Predictive modelling and high-performance enhancement smart thz antennas for 6 g applications using regression machine learning approaches

Abstract

This research introduces a novel design for a graphene-based multiple-input multiple-output (MIMO) antenna, specifically developed for sixth generation (6G) terahertz (THz) applications. The proposed antenna demonstrates multi-resonant behavior across three broad bandwidths: 2.479 THz (1.00–3.49 THz), 0.516 THz (3.64–4.156 THz), and 1.694 THz (4.50–6.20 THz). It achieves a peak gain of 13.41 dB, exceptional isolation of −34.2 dB, and a high efficiency of up to 90%. The design was developed using CST Studio Suite and validated through an RLC equivalent circuit model in ADS. This ensured accurate representation of its electromagnetic behavior. To improve the predictive capabilities of the antenna design process, five supervised regression-based machine learning (ML) models were employed. The models used were Extra Trees, Random Forest, Decision Tree, Ridge Regression, and Gaussian Process Regression. Among these, the Extra Trees Regression model delivered the highest prediction accuracy, achieving a mean absolute error (MAE) of 2.51%, a mean squared error (MSE) of 0.44%, and an R² score of 98.91%. The ML models were trained using a comprehensive dataset generated from parametric variations in patch, feed, substrate, and slot geometries. The proposed antenna leverages the advanced properties of graphene to deliver outstanding performance in gain, bandwidth, efficiency, and diversity metrics. It achieves a diversity gain of 9.9993 and an envelope correlation coefficient (ECC) as low as 0.00013. The integration of machine learning and RLC modeling reduces simulation time and improves design optimization. This creates a robust framework for future 6G wireless and biomedical THz applications.

Introduction

The ongoing advancement of wireless technologies has led to an increasing demand for antennas that can operate efficiently within the terahertz (THz) frequency range^1,2. The spectrum ranging from 0.1 to 10 THz is anticipated to be pivotal in facilitating sixth-generation (6G) wireless networks³. The THz band offers extensive bandwidth capabilities, significantly exceeding those of 5G systems, and enables ultra-high data rates, minimal latency, and enhanced energy efficiency⁴. The distinctive properties of THz frequencies render them appropriate for advanced applications, including real-time holographic communication, tactile internet, and massive machine-type communication (mMTC)^5,6,7. These applications require data transmission in the terabit-per-second (Tbps) range, which is attainable via the expanded spectrum available in the THz band⁸. Nonetheless, the practical implementation of THz systems presents numerous challenges, especially in antenna design, attributable to heightened propagation losses and susceptibility to environmental influences.

Microstrip patch antennas are renowned for their compact design, ease of fabrication, and cost efficiency. These attributes render them suitable candidates for THz communication systems^9,10,11. However, their performance deteriorates at elevated frequencies due to significant conductor and dielectric losses, resulting in a constrained bandwidth and diminished radiation efficiency. Slotted patch antenna designs have been introduced to enhance these performance limitations¹². The incorporation of slots into the patch configuration improves impedance matching and radiation properties, thereby broadening the bandwidth and enhancing overall efficiency. This alteration enables the antenna to accommodate the requirements of THz communication environments more effectively¹³. Simultaneously, Multiple-Input Multiple-Output (MIMO) technology has become an essential element for forthcoming wireless systems¹⁴. MIMO increases system capacity by enhancing spectral efficiency and spatial diversity and facilitating beamforming techniques¹⁵. MIMO architectures, when combined with slotted THz patch antennas, can provide exceptionally reliable and high-speed communication, even in intricate propagation environments. The integration of MIMO with slotted patch antennas optimizes data throughput and mitigates the significant channel fading and signal degradation typically encountered at terahertz (THz) frequencies^16,17. This integrated approach offers considerable potential for developing robust and efficient communication systems that meet the stringent demands of 6G networks¹⁸. Novel materials, such as graphene, are utilized to enhance the performance of MIMO-integrated THz antennas further. As wireless technology advances into the terahertz spectrum, graphene provides a revolutionary benefit in antenna design. Its exceptional electrical conductivity, mechanical flexibility, and thermal stability facilitate the creation of antennas that are both compact and low-loss while also accommodating extensive bandwidths¹⁹. These attributes are ideally suited to the specifications of 6G systems, where rapid communication and device miniaturization are essential.

The proposed antenna exhibits multi-resonant behavior at 1.5, 3.26, 3.9, and 4.92 THz, with each mode strongly supported by a wide range of studies across adjacent terahertz frequency domains²⁰. At the lower end of the spectrum, triangular graphene antennas operating near 1.5 THz (within the 1–2 THz band) have demonstrated high-gain characteristics suitable for applications in spectroscopic analysis, Doppler radar, and biomedical imaging²¹. This study is further complemented by MIMO antenna systems, which function effectively across 0.9–1.68 THz, confirming their compatibility with sixth-generation (6G) wireless networks that demand high capacity and low latency²². Operation within the 1.41–3.0 THz range has been validated for use in explosive detection, near-field communication, and threat sensing, particularly due to the availability of wide impedance bandwidths in this band²³. In addition, environmental sensing in the 1.08–1.8 THz band is made possible through microfluidic-integrated antennas capable of detecting water contamination, thereby validating the viability of low-THz systems in smart monitoring applications²⁴. The 3.26 THz resonance in the proposed antenna aligns with performance peaks reported in the 2.88–4.13 THz range, where antennas have been successfully employed in biomedical sensing, IoT, multimedia transmission, imaging, and 6G communication²⁵. This resonance is also supported by a broader family of multi-resonant structures that operate at 1.39, 3.26, 4.72, 5.96, and even higher THz frequencies, emphasizing the relevance of this spectral region for high-speed, multi-channel THz communication systems²⁶. Broadband operation extending from 2.25 to 6.0 THz further substantiates the antenna’s ability to support next-generation imaging, sensing, and high-throughput wireless systems²⁷. At 3.9 THz, the antenna exhibits an ultra-low return loss of approximately –46 dB, confirming its efficiency in high-integrity data transmission scenarios²⁸. The upper resonant mode at 4.92 THz is strongly corroborated by radiator-type antennas operating between 4.71 and 5.85 THz, where far-field and diversity performance have validated their suitability for wireless power transfer and energy harvesting²⁹. Twin-band performance in the 3.3–3.98 THz and 4.9–5.45 THz ranges further reinforces the proposed antenna’s potential in MIMO-based biomedical THz applications³⁰. Additionally, antennas operating in the 4.55–5.85 THz band have been demonstrated to support directional, high-diversity communication within indoor wireless LAN systems³¹. High-isolation performance has also been validated in polarization-diverse two-port antenna configurations spanning the 4.9–6.3 GHz range, confirming the applicability of this band for robust multi-user communications³². Moreover, wideband terahertz antennas covering the 1.98–14.5 THz range provide further evidence of their suitability for adaptive sensing and sixth-generation wireless technologies³³. The 5.0 THz region is specifically supported by environmental sensing systems operating near 5.046 THz, which have demonstrated effectiveness in monitoring temperature, humidity, and pollutants³⁴. Wearable and military-grade antenna solutions designed for flexible operation in the 2.2–2.4 THz, 3.15–3.5 THz, and 4.6–4.9 THz ranges also reinforce the practicality of mid-band THz deployment in defense and field-based applications³⁵. The direct biomedical application of the 4.92 THz resonance has been validated through the successful implementation in THz imaging and diagnostic technologies³⁶. Finally, circular nano-patch antennas exhibiting resonances at 3.26, 4.69, 5.64, and 6.95 THz further validate the design’s relevance to terahertz optical transmission and high-capacity 5G/6G communications, confirming the antenna’s alignment with future data-intensive networking environments³⁷. Collectively, these validations affirm the efficacy and domain relevance of your antenna’s resonance characteristics within the terahertz spectrum.

