Leveraging Massive MIMO for Enhanced Energy Efficiency in High-Density IoT Networks

Abstract

Maximizing energy efficiency (EE) in massive multiple-input multiple-output (MIMO) systems, while supporting the rapid expansion of Internet of Things (IoT) devices, is a critical challenge. In this paper, we delve into the intricate operations geared toward enhancing EE in such complex environments. To effectively support a multitude of IoT devices, we adopt a strategy of heavy reference signal (RS) reuse, and in this circumstance, we formulate the EE metrics and their corresponding inverses to determine pivotal operational parameters. These EE-centric parameters encompass factors such as the number of service antennas in the base station (BS), the number of IoT devices, and permissible coverage extents. Our objective is to calibrate these parameters to meet a predefined EE threshold, ensuring optimal system performance. Additionally, we recognize the indispensable role of Peak-to-Average Power Ratio (PAPR) reduction techniques, particularly in multicarrier systems, to further enhance EE. As such, we employ clipping-based PAPR reduction methods to mitigate signal distortions and bolster overall efficiency. Theoretical EE metrics are derived based on formulated signal-to-interference-plus-noise ratios (SINRs), yielding insightful closed-form expressions for the operational parameters. Leveraging two distinct EE metric models, we undertake parameter determinations, accounting for the levels of approximation. Intriguingly, our analysis reveals that even simplified models exhibit remarkable applicability in real-world scenarios, with a minimal margin of error. The results not only underscore the practical applicability of our theoretical constructs but also highlight the potential for significant EE enhancements in massive MIMO systems, thereby contributing to sustainable evolution in the IoT era.

1. Introduction

The extensive literature on massive multiple-input multiple-output (MIMO) systems underscores their applicability in the realm of massive Internet of Things (IoT) networks. These systems have demonstrated remarkable efficacy in various contexts [1,2,3,4,5,6,7,8]. Particularly noteworthy is their utility in supporting uplink data gathering systems within single-cell massive IoT connectivity scenarios, where the imperative lies in maintaining moderate data rates and energy efficiency (EE) while minimizing latency [4,5,6]. Given that IoT devices often operate under constrained battery resources, the quest for energy-efficient operations in massive MIMO systems is of paramount importance [4]. Moreover, the advent of ultra-reliable low-latency communication technologies has ushered in a new era of wireless communication, catering to diverse applications such as cyber-physical systems, remote surgery, and autonomous vehicles. In this context, the development of high EE operations, coupled with stringent requirements for reliability and low latency, emerges as a pressing concern in the IoT era.

1.1. Prior Works

As evidenced by the extensive works, enhancing EE in massive MIMO systems has been a focal point for investigation [9,10,11,12,13,14,15,16,17,18]. For instance, Prasad et al. [9] identified several avenues for maximizing EE gains in massive MIMO systems. Strategies such as implementing low-complexity operations at the base station (BS), augmenting the number of service antennas, and mitigating RF chain power consumption and power amplifier losses were explored, laying the groundwork for further research endeavors. Additionally, they delved into EE maximization schemes tailored for hybrid massive MIMO systems, where these systems coexist with millimeter-wave and heterogeneous networks, thereby broadening the spectrum of potential solutions. In a work by Bjornson et al. [10], the focus was on the uplink of a multiuser MIMO system, where multicell BSs are distributed according to a homogeneous Poisson point process. The authors posed a problem of uplink EE maximization and addressed it by considering parameters such as transmit power levels, the number of BS antennas, user equipments (UEs) per cell, BS density, and the reference signal (RS) or pilot reuse factor. Leveraging a massive MIMO scheme, where multiple UEs are multiplexed, led to the achievement of maximal EE by reducing the energy cost per UE. Through analysis, it was observed that reducing cell size ensures high EE, while the benefits from increasing BS density reach a saturation point when circuit power consumption outweighs radiation-related power consumption. In another investigation by Ngo et al. [11], the total EE of a time-division duplex (TDD)-based cell-free massive MIMO system was explored. This system features all access points with multiple antennas cooperating through a backhaul network to transmit signals collectively to all UEs using shared resources. The authors employed maximum ratio (MR) precoding for transmission and proposed an optimal power control algorithm under various power constraints. The power control problem was addressed through the approximation of a sequence of second-order cone programs. Notably, the work also examined the impact of backhaul power consumption on EE, revealing its significant influence, particularly in scenarios with a large number of APs. In Zhang et al.’s work [12], the focus was on enhancing the EE of an amplify-and-forward relay network, where multiple pairs of full-duplex UEs exchanged information through a massive antenna full-duplex relay. The authors proposed four power-scaling schemes with linear precoding techniques for relay processing, aiming to mitigate inter-user interferences. They demonstrated that equipping the relay with massive MIMO capabilities led to interference reduction by optimizing transmit powers at the sources and the relay. Furthermore, they assessed EE performance using a practical power consumption model and investigated the impact of relay antenna numbers on EE. He et al. [13] proposed a dynamic power control scheme for massive MIMO to alleviate interference between cellular and device-to-device (D2D) communications. They developed an analytical framework based on the proposed power control to evaluate both spectral efficiency (SE) and EE. The study highlighted that additional antennas in massive MIMO systems could enhance SE and EE for cellular UEs without affecting D2D communication. Verenzuela et al. [14] analyzed the uplink of a multicell massive MIMO network employing MR combining. By comparing regular RS and superimposed RS in terms of SE and EE, they derived closed-form achievable rates under practical BS deployment scenarios. Their findings suggested that while superimposed RS reduced RS contamination, it also introduced additional coherent and noncoherent interference, resulting in comparable SE and EE to random RS in practical settings. In You et al.’s investigation [15], EE optimization for single-cell downlink transmission in massive MIMO systems was explored using statistical channel state information (CSI) available at the BS. The study delved into deriving an optimal transmission strategy for EE maximization and proposed a power allocation scheme based on optimal transmit covariance matrices to enhance the EE of downlink massive MIMO systems. In the work by Rajput et al. [16] on massive MIMO IoT networks, each sensor, equipped with a single antenna, communicates with a fusion center that is outfitted with a large antenna array. Each sensor performs linear noisy observations of an underlying random quantity, processes this information, and transmits a precoded version of the observation over a fading wireless channel to the fusion center for efficient aggregation. Given that these sensors are typically battery-operated, optimizing EE is paramount. Additionally, mean square error (MSE)-based quality of service (QoS) constraints were applied to ensure accurate estimation of the random quantities at the fusion center. To address the resulting nonconvex EE maximization problem, the authors employed quadratic transform theory, and utilized a first-order Taylor series approximation to linearize nonconvex terms, achieving a viable solution. In Mahmoud et al. [17], the authors explored the application of Cell-Free massive MIMO in indoor industrial settings, positioning it as a foundational technology for enhancing reliability and SE in 5G and beyond. To address varied cooperation levels between access points and central processing units, they developed two implementation schemes: centralized and distributed. They proposed an effective AP selection method and a RS assignment scheme to alleviate RS contamination, a significant factor in these environments. The study identified key influences on uplink SE, such as vertical distance and RS contamination, and introduced a scalable Cell-Free massive MIMO model that reduces system complexity without degrading SE. Numerical results validated the advantages of centralized Cell-Free massive MIMO, highlighting notable improvements in both uplink and downlink SE. In Younas et al. [18], the authors proposed a novel framework for connecting IoT edge networks using 3D massive MIMO technology. The work introduced an improved regularized zero-forcing (ZF) precoding scheme specifically designed for single-cell systems with imperfect channel state information. This approach enables multiple single-antenna IoT devices to communicate simultaneously using shared time-frequency resources, potentially enhancing spectrum efficiency. The authors provided a comprehensive mathematical model for their proposed scheme and conducted comparative analyses against conventional ZF and regularized ZF techniques in large antenna regimes. Simulation results demonstrated superior performance of the proposed method in terms of achievable rates as the number of antennas increases, particularly with respect to bandwidth efficiency. The work examined various parameters including the number of antennas, angular spread, and cumulative distribution function (CDF). This work contributed to the advancement of IoT connectivity solutions by offering a framework that potentially maximizes device connectivity while improving bandwidth utilization in 3D massive MIMO systems. These works collectively contribute to the understanding and advancement of EE optimization strategies in various massive MIMO communication scenarios.

