Performance prediction and optimization of a high-efficiency tessellated diamond fractal MIMO antenna for terahertz 6G communication using machine learning approaches

Abstract

This paper introduces the design and exploration of a compact, high-performance multiple-input multiple-output (MIMO) antenna for 6G applications operating in the terahertz (THz) frequency range. Leveraging a meta learner-based stacked generalization ensemble strategy, this study integrates classical machine learning techniques with an optimized multi-feature stacked ensemble to predict antenna properties with greater accuracy. Specifically, a neural network is applied as a base learner for predicting antenna parameters, resulting in increased predictive performance, achieving R², EVS, MSE, RMSE, and MAE values of 0.96, 0.998, 0.00842, 0.00453, and 0.00999, respectively. Utilizing regression-based machine learning, antenna parameters are optimized to attain dual-band resonance with bandwidths of 3.34 THz and 1 THz across two bands, ensuring robust data throughput and communication stability. The antenna, designed with dimensions of 70 × 280 μm², demonstrates a maximum gain of 15.82 dB, excellent isolation exceeding − 32.9 dB, and remarkable efficiency of 99.8%, underscoring its suitability for high-density, high-speed 6G environments. The design methodology integrates CST simulations and an RLC equivalent circuit model, substantiated by ADS simulations, with comparable reflection coefficients validating the accuracy of the models. With its compact footprint, broad bandwidth, and optimized isolation and efficiency, the proposed MIMO antenna is positioned as an ideal candidate for future 6G communication applications.

Introduction

Existing GHz-based wireless communication networks are becoming increasingly overwhelmed by the exponential growth in global data consumption. This strain renders them incapable of meeting the escalating demands of data-driven applications, including the Internet of Things (IoT), immersive virtual experiences, and artificial intelligence (AI)¹. To address these limitations, researchers are turning to 6G, the next generation of mobile networks, which seeks to revolutionize communication by leveraging the underutilized terahertz (THz) spectrum ranging from 0.1 to 10 THz². While GHz frequencies are oversaturated and inadequate for future performance requirements, the THz band offers vast potential to enable ultra-high-speed data rates, massive device connectivity, and significantly reduced latency³. Furthermore, the shorter wavelengths of THz waves support higher data transmission rates and facilitate the development of compact, efficient antennas capable of accommodating novel system architectures⁴. Harnessing the untapped capacity of the THz spectrum, researchers aim to develop innovative solutions to meet the needs of future communication systems. Among these, the integration of Multiple Input Multiple Output (MIMO) technology with THz communication is a significant advancement. MIMO technology enhances wireless network performance by employing multiple antennas to simultaneously transmit and receive signals, resulting in higher data throughput and reliability⁵. This THz-MIMO combination provides the ultra-high-speed and low-latency communication necessary to support dense device networks and real-time data processing required in 6G applications⁶. Its potential is particularly evident in IoT, where quick, reliable, and energy-efficient communication is vital for enabling a large number of connected devices⁷.

Beyond IoT, the application of THz-MIMO technology extends to various high-speed, data-intensive industries. For instance, satellite communication systems stand to benefit significantly from the high-frequency, high-bandwidth properties of the THz spectrum, ensuring secure and efficient data transmission over long distances⁸. In the transportation sector, autonomous vehicles rely on THz-MIMO networks for the ultra-fast data exchanges essential for navigation and safety^9,10. Similarly, next-generation data centers can leverage the rapid data handling capabilities of THz frequencies to support the computational demands of AI and machine learning models¹¹. In healthcare, wearable devices equipped with THz-enabled sensors can transmit patient health data in real-time, enabling remote diagnostics and timely interventions even in resource-constrained settings^12,13. Smart city infrastructure depends on THz-MIMO for applications such as intelligent traffic control, utility monitoring, and energy management, fostering safer and more responsive urban environments¹². Recent advancements, such as the integration of metamaterials, have been instrumental in enhancing the performance of THz antennas by improving parameters like gain, bandwidth, and radiation efficiency, making them more effective for diverse applications. Moreover, these metamaterial-based designs have enabled improved control over electromagnetic wave propagation, contributing to better signal quality and reduced losses¹³. The adoption of polymer-based flexible and compact materials has also revolutionized THz antenna design, offering lightweight, durable, and adaptable solutions ideal for wearable and portable devices. These materials not only provide mechanical flexibility but also ensure high efficiency and ease of fabrication, supporting a wide range of use cases across industries¹⁴. By addressing the bandwidth and latency constraints of GHz frequencies, THz-MIMO technology not only enhances existing wireless systems but also lays the foundation for scalable, future-ready communication networks. This integration is a critical enabler of 6G, poised to meet the evolving demands of a data-driven world and revolutionize connectivity for both individuals and industries¹⁵.