Recent studies have extensively documented the progress in THz communication and MIMO antenna technologies, highlighting their critical role in advancing 6G technology^38,39. The proposed antenna design is evaluated against various referenced designs, underscoring its superior performance in terms of bandwidth, gain, isolation, efficiency, and diversity metrics. The design in¹² operates at dual resonances of 0.445 THz and 0.540 THz with Limited bandwidths of 0.021 THz and 0.036 THz, respectively. Despite achieving moderate gain (7.9 dB) and efficiency (85.64%), it suffers from low isolation (−20 dB) and high ECC (0.07), which indicates poor mutual decoupling and limits its MIMO performance. Its relatively large footprint (12.39 λ₀ × 7.96 λ₀) µm² also constrains integration into compact systems. The antenna in⁴⁰resonates at 7.5 THz and offers a wide bandwidth of 5.9 THz, making it suitable for broadband communication. However, its moderate gain (8.3 dB) and weak isolation (−16 dB) reduce its reliability for high-speed data transmission. Additionally, the lack of efficiency data and absence of ML or RLC integration limit its adaptability in advanced systems. Reference⁴¹presents a design resonating at 2.2 THz with a 0.78 THz bandwidth and high efficiency (92%). Nonetheless, the gain is low (4.4 dB), and isolation remains weak (−20 dB), while ECC (0.006) suggests only modest decoupling. Though the board is compact (0.498 λ₀ × 0.52 λ₀) µm², the absence of ML optimization and diversity gain data restricts its viability in high-performance environments. In⁴², the antenna resonates at 0.395 THz and 0.629 THz with very narrow bandwidths of 0.01 THz and 0.025 THz. While efficiency is excellent (92.48%) and isolation is satisfactory (−20 dB), low gain (5.17 dB), limited bandwidth, and relatively large board dimensions (4.98 λ₀ × 5.97 λ₀) µm² diminish its competitiveness for compact and high-throughput THz systems. The design in⁴³ resonates at 0.128 THz and 0.178 THz with extremely narrow bandwidths (0.004 THz and 0.0061 THz). Although it shows excellent isolation (−25 dB), high efficiency (90%), and superior diversity performance (DG: 9.999 dB), the low bandwidth and absence of ML or RLC integration make it inadequate for real-world broadband THz applications. In⁴⁴, the antenna operates at 0.472 THz with a 0.435 THz bandwidth. Despite its strong diversity gain (DG: 9.99 dB) and adequate isolation (−20 dB), it underperforms in gain (3.9 dB) and suffers from a high ECC (0.458), indicating substantial inter-element coupling. Its board size (2.99 λ₀ × 1.49 λ₀) µm² further limits suitability for miniaturized applications. Reference⁴⁵ functions at 0.654 THz with a narrow bandwidth (0.04 THz) but delivers high gain (11.3 dB), strong isolation (≥25 dB), and excellent diversity characteristics (ECC: 0.003, DG: 9.99 dB). However, its low efficiency (76.45%) and large physical size (5.97 λ₀ × 10.95 λ₀) µm² severely impair its integration into compact systems and raise power consumption concerns. The design in⁴⁶ resonates at 0.514 THz with a bandwidth of 0.4 THz and achieves favorable isolation (−25 dB), efficiency (85.24%), and diversity gain (DG: 9.99 dB). Still, its moderate gain (5.49 dB), coupling (ECC: 0.015), and board dimensions (2.99 λ₀ × 1.49 λ₀) µm² restrict its effectiveness in space-constrained and high-performance MIMO systems.

The proposed antenna represents a notable enhancement, functioning at multiple resonances of 1.492 THz, 3.26 THz, 3.923 THz, and 4.924 THz, encompassing a broad frequency range of 1–3.49 THz, 3.64–4.156 THz, and 4.50–6.20 THz, respectively. The design attains an extraordinarily broad bandwidth of 2.479 THz, 0.5156 THz, and 1.694 THz, surpassing current studies in bandwidth coverage. A gain of 13.41 dB and an efficiency of 90% signify a design that is both highly efficient and powerful. The proposed design guarantees robust isolation of −34.2 dB, markedly diminishing interference. The ECC value of 0.00013 and DG of 9.9993 dB affirm exceptional MIMO performance. The proposed antenna distinguishes itself from previous designs by incorporating machine learning techniques and RLC components, thereby providing adaptability and enhanced performance. The diminutive board dimensions of (0.498 λ₀ × 1.742 λ₀) µm² further augment its integration potential. The proposed design offers an optimized solution for next-generation THz communication systems due to its superior bandwidth, high gain, strong isolation, and innovative application of ML and RLC components. Overall, the results presented in Table 1 demonstrate that the proposed antenna achieves superior performance across key indicators, which positions it as a frontrunner in the continuous development of antenna technology.

Table 1 Antenna design parameter.

Research gap and contributions

The rapid advancement of terahertz (THz) technology has created significant opportunities for both 6G wireless connectivity and advanced biomedical diagnostics. Nonetheless, current THz MIMO antenna designs encounter ongoing difficulties in concurrently attaining high gain, efficiency, and compactness—essential criteria for incorporation into next-generation communication systems and biomedical instruments^47,48. Contemporary methodologies predominantly rely on numerical simulations, often lacking a circuit-level analytical framework that could enhance theoretical understanding and provide greater design versatility. The design process is frequently impeded by protracted full-wave simulations and the limited use of machine learning techniques, which constrain optimization capabilities and predictive performance modelling.

This study’s principal contributions in addressing these difficulties are summarized as follows:

• 1. Creation of an RLC Equivalent Circuit Model: An RLC equivalent circuit model is presented to represent the antenna’s electromagnetic behavior analytically. This paradigm aids theoretical comprehension and is corroborated by full-wave simulation findings.

• 2. Compact, high-performance MIMO antenna design: In contrast to several previous antennas with larger board dimensions^{29,30,32,33,34,35,36}, the proposed design achieves a compact footprint of 0.498 λ₀ × 1.742 λ₀ while simultaneously providing a wider bandwidth, higher gain (up to 13.41 dB), strong isolation (−34.2 dB), and an extremely low envelope correlation coefficient (ECC). This combination surpasses many earlier designs that typically involve a trade-off between size and performance.

• 3. Integration of machine learning for design optimization: Unlike conventional optimization approaches in the literature^{30,31,32,33,34,35,36} that depend on computationally intensive simulation iterations, our method employs regression-based machine learning to efficiently predict and refine design parameters. This significantly reduces optimization time, speeds up convergence, and enables faster prototyping with potential for real-time adaptability.

• 4. Attaining Enhanced Electromagnetic Efficacy: The suggested antenna demonstrates enhanced gain, efficiency, and compactness, effectively overcoming significant Limits in existing THz antenna designs. These enhancements render the architecture highly suitable for high-data-rate 6G applications and biomedical applications, such as cancer detection and tissue characterization.

Properties of graphene material

The fundamental structure of graphene exhibits a captivating hexagonal arrangement of carbon atoms, creating a one-atom-thick honeycomb pattern. This exceptional material is part of the carbon allotrope family, which encompasses graphite, carbon nanotubes, and fullerenes, but is distinguished by its remarkable electrical properties⁴⁹. The sp²-hybridised carbon bonds in this atomic monolayer result in remarkable electron mobility, making graphene the premier electrical conductor for advanced applications⁵⁰. The intrinsic properties of graphene are especially advantageous in antenna design, as its conductivity can be accurately regulated by modifying essential parameters, usually utilizing a chemical potential (μc) of 0.5 eV and a relaxation time (τ) of 1 ps for a 0.01 μm-thick layer in simulation settings⁵¹. This precise tunability enables the development of high-performance antennas that operate within the 0.1-10 THz spectrum, presenting exceptional prospects for miniaturized, reconfigurable wireless devices that maintain excellent radiation properties while advancing the limits of traditional antenna technology^52,53.