1.2. Contributions

In this paper, we introduce innovative and straightforward EE enhancement techniques in massive MIMO systems, aimed at facilitating the deployment of massive IoT networks. Our focus is on operational strategies to elevate EE in massive MIMO systems amid the challenge of accommodating a substantial number of IoT devices. Unlike much of the existing literature that predominantly addresses configurations where the number of service antennas significantly surpasses that of IoT devices, our work addresses scenarios where IoT devices vastly outnumber the service antennas at the BS [19,20,21]. Prior works have demonstrated the feasibility of supporting an extensive amount of IoT devices—far exceeding the number of BS antennas—through massive MIMO systems employing MR processing [22]. Orthogonal Frequency Division Multiplexing (OFDM) signals, renowned for their high SE, are commonly integrated within various IoT services alongside massive MIMO systems. To enhance the operational efficiency of power amplifiers (PAs) within this context, we advocate for the adoption of the clipping Peak-to-Average Power Ratio (PAPR) reduction technique. This method stands out due to its simplicity and effectiveness in PAPR mitigation, thereby aligning well with the needs of systems requiring low power consumption. We formulate two kinds of signal-to-interference plus noise ratio (SINR) expressions, and use them to present the EE metrics. Rather than using extensive search algorithms requiring time consumption and high complexity, using the inverse of EE metrics, we provide the expressions for the parameters, such as the number of IoT devices, the number of service antennas, allowable coverage, and uplink transmission power. It is shown that the derived parameters satisfy the predetermined EE thresholds. Based on the analysis, we also provide the EE-oriented operation for the massive MIMO under the situation of supporting massive IoT devices. Our analysis culminates in a comprehensive set of EE-oriented operational guidelines for massive MIMO systems designed to support an extensive IoT ecosystem. This framework not only addresses the pressing need for EE optimization in the face of growing IoT device densities but also contributes to the sustainable and efficient expansion of IoT networks.

A similar approach utilizing the inverse of EE metric was proposed in [23]. The primary departure lies in the derivation of equations and operational strategies, particularly in the context of massive MIMO systems. While previous work [23] focused on conventional massive MIMO operation, characterized by a surplus of service antennas at the BS compared to the distributed IoT devices, our study addresses the scenario of massive IoT deployments, where the number of IoT devices significantly exceeds the number of service antennas. Moreover, our contribution extends beyond previous endeavors by providing simplified approximations for each parameter. These approximations offer a balance between accuracy and computational complexity, facilitating practical implementation while yielding precise results.

We summarize the main contributions of this paper as follows.

• We develop expressions for the SINRs pertinent to massive MIMO systems integrated with extensive IoT connectivity. Leveraging MR processing and clipping PAPR reduction techniques, we delineate two distinct SINR models: Model I and Model II. Notably, Model II offers a simplified framework compared to Model I, yet through appropriate operational adjustments, it closely approximates the accuracy of its counterpart. Remarkably, the simplified model demonstrates robust applicability to real-world systems with minimal deviation.
• Utilizing the derived SINRs, we define two models of EE metrics, denoted as Model I and Model II. These metrics serve as pivotal tools for assessing the EE performance of massive MIMO-OFDM systems. Subsequently, we furnish closed-form expressions for critical operational parameters, including the number of service antennas, clipping ratio levels, and allowable coverage. Moreover, we devise operational schemes aimed at achieving satisfactory EE levels surpassing predefined thresholds.
• Numerical results obtained from comprehensive simulations corroborate our analytical findings. These results elucidate the dynamic interplay between system performance and parameter variations across different scenarios, thereby offering invaluable insights for system design and optimization.

This work advances the field of massive MIMO research by addressing critical, previously unexplored aspects essential for the sustainable evolution of IoT networks. We employ a novel approach involving intensive RS reuse to support the rapidly expanding scale of IoT device proliferation—a significant challenge not adequately addressed in the current massive MIMO literature. Additionally, we introduce EE metrics and their inverse counterparts, specifically designed for IoT-centric massive MIMO systems. This methodology enables precise calibration of operational parameters, critical to managing energy demands in large-scale IoT deployments. Furthermore, we uniquely integrate the EE threshold with PAPR reduction, tackling a major challenge in multicarrier systems often overlooked in massive MIMO works. Our work demonstrates that the simplified EE model achieves high accuracy in real-world scenarios, bridging theoretical constructs with practical application—an often neglected aspect in the existing literature. Recently, a strategy of enhancing massive IoT connectivity using massive MIMO was proposed in [2]. While the work in [2] primarily explores power control strategies such as channel inversion power control and max-min fairness power control, our current work concentrates on achieving EE threshold with RS reuse and PAPR reduction techniques. We introduce novel EE metrics and their inverses, which are not present in [2]. This allows for more precise calibration of operational parameters in IoT-centric massive MIMO systems.

To contextualize the practical relevance of massive MIMO technology integrated with OFDM, it is valuable to highlight key commercial implementations. Numerous telecommunications companies and infrastructure providers have successfully incorporated massive MIMO with OFDM in real-world networks, yielding substantial improvements in SE, data rates, and network reliability. For instance, 5G deployments by leading carriers have adopted massive MIMO combined with OFDM to support the diverse requirements of enhanced mobile broadband, massive IoT, and ultra-reliable low-latency communications. This integration enhances spatial multiplexing, allowing network operators to serve a larger number of IoT devices simultaneously. Additionally, massive MIMO-OFDM technology has been instrumental in fixed wireless access solutions, extending high-speed internet connectivity to rural and underserved regions, where traditional wired infrastructure may not be feasible. The widespread adoption of these technologies in commercial 5G networks underscores the importance of continued research in optimizing Massive MIMO systems for EE, particularly as we move towards more dense IoT ecosystems.

The importance of EE in massive MIMO systems has markedly increased, driven by the rising demands of densely populated IoT networks. The vast volume of connected devices in such settings places significant pressure on energy resources, revealing limitations in traditional massive MIMO energy management methods, which are typically designed for lower-density environments. As device density intensifies, challenges with interference and power management become more pronounced, making EE optimization essential for achieving scalable and sustainable network deployments. While prior research provides foundational insights on EE in massive MIMO, it often lacks specialized solutions for conditions involving extensive RS reuse and PAPR constraints—both crucial in dense IoT networks. These unique conditions demand tailored strategies to control power consumption while maintaining signal quality. This work addresses these research gaps by deriving closed-form EE metrics in the context of RS reuse and introducing practical PAPR reduction techniques, thereby optimizing energy usage without compromising system performance.

1.3. Organization, List of Abbreviations, and Notations

This paper is organized as follows. Section 2 elucidates the system model, providing a comprehensive overview of its architecture and functionality. In Section 3, we delve into the analysis concerning the EE oriented operation of massive MIMO systems catering to extensive IoT connectivity. This section encompasses the formulation of SINRs and EE metrics, alongside the derivation of closed-form expressions for critical parameters essential for swift parameter determinations aimed at averting system outage. Furthermore, methodologies for achieving high EE operations are delineated. Section 4 presents numerical results obtained through comprehensive simulations, providing empirical validation of our analytical findings. Finally, Section 5 encapsulates the concluding remarks, summarizing key insights gleaned from our work. A lists of the abbreviations used in this paper are shown in Abbreviations section.

Notation: $\mathbb{E}[·]$ denotes the expectation operator and $\mathbb{V}[·]$ denotes the variance operator. Boldface characters represent vectors. For a matrix A, ${\left(A\right)}^T$, ${\left(A\right)}^{*}$, and ${\left(A\right)}^{†}$ denote its transpose, conjugate (without transposition), and conjugate transpose, respectively. $\mathbb{R}$ denotes the set of real numbers and $\mathbb{C}$ denotes the set of complex numbers. ${\mathbb{R}}_{+}$ denotes the set of positive real numbers and ${\mathbb{R}}_{0+}=\left\{0\right\}\cup {\mathbb{R}}_{+}$ denotes the set of nonnegative real numbers. For any integer $k>0$, ${\mathbb{R}}^k$ represents k-dimensional space, and for any integer $n>0$, ${\mathbb{R}}^{k\times n}$ represents the space of $k\times n$ real-valued matrices. Replacing $\mathbb{R}$ with $\mathbb{C}$ denotes the corresponding complex spaces. For matrix A, $\mathrm{tr}\left(A\right)$ denotes its trace. ${\Vert ·\Vert }_{\infty }$ and ${\Vert ·\Vert }_2$ represent the $l_{\infty }$-norm and $l_2$-norm of a vector, respectively.

