power control, and link matching jointly. We took WSP as the objective function and formulated a MINLP problem. Our two-stage solution decomposes the MINLP problem into tractable subproblems. First, we handle weighted bipartite graph matching, then address convex optimization for spectrum allocation and power control between candidate PL-SL pairs. We leveraged Newton’s method for spectrum allocation and power control while guaranteeing the required QoS of each link. The link matching was determined at the central controller via the KM algorithm. Moreover, we presented heuristic approaches to select the initial value for Newton’s method. The simulation results demonstrated that the proposed solutions achieve optimal performance at a very fast speed.