Designing efficient THz antennas for MIMO applications is essential for advancing high-frequency communication systems, where critical parameters like bandwidth, gain, isolation, efficiency, and ECC must be optimized. Each design contributes uniquely to THz technology development, though many face limitations that impact their adaptability and performance in next-generation applications. The study in¹⁶ employs an RT/Duroid 6010 substrate for a 2 × 2 MIMO antenna resonating at 8.84 THz. This design achieves 8.2 dB gain and isolation of -22.26 dB within a bandwidth of 0.0404 THz. The ECC of 0.0005 indicates a low correlation between ports, but the narrow bandwidth limits its capacity for broader applications in THz communication. The author in¹⁷ presents a 2 × 2 antenna on a polyimide substrate operating at 2.2 THz with 0.78 THz bandwidth, 4.4 dB gain, 92% efficiency, and an ECC of 0.006. Despite its high efficiency, the modest gain and bandwidth may limit its use in high-capacity applications. In¹⁸, a 2 × 2 MIMO antenna constructed with a Rogers RO4835-T substrate achieves 4.4 dB gain and isolation of -17 dB across a bandwidth of 0.114 THz. While the efficiency of 94% supports energy-efficient performance, the narrow bandwidth could limit the antenna’s ability to handle high-speed data across wide THz channels, and the moderate isolation may allow some interference in densely packed MIMO systems. The design in¹⁹ presents a 1 × 2 MIMO antenna with a polyimide substrate, offering a bandwidth of 0.4 THz, a low gain of 5.49 dB, and efficiency of 85.24%. However, its isolation of -25 dB reduces mutual coupling, and the low gain and ECC of 0.015 may impact diversity gain and signal strength. In²⁰, A polyimide-based 1 × 2 MIMO antenna with 90% efficiency operates at 0.128 and 0.178 THz with narrow bandwidths (0.00598 and 0.00613 THz) and low gain of 6.24 dB. However, its limited bandwidth, modest gain, and ECC of 0.012 may restrict its suitability for wideband THz and MIMO applications²¹. uses a polyimide substrate in an 8-port antenna configuration with a bandwidth of 0.78 THz and a gain of 7.5 dB. The system’s impressive 98% efficiency indicates little power loss. However, the ECC of 0.01 and the moderate gain may limit its signal quality. This design’s usage of machine learning (ML) helps improve performance by modifying important parameters. The design in²² features a 2 × 2 polyimide-based configuration, achieving a gain of 15 dB, isolation of -32.04 dB, 1 THz bandwidth, 97.4% efficiency, and an ECC of 0.0002. By incorporating ML and RLC modeling, it ensures exceptional MIMO performance, though its larger dimensions may limit suitability for compact applications. The antenna in²³ operates at 0.73 THz on a polyimide substrate, achieving a 4.331 THz bandwidth, 13.3 dB gain, -27 dB isolation, 95% efficiency, and an ECC of 0.0002. Incorporating ML and RLC modeling, it ensures strong signal quality, bandwidth, and efficiency, though slight efficiency improvements could enhance power management.