Design methodology

Addressing spectral efficiency, data throughput, and adaptive functionality in the terahertz (THz) spectrum is crucial for 6G communications, necessitating a shift from single-element antennas to advanced multiple-input multiple-output (MIMO) architectures, as shown in Fig. 1. This transition necessitates structural innovations and the use of advanced materials. This study introduces a THz MIMO antenna system utilizing graphene as its foundation. Through the optimization of surface plasmon confinement and impedance matching, the system leverages graphene’s plasmonic characteristics, exceptional electron mobility (>200,000 cm²/V·s), bias-tunable conductivity, and additional benefits. This design, constructed on a 6.6 polyimide substrate (dielectric constant: 3.5, loss tangent: 0.0027), ensures minimal dielectric loss, thermal stability, and mechanical durability⁵⁴. Incorporated heat dissipation channels mitigate thermal degradation, facilitating reliable performance at elevated frequencies. Enhancing spatial diversity and channel capacity is accomplished by strategically placing engineered decoupling structures to increase isolation while maintaining compactness, thereby mitigating the mutual coupling characteristic of MIMO configurations. The synergistic integration of graphene and polyimide enables high radiation efficiency and gain throughout the THz spectrum, facilitates reconfigurable operation, and allows for miniaturization and reduced ohmic loss. This study establishes the foundation for next-generation THz MIMO systems by optimizing materials, plasmonic, and decoupling techniques to fulfil the performance and scalability demands of 6G networks.

Fig. 1

Evolutionary progression of single element to MIMO antenna design.

Single element antenna

Our single-element antenna’s evolution starts with a thorough design approach aimed at maximizing performance by means of meticulous material selection and dimensional accuracy. This first phase uses known electromagnetic principles to compute the microstrip patch parameters, forming the basis of our antenna system. The design process uses these basic equations to identify the best antenna size.

$$\lambda =\frac{c}{f}$$

(1)

$${\varepsilon }_{eff}=\left(\frac{{\varepsilon }_{r}+1}{2}+\frac{{\varepsilon }_{r}-1}{2}\right)\times {(1+12\times \frac{h}{W})}^{-0.5}$$

(2)

$$W=\frac{c}{2f}\times \sqrt{\frac{2}{{\varepsilon }_{r}+1}}$$

(3)

$$L=\frac{c}{\left(2f\sqrt{\left({\varepsilon }_{eff}\right)}\right)}-2\times \Delta L$$

(4)

$$\Delta L=0.412h\times \frac{\frac{{\varepsilon }_{eff}+0.3}{\frac{w}{h}+0.264}}{\frac{{\varepsilon }_{eff}-0.258}{\frac{w}{h}+0.8}}$$

(5)

The design of a single-element microstrip patch antenna begins with calculating its critical dimensions using a systematic approach. First, the operating wavelength (λ) is determined using Eq. 1, where c is the speed of light and f is the lowest resonant frequency. Next, the width (W) of the patch is calculated using Eq. 3, ensuring optimal radiation efficiency. The effective dielectric constant (ε_eff), which accounts for fringing fields, is derived from Eq. 2, where ε_r is the substrate’s relative permittivity and h is its thickness. The patch length (L) is then computed using Eq. 4, where ΔL represents the length extension due to fringing effects, given by Eq. 5. These equations ensure precise tuning of the antenna’s dimensions, enabling efficient performance at the target frequency while accounting for substrate properties and electromagnetic edge effects. The final design is further refined through simulation and experimental validation to achieve optimal impedance matching and radiation characteristics.

Graphene serves as the material for both the patch and ground elements, each possessing a consistent thickness of 3 micrometers. The antenna dimensions are 100 × 100 μm², featuring a patch of 75 × 88 μm² with slots. The configuration comprises a central small horizontal elliptical slot, two vertically aligned elliptical slots flanking the central area, and three star-shaped slots positioned in the upper section. Two insets border the feedline, and the ground features a triangular slot (resembling a pyramid-shaped window) at the base, partitioned into three smaller apertures by two slender vertical rectangular bars positioned one unit apart, with the remainder of the ground remaining intact. The design modifications aim to enhance gain, bandwidth, and reflection coefficient. The antenna is modelled in CST Studio Suite to evaluate performance metrics, including return loss, radiation pattern, and gain, offering insights into its behavior across different operating conditions. Figure 2 illustrates the comprehensive geometry, encompassing all features. Table 1 provides a comprehensive breakdown of various dimensions in micrometers, along with their respective acronyms.

Fig. 2

Top and bottom views of single-element configuration.

Evolution of the single-element antenna

The evolution of the single-element antenna design unfolds through four distinct stages, as shown in Fig. 3, each marked by iterative refinements aimed at optimizing performance. In Stage 1, the design begins with a simple rectangular patch element, incorporating two insets on either side of the feedline to enhance performance. However, simulation results fail to align with the intended specifications, showing no discernible resonant frequency, as depicted in Fig. 4(b). To address these shortcomings, Stage 2 introduces a more refined structure featuring a central small horizontal elliptical slot flanked by two vertically aligned elliptical slots. This modification is intended to improve the antenna’s characteristics. While simulations reveal some enhancements compared to the initial design, the results still fall short of the desired performance. This result confirms the presence of resonant frequencies, but the return loss remains inadequate. Additionally, as shown in Fig. 4(b), the efficiency reaches a maximum of 72%, indicating progress but leaving room for further optimization. In Stage 3, the ground plane is modified to include a triangular slot at the base, resembling a pyramid-shaped window. This slot is further divided into three smaller apertures by two slender vertical rectangular bars. These modifications yield significant improvements, leading to an enhanced efficiency of 81%, as shown in Fig. 4(c), and a maximum gain of 8.5 dB, as illustrated in Fig. 4(a); however, they fail to improve the resonant frequency. However, while efficiency improves, the gain remains largely consistent with the previous stage. For the final stage, Stage 4, from another stage, yields five resonant frequencies at 1.5 THz, 2.5 THz, 3.3 THz, 4 THz, and 4.9 THz, along with a substantial bandwidth and an efficiency of 84%. Moreover, the gain reaches an impressive 8.93 dB, marking a significant improvement. Through this systematic and iterative process of refinement, the single-element antenna evolves from a basic structure into a highly optimized design that meets the required specifications. Each stage builds upon the insights gained from the previous iteration, demonstrating the effectiveness of progressive design enhancements in achieving the final, optimized outcome.

Fig. 3

Evolutionary advancement of single element antenna design.

Fig. 4

Performance indicators across various stages in the evolution of single-element antennas: (a) Gain; (b) Reflection Coefficient; and (c) Efficiency.

Substrate material analysis

This work investigates the impact of substrate material choice on the performance of a graphene-based antenna, where graphene serves as both the radiating patch and the ground plane. The interaction between the substrate and antenna characteristics is analyzed by evaluating two distinct substrate materials: silicon⁵⁵ and polyimide⁵⁶. The performance metrics, including gain and reflection coefficient, are depicted in Fig. 5(a) and b for each material. In contrast, silicon accommodates multiple resonances but is constrained by narrow bandwidth, moderate return loss, and inferior gain from the proposed material. Conversely, polyimide offers outstanding performance, featuring a broad bandwidth (1 to 1.9) THz and (2 to 4.23) THz, significant return loss, and five distinct resonant frequencies. Furthermore, it attains a maximum gain of 8.93 dB within the operational bandwidth, exceeding that of silicon. The results indicate that substrate selection is crucial for enhancing the performance of graphene antennas. Polyimide demonstrates superior efficacy among the evaluated materials, providing optimal enhancements in bandwidth, gain, and impedance matching without notable disadvantages, thereby establishing it as the preferred substrate for high-efficiency THz antennas.

Fig. 5

Antenna performance with various substrate materials: (a) Gain; (b) Reflection coefficient.