2. System Model

We investigate an uplink massive MIMO-OFDM system characterized by M transmission or service antennas at the BS and K IoT devices. Notably, the number of IoT devices, denoted as K, significantly exceeds the number of service antennas, represented by M. Assuming sufficient guard interval such that inter-symbol interference is eliminated, the received signal vector for the nth subcarrier, denoted as $\mathbf{y}\in {\mathbb{C}}^M$, can be effectively expressed in a conventional manner within the spatial domain vector framework. Receiver (RX) processing is commonly employed to mitigate inter-user interference (IUI). By employing a RX processing vector for the kth UE or IoT device, denoted as ${\mathbf{v}}_k\in {\mathbb{C}}^M$, the received signal can be expressed as follows:

$$\begin{array}{cc}\hfill {\mathbf{v}}_k^{†}\mathbf{y}& =\sum _{i=1}^K{\mathbf{v}}_k^{†}{\mathbf{h}}_i\sqrt{p_{u,i}}{s}_i+{\mathbf{v}}_k^{†}\mathbf{n}\hfill \\ \hfill & ={\mathbf{v}}_k^{†}{\mathbf{h}}_k\sqrt{p_{u,k}}{s}_k+\sum _{{\displaystyle \genfrac{}{}{0pt}{}{i=1\hfill }{i\ne k\hfill }}}^K{\mathbf{v}}_k^{†}{\mathbf{h}}_i\sqrt{p_{u,i}}{s}_i+{\mathbf{v}}_k^{†}\mathbf{n},\hfill \end{array}$$

(1)

where ${\mathbf{h}}_i\in {\mathbb{C}}^M$ is the channel vector, $p_{u,i}\in {\mathbb{R}}_{0+}$ is the allocated uplink power, and $\mathbf{n}$ is the additive white Gaussian noise (AWGN) vector. It has mean zero and variance ${\sigma }_{\mathsf{UL}}^2$ (i.e., $\mathbf{n}\sim \mathcal{C}\mathcal{N}(0_M,{\sigma }_{\mathsf{UL}}^2{I}_M)$). In the rich scattering situation, the channel becomes uncorrelated Rayleigh fading which can be represented as ${\mathbf{h}}_i\sim \mathcal{C}\mathcal{N}(0_M,{\beta }_i{I}_M)$, where ${\beta }_i$ is the large-scale fading coefficient for ith IoT device, encapsulating the path loss characteristic. Concurrently, the small-scale fading dynamics are captured through a Gaussian statistical model, which accurately reflects the rapid fluctuations in signal amplitude due to multipath scattering. The system adheres to a power constraint for the uplink signal transmission, formalized mathematically as follows:

$$\parallel \mathbb{E}[{\mathbf{s}}^{*}\circ \mathbf{s}]{\parallel }_{\infty }\le 1,$$

(2)

where the operation ∘ denotes component-wise multiplication, manifesting in the operation on signal vector $\mathbf{s}$, which comprises the transmitted signals from all K IoT devices formatted as $\mathbf{s}={[s_1{s}_2\cdots s_K]}^T$. This constraint ensures that the expected infinite norm of the component-wise product of the conjugated signal vector and the signal vector itself remains bounded, thereby regulating the maximum power transmitted by the IoT devices in the uplink channel.

The PAPR in the OFDM time domain signal can show the level of variation for the OFDM signal, and the signal is generally oversampled to catch the PAPR of the analog signal. In OFDM systems, the PAPR of the time-domain signal serves as an indicator of the extent of signal fluctuation. To accurately represent the PAPR characteristic of the corresponding analog signal, the OFDM signal is typically subjected to oversampling. This process is imperative for capturing the true essence of the signal’s dynamics, as it allows for a more precise approximation of the analog signal’s PAPR by examining its discrete counterpart at a higher resolution. The PAPR of the oversampled OFDM signal associated with the kth UE can be expressed as follows:

$$\mathsf{PAPR}={\displaystyle \frac{LJ·{∥\tilde{\mathbf{x}}∥}_{\infty }^2}{\mathbb{E}\left[{∥\tilde{\mathbf{x}}∥}_2^2\right]}},$$

(3)

where $\tilde{\mathbf{x}}\in {\mathbb{C}}^{LJ}$ is the oversampled OFDM signal in time domain, and J is the FFT size, L is the oversampling factor, where generally $L=4$. In the uplink signal, each IoT device performs the clipping operation. After clipping, the output signal is:

$${\widehat{x}}_t=\left\{\begin{array}{cc}{\tilde{x}}_t,\hfill & |{\tilde{x}}_t|\le A_{\mathsf{MAX}}\\ A_{\mathsf{MAX}}{e}^{j{\theta }_n},\hfill & |{\tilde{x}}_t|>A_{\mathsf{MAX}},\end{array}\right.$$

(4)

where ${\tilde{x}}_t\in \mathbb{C}$ is the tth time domain input OFDM signal to the clipping process, $A_{\mathsf{MAX}}\in {\mathbb{R}}_{0+}$ is the maximum permissible amplitude of signal, and if the input signal amplitude is higher than $A_{\mathsf{MAX}}$, the amplitude is limited to $A_{\mathsf{MAX}}$. ${\theta }_n$ represents the phase of the input OFDM signal, and regardless of the clipping process, it is not changed. To perform the clipping process, we need to define the level of clipping. The level of clipping can be set based on the definition of the clipping ratio (CR), $\nu$:

$$\nu ={\displaystyle \frac{A_{\mathsf{MAX}}}{\sqrt{P_{\mathsf{IN}}}}},$$

(5)

where $P_{\mathsf{IN}}\in {\mathbb{R}}_{0+}$ is the average power of input OFDM signal. Once the clipping process is conducted in the time domain, the clipping noise is added to all the subcarriers of the OFDM signal in the frequency domain. It is known that the clipped signal vector in the time domain is [24]:

$$\widehat{\mathbf{x}}=\alpha \circ \tilde{\mathbf{x}}+\tilde{\mathbf{d}}$$

(6)

where $\alpha \in {\mathbb{R}}^{LJ}$ is the attenuation factor vector and $\tilde{\mathbf{d}}\in {\mathbb{C}}^{LJ}$ is the distortion noise vector in time domain. $\alpha$ can be estimated and approximately recovered. Due to this reason, we will focus on the clipping noise in this paper. It is noteworthy that the compensation of the attenuation factor causes noise enhancement, but it can be neglected in the region of interest that is relatively high $\nu$.

The RS is heavily reused to support massive IoT devices simultaneously. In this context, we adopt the notation $(n,g)$ with double subscript to denote the nth IoT device within the gth group for the sake of simplicity. Here, G signifies the total number of groups, and N denotes the quantity of IoT devices in each group. To mitigate interference within each group, mutually orthogonal RS are deployed. However, across different groups, the reuse of RS leads to potential RS contamination. We organize all IoT devices into an $N\times G$ matrix, where $G=\lceil K/N\rceil$, accommodating the requisite number of groups based on the total count K of IoT devices:

$$B_{ch}=\left(\begin{array}{cccc}{\beta }_{1,1}& {\beta }_{1,2}& \cdots & {\beta }_{1,G}\\ {\beta }_{2,1}& {\beta }_{2,2}& \cdots & {\beta }_{2,G}\\ ⋮& ⋮& ⋮& ⋮\\ {\beta }_{N,1}& {\beta }_{N,2}& \cdots & {\beta }_{N,G}\end{array}\right),$$

(7)