As shown in Table 1, the proposed antenna addresses several performance limitations seen in prior designs through RLC circuit modeling and machine learning (ML). These techniques enable optimal parameter tuning, resulting in a 1 × 2 MIMO configuration that operates across 5.03, 6.2, 7.77, and 9.07 THz with an expansive bandwidth of 3.34 THz and 1 THz. This antenna achieves 15.82 dB gain and isolation of -32.9 dB, combining high signal strength and effective isolation to reduce port-to-port interference. Additionally, the efficiency of 99.8% ensures minimal energy loss, and an ECC of 0.0003 supports exceptionally low signal correlation, enhancing MIMO performance by reducing the potential for interference. The DG of 9.9997 further underscores its effectiveness in providing diversity gain. By integrating ML and RLC modeling, this antenna achieves an advanced balance of gain, isolation, bandwidth, and efficiency. It is a powerful solution for next-generation THz communication applications demanding high performance and adaptability.

Table 1 Performance comparison with related works.

Design methodology

In our quest to advance antenna technology for future 6G applications, our design journey advances from a single-element antenna to a Multiple-Input Multiple-Output (MIMO) configuration, as illustrated in Fig. 1. The aim of this advancement is to improve wireless communication systems’ performance, effectiveness, and flexibility. In order to meet the fast-growing needs for higher data rates, lower latency, and more network capacity in 6G networks, we can take advantage of spatial diversity and multiplexing developments by switching from a single-element antenna to a MIMO configuration²⁴. Our deliberate choice of materials for the antenna components is a crucial part of this evolution. Copper is used for grounding, further enhancing conductivity and electromagnetic qualities that promote effective signal grounding and radiation efficiency, while graphene, which is prized for its exceptional conductivity, mechanical robustness, and flexibility, is employed as the patch material²⁵. Furthermore, the substrate is polyimide, which was chosen due to its advantageous dielectric characteristics, including a low-loss tangent of 0.0027 and a dielectric constant of 3.5. At 10 micrometers in thickness, the polyimide substrate minimizes energy loss and signal attenuation while allowing for optimal signal propagation. Our antenna design makes use of these material qualities as part of a comprehensive strategy to create high-performance wireless communication systems that meet 6G requirements.

Fig. 1

Evolutionary progression of antenna design; (a) single element antenna; (b) MIMO antenna.

Single element antenna

To build our single-element antenna as the first stage in our antenna’s evolution, we utilize a structured approach focused on enhancing its performance and functionality. This design progression starts by carefully choosing the materials and defining the dimensions for best results. Here’s the equation used to determine the patch’s size²⁶.

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

(1)

$$\:{{\upepsilon}}_{\text{e}\text{f}\text{f}}=\left(\frac{{{\upepsilon}}_{r}+1}{2}+\frac{{{\upepsilon}}_{r}-1}{2}\right)\times\:{(1+12\times\:\frac{h}{W})}^{-0.5}\:$$

(2)

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

(3)

$$\:\text{L}=\frac{\text{c}}{\left(2f\sqrt{\left({{\upepsilon}}_{\text{e}\text{f}\text{f}}\right)}\right)}-2\times\:{\Delta\:}\text{L}$$

(4)

$$\:{\Delta\:}\text{L}=0.412\text{h}\times\:\frac{\frac{{{\upepsilon}}_{\text{e}\text{f}\text{f}}+0.3}{\frac{w}{h}+0.264}}{\frac{{{\upepsilon}}_{\text{e}\text{f}\text{f}}-0.258}{\frac{w}{h}+0.8}}$$

(5)

The dimension of fw​ is determined by tuning it to achieve the desired characteristic impedance (Zc​) using Eq. 6²⁷.

$$Zc\left\{ {\begin{array}{*{20}c} {\frac{{60}}{{\sqrt {\upepsilon_{{reff}} } }}ln\left[ {\frac{{8{\text{~}}h}}{{f\omega }} + \frac{{fw}}{{4{\text{~}}h}}} \right]} & {\frac{{fw}}{h} \le h\:\:\:\:\:\left( {6.1} \right)} \\ {\frac{{120\pi }}{{\sqrt {\upepsilon_{{reff}} } \:\left[ {\frac{{fw}}{h} + 1.393 + 0.667\:ln\left( {\frac{{fw}}{h} + 1.444} \right)} \right]}}} & {\frac{{fw}}{h} > 1\:\:\:\:\:\left( {6.2} \right)} \\ \end{array} } \right.$$

The length of the feed line is calculated using Eq. 7²⁷.