Parametric analysis

Substrate thickness (st)

In the parametric analysis of our antenna, we investigate the effect of varying substrate thickness (st) while maintaining all other design parameters constant. This analysis aims to assess its influence on key performance metrics, including return loss, bandwidth, and gain. When the substrate thickness is increased to 9 μm, a notable reduction in return loss and bandwidth is observed, as depicted in Fig. 6(b). Additionally, this adjustment results in a decrease in gain, as shown in Fig. 6(a). Such a reduction in return loss and bandwidth is undesirable for THz applications, making this thickness value unsuitable despite its bandwidth enhancement. Furthermore, the gain performance does not meet the desired criteria. Conversely, when the substrate thickness is decreased to 3 μm, the reflection coefficient worsens, although the gain remains comparable to that observed at the increased thickness. However, in both cases of thickness variation, the return loss remains unsatisfactory, further emphasizing the antenna’s sensitivity to variations in substrate thickness. This analysis underscores the critical role of substrate thickness in determining the antenna’s resonant frequency, return loss, bandwidth, and gain. Among the tested values, the proposed substrate thickness of 6.6μm demonstrates the most optimal performance, making it the preferred choice for achieving the desired antenna characteristics.

Fig. 6

Impact of substrate thickness variation on antenna performance (a) Gain, (b) Reflection coefficient.

Patch thickness (t)

Patch thickness is a crucial determinant that affects antenna performance, especially in terahertz (THz) applications—departing from the prescribed patch thickness results in significant alterations to antenna performance. For example, augmenting the patch thickness beyond the optimal design specification leads to the appearance of two separate resonant frequencies at 3.8 THz and 5.4 THz, as depicted in Fig. 7(b). This configuration provides a broad bandwidth but exhibits inadequate gain and return loss performance. The gain curve is notably inferior to the proposed curve, as illustrated in Fig. 7(a). Although the bandwidth is satisfactory, the diminished return loss hurts the antenna’s overall efficiency. In contrast, decreasing the patch thickness from the suggested value results in a singular resonance at 6.3 THz and an enhanced return loss of −33 dB. This configuration demonstrates the lowest gain curve of all tested setups, rendering it unsuitable for practical THz applications. These findings indicate that modifying the patch thickness—whether by increasing or decreasing it beyond the prescribed value adversely affects critical performance metrics. The suggested patch thickness attains an ideal equilibrium, offering the most favorable compromise among return loss, bandwidth, and gain. Consequently, the accurate optimization and regulation of patch thickness are crucial in designing effective THz antennas.

Fig. 7

Impact of patch and ground thickness variation on antenna performance (a) Gain; (b) Reflection coefficient.

Feed width (fw)

The width of the feed Line has a substantial impact on critical antenna performance parameters, including the reflection coefficient, gain, efficiency, and operational bandwidth. An optimally designed microstrip feed Line guarantees effective energy transfer, thus improving overall antenna performance. With a feed Line width of 10 µm, the antenna demonstrates two separate resonant frequencies at 2.4 THz and 3.37 THz, corresponding to return losses of −36 dB and −23 dB, respectively. This configuration exhibits an extensive operational bandwidth of approximately 2.25 THz, spanning from 1.95 THz to 4.2 THz, as shown in Fig. 8(b). Nonetheless, the maximum gain under this condition is constrained to 6.5 dB, which is inadequate for THz applications. Augmenting the feed Line width to 30 µm leads to a decline in the reflection coefficient, whereas the gain remains relatively constant in comparison to the 10 µm scenario. Conversely, when the suggested feed Line width of 50 µm is utilized, the antenna attains five resonant frequencies with enhanced return loss and bandwidth. The maximum gain reaches 8.93 dB (within the bandwidth), the highest of all evaluated configurations, as illustrated in Fig. 8(a), making it the most efficient design. These findings underscore the critical importance of feed line width in enhancing antenna performance. The proposed configuration achieves an exceptional reflection coefficient, enhanced gain, and broad bandwidth, demonstrating its effectiveness for THz applications. This underscores the importance of meticulous feed line engineering to achieve an optimal balance among gain, bandwidth, and efficiency in high-performance THz antenna designs.

Fig. 8

Impact of feed line width variation on antenna performance (a) Gain; (b) Reflection coefficient.

MIMO antenna design

The development of graphene-based microstrip MIMO patch antennas is detailed in this section. These antennas are constructed by extending the single-element design into a 2-port MIMO configuration. This design capitalizes on the advantages of spatial diversity, which results in enhanced communication security, increased channel capacity, effective utilization of multipath environments, and optimized signal reception and interference suppression⁵⁷. To achieve an optimal MIMO antenna design, careful attention was given to the alignment of antenna elements, incorporating a double decoupling structure composed of graphene material with a height of 8 μm to significantly enhance isolation performance. Figure 9 depicts three distinct configurations utilized in the antenna design. Every configuration employs the identical patch and ground structure as the single-element antenna to guarantee performance consistency. Uniform edge-to-edge spacing of 150 μm is upheld across all configurations to guarantee accurate alignment and optimal separation between elements. In the initial configuration (a) (Ant 1), illustrated in Fig. 9(a), the two elements are arranged adjacently with a 0-degree orientation. The second configuration (b) (Ant 2), illustrated in Fig. 9(b), positions the elements in a 180-degree orientation, with the second element inverted in relation to the first. In the third configuration (c) (Ant 3), depicted in Fig. 9(c), the elements are oriented at 90 degrees and arranged adjacently with a perpendicular alignment. The performance of the three types of MIMO configuration antennas is evaluated in terms of resonant frequencies, return losses, impedance bandwidths, and isolation. Antenna 1 exhibits four resonances at 1.47 THz, 2.47 THz, 4.05 THz, and 5.00 THz, with corresponding return losses of −32 dB, −26 dB, −30 dB, and −36 dB, respectively, and achieves a 1.9 THz bandwidth within the 2.4–4.3 THz range where S₁₁ remains below −10 dB. Antenna 2 operates at five resonances—2.57 THz (−28 dB), 3.29 THz (−27 dB), 3.90 THz (−37 dB), 5.00 THz (−37.7 dB), and 5.90 THz (−34.5 dB)—and maintains a similar 1.9 THz bandwidth from 2.3 to 4.2 THz under the −10 dB threshold. In contrast, Antenna 3 delivers enhanced performance with six well-defined resonances at 1.49 THz (−56.20 dB), 2.50 THz (−24 dB), 3.26 THz (−32.09 dB), 3.92 THz (−31.02 dB), 4.92 THz (−46.05 dB), and 5.78 THz (−24 dB). It supports three distinct wideband regions where S₁₁ stays below −10 dB: 1.00–3.49 THz (2.49 THz), 3.69–4.16 THz (0.47 THz), and 4.50–6.20 THz (1.70 THz). The performance analysis, informed by the reflection coefficient Fig. 10(a), indicates that Antenna 3 demonstrates the most advantageous characteristics. Furthermore, the mutual coupling results depicted in Fig. 10(b) demonstrate that Antenna 3 attains the lowest mutual coupling of −34.2 dB throughout the entire frequency band, surpassing Antenna 1 (−25.88 dB) and Antenna 2 (−16.27 dB). These results identify Antenna 3 as the most efficient configuration, providing enhanced isolation and reflection capabilities. The comparative analysis offers essential insights into the impact of element orientation on MIMO performance, directing the choice of Antenna 3 as the most suitable design for the intended application.

Fig. 9

A comparison of three different antenna configurations: (a) Ant-1;(b) Ant-2; (c) Ant-3.

Fig. 10

Comparative analysis of performance across three MIMO antenna configurations, (a) S11; (b) S21.