Then, from (1), after the clipping process in the time domain, the received signal at nth UE in gth group is shown as:

$$\begin{array}{cc}\hfill {\mathbf{v}}_{n,g}^{†}\mathbf{y}& =\sum _{g^{\prime }=1}^G\sum _{n^{\prime }=1}^N{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n^{\prime },g^{\prime }}\sqrt{p_u{\eta }_{n^{\prime },g^{\prime }}}{\widehat{s}}_{n^{\prime },g^{\prime }}+{\mathbf{v}}_{n,g}^{†}\mathbf{n}\hfill \\ \hfill & =\underset{\mathrm{DS}}{\underbrace{\mathbb{E}\left[{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n,g}\sqrt{p_u{\eta }_{n,g}}\right]s_{n,g}}}+\underset{\mathrm{CDDS}}{\underbrace{{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n,g}\sqrt{p_u{\eta }_{n,g}}{d}_{n,g}}}\hfill \\ \hfill & +\underset{\mathrm{IFSRS}}{\underbrace{\sum _{{\displaystyle \genfrac{}{}{0pt}{}{g^{\prime }=1\hfill }{g^{\prime }\ne g\hfill }}}^G{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n,g^{\prime }}\sqrt{p_u{\eta }_{n,g^{\prime }}}{s}_{n,g^{\prime }}}}\hfill \\ \hfill & +\underset{\mathrm{CDSRS}}{\underbrace{\sum _{{\displaystyle \genfrac{}{}{0pt}{}{g^{\prime }=1\hfill }{g^{\prime }\ne g\hfill }}}^G{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n,g^{\prime }}\sqrt{p_u{\eta }_{n,g^{\prime }}}{d}_{n,g^{\prime }}}}\hfill \\ \hfill & +\underset{\mathrm{IFDRS}}{\underbrace{\sum _{g^{\prime }=1}^G\sum _{{\displaystyle \genfrac{}{}{0pt}{}{n^{\prime }=1\hfill }{n^{\prime }\ne n\hfill }}}^N{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n^{\prime },g^{\prime }}\sqrt{p_u{\eta }_{n^{\prime },g^{\prime }}}{s}_{n^{\prime },g^{\prime }}}}\hfill \\ \hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\underset{\mathrm{CDDRS}}{\underbrace{\sum _{g^{\prime }=1}^G\sum _{{\displaystyle \genfrac{}{}{0pt}{}{n^{\prime }=1\hfill }{n^{\prime }\ne n\hfill }}}^N{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n^{\prime },g^{\prime }}\sqrt{p_u{\eta }_{n^{\prime },g^{\prime }}}{d}_{n^{\prime },g^{\prime }}}}+\underset{\mathrm{RN}}{\underbrace{{\mathbf{v}}_{n,g}^{†}\mathbf{n}}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\underset{\mathrm{DSOUC}}{\underbrace{{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n,g}\sqrt{p_u{\eta }_{n,g}}{s}_{n,g}-\mathbb{E}\left[{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n,g}\sqrt{p_u{\eta }_{n,g}}\right]s_{n,g}}}.\hfill \end{array}$$

(8)

DS denoting the desired signal, and various sources of distortion and interference that can degrade the quality of this signal. CDDS represents the clipping distortion directly associated with the desired signal, arising when the signal’s amplitude exceeds a predefined threshold. IFSRS refers to interference caused by other UEs transmitting over the same RS as the UE of interest. Correspondingly, CDSRS symbolizes the clipping distortion emanating from these co-channel UEs. Furthermore, IFDRS describes interference from UEs that are allocated different RSs than the UE of interest. CDDRS captures the clipping distortion introduced by these cross-channel UEs. RN represents the received noise in the system, incorporating both external and internal noise sources affecting the signal reception. DSOUC stands for the desired signal as transmitted over an unknown channel, indicating the uncertainties and variations in the propagation environment that can affect the signal’s integrity. The notation $d_{n,g}$ is introduced to specify the clipping distortion noise observed in the frequency domain, attributed to the nth IoT devices within the gth group. The term $p_u$ specifies the initial uplink transmission power from a UE, while ${\eta }_{n,g}$ denotes the uplink power control factor applied to the nth IoT devices within the gth group to manage their transmission power effectively. This comprehensive framework highlights the multifaceted nature of signal quality assessment in communication systems, addressing both signal distortions and interferences alongside operational parameters like power control.

A block diagram for (8) is shown in Figure 1.

With the appropriate oversampling and enough number of subcarriers, $d_{n,g}$ is [23]:

$$d_{n,g}\sim \mathcal{CN}(0,ϵ(e^{-{\nu }^2}-\sqrt{\pi }\nu ·erfc\left(\nu \right))).$$

(9)

where $erfc(·)$ is the complementary error function and $ϵ$ is used to reflect the nonlinearity of oversampling. We use $ϵ=2.85$ when $\nu$ [dB] is larger than 3 dB. If $\nu$ [dB] is smaller than 3 dB, we use $ϵ=3.85$.

3. Determination of Parameters

The expedient determination of system parameters plays a critical role in ensuring its optimal functionality, particularly in environments requiring minimal delays. This section elucidates the strategies for parameter identification and outlines methodologies aimed at enhancing EE within the realms of massive MIMO-OFDM systems that incorporate extensive IoT connectivity. A foundational step involves delineating the SINR for the massive MIMO-OFDM framework. In this context, the reduction of the PAPR emerges as a pivotal concern, necessitating the adoption of clipping schemes. The expression in (8) reveals the complexity introduced by interference and clipping distortion components. To articulate the SINR formulation accurately, it becomes imperative to ascertain the power or variance associated with each constituent in (8). Consequently, leveraging the foundation laid out in (8), the SINR can be expressed as presented in (10):

$$\begin{array}{cc}\hfill & {\mathsf{SINR}}_{n,g}^{\mathtt{UL}}=\hfill \\ \hfill & {\displaystyle \frac{\mathbb{V}\left[\mathsf{DS}\right]}{\mathbb{V}\left[\mathsf{CDDS}\right]+\mathbb{V}\left[\mathsf{IFSRS}\right]+\mathbb{V}\left[\mathsf{CDSRS}\right]+\mathbb{V}\left[\mathsf{IFDRS}\right]+\mathbb{V}\left[\mathsf{CDDRS}\right]+\mathbb{V}\left[\mathsf{DSOUC}\right]+\mathbb{V}\left[\mathsf{RN}\right]}},\hfill \end{array}$$

(10)

where

$$\mathbb{V}\left[\mathsf{DS}\right]=\mathbb{V}\left[\mathbb{E}\left[{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n,g}\sqrt{p_u{\eta }_{n,g}}\right]s_{n,g}\right],$$

$$\mathbb{V}\left[\mathsf{CDDS}\right]=\mathbb{V}\left[{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n,g}\sqrt{p_u{\eta }_{n,g}}{d}_{n,g}\right],$$

$$\mathbb{V}\left[\mathsf{IFSRS}\right]=\mathbb{V}\left[\sum _{g^{\prime }=g}^G{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n,g^{\prime }}\sqrt{p_u{\eta }_{n,g^{\prime }}}{s}_{n,g^{\prime }}\right],$$

$$\mathbb{V}\left[\mathsf{CDSRS}\right]=\mathbb{V}\left[\sum _{g^{\prime }=1}^G{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n,g^{\prime }}\sqrt{p_u{\eta }_{n,g^{\prime }}}{d}_{n,g^{\prime }}\right],$$

$$\mathbb{V}\left[\mathsf{IFDRS}\right]=\mathbb{V}\left[\sum _{g^{\prime }=1}^G\sum _{{\displaystyle \genfrac{}{}{0pt}{}{n^{\prime }=1\hfill }{n^{\prime }\ne n\hfill }}}^N{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n^{\prime },g^{\prime }}\sqrt{p_u{\eta }_{n^{\prime },g^{\prime }}}{s}_{n^{\prime },g^{\prime }}\right],$$

$$\mathbb{V}\left[\mathsf{CDDRS}\right]=\mathbb{V}\left[\sum _{g^{\prime }=1}^G\sum _{{\displaystyle \genfrac{}{}{0pt}{}{n^{\prime }=1\hfill }{n^{\prime }\ne n\hfill }}}^N{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n^{\prime },g^{\prime }}\sqrt{p_u{\eta }_{n^{\prime },g^{\prime }}}{d}_{n^{\prime },g^{\prime }}\right],$$

$$\mathbb{V}\left[\mathsf{DSOUC}\right]={\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n,g}\sqrt{p_u{\eta }_{n,g}}{s}_{n,g}-\mathbb{E}\left[{\mathbf{v}}_{n,g}^{†}{\mathbf{h}}_{n,g}\sqrt{p_u{\eta }_{n,g}}\right]s_{n,g},\mathrm{and}\mathbb{V}\left[\mathsf{RN}\right]=\mathbb{V}\left[{\mathbf{v}}_{n,g}^{†}\mathbf{n}\right].$$

The TDD approach is commonly employed to mitigate RS overhead. Within this framework, the number of resource elements allocated for each coherence interval is given by ${\tau }_c=B_c{T}_c$, where $T_c$ represents the coherence time and $B_c$ denotes the coherence bandwidth. Within each coherence interval, ${\tau }_p$ slots are dedicated to uplink RS transmission, while the remaining ${\tau }_c-{\tau }_p$ slots are allocated for uplink data transmission. Consequently, the overhead incurred by uplink RS amounts to ${\tau }_p/{\tau }_c$. Then, the uplink SE is:

$${\mathsf{SE}}^{\mathtt{UL}}=\sum _{i=1}^{{\chi }^{\mathrm{u}}}{\zeta }^{\mathrm{u}}\left(1-{\displaystyle \frac{{\tau }_p}{{\tau }_c}}\right){log}_2(1+{\mathsf{SINR}}_i^{\mathtt{UL}}),$$

(11)

where ${\mathsf{SINR}}_i^{\mathsf{UL}}$ signifies the uplink SINR for the ith UE, ${\chi }^{\mathrm{u}}$ represents the count of simultaneously supported IoT devices for uplink transmission, and ${\zeta }^{\mathrm{d}}$ and ${\zeta }^{\mathrm{u}}$ denote the proportions of downlink and uplink data transmission, respectively.