$$\:{L}_{0}=\frac{1}{4}{\lambda\:}_{g}$$

(7)

Where, $\:{\lambda\:}_{g}=\frac{c}{{f}_{r}\sqrt{{\upepsilon}_{r}}}$

The dimensions of the ground plane are determined by the Equations below²⁷:

$$Lg = 6~h + L$$

(8)

$$Wg = 6~h + W$$

(9)

In the given context, the symbols used denote the following quantities: λ denotes the wavelength, c signifies the speed of light, ’f ’denotes frequency, ε_r means a dielectric constant, the Effective Dielectric Constant is represented by ε_eff.

The antenna in Fig. 2 is designed with compact sizes of 70 × 70 μm², tailored for high-frequency applications. At its core, a graphene patch measuring 60 × 55 μm² serves as the primary radiating element, selected for its excellent conductivity and flexibility, which supports effective signal transmission. The polyimide substrate on which this patch is placed and the copper ground plane on the other side are both evenly 0.1 μm thick.

Fig. 2

Top and bottom views of the single-element configuration.

The patch’s structure comprises two rectangular frames placed at the center, intended to increase radiation characteristics and optimize precise aspects like gain and bandwidth. Additionally, numerous circular slots are symmetrically implanted across the patch, creating an intricate pattern that further augments the antenna’s bandwidth and radiation properties by optimizing current distribution on the patch surface. Spreading from the lower part of the patch is a narrow feedline that attaches the antenna to the signal source, ensuring efficient energy transfer.

On the ground plane side, a defective slot is deliberately incorporated to help shape the radiation pattern and improve the antenna’s gain and bandwidth. The ground plane also shows a complex, repeating lattice pattern, which potentially increases the electromagnetic properties of the ground structure, diminishing loss and enhancing performance metrics like the reflection coefficient. The choice of materials further supports the antenna’s functionality; Copper’s steady grounding qualities guarantee effective operation, while graphene’s strong conductivity improves signal radiation.

This design, with its amalgamation of rectangular frames, circular slots, and a patterned ground plane, is simulated using CST software to evaluate characteristics. This antenna is ideal for sophisticated high-frequency communication applications due to its complex structure and specific modifications.

Development of the single-element antenna

The development of the single-element antenna design unfolds through a series of iterative enhancements, with the goal of achieving the best possible resonant performance at every level²⁸. This iterative development is illustrated in Fig. 3. The outcome of each step is shown in Fig. 4 where Fig. 4a shows the reflection coefficient and Fig. 4b shows the gain. At each step, the consequences of simulations are carefully examined to detect limitations, enabling targeted improvements to the antenna’s structure.

Fig. 3

Evolutionary progression of the single element antenna design.

Fig. 4

Performance metrics of single element antenna evolution: (a) S11; and (b) gain.

In the first stage, the design initiates with a rectangular patch element that incorporates circular slots distributed across its surface. The circular slots are presented with the goal of influencing the antenna’s resonant behavior and enhancing its performance. This initial design attains dual-band operation at frequencies of 4.9 THz and 6.2 THz. However, it is lacking in important performance areas. The return loss is insufficient, and the bandwidth at both frequencies is narrow, as shown in Fig. 4a. Furthermore, as Fig. 4b shows, the gain stays modest, limiting the antenna’s applicability for effective 6G applications. These boundaries highlight the need for further adjustments to improve the design.

In order to resolve the problems identified in the first stage, the design advances to a second stage, where a rectangular frame is inserted at the center of the patch, surrounding the circular slot pattern. This adjustment aims to augment the antenna’s resonance and enhance its gain. Simulations of this updated design suggest some improvements over the initial configuration. However, despite these improvements, the bandwidth and gain remain inadequate, suggesting that further modifications are necessary to meet the target specifications. This brings the design process to the next step, when more improvements are investigated.

In the third stage of the design development, substantial alterations are made to the patch element to further optimize its performance. A second rectangular frame is established at the center of the patch, creating a more complex geometry that facilitates improved resonant characteristics. Following these modifications and subsequent simulations, notable improvements are detected. Specifically, this design iteration achieves three distinct resonant frequencies, accompanied by an improved gain of 8 dB, as illustrated in Fig. 4b. However, while these augmentations mark a noteworthy progression, the return loss and bandwidth remain largely consistent with the previous stage, representing room for further optimization.