Flow chart of antenna design methodology

The MIMO antenna design and evaluation process is a cyclical workflow that commences with the specification of system requirements, including frequency range, gain, and application-specific criteria. An individual-element antenna is initially designed and simulated in CST Microwave Studio, with critical parameters such as S11 (return loss), gain, and efficiency assessed. Should performance be inadequate, parametric sweeps and material optimizations will be performed until an optimized single-element design is achieved. The design is subsequently expanded to a two-port MIMO configuration, wherein essential parameters such as S21 (mutual coupling), ECC (Envelope Correlation Coefficient), DG (Diversity Gain), and radiation patterns are examined. Should the results be unsatisfactory, additional decoupling methods (such as defected ground structures and parasitic elements) or substrate enhancements (including low-loss materials and EBG structures) will be implemented. Upon validation of MIMO performance, a corresponding RLC circuit model is constructed in ADS for system-level integration. The flowchart presented in Fig. 11 illustrates the step-by-step progression of the design process, beginning with the initial single-element configuration, progressing through the development of the MIMO antenna system, and culminating in the implementation of the equivalent RLC circuit model. The design approach is finalized, having successfully validated across both electromagnetic and circuit simulation environments, marking the completion of the overall antenna development framework.

Fig. 11

A flowchart of single-element to MIMO antenna development with RLC circuit representation.

Result analysis of proposed MIMO antenna

Reflection coefficient

The reflection coefficient, denoted as the S₁₁ parameter, is a key metric for evaluating antenna performance. It indicates the efficiency of power transfer between the transmission line and the antenna. It measures the ratio of reflected to incident radio frequency (RF) power at the feed point, usually expressed in decibels (dB)⁵⁸. A lower (more negative) S₁₁ value signifies superior impedance matching and reduced power reflection. This leads to improved radiation efficiency and minimal transmission losses. In practical applications, a S₁₁ value under −10 dB is typically deemed acceptable, signifying efficient antenna performance. The proposed microstrip patch antenna demonstrates excellent electromagnetic performance with six distinct resonant frequencies. These occur at 1.49 THz, 2.50 THz, 3.26 THz, 3.92 THz, 4.92 THz, and 5.78 THz, as shown in Fig. 12(a). Each resonance point exhibits significant return loss, with pronounced minima recorded at −56.20 dB (1.49 THz), −24 dB (2.50 THz), −32.09 dB (3.26 THz), −31.02 dB (3.92 THz), −46.05 dB (4.92 THz) and −24 dB (5.78 THz). The significant troughs in the S₁₁ curve validate effective resonance and efficient radiation at the specified frequencies, confirming the antenna’s appropriateness for high-frequency applications. Furthermore, the antenna offers extensive impedance bandwidths—characterized as frequency ranges where S₁₁ is maintained below −10 dB—recorded at 2.49 THz (1.00–3.49 THz), 0.47 THz (3.69–4.16 THz), and 1.70 THz (4.50–6.20 THz). The extensive operational bandwidths underscore the antenna’s capacity for efficient multi-band operation, rendering it ideal for various terahertz communication and sensing applications.

Fig. 12

Proposed MMO Antenna: (a) Reflection coefficient;(b) Transmission coefficient.

Transmission coefficient

In MIMO (Multiple-Input Multiple-Output) antenna systems, isolation is a crucial factor. It is quantified by the transmission coefficient (S₂₁ or S₁₂) and measures the level of electromagnetic coupling between antenna elements⁵⁹. This metric signifies the extent to which the transmission or reception of one antenna disrupts neighboring elements. Robust isolation guarantees minimal mutual coupling. This maintains the autonomy of communication channels and enhances system performance in terms of capacity, reliability, and signal clarity. Fig. 12(b) demonstrates that the engineered MIMO antenna achieves a remarkable minimum isolation of −34.2 dB between its radiating elements. This significant degree of isolation indicates minimal electromagnetic interaction, which is essential for achieving low channel correlation. The antenna enhances spectral efficiency, facilitates increased data throughput, and ensures reliable performance by reducing inter-element interference. This positions it as a prime candidate for sophisticated MIMO applications. The antenna optimizes spectral utilization, facilitating increased data speeds and enhanced performance, which are crucial benefits for high-demand MIMO applications.

Gain and efficiency

Antenna gain is a crucial element in evaluating the effectiveness of Multiple-Input Multiple-Output (MIMO) systems. It directly impacts signal coverage, strength, and overall data throughput. It delineates the antenna’s capacity to focus emitted power in a particular direction, hence augmenting the effective radiated power while mitigating interference from unwanted directions⁶⁰. Elevated gain values are associated with more concentrated radiation patterns. These are essential for achieving long-range communication, stable connections, and high link reliability. Fig. 13 depicts the simulated gain performance of the proposed MIMO antenna. The antenna demonstrates a maximum gain of 13.41 dB, with significant gain measurements at critical resonance frequencies: 4.83 dB at 1.492 THz, 9.15 dB at 3.26 THz, 8.61 dB at 3.93 THz, and 11.04 dB at 4.92 THz. These results highlight the antenna’s ability to maintain directional radiation across multiple frequency bands, which is essential for high-frequency, high-data-rate wireless applications. The continually elevated gain throughout the operational spectrum confirms the antenna’s appropriateness for advanced MIMO systems. This facilitates improved spectral efficiency and resilient performance in next-generation communication networks.

Fig. 13

Proposed MIMO antenna gain and efficiency.

Radiation efficiency is a vital performance metric in antenna design, particularly for MIMO systems operating at terahertz frequencies, where power management and integration density are paramount concerns. It indicates the antenna’s ability to convert input power into radiated energy while minimizing losses due to conductor resistance, dielectric absorption, and surface waves. In high-frequency, compact MIMO systems, maintaining high radiation efficiency is crucial for minimizing power consumption, improving thermal stability, and ensuring dependable system performance⁶¹. Figure 13 illustrates that the proposed antenna demonstrates exceptional radiation efficiency throughout its operational bandwidth, achieving peak values of up to 90%. This great efficiency signifies minimal internal losses and efficient electromagnetic energy conversion. It results from optimized geometry, material choices, and impedance matching. This performance guarantees that the antenna can facilitate steady, energy-efficient operation throughout a broad frequency spectrum, making it a compelling choice for use in next-generation terahertz MIMO communication systems.

ECC and DG

The Envelope Correlation Coefficient (ECC) is a crucial statistic for quantifying the correlation between the radiation patterns of antenna elements in a MIMO system. It quantifies the degree of autonomous operation of the antenna elements for signal reception and transmission. An optimal ECC value is ideally close to zero. This reflects minimal coupling between antenna elements and low correlation. Such characteristics enable higher diversity performance and greater signal decorrelation. They also contribute to improved overall system reliability⁶².The ECC is generally computed using the far-field radiation patterns of the antenna elements, utilizing the following expression⁶³:

$$\text{ECC}=\frac{{\left|{\int }_{4\uppi }^{ } \left[{\text{E}}_{1}\left(\uptheta ,\varphi \right)*{\text{E}}_{2}(\uptheta ,\varphi )\right]\text{d}\Omega \right|}^{2}}{{{\int }_{4\uppi }^{ } \left|{\text{E}}_{1}(\uptheta ,\varphi \right|}^{2}\text{d}\Omega {{\int }_{4\uppi }^{ } \left|{\text{E}}_{2}(\uptheta ,\varphi \right|}^{2}\text{d}\Omega }$$

(6)

Figure 14(a) illustrates that the calculated ECC for the proposed MIMO antenna consistently remains below 0.00013 throughout the assessed frequency range. This remarkably low number validates the antenna’s superior diversity performance and suggests negligible pattern correlation among the elements. This behavior enhances signal integrity, ensures robust isolation, and improves reliability in multipath settings, rendering the antenna exceptionally appropriate for high-capacity MIMO communication systems.

Fig. 14

Proposed MIMO antenna: (a) ECC; (b) DG.