In scenarios where there is a substantial number of IoT devices compared to the available service antennas (i.e., $M\ll K$), zero-forcing (ZF) processing fails to offer substantial performance enhancements [19]. Hence, in this paper, we employ MR processing.

Then, (10) can be further derived as:

$${\mathsf{SINR}}_{n,g}^{\mathtt{MR},\mathtt{UL}}={\displaystyle \frac{M{p}_{\mathrm{u}}{\beta }_{n,g}{\gamma }_{n,g}{\eta }_{n,g}}{{\mathsf{DN}}_1+{\mathsf{DN}}_2+{\mathsf{DN}}_3+{\mathsf{DN}}_4+{\mathsf{DN}}_5+{\sigma }_{\mathsf{UL}}^2}}$$

(12)

where

$${\mathsf{DN}}_1=M{p}_{\mathrm{u}}{\beta }_{n,g}{\gamma }_{n,g}{\eta }_{n,g}{D}_{n,g},$$

$${\mathsf{DN}}_2=p_{\mathrm{u}}\sum _{g^{\prime }=1}^G\sum _{n^{\prime }=1}^N{\beta }_{n^{\prime },g^{\prime }}{\eta }_{n^{\prime },g^{\prime }},$$

$${\mathsf{DN}}_3=p_{\mathrm{u}}\sum _{g^{\prime }=1}^G\sum _{n^{\prime }=1}^N{\beta }_{n^{\prime },g^{\prime }}{\eta }_{n^{\prime },g^{\prime }}{D}_{n^{\prime },g^{\prime }},$$

$${\mathsf{DN}}_4=M{p}_{\mathrm{u}}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{g^{\prime }=1\hfill }{g^{\prime }\ne g\hfill }}}^G{\beta }_{n,g^{\prime }}{\gamma }_{n,g^{\prime }}{\eta }_{n,g^{\prime }},$$

$${\mathsf{DN}}_5=M{p}_{\mathrm{u}}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{g^{\prime }=1\hfill }{g^{\prime }\ne g\hfill }}}^G{\beta }_{n,g^{\prime }}{\gamma }_{n,g^{\prime }}{\eta }_{n,g^{\prime }}{D}_{n,g^{\prime }}.$$

The derivation is in Appendix A. Here, ${\gamma }_{n,g}$ is the channel estimation quality indicator (CEQI) for nth UE in gth group with MMSE estimation, and the value of ${\gamma }_{n,g}$ is between zero and one [19,25,26,27]:

$${\gamma }_{n,g}={\displaystyle \frac{p_u{\eta }_{n,g}{\tau }_p{\beta }_{n,g}}{{\sigma }_{\mathsf{UL}}^2+{\displaystyle \sum _{g^{\prime }=1}^G}{p}_u{\eta }_{n,g^{\prime }}{\tau }_p{\beta }_{n,g^{\prime }}}},$$

(13)

With an appropriate uplink channel inverse power control, we can choose ${\eta }_{n,g}=\beta /{\beta }_{n,g}$, and then all the RX SNRs become the same:

$${\gamma }_{n,g}\equiv \gamma ={\displaystyle \frac{p_u{\eta }_{n,g}{\tau }_p{\beta }_{n,g}}{{\sigma }_{\mathsf{UL}}^2+\sum _{g^{\prime }=1}^G{p}_u{\eta }_{n,g^{\prime }}{\tau }_p{\beta }_{n,g^{\prime }}}}={\displaystyle \frac{{\tau }_p{p}_u\beta }{{\sigma }_{\mathsf{UL}}^2+{\tau }_p{p}_{u}G\beta }},$$

(14)

Typically, RS power is boosted for stable channel estimation. Thus, (12) is further simplified with the following approximation:

$${\gamma }_{n,g}\equiv \gamma ={\displaystyle \frac{{\tau }_p{p}_u\beta }{{\sigma }_{\mathsf{UL}}^2+{\tau }_p{p}_{u}G\beta }}\approx {\displaystyle \frac{1}{G}},$$

(15)

Then, we can simplify (12) as (16),

$$\begin{array}{cc}\hfill {\mathsf{SINR}}_{n,g}^{\mathtt{MR},\mathtt{UL}}& \equiv {\mathsf{SINR}}^{\mathtt{MR},\mathtt{UL}}\hfill \\ \hfill & ={\displaystyle \frac{M{p}_{\mathrm{u}}\beta \frac{1}{G}}{M{p}_{\mathrm{u}}\beta D\frac{1}{G}+NG{p}_{\mathrm{u}}+NG{p}_{\mathrm{u}}D+M{p}_{\mathrm{u}}\beta (G-1)\frac{1}{G}+M{p}_{\mathrm{u}}\beta D(G-1)\frac{1}{G}+{\sigma }_{\mathsf{UL}}^2}},\hfill \end{array}$$

(16)

We assume that the same clipping ratio is applied to all IoT devices.

Observation 1.

In the scenario of massive MIMO supporting extensive IoT connectivity, under appropriate power control, the CEQI can be effectively approximated by the reciprocal of the RS reuse factor, denoted as $1/G$, where G represents the RS reuse factor.

With enough allocation of TX power $p_u$, from (16), we can formulate SINR as follows:

$$\begin{array}{cc}\hfill {\mathsf{SINR}}_{\mathsf{Approx}}^{\mathtt{MR},\mathtt{UL}}& ={\displaystyle \frac{M\left(\frac{1}{G}\right)}{MD\left(\frac{1}{G}\right)+NG+NGD+M(G-1)\left(\frac{1}{G}\right)+MD(G-1)\left(\frac{1}{G}\right)}}\hfill \\ \hfill & ={\displaystyle \frac{M}{MD+N{G}^2+N{G}^{2}D+M(G-1)+MD(G-1)}}\hfill \\ \hfill & ={\displaystyle \frac{M}{N{G}^2+M(G-1)+\left(M+N{G}^2+M(G-1)\right)D}}.\hfill \end{array}$$

(17)

Surprisingly, the SINR can be expressed in a significantly simplified form, which will be demonstrated to align closely with the outcomes of simulations in the subsequent section. The numerator originates from the channel gain, while the first term of the denominator accounts for inter-user interference (IUI), the second term relates to inter-group interference (IGI) resulting from RS collision, and the third term corresponds to the clipping distortion contributed by all IoT devices.

Observation 2.

The channel gain is observed to be directly proportional to $M/G$. Meanwhile, the interference scales with K, and the clipping distortion exhibits a dependency on both $M/G$ and K, highlighting the intricate relationship between system parameters and network performance.

Utilizing (16) and (17), we can now formulate two EE metrics using (16) and (17). Researchers often define EE as a rate that is divided by power consumption. For analytical tractability, it is pivotal to articulate a simplified yet effective power consumption model. We only consider the transmission power consumption of UEs, and represented it as $NG\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)$, where $P_{\mathsf{PA}}^{\mathtt{UL}}$ is the uplink PA power consumption and $P_{\mathsf{C}}^{\mathtt{UL}}$ is the rest of power consumption [28]. The relation between $p_{u,k}$ and $P_{\mathsf{PA}}$ is $p_{u,k}={\mu }_k{P}_{\mathsf{PA}}$, where ${\mu }_k$ is the power efficiency of uplink PA.