In the final stage of the antenna design development, a critical enhancement is applied to address the remaining challenges of bandwidth and gain. This improvement involves the strategic introduction of a defective ground slot, which significantly alters the ground plane of the antenna, resulting in a dual-band antenna with four resonant frequencies and promising return loss characteristics, enveloped by a broad bandwidth of 1.7 THz in the first band and 1.3 THz in the second. Additionally, this last stage achieves a significant gain of 10.87 dB, which is essential for applications in next-generation communication technologies.

Through these iterative stages of modification and optimization, the design of the single-element antenna progresses from a basic configuration to a complex, optimized structure that meets the specified performance requirements. Each stage builds upon insights gained from the previous iterations, representing the iterative nature of the design procedure in achieving the final anticipated outcome.

These values represent the ideal setting of the antenna, all measurements are in micrometers, sw = 70, sl = 70, pw = 50, pl = 48, fw = 10, fl. = 16.

Parametric analysis

In antenna design, parametric study is a vital approach to understanding how changes in specific parameters impact performance. By systematically adjusting aspects such as size, materials, or configuration, researchers can gather key insights into how these adjustments affect key performance indicators like bandwidth, return loss, and radiation properties²⁹. This section explores a crucial and complex aspect of the antenna design, emphasizing the quantifiable results that may be obtained with even small changes to parameter dimensions.

Substrate thickness (St)

In our antenna’s parametric study, we specifically adjusted the substrate thickness (st) while keeping all other design parameters constant to assess its impact on antenna performance. As seen in Fig. 5a, a dual-band response is observed when the substrate thickness is extended above the suggested amount. This response shows an acceptable loss in the first band but inferior performance in the second band. The gain remains fairly good shown in Fig. 5b; however, the bandwidth in both bands is quite narrow. Conversely, reducing the substrate thickness results in an expansion of bandwidth, but the return loss further decreases, and the gain also declines. This analysis underscores the substantial impact of substrate thickness on antenna bandwidth, while highlighting the challenge of achieving satisfactory performance.

Fig. 5

Performance variation with changes in substrate thickness: (a) S11; (b) Gain.

Outer radius of circle (r1)

In examining the effect of changing the outer radius of the circle on antenna performance, we find that increasing the outer radius beyond the proposed design parameter does not yield favorable resonance frequencies or reflection coefficients, as shown in Fig. 6a. Under these conditions, the maximum gain observed is 8.2, as indicated in Fig. 6b. Similarly, reducing the outer radius also fails to produce favorable resonance frequencies. However, in this case, the resulting bandwidth is narrower than that of the proposed single antenna design, and the gain experiences an additional decrease. These findings emphasize the importance of optimizing the substrate length in antenna design, underscoring the need for a careful balance between return loss and bandwidth to enhance overall antenna performance.

Fig. 6

Performance variation with changes in Outer Radius of Circle: (a) S11; (b) Gain.

Patch thickness (t)

Investigating the influence of varying patch thickness on antenna performance reveals significant findings. Increasing the patch thickness above the proposed design parameter results in only moderate return loss, as shown in Fig. 7a, which falls short of desired performance levels. The bandwidth under this condition is also limited, indicating restricted frequency coverage. Conversely, reducing the patch thickness from the suggested value fails to produce favorable resonance frequencies, and the gain remains insufficient to meet performance standards as illustrated in Fig. 7b. These finding highlight the complex relationship among patch thickness and antenna performance, underscoring the challenge of balancing bandwidth and return loss. Overall, these explanations stress the importance of carefully evaluating patch thickness to optimize return loss and bandwidth, thereby improving the antenna’s overall performance.

Fig. 7

Performance variation with changes in patch thickness: (a) S11; (b) Gain.