Diversity Gain (DG) is a critical metric for MIMO antenna system efficacy, signifying the enhancement in signal quality obtained via many antennas operating in uncorrelated or weakly correlated fading channels. Through the utilization of geographical diversity, DG fortifies link robustness, diminishes the probability of signal deterioration caused by multipath fading, and promotes overall system reliability and coverage⁶⁴.The extent of diversity gain achievable in a MIMO system is fundamentally linked to the Envelope Correlation Coefficient (ECC), a metric that indicates the similarity of radiation patterns among various antenna elements. Reduced ECC values correlate with increased DG, underscoring the necessity of avoiding reciprocal correlation to attain optimal diversity. The mathematical link between ECC and DG is articulated as⁶⁵:

$$\text{DG}=10\sqrt{1-{\text{ECC}}^{2}}$$

(7)

Figure 14(b) demonstrates that the proposed MIMO antenna attains exceptional diversity performance, exhibiting a minimum simulated DG value of 9.9993 throughout the working frequency range. This nearly optimal outcome signifies exceptional isolation and pattern decorrelation among the antenna elements, guaranteeing robust resistance to signal fading. The significant diversity gain validates the antenna’s efficacy in ensuring steady and efficient communication, rendering it exceptionally appropriate for implementation in contemporary high-capacity MIMO systems.

Surface current distribution

The surface current distribution is essential for analyzing the radiating characteristics of the antenna structure⁶⁶. For the single-element antenna configuration, as shown in Fig. 15(a), the maximum surface current reaches 15370.2 A/m. The current is most concentrated along the central horizontal bar of the antenna and the nearby star-shaped slots at the top, indicating their significant contribution to radiation. The two elongated diamond-shaped slots on either side also exhibit moderate current intensity, suggesting their supporting role in tuning or coupling the fields. The current is comparatively weaker near the outer edges of the ground plane and feed area, which primarily act as structural and excitation supports rather than as strong radiating elements. This distribution highlights the importance of the central and star-shaped features in defining the antenna’s electromagnetic response.

Fig. 15

Surface current distribution: (a) Single-element antenna; (b) MIMO antenna at port 1; (c) MIMO antenna at port 2.

In the MIMO configuration, the interaction between elements and mutual coupling must be considered. When port 1 is excited Fig. 15(b), the active antenna element shows a surface current pattern similar to the single-element case, with strong currents localized around the central bar and the trio of star-shaped slots. The diamond slots again show moderate current levels, reinforcing their involvement in field shaping. Meanwhile, the second antenna, which is terminated with a matched load, shows minimal surface current activity, indicating effective isolation and low mutual coupling. The decoupling structure placed between the antennas exhibits a weak yet noticeable current distribution, suggesting it plays a role in suppressing undesired interactions.

Conversely, when port 2 is excited and port 1 is terminated Fig. 15(c), a similar behavior is observed in a mirrored fashion. The active antenna displays strong current concentrations along the central bar and top star slots, affirming their primary radiating role. The current distribution on the first antenna remains very low, verifying the MIMO design’s capacity to suppress mutual coupling. In both excitation scenarios, the decoupling structure functions effectively, maintaining minimal interference and preserving individual antenna performance. The observed results confirm that the complex slot arrangement and strategic element positioning are crucial for achieving desirable radiation and isolation properties in the MIMO configuration.

Radiation pattern

The radiation pattern of an antenna is a fundamental parameter that defines how electromagnetic energy is distributed in space as a function of direction⁶⁷. At an operating frequency of 4.9 THz, the radiation characteristics of the designed antenna were carefully analyzed in both the E-plane and H-plane configurations to evaluate directional performance as shown in Fig. 16.

Fig. 16

Radiation pattern simulation of the proposed MIMO antenna.

In the E-plane, which corresponds to the electric field distribution and typically includes the direction of maximum radiation, the antenna exhibits strong directive behavior. Specifically, along the azimuthal angle ϕ = 0°, the main lobe of the electric field reaches a peak magnitude of 16.5 dB V/m, indicating a concentrated energy transmission in this direction. Additionally, the half-power beam-width (HPBW) at this plane is measured to be 43.2°, suggesting a moderately narrow beam that enhances the antenna’s directivity and is beneficial for focused signal transmission. Furthermore, at the elevation angle θ = 90°, the HPBW is slightly narrower, measured at 38.5°, indicating a tighter radiation focus in that orientation.

The H-plane, which represents the magnetic field distribution and is typically orthogonal to the E-plane, was examined along ϕ = 90°. The observed HPBW here is 44.3°, reflecting a broader radiation spread compared to the E-field pattern. This broader beam-width in the H-plane could be advantageous in applications requiring wider area coverage. Overall, the antenna exhibits well-balanced radiation characteristics, making it suitable for high-frequency THz applications where precision and beam control are essential.

RLC equivalent circuit

The performance of the proposed MIMO antenna is delineated by an equivalent RLC circuit model that encapsulates its electromagnetic behavior through resistive (R), inductive (L), and capacitive (C) components. Inductance (L) signifies the capacity to store magnetic energy, capacitance (C) indicates the ability to store electric energy, and resistance (R) illustrates energy dissipation through radiation and conduction^60,68. A detailed impedance study was conducted utilizing CST Studio Suite and Keysight’s Advanced Design System (ADS) to extract these characteristics precisely. The RLC model, although simplifying the intricate current distributions in the antenna, offers significant insights into its operational features and correlates effectively with full-wave CST simulations. The modeling commenced with CST simulations to evaluate return loss (S11), impedance, and resonance characteristics. Data from CST was exported in Touchstone (.s2p) format and subsequently loaded into ADS for circuit-level modeling employing a bottom-up methodology. Each resonance detected in the CST S11 curve was modeled in ADS as a parallel RLC branch, with starting parameters computed using the conventional formulas:

$${\text{f}}_{0}=\frac{1}{2\pi \sqrt{\text{LC}}}$$

(8)

$$\text{BW}=\frac{\text{R}}{2\pi \text{L}}$$

(9)

The geometric characteristics of the proposed antenna structure were rigorously mapped into equivalent lumped circuit components to enable accurate circuit-level modeling. Specifically, structural discontinuities such as elliptical and star-shaped slots, which introduce localized capacitive effects and disrupt current flow, were modeled as lumped capacitors. These capacitors effectively emulate the electrostatic storage and resonance-shifting behavior induced by the slots. Metallic regions, including the radiating patch and conductive interconnects, were represented as inductors to capture the magnetic energy storage associated with current-carrying conductive paths. Resistors were incorporated to represent both ohmic losses arising from material conductivity and radiation losses due to energy dissipation into free space. The final equivalent circuit diagram of the proposed MIMO antenna is presented in Fig. 17. The resulting equivalent model accommodates six discrete resonance frequencies corresponding to the operational bandwidths of interest. For Antenna 1, the first resonance frequency is realized through the parallel combination of R1, L1, and C1; the second resonance frequency is achieved using R2, L2, and C2; and R3, L3, and C3 define the third resonance frequency. Similarly, the fourth, fifth, and sixth resonance frequencies are determined by the respective combinations of R4–L4–C4, R5–L5–C5, and R6–L6–C6. This structured configuration enables fine-tuning and independent control of each resonant point within the antenna’s multi-band design, thereby supporting optimized impedance matching and enhanced bandwidth efficiency. To enhance accuracy, capacitive elements were strategically assigned to specific geometrical features of the antenna: the central horizontal elliptical slot was represented by C9; C7 and C8 modeled the left and right vertically aligned elliptical slots, respectively; C10, C11, and C12 represented the three-upper star-shaped slots; and the inset slots located symmetrically on either side of the feedline were modeled using C13 and C14. The feedline, serving as the primary transmission interface, was modeled with a combination of lumped resistance (Rx), inductance (Lx), and capacitance (Cx), effectively capturing its impedance profile and transmission characteristics over the frequency spectrum.