Utilizing (16), we derive the expression for ${\mathsf{EE}}_{\mathsf{MR}}^{\mathtt{UL}}$, as delineated in (18):

$${\mathsf{EE}}_{\mathsf{MR}}^{\mathtt{UL}}={\displaystyle \frac{\delta NG{log}_2\left(1+\frac{M{p}_{\mathrm{u}}\beta \frac{1}{G}}{{\sigma }_{\mathsf{UL}}^2+M{p}_{\mathrm{u}}\beta D\frac{1}{G}+NG{p}_{\mathrm{u}}+NG{p}_{\mathrm{u}}D+M{p}_{\mathrm{u}}\beta (G-1)\frac{1}{G}+M{p}_{\mathrm{u}}\beta D(G-1)\frac{1}{G}}\right)}{NG\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}},$$

(18)

where $\delta =B·{\zeta }^{\mathrm{u}}\left(1-{\displaystyle \frac{{\tau }_p}{{\tau }_c}}\right)$. Here, B indicates the signal bandwidth. This particular EE metric is referred to as Model I. We assume that $K=NG$ for simplicity. Next, using (17), we formulate (19) and we call this EE metric model II:

$$\begin{array}{cc}\hfill & {\mathsf{EE}}_{\mathsf{MR},\mathsf{Approx}}^{\mathtt{UL}}={\displaystyle \frac{\delta NG{log}_2\left(1+\frac{M}{N{G}^2+M(G-1)+\left(M+N{G}^2+M(G-1)\right)D}\right)}{NG\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}},\hfill \end{array}$$

(19)

In the context of massive MIMO, antenna selection can be executed by establishing the number of service antennas, denoted as M. It has been established that random antenna selection can yield satisfactory performance with a sufficient number of antennas in massive MIMO [29]. Given the EE threshold, ${\mathsf{EE}}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}$, and other parameters, the determination of M for antenna selection is based on the closed-form expression (20). This scenario applies when utilizing Model I.

$${\widehat{M}}_{\mathsf{MR}}={\displaystyle \frac{-G\left(\beta p_{u}NG+\beta p_{u}DNG+{\sigma }_{\mathsf{UL}}^2\right)\left(-1+2^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}\right)}{\beta p_u\left(-G-DG-2^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}+G{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}+DG{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}\right)}},$$

(20)

Additionally, for antenna selection, a simplified approach can be adopted by leveraging the approximation model introduced in Equation (19). This model facilitates the determination of M, as outlined in Equation (21). This pertains to the scenario when employing Model II:

$${\widehat{M}}_{\mathsf{MR},\mathsf{Approx}}={\displaystyle \frac{-N{G}^2\left(1+D\right)\left(-1+2^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}\right)}{-G-DG-2^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}+G{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}+DG{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}}},$$

(21)

K and D can also be formulated in a similar manner, which are shown as follows:

$${\widehat{K}}_{\mathsf{MR}}={\displaystyle \frac{\beta p_{u}GM+\beta p_{u}DGM+G{\sigma }_{\mathsf{UL}}^2+\beta p_{u}M{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}-\beta p_{u}GM{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}-\beta p_{u}DGM{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}-G{\sigma }_{\mathsf{UL}}^2{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}}{\beta p_{u}G\left(1+D\right)\left(-1+2^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}\right)}},$$

(22)

$${\widehat{K}}_{\mathsf{MR},\mathsf{Approx}}={\displaystyle \frac{GM+DGM+M{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}-GM{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}-DGM{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}}{G\left(1+D\right)\left(-1+2^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}\right)}},$$

(23)

$${\widehat{D}}_{\mathsf{MR}}={\displaystyle \frac{\beta p_{u}N{G}^2+\beta p_{u}GM+G{\sigma }_{UL}^2-\beta p_{u}N{G}^2{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}+\beta p_{u}M{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}-\beta p_{u}GM{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}-G{\sigma }_{\mathsf{UL}}^2{2}^{\left(\frac{E{E}_{MR}^{UL}\left(p_u/\mu +P_C^{UL}\right)}{\delta }\right)}}{\beta p_{u}G\left(NG+M\right)\left(-1+2^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}\right)}},$$

(24)

$${\widehat{D}}_{\mathsf{MR},\mathsf{Approx}}={\displaystyle \frac{N{G}^2+GM-N{G}^2{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}+M{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}-GM{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}}{G\left(NG+M\right)\left(-1+2^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}\right)}},$$

(25)

Regarding the parameters $\beta$ and $p_u$, it is not feasible to derive their values using Model II, as this model does not incorporate $\beta$ and $p_u$ within its framework. Therefore, in such instances, only Model I is employed for determining these parameters. The respective expressions for $\beta$ and $p_u$ are denoted as (26) and (27):

$${\widehat{\beta }}_{\mathsf{MR}}={\displaystyle \frac{-{}_G\sigma _{UL}^2\left(-1+2^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}\right)}{p_u\left(-N{G}^2-DN{L}^2-GM-DGM+N{G}^2{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}+DN{G}^2{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}-M{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}+GM{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}+DGM{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}\right)}},$$

(26)

$${\widehat{p}}_{u,\mathsf{MR}}={\displaystyle \frac{-{}_G\sigma _{UL}^2\left(-1+2^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}\right)}{\beta \left(-N{G}^2-DN{G}^2-GM-DGM+N{G}^2{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}+DN{G}^2{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}-M{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}+GM{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}+DGM{2}^{\left(\frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }\right)}\right)}}.$$

(27)

Observation 3.

Model II does not provide valid expressions for $\beta$ and $p_u$; therefore, it is necessary to utilize model I to determine these parameters.

Using (26), it is straightforward to get the allowable coverage to satisfy the EE threshold. Utilizing the ETSI path loss model, the estimated coverage ${\widehat{\Omega }}_{\mathsf{MR}}$ is:

$${\widehat{\Omega }}_{\mathsf{MR}}={10}^{\left({\displaystyle \frac{-10{log}_{10}{\widehat{\beta }}_{,\mathsf{MR}}-128.1}{37.6}}\right)}.$$

(28)

The derived closed-form expressions for the parameter estimations are summarized in

Table 1. Derived parameters based on model I and model II.

Model I	M	${\widehat{M}}_{\mathrm{MR}}=\frac{-G\left(\beta p_{u}NG+\beta p_{u}DNG+{\sigma }_{\mathrm{UL}}^2\right)\left(-1+{\Phi }_{\mathrm{MR}}\right)}{\beta p_u\left(-G-DG-{\Phi }_{\mathrm{MR}}+G{\Phi }_{\mathrm{MR}}+DG{\Phi }_{\mathrm{MR}}\right)}$
	K	${\widehat{K}}_{\mathsf{MR}}={\displaystyle \frac{\beta p_{u}GM+\beta p_{u}DGM+G{\sigma }_{\mathsf{UL}}^2+\beta p_{u}M{\Phi }_{\mathsf{MR}}-\beta p_{u}GM{\Phi }_{\mathsf{MR}}-\beta p_{u}DGM{\Phi }_{\mathsf{MR}}-G{\sigma }_{\mathsf{UL}}^2{\Phi }_{\mathsf{MR}}}{\beta p_{u}G\left(1+D\right)\left(-1+{\Phi }_{\mathsf{MR}}\right)}}$
	D	${\widehat{D}}_{\mathsf{MR}}={\displaystyle \frac{\beta p_{u}N{G}^2+\beta p_{u}GM+G{\sigma }_{\mathsf{UL}}^2-\beta p_{u}N{G}^2{\Phi }_{\mathsf{MR}}+\beta p_{u}M{\Phi }_{\mathsf{MR}}-\beta p_{u}GM{\Phi }_{\mathsf{MR}}-G{\sigma }_{\mathsf{UL}}^2{\Phi }_{\mathsf{MR}}}{\beta p_{u}G\left(NG+M\right)\left(-1+{\Phi }_{\mathsf{MR}}\right)}}$
	$p_u$	${\widehat{p}}_{u,\mathsf{MR}}={\displaystyle \frac{-{}_G\sigma _{\mathsf{UL}}^2\left(-1+{\Phi }_{\mathsf{MR}}\right)}{\beta \left(-N{G}^2-DN{G}^2-GM-DGM+N{G}^2{\Phi }_{\mathsf{MR}}+DN{G}^2{\Phi }_{\mathsf{MR}}-M{\Phi }_{\mathsf{MR}}+GM{\Phi }_{\mathsf{MR}}+DGM{\Phi }_{\mathsf{MR}}\right)}}$
Model II	M	${\widehat{M}}_{\mathsf{MR},\mathsf{Approx}}={\displaystyle \frac{-N{G}^2\left(1+D\right)\left(-1+{\Phi }_{\mathsf{MR}}\right)}{-G-DG-{\Phi }_{\mathsf{MR}}+G{\Phi }_{\mathsf{MR}}+DG{\Phi }_{\mathsf{MR}}}}$
	K	${\widehat{K}}_{\mathsf{MR},\mathsf{Approx}}={\displaystyle \frac{GM+DGM+M{\Phi }_{\mathsf{MR}}-GM{\Phi }_{\mathsf{MR}}-DGM{\Phi }_{\mathsf{MR}}}{G\left(1+D\right)\left(-1+{\Phi }_{\mathsf{MR}}\right)}}$
	D	${\widehat{D}}_{\mathsf{MR},\mathsf{Approx}}={\displaystyle \frac{N{G}^2+GM-N{G}^2{\Phi }_{\mathsf{MR}}+M{\Phi }_{\mathsf{MR}}-GM{\Phi }_{\mathsf{MR}}}{G\left(NG+M\right)\left(-1+{\Phi }_{\mathsf{MR}}\right)}}$
	$p_u$	$\mathrm{N}/\mathrm{A}$