Patch analysis

A careful material selection process was conducted in an effort to improve antenna performance for THz applications, taking into account copper³⁰, graphene³¹, and silver³² for both patch and ground components because of their excellent conductivity qualities. Diverse material combinations were carefully examined through methodical research and rigorous evaluation in order to determine their impact on crucial antenna factors like impedance matching, radiation efficiency, and signal transmission characteristics. Following a thorough examination, a carefully selected combination that demonstrated exceptional performance emerged as the front-runner. This section explores the complex rationale behind the choice of materials for the patch and ground parts, clarifying the performance measurements and subtle factors that were crucial in forming the ultimate choice.

• 1. Variation of ground material with fixed copper patch element:

  Primarily, copper was selected as the patch element due to its broadly recognized conductivity. The subsequent analysis involved systematically varying the ground materials to assess their influence on antenna behavior, as illustrated in Fig. 8a,b, which depict the reflection coefficient and gain analysis respectively. When copper served as the ground material, the antenna showed dual-band characteristics, resonating at 4.8 THz and 6.1 THz, with bandwidths of 0.3 THz and 1.1 THz, respectively. Shifting to silver as the ground material yielded a similar dual-band response, with bandwidths of 0.48 THz and 0.91 THz, although the gains, at about 8.7 dB, remained suboptimal. Again, using graphene as the ground material caused in a single resonance frequency with strong return loss and a broad bandwidth of 1.25 THz. However, the gain, around 6.9 dB, was still lower than preferred. These findings underscore the critical importance of material compatibility in achieving optimal antenna performance.

  Fig. 8

  

  Effect of ground material variation on performance of the antenna with fixed copper element: (a) S11; (b) gain.

• 2. Variation of ground material with fixed silver patch element:

  In this stage, silver was selected as the patch element due to its inherent conductivity advantages, and different ground materials were explored to evaluate their effect on antenna performance. Figure 9a presents the reflection coefficient and Fig. 9b showed the gain from this analysis. Using copper as the ground material yielded a single-band response with a resonance frequency centered around 8 THz and a bandwidth of 1.1 THz. Despite the consistent resonance characteristics, the resulting gain of about 6.4 dB fell short of the desired levels. When silver was used as the ground material, the antenna achieved a single resonance frequency with a broader bandwidth of 1.23 THz. However, the gain remained low, at around 5.8 dB. Similarly, integrating graphene as the ground material created a single band with a broad bandwidth of 1.15 THz, but the gain was still suboptimal at only 6.01 dB. These finding highlight the complexities involved in choosing optimal materials for enhanced antenna performance.

  Fig. 9

  

  Effect of ground material variation on performance of the antenna with fixed silver element: (a) S11; (b) gain.

• 3. Variation of ground material with fixed graphene patch element:

  The final section focused on investigating different ground materials while employing graphene as the fixed patch element, which was selected due to its remarkable electrical characteristics. The reflection coefficients and gain for this analysis are shown in Fig. 10a,b respectively. With two resonance frequencies in each band and improved reflection coefficients, the dual-band response produced by using copper as the ground material has significant bandwidths of 1.8 THz for the first band and 1.3 THz for the second. Interestingly, the gain peaked at 10.82 dB, demonstrating the substantial influence of material synergy on antenna performance. Although there was a minor decrease in both reflection coefficients and gain, using silver as the ground material resulted in two resonances within a wider 1.5 THz bandwidth. Finally, using graphene as the ground material also resulted in a dual-band response with improved reflection coefficients but reduced bandwidth and gain. These findings underscore the intricate trade-offs inherent in material selection for optimizing antenna performance.

  Fig. 10

  

  Effect of ground material variation on performance of the antenna with fixed graphene element: (a) S11; (b) gain.

The performance outcomes for each patch and ground material amalgamation are shown in Table 2. According to a thorough analysis, the best outcomes for the antenna design are obtained when graphene is used as the patch element and copper is used as the ground element. Stronger resonance and effective signal transmission are fostered by graphene’s unique electrical properties, which include high conductivity and flexibility, which make it perfect for the patch element. Because of its exceptional conductivity and stability, copper improves grounding and reduces signal loss, which improves antenna performance overall. As a result, the ideal equilibrium between graphene and copper is achieved, leading to higher performance for terahertz applications.