Fig. 17

Equivalent circuit of the proposed MIMO antenna.

In constructing the MIMO configuration, a second identical antenna element was added, and the same RLC modeling approach was applied to preserve the integrity of individual element behavior. Given the proximity of the antenna elements in the MIMO array, mutual coupling effects became a critical consideration. To mitigate these effects and improve port-to-port isolation, two graphene-based isolation walls were introduced between the antenna elements. In the equivalent circuit, the graphene-based isolation walls were modeled as parallel combinations of inductance and resistance—specifically, L15 in parallel with R7 for the left wall and L16 in parallel with R8 for the right wall—effectively representing the electromagnetic absorption and shielding behavior characteristic of graphene materials. This enhanced RLC equivalent model not only enables accurate prediction of individual antenna performance but also provides insight into interaction dynamics within the MIMO environment. To validate the accuracy and fidelity of the proposed model, return loss (S11) results obtained from the ADS-based circuit simulations were directly compared with full-wave simulation results from CST Studio Suite. The comparative analysis revealed a high degree of correlation, thereby confirming the reliability of the equivalent circuit approach. The complete set of extracted R, L, and C parameter values used in the model is comprehensively presented in Table 2. At the same time, Fig. 18 illustrates the comparative S11 performance between the two modeling platforms.

Table 2: RLC equivalent circuit parameter values.

Fig 18.

Reflection coefficient of CST and equivalent circuits in ADS.

Machine learning (ML) based optimization

Antenna design has been significantly enhanced through the use of machine learning (ML), which surpasses traditional electromagnetic (EM) approaches like the finite element method (FEM) and finite difference time domain (FDTD) techniques⁶⁹. Unlike conventional methods that are computationally intensive and can take days for each simulation, ML models can quickly predict an antenna’s behavior after training. This rapid predictive capability is crucial for industries such as telecommunications, aerospace, and the Internet of Things (IoT), which rely on swift prototyping⁷⁰. Furthermore, ML excels at autonomously exploring extensive design spaces, effectively reducing the manual tuning and iterative simulations commonly associated with traditional techniques. Compared to the idealized material characteristics, fabrication tolerances, and environmental factors considered by traditional solvers, ML models demonstrate greater resilience to real-world variances⁷¹. In addition, ML facilitates multi-objective optimization, streamlining the design process by simultaneously optimizing various design factors, including size, bandwidth, and efficiency. In summary, ML enhances computational speed, fosters design efficiency and robustness, and optimizes antenna development.

Preparing data sets

The diagram in Fig. 19 illustrates the use of machine learning to predict antenna efficiency by examining essential design parameters of various components. These components include the Patch (width, length, and slots), Ground (width, length, and thickness), Feed (width, inset width, and inset length), and Substrate (width, length, and thickness). A machine learning model is employed to optimize antenna performance through this analysis. The Rule of Thumb for Small Datasets or simple models suggests that when working with a relatively small dataset, a good starting point for the number of data points (observations or rows) should be roughly 10 times the number of features (variables) in your dataset. This guideline is designed to ensure that enough data is available for the model to learn the relationships between the features and the target outcome, preventing issues Like overfitting. For instance, if 13 input features are present in the dataset, it is suggested that at least 130 data points should be aimed for. In this manuscript, a total of 130 data points have been employed.

Fig. 19

ML model for the proposed work.

Methodology of ml in antenna

The diagram illustrates the methodology for designing and optimizing a Dual-Port MIMO antenna using machine learning techniques. The initial phase involves the design of the antenna, considering multiple criteria such as width, length, substrate, and slot specifications. Once the antenna design is finalized, a dataset is generated and divided into training (80%) and testing (20%) subsets as depicted in Fig. 20. Subsequently, machine learning algorithms—including Decision Trees (DT), Ridge Regression (RR), Extra Tree Regression (ETR), Gaussian Process Regression (GPR), and Random Forest Regression (RFR)—are applied to the dataset. The most accurate machine learning model is then selected to predict the performance of the antenna design.

Fig. 20

The flowchart of the ML for the proposed work.

Selection of algorithm

Using regression analysis based on different antenna parameters, the suggested method predicts important antenna features as gain, bandwidth, efficiency, and return loss. The accuracy and performance of the design are improved using an ensemble method that combines five advanced machine learning regression models⁷². By incorporating both linear and non-linear correlations, this multi-model approach improves prediction accuracy and reveals intricate patterns that conventional regression methods frequently fail to detect. Users are also helped in selecting the most trustworthy models for accurate antenna performance prediction by the systematic model selection procedure.

Gaussian process regression (GPR)

Gaussian Process Regression (GPR) is a probabilistic, non-parametric regression technique that excels at modelling complex, non-linear relationships. It operates on the premise that the data can be represented as a Gaussian process—a collection of random variables that share a joint Gaussian distribution. GPR is capable of making predictions while also quantifying uncertainty, making it well-suited for applications requiring confidence levels. Its flexibility and ability to learn from small datasets make it particularly advantageous in antenna design and optimization. This graph compares the simulated efficiency percentages and the anticipated values across a range of forecasts generated through Gaussian regression as shown in Fig. 21. The blue line represents the simulated efficiency, while the red dashed line depicts the projected values. The graph shows that the anticipated efficiency does not match the simulated results, with many deviations at certain points. These discrepancies suggest potential areas for enhancing the model or further modifications to improve accuracy⁷³.

Fig 21:

Simulated vs predicted Efficiency using Gaussian Process Regression.

Ridge regression

Ridge Regression, a linear regression technique, addresses multicollinearity by incorporating an L2 regularization factor into the cost function. This penalty term effectively reduces the magnitude of the coefficients, thereby mitigating overfitting and improving generalization to new data. It is particularly effective when dealing with a large number of correlated predictors, making it an excellent choice for high-dimensional datasets. This graph juxtaposes the simulated and anticipated efficiency percentages derived by Ridge regression. The blue line denotes the simulated data, whereas the red dashed line illustrates the expected values as shown in Fig. 22. As with the preceding plot, the projected values slightly match the simulated efficiency but with volatility, especially at elevated prediction levels. These disparities underscore the model’s efficacy and the aspects where Ridge regression may require additional refinement to more accurately reflect the efficiency trends⁷⁴.

Fig. 22

Simulated vs predicted efficiency using ridge regression.

Random forest regression

Random Forest Regression is an ensemble learning technique that combines multiple decision trees to enhance predictive accuracy and reduce the risk of overfitting. Each tree is created using a random subset of the data and a randomly selected set of features, with the final prediction determined by averaging the outputs of all the trees. This inherent randomness helps to lessen variation and improves the model’s ability to generalize to new data. Random Forest is particularly adept at uncovering complex, non-linear relationships and is resilient to noisy data, making it a popular choice for regression tasks across a wide range of applications. This graph illustrates the comparison between the simulated efficiency percentages and the projected values using the Random Forest Regressor model. The blue line represents the simulated efficiency, while the red dashed line indicates the predicted values. The forecasts closely follow the overall trend of the simulated data, with only minor deviations at certain points, as illustrated in Fig. 23. This suggests that the Random Forest Regressor effectively captures the efficiency pattern, although further adjustments may be needed to reduce discrepancies, especially at higher prediction values⁷⁵.

Fig. 23

Simulated vs predicted efficiency using random forest regression.