, where ${\Phi }_{\mathsf{MR}}=2^{\left({\displaystyle \frac{E{E}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}\left(P_{\mathsf{PA}}^{\mathtt{UL}}+P_{\mathsf{C}}^{\mathtt{UL}}\right)}{\delta }}\right)}$.

In the downlink scenario, the process is almost similar, but it is crucial to note that the power consumption model for the downlink differs from that of the uplink. The simplified downlink power consumption model is expressed as: $P_{\mathsf{SUM}}^{\mathtt{DL}}=P_{\mathsf{PA}}^{\mathtt{DL}}+M·P_{\mathsf{C}}^{\mathtt{DL}}+P_{\mathsf{F}}^{\mathtt{DL}}$, where $P_F^{\mathtt{DL}}$ denotes the fixed power consumption that is not proportionate to M. Essentially, the power consumption is directly proportional to the number of service antennas, M, indicating that an increase in M does not invariably lead to advantageous outcomes.

4. Numerical Results and Discussion

In this section, we present the numerical results. The simulation parameters are given in

Table 2. Simulation parameters.

Parameter	Value
Signal Bandwidth, B	20 MHz
Initial Clipping Ratio, $\nu$	5 dB
Path Loss Model	ETSI
Coherence Time, $T_c$	50 ms
Coherence Bandwidth, $B_c$	180 kHz
RX Processing	MR

.

We use 20 MHz system bandwidth, 50 ms coherence time, and 180 kHz coherence bandwidth. As a path loss model, we use the ETSI urban-macro path loss model of 2 GHz carrier frequency.

In Figure 2, we present the relationships between EE and several important parameters, M, K, $\nu$, and $p_u$. Simulation results are depicted as red ‘∗’ dotted lines. The theoretical analysis aligns well with the simulation results. Figure 2a illustrates EE versus M when $K=8400$. Assuming 3GPP-based systems, there are 168 available resource symbols in 1 ms, totaling 8400 available resource symbols in 50 ms [30,31,32,33,34,35]. Half of the available resource symbols are used for uplink RS at the maximum, resulting in ${\tau }_p=N=4200$ with $T_c=50$ ms. Since $N=4200$ when $T_c=50$ ms, $G=2$ in this case. We utilize $P_{\mathsf{C}}^{\mathtt{UL}}=0.5$ W. As M increases, EE also increases. In the power consumption model presented in this paper, $P_{\mathsf{C}}^{\mathtt{UL}}$ accounts for a larger portion than $P_{\mathsf{PA}}^{\mathtt{UL}}$; hence, with high $\nu$, the EE performance is insufficient. There is little difference in EE performance between $\nu$ = 3 dB to 8 dB. Figure 2b displays the EE versus K when $M=500$. With high K, both IUI and clipping distortion noise increase due to heavy RS reuse. In the denominator of EE, power consumption also rises as K increases. It is important to note that RS reuse commences at $K=4200$.

Remark 1. 

EEdemonstrates an increasing trend with the augmentation of M. Conversely, an exponential decline inEEis observed upon the initiation of RS reuse as K increases.

Figure 2c illustrates the relationship between EE and $\nu$ for the case of $M=500$ and $K=8400$. The optimal values of $\nu$ exist and they are subject to change with variations in M and K. The performance of $\nu$ is contingent upon the values of M and K. In the power consumption model employed in this study, light clipping may prove beneficial in enhancing the EE. Figure 2d depicts the variation of EE with respect to $p_u$ for the scenario where $M=500$ and $K=8400$. There exists a specific criterion for achieving sufficient EE or SE performance based on the value of $p_u$. In the given model, in order to maximize EE, the value of $p_u$ should exceed 0.1 mW to attain satisfactory performance.

Remark 2. 

Optimal points exist for both ν and $p_u$, and these points can vary depending on different situations. Specifically, if $p_u$ falls below a certain threshold, neither the EE nor the SE performances are deemed acceptable.

In Figure 3, we illustrate the parameters derived from closed-form expressions, with simulation results depicted through red asterisk (‘∗’) dotted lines. Model I employs the formulation provided in (18), whereas Model II utilizes the approximation of the SINR as outlined in (19). The comparison between these models and the simulation results demonstrates a noteworthy concordance, indicating the reliability of the approximations used in Model II as the exact formulations of Model I.

Figure 3a illustrates the relationship between M and $\nu$, along with the corresponding EE based on (20) and (21). Here, we set $K=8400$ and ${\mathsf{EE}}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}$ = 2 Mbps/W. Both model I and model II cases satisfy the required EE thresholds. The determined value of M exhibits an inverse characteristic to the EE versus $\nu$, as demonstrated in Figure 2a. Furthermore, Figure 3b depicts the relationship between K and $\nu$, along with the associated EE based on (22) and (23), with ${\mathsf{EE}}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}$ = 2 Mbps/W. The determined value of K demonstrates similar behavior to the EE versus $\nu$ relationship shown in Figure 2b.

Remark 3. 

The established value of M exhibits an inverse relationship with the EE versus ν, while the determined value of K demonstrates a similar characteristic to the EE versus ν.

Figure 3c displays $\nu$ versus M, along with the associated EE based on (24) and (25), with $K=8400$ and ${\mathsf{EE}}_{\mathsf{th}}^{\mathtt{MR}}$ = 2 Mbps/W. Once D is determined, $\nu$ can be directly derived using the inverse function of (9). In this scenario, $\nu$ decreases as M increases. The EE values fulfill the entire range of M, and decreasing $\nu$ consistently leads to reduced power consumption. Moreover, Figure 3d illustrates $\nu$ versus K, and the related EE based on Equations (24) and (25), with $M=500$ and ${\mathsf{EE}}_{\mathsf{th}}^{\mathtt{MR}}$ = 0.5 Mbps/W. Here, we lower the EE threshold to accommodate the complete range of K. As K increases, the reuse of resource becomes more intensive, necessitating a reduction in the EE threshold. Figure 3e,f present $\Omega$ versus K and M, along with the associated EEs based on (26). In this case, there is no approximated expression as the approximated model does not encompass $p_u$ and $\beta$. The permissible coverage can be readily determined based on (26).

Remark 4. 

Coverage determination can be guided by (26). To enhance coverage, it is advisable to augment M. Conversely, an increase in K leads to a reduction in coverage. This delineates a clear strategy for adjusting system parameters to achieve desired coverage objectives, emphasizing the trade-off between K and M in network design.

In Figure 4, we illustrate the variations in EE, $\nu$, $p_u$, $\Omega$, and the total clipping distortion noise (CDN) across specified parameters through a three-dimensional plot. This visualization provides a comprehensive overview of how these key metrics interact within the system’s parameter space. For the detailed analysis of total CDN, Figure 4g,h employ the following equation to quantify its variation with respect to other parameters, offering deeper insights into the impact of system configuration on clipping distortion noise:

$${\Theta }_{\mathsf{MR}}^{\mathtt{UL}}=\left(M{p}_{\mathrm{u}}\beta \gamma +NG{p}_{\mathrm{u}}+M{p}_{\mathrm{u}}\beta (G-1)\gamma \right)D/{\sigma }_{\mathsf{UL}}^2,$$

(29)

It is observed that ${\Theta }_{\mathsf{MR}}^{\mathtt{UL}}$ are proportional to the M, K, and $\nu$.

Remark 5. 

The total noise resulting from clipping distortion exhibits a direct proportionality to M, K, and ν.