Table 2 Comparison of antenna performance for different material combinations of patch and ground elements.

Development of the MIMO antenna design

The graphene-based microstrip MIMO patch antennas are depicted in the following section. Our MIMO antenna design methodology focuses on developing a 2-port MIMO configuration, starting with the single-element antenna as a foundational structure. This approach improves signal reception by leveraging spatial diversity, reducing interference, increasing capacity, exploiting multipath, and enhancing security³³. To create an optimized MIMO configuration, we first determine the orientation of antenna elements to achieve a high level of isolation.

Three configurations, shown in Fig. 11, were evaluated, with the patch and ground structures consistent with the single-element antenna to ensure uniform performance. A 100 μm edge-to-edge spacing was maintained across all configurations, ensuring proper alignment and element separation. In the first configuration (Ant. 1), the two antenna elements are aligned side by side at a 0-degree orientation, as illustrated in Fig. 11a. In the second setup (Ant. 2), shown in Fig. 11b, the elements are flipped to a 180-degree orientation relative to the first element. Lastly, the third configuration (Ant. 3), shown in Fig. 11c, also places the elements at a 180-degree orientation, aligned side by side.

Fig. 11

Comparative analysis of MIMO antenna configurations, (a) Ant 1; (b) Ant 2; (c) Ant 3(Proposed).

Figure 12 shows the outcome of each configuration where Fig. 12a shows the reflection coefficient and Fig. 12b shows the gain. Antenna 1 achieved a triple-band response; however, each band displayed narrow bandwidths, with high return losses, which reduces efficiency. Additionally, the mutual coupling for Antenna 1 was relatively high at -4.4 dB, indicating significant interference between elements. Antenna 2 also achieved a triple-band response, with Band 3 showing an improved, broader bandwidth of 2 THz. However, the bandwidths in the other two bands remained narrow, and the return loss across all bands was suboptimal. The mutual coupling was better than Antenna 1, at -22.68 dB, suggesting moderate isolation between antenna elements.

Fig. 12

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

Antenna 3 exhibited the most favorable dual-band performance. In the first band, it achieved three resonance frequencies within a wide 3.34 THz bandwidth, while the second band offered a broad 1.05 THz bandwidth. Antenna 3 also demonstrated the lowest mutual coupling at -32.9 dB, indicating excellent isolation and minimal interference between elements. These results affirm Antenna 3 as the optimal configuration, balancing broad bandwidth, low return loss, and high isolation, making it the most suitable for our MIMO antenna application.

Reflection coefficient

The reflection coefficient, often represented as S11, measures the amount of signal reflected back from the antenna relative to the incident signal, indicating how well the antenna is matched to its operating frequency³⁴. A low reflection coefficient is desirable as it means most of the signal is being transmitted rather than reflected, contributing to efficient operation³⁵. For the proposed MIMO antenna, the reflection coefficient shows exceptional performance, achieving dual-band operation with strong resonance characteristics. In the first band, the antenna resonates at three distinct frequencies: 5.03 THz, 6.2 THz, and 7.77 THz, achieving return losses of -39.7 dB, -29.9 dB, and − 55.249 dB, respectively. This band spans a wide operating range from 4.9 THz to 8.24 THz, resulting in a substantial bandwidth of 3.34 THz, which enhances the antenna’s ability to support wide-frequency applications. The second band resonates at 9.07 THz with a return loss of -25 dB, covering a frequency range from 8.5 THz to 9.5 THz and yielding an additional bandwidth of 1 THz. Figure 13 illustrates the reflection coefficient across these resonant frequencies and bands.

Fig. 13

Reflection Coefficient of the proposed MIMO antenna.

Transmission coefficient

The transmission coefficient (or S21 for a two-port system) measures the signal transfer between antenna elements, directly impacting mutual coupling in MIMO systems. Low transmission coefficients (high isolation) are ideal as they indicate reduced interference between elements, essential for optimal MIMO performance³⁶. The proposed antenna achieves a high minimum isolation of -32.9 dB, with the maximum isolation reaching − 55 dB, indicating excellent inter-element isolation and reliable performance in multi-antenna configurations. Figure 14 illustrates the transmission coefficient of the proposed MIMO antenna, showcasing its effective isolation characteristics.