Decision tree regression

Decision Tree Regression is a non-linear predictive modelling method that organizes data by branching based on feature values, resulting in a tree structure. The method chooses the feature that divides the data best at each node to reduce target variable variation. Recursion continues until the tree reaches its maximum depth or the data cannot be separated. Decision Tree Regression is straightforward to read and handles numerical and categorical data well, but it can overfit if not pruned or regularized. Many regression projects use it because of its simplicity and interpretability. This graph illustrates the comparison between the simulated and predicted efficiency percentages utilizing the Decision Tree Regressor model. The blue line denotes the simulated data, whereas the red dashed line signifies the predicted values. While the predicted values generally align with the trend of the simulated efficiency, they exhibit more pronounced deviations, particularly at the peaks. This suggests that the Decision Tree Regressor successfully captures the underlying patterns as depicted in Fig. 24. Still, its predictions may be overly sensitive to certain fluctuations, resulting in less smooth predictions compared to other models⁷⁶.

Fig. 24

Simulated vs predicted efficiency using decision tree regression.

Extra tree regression

The ensemble learning technique known as Extra Tree Regression, constructs multiple decision trees by utilizing random feature selections and sample splits. Unlike traditional decision trees that determine the best split based on information gain, Extra Trees randomly select split locations. The graph presents a comparison between Simulated efficiency percentages (represented by the solid blue line) and predicted efficiency percentages (illustrated by the dashed red line) using the Extra Tree Regressor model. The x-axis is labelled"Prediction (Extra Tree Regressor),"indicating the index or sequence of predictions, while the y-axis is titled"Efficiency %,"covering a range from approximately 88.5% to 92% as shown in Fig. 25. The general pattern of the simulated and forecasted values closely aligns with one another, demonstrating a strong correlation⁷⁷.

Fig. 25

Simulated vs predicted efficiency using extra tree regression.

The submitted image illustrates in Fig. 26 a comparison of Error Metrics (MAE, MSE, and RMSE) across various regression models, assessed by the accuracy of their predictions relative to the actual values. These error metrics assess the effectiveness of each model in terms of forecast accuracy and the degree of divergence from actual values.

Fig. 26

Error matrix bar chart.

Comparison of mean absolute error (MAE)

For MAE, Extra Trees Regression stands out with a low error of 0.44%, showing superior accuracy in forecasting efficiency values. The projected values closely match the actual values, with only a negligible average error. In contrast, Ridge Regression has a much higher MAE of 21.07%, making its predictions imprecise. Similarly, Gaussian Process Regression has a Mean Absolute Error of 21.98%, reflecting inadequate predictive performance. Random Forest Regression has a significant MAE of 14.25%, indicating moderate accuracy. Decision Tree Regression performs better with an MAE of 4.29%, but still falls short of Extra Trees.

Comparison of mean squared error (MSE)

The Extra Trees Regression model has exceptional performance with a low Mean Squared Error (MSE) of 0.01%, underscoring its accuracy and indicating that Extra Trees Regression shows exceptional performance with a low Mean Squared Error (MSE) of 0.01%, highlighting its accuracy and small discrepancies between predicted and actual values. Decision Tree Regression has a slightly higher MSE of 2.08%, indicating modest prediction inaccuracies. Ridge Regression and Gaussian Process Regression have much higher MSE values of 11.21% and 12.48%, showing greater variance between their predictions and the actual data. Random Forest Regression has an MSE of 6.88%, placing it between Extra Trees and the other models. with a remarkably low RMSE of 0.44%. This low value signifies that, on average, the projected values are quite close to the actual values. Decision Tree Regression exhibits satisfactory performance with an RMSE of 8.32%, although it remains inferior to Extra Trees. Conversely, Ridge Regression and Gaussian Process Regression exhibit markedly elevated RMSE values of 33.48% and 35.39%, respectively. The elevated RMSE scores indicate significant divergence between the anticipated and actual values, signifying suboptimal model performance. Random Forest Regression, exhibiting an RMSE of 26.23%, demonstrates moderate accuracy, surpassing Ridge and Gaussian Process while falling short of Extra Trees. Comprehensive Evaluation and Summary:

In summary, Extra Trees Regression demonstrates optimal performance across all evaluated error metrics, including MAE, MSE, and RMSE. Its consistently low error values indicate high reliability and precision for this dataset. Ridge Regression and Gaussian Process Regression exhibit the largest errors, indicating limited suitability for this predictive task. Random Forest Regression and Decision Tree Regression perform moderately well, with Random Forest slightly outperforming Decision Tree Regression. However, neither achieves the accuracy of Extra Trees Regression. Consequently, Extra Trees Regression is identified as the most suitable model for accurate predictions with minimal error.

The image provides a visual comparison of the accuracy of five regression models based on two key metrics: R² (R-Squared) and Variance Score (Var Score) as illustrated in Fig. 27. These metrics are crucial for evaluating the goodness of fit of a model and its capacity to account for the variance in the dataset.

Fig. 27

Performance accuracy bar chart.

R² (R-squared) comparison

The R² score shows how well the model’s predictions match the actual values by measuring the proportion of variance in the target variable explained by the features. Extra Trees Regression leads with an R² score of 98.91%, explaining nearly 99% of the variance and demonstrating high predictive accuracy. Decision Tree Regression follows with an R² score of 90.65%, showing good predictive ability but less accuracy than Extra Trees. Random Forest Regression has an R² score of 83.10%, capturing a significant portion of the variance but leaving room for improvement. Gaussian Process Regression has an R² score of 69.32%, indicating it struggles to explain the variance compared to the other models. Ridge Regression is the weakest, with an R² score of 72.46%, showing it is less effective at explaining the variance.

Variance score (var score) comparison

The Variance Score measures the proportion of variance in the true values captured by the model. Like R², higher values indicate better fit. Extra Trees Regression leads with a Variance Score of 98.94%, closely matching its R² score and confirming its strong performance. Decision Tree Regression follows with 92.14%, showing it is effective but outperformed by Extra Trees. Random Forest Regression has a Variance Score of 84.66%, performing better than Ridge and Gaussian Process but behind Decision Trees and Extra Trees. Gaussian Process Regression has a Variance Score of 70.46%, showing it does not capture the variance as well as the other models. Ridge Regression has the lowest score at 73.25%, confirming it struggles most to fit the data.

The chart shows that Extra Trees Regression is the top performer in both R² and Variance Score, making it the most accurate model tested. Decision Tree Regression also performs well but is slightly less effective. Random Forest Regression offers moderate accuracy, while Gaussian Process Regression and Ridge Regression are the least effective in both accuracy and precision metrics. These results suggest Extra Trees Regression is the most suitable model for this dataset, offering the best balance of predictive accuracy and variance explanation Table 3.

Table 3 Performance comparison with related works.

Conclusion

This research presents a high-performance graphene-based MIMO antenna specifically designed for sixth-generation (6G) terahertz (THz) communication systems. It integrates electromagnetic simulation, circuit modeling, and machine learning (ML)-driven optimization. The proposed antenna showcases exceptional operational features, including an extensive bandwidth ranging from 1.00 to 6.20 THz, a peak gain of 13.41 dB, radiation efficiency of up to 90%, and robust port isolation of −34.2 dB. These attributes validate the antenna’s suitability for next-generation wireless applications that demand high data rates, low latency, and compact designs. Utilizing an RLC equivalent circuit model in ADS, this work provides a theoretical framework that closely matches full-wave CST simulation results, thereby enhancing the antenna’s reliability and system-level integration. Additionally, five supervised regression ML algorithms were employed to predict antenna efficiency, with the Extra Trees Regression (ETR) model emerging as the most effective, achieving the lowest error metrics and the highest prediction accuracy (R² = 98.91%). By integrating slotted graphene-based patch designs with MIMO configurations, advanced decoupling structures, and ensemble ML models, this study offers a comprehensive methodology that significantly enhances design accuracy, reduces computational time, and enables predictive tuning of antenna parameters. The proposed framework holds great potential for a wide array of 6G THz applications, including real-time holographic communication, biomedical sensing, and ultra-dense wireless networks. Future research will focus on experimental prototyping, real-world testing, and expanding ML frameworks to incorporate deep learning for end-to-end antenna performance prediction and adaptive control.