In Figure 5, we display the cumulative distribution function (CDF) of EE. The dotted line represents the scenario where we determined the allowable coverage $\Omega$ using ${\mathsf{EE}}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}$ = 1.5 Mbps/W, $M=500$, $K=8400$. We utilize (28) to obtain $\Omega$ = 480 m. With the determined $\Omega$, the average EE is calculated to be 1.5 Mbps/W. On the other hand, the solid line corresponds to the case where we determined the value of M with ${\mathsf{EE}}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}$ = 2 Mbps/W, $K=8400$. Using (20), we obtain M = 513. With the determined value of M, the average EE is calculated to be 2 Mbps/W. It is important to note that ${\mathsf{EE}}_{\mathsf{MR},\mathsf{th}}^{\mathtt{UL}}$ represents the average EE of all IoT devices, with 50% of IoT devices having a higher EE than the threshold and 50% having a lower EE than the threshold. It is crucial to consider that in order to set the EE to prevent an outage, there must be sufficient EE margin. For instance, in the case of the dotted line, the EE threshold should be set below 1 Mbps/W to fully prevent the EE outage. Similarly, in the case of the solid line, the EE threshold should be set below 1.5 Mbps/W to fully prevent the EE outage.

Remark 6. 

The EE threshold represents the average EE that meets the EE requirements for 50% of IoT devices. To prevent an outage, it should be established with an adequate EE margin.

We analyze the computational complexity of the proposed scheme, presenting the number of required multiplications, additions, and divisions, summarized in

Table 3. Complexity analysis of the proposed scheme.

Classification	Model I			Model II
	$\vartheta$	$\varsigma$	$\pi$	$\vartheta$	$\varsigma$	$\pi$
M	19	11	9	11	10	9
K	31	13	11	15	10	9
D	30	13	11	14	10	9
${\rho }_u$	26	15	13	N/A	N/A	N/A

. In the table, $\vartheta$ denotes the number of multiplications, $\varsigma$ the number of additions, and $\pi$ the number of divisions. As mentioned, it indicates that model I generally demands a higher computational complexity compared to model II.

The computational complexity of the proposed scheme is bounded, even with increases in network size, as the necessary parameters can be efficiently derived from closed-form equations. For a LUT-based approach, where precalculated values are stored in memory, complexity can be approximated as $\mathcal{O}\left(n\right)$, where n is the data size. Since finding parameters that meet a given EE threshold is inherently nonlinear, a LUT is an effective approach for identifying near-optimal values. Iterative search methods may offer comparable complexities. Techniques such as binary search, jump search, interpolation search, and Fibonacci search all provide different complexity trade-offs. For instance, the jump search operates at $\mathcal{O}\left(\sqrt{n}\right)$, while binary search and Fibonacci search achieve $\mathcal{O}(logn)$. Although these algorithms have lower complexity, they still scale with data size n.

To bridge theoretical insights with practical and commercially viable implementations of massive MIMO, we refer current commercial deployments in 5G and early-stage 6G that incorporate similar EE-optimized configurations to enhance system capacity and sustainability. In commercially deployed 5G New Radio systems, massive MIMO configurations are frequently utilized to improve throughput and connectivity, especially in high-density urban areas. For example, 3GPP standards support 168 resource elements in 1 ms TDD frames, with a maximum coherence interval of 8400 resource elements over a 50 ms duration. This setup enhances energy-efficient communication by optimizing resource allocation, employing dynamic beamforming techniques, and maximizing SE in multiuser environments. Many vendors integrate comparable energy-aware configurations into their BSs, utilizing adaptive antenna arrays to optimize both coverage and power consumption. For instance, in the 3.5 GHz band, several vendors employ 64T64R massive MIMO configurations. Based on our optimized parameters, adjusting the number of active antennas according to traffic load (e.g., reducing from 64 to 32 during low-demand periods) can yield significant EE improvements with minimal impact on performance. Emerging 6G commercial trials further emphasize these energy-efficient parameters, particularly to support IoT and ultra-reliable low-latency communication requirements. Vendors are exploring EE-focused beamforming strategies in experimental massive MIMO setups, targeting extended IoT device battery life while maximizing coverage range. These advancements reflect an industry shift towards implementing adaptive, resource-efficient massive MIMO configurations that align with our proposed EE optimization framework.

The findings of this work provide insights that are directly applicable to the design and optimization of massive MIMO networks, particularly in dense IoT environments where EE and system scalability are paramount. Several practical applications can benefit from the optimized strategies and metrics developed in this work. The methodologies we propose for high EE can be directly applied to the design and optimization of next-generation wireless networks. By calibrating operational parameters such as the number of service antennas and coverage extents, network operators can enhance performance while minimizing energy consumption. The simplified EE models derived in our work demonstrate remarkable applicability in real-world scenarios. This allows network planners to make informed decisions that balance EE with service quality in diverse environments, such as urban areas with high IoT device density. Energy consumption becomes a critical concern in IoT deployments, and our work provides actionable insights that contribute to sustainable practices in technology development. This is particularly relevant for industries aiming to reduce their carbon footprint while maintaining operational efficiency. The derived EE metrics allow network operators in urban and smart city applications to optimize the number of service antennas based on device density and required coverage area. This approach can be used in practice to significantly reduce power consumption while maintaining service quality across large numbers of IoT devices, such as sensors and smart meters. By dynamically adjusting antenna arrays, network planners can achieve high EE levels without sacrificing connectivity, which is essential for sustainable urban infrastructure. The clipping-based PAPR reduction techniques explored in this work can be directly implemented in the signal processing units of BS equipment. These methods offer equipment manufacturers guidelines for developing more power-efficient PAs tailored to dense IoT networks. In industrial applications, such as smart manufacturing and logistics, the ability to support a high density of IoT devices within a limited coverage area is critical. The heavy RS reuse strategies discussed in this work can be applied to allocate spectral resources effectively in these settings, allowing for simultaneous connectivity of numerous devices while minimizing interference and conserving power. The closed-form solutions for SINR can aid in configuring network settings that ensure reliability and energy savings in environments that require high device density. Given the increased regulatory focus on reducing the carbon footprint of telecommunication networks, the EE improvements demonstrated in this work can contribute to sustainable practices in massive MIMO deployments. By reducing power requirements through optimized operational parameters, network operators can achieve compliance with environmental standards, particularly in regions where green energy policies are enforced. The proposed methods align with initiatives aimed at reducing overall energy consumption, helping telecommunication providers meet energy targets while supporting growing IoT infrastructure. These applications showcase the relevance and utility of our findings across various real-world scenarios, highlighting the practical value of EE-centric designs in massive MIMO for IoT expansion.

Building upon the findings of this work, several promising avenues for future research emerge to advance the application and understanding of EE in massive MIMO systems, especially within the rapidly evolving IoT landscape. Integrating machine learning techniques offers a dynamic approach to EE management, enabling adaptive optimization of parameters such as antenna activity, power allocation, and resource blocks in response to real-time network conditions. This adaptive approach can enhance system performance and resilience under varying IoT loads. Additionally, as distributed architectures like cell-free massive MIMO gain prominence, applying the EE framework in these contexts is a valuable next step. Examining the effects of user-centric clustering, distributed beamforming, and cooperative signal processing on EE is particularly relevant for scalable IoT applications. Furthermore, exploring the intersection of EE optimization with emerging technologies—such as millimeter-wave communication, network slicing, and edge computing within massive MIMO for IoT—offers a forward-looking research direction. Extending this EE framework to accommodate heterogeneous IoT environments, where devices exhibit diverse capabilities, energy constraints, and quality of service requirements, would further strengthen its adaptability and practical impact.

5. Conclusions

In this paper, we developed an efficient approach for determining key operational parameters to maximize EE in massive MIMO-OFDM systems for large-scale IoT networks, particularly under conditions of heavy RS reuse. Our focus was on minimizing power consumption through the application of clipping-based PAPR reduction techniques. We began by deriving two SINR expressions based on varying approximation levels, which were then used to formulate two distinct EE models. Rather than relying on computationally expensive search algorithms, we proposed a novel approach that leverages the inverse of EE metrics to streamline the parameter determination process. Notably, the simplified EE metric demonstrated broad applicability with negligible performance degradation. Our simulation results confirmed the effectiveness of the proposed parameter optimization schemes, offering a practical solution for enhancing EE in massive MIMO systems, thus contributing to the advancement of energy-efficient IoT networks.