Fig. 14

Transmission Coefficient of the proposed MIMO antenna.

Gain and efficiency

Gain and efficiency are key indicators of antenna performance. Gain represents the antenna’s ability to direct energy in a specific direction, while efficiency indicates how effectively the antenna converts input power into radiated energy^37,38. High values are preferred for both gain and efficiency, as they contribute to effective and powerful signal transmission. As shown in Fig. 15, the proposed antenna achieves a maximum gain of 15.4 dB and 15.8 dB across the respective bands, supporting strong signal strength. The efficiency reaches an impressive 99.8% and 99% across both bands, ensuring minimal energy loss during transmission and confirming the high efficiency of the antenna design.

Fig. 15

Efficiency and Gain of the proposed MIMO antenna.

Diversity performance analysis

In MIMO systems, diversity performance parameters, including the Envelope Correlation Coefficient (ECC) and Diversity Gain (DG), play a crucial role in evaluating the effectiveness of multi-element antennas.

Envelope correlation coefficient (ECC)

ECC quantifies the level of correlation between signals received from different antenna elements. A low ECC value, ideally below 0.5, is desirable because it indicates reduced signal correlation, which enhances the independence of the signals and improves the diversity performance of the antenna³⁹. ECC is calculated using the equation.

$$ECC=\frac{{\left|{\int\:}_{4\pi\:}^{\:}\:\left[{E}_{1}\left(\theta\:,\phi\:\right)\text{*}{E}_{2}(\theta\:,\phi\:)\right]d\varOmega\:\right|}^{2}}{{{\int\:}_{4\pi\:}^{\:}\:\left|{E}_{1}(\theta\:,\phi\:\right|}^{2}d\varOmega\:\:{{\int\:}_{4\pi\:}^{\:}\:\left|{E}_{2}(\theta\:,\phi\:\right|}^{2}d\varOmega\:}$$

(10)

For the proposed antenna, the ECC value is an exceptionally low 0.0003 as shown in Fig. 16, indicating minimal correlation between the antenna elements and validating the antenna’s capability for effective spatial diversity in high-frequency applications.

Fig. 16

Envelop Correlation Coefficient (ECC) of the proposed MIMO antenna.

Diversity gain (DG)

DG measures the enhancement of signal reliability provided by the MIMO system in multipath environments, with higher values indicating better diversity gain. DG is ideally close to 10 and is calculated based on ECC by the formula⁴⁰.

$$\:\text{D}\text{G}=10\sqrt{1-{\text{E}\text{C}\text{C}}^{2}}$$

(11)

The proposed MIMO antenna achieves an impressive DG value of 9.9997, which is near the theoretical maximum, signifying superior diversity performance and robustness in signal reliability. Simulated DG are illustrated in Fig. 17, clearly demonstrating the antenna’s excellent suitability for MIMO applications by offering minimal signal correlation and high signal reliability, essential for efficient performance in dual-band THz frequencies.

Fig. 17

Diversity Gain (DG) of the proposed MIMO antenna.

Total active reflection coefficient (TARC)

TARC (Total Active Reflection Coefficient), an important measure in multi-antenna systems such as MIMO, indicates the degree of correlation between the signals transmitted or received power by different antennas, within the system⁴¹. A TARC value below 0 dB signifies a low signal correlation, representing an ideal scenario.

The below equation can be used to determine TARC:

$$\:TARC=\sqrt{\frac{{\left(\left|{s}_{aa}|+|{s}_{ab}{e}^{\phi\:}\right|\right)}^{2}+{\left(\left|{s}_{ba}|+|{s}_{bb}{e}^{\phi\:}\right|\right)}^{2}}{2}}$$

(13)

Where a and b represent the ports 1 and 2 and $\:\phi\:$ is the phase difference. The simulated outputs of TARC for the proposed antenna are shown $\:\le\:$-15 dB in Fig. 18.

Fig. 18

Simulated TARC of the proposed MIMO Antenna.

Surface current

Antenna current distribution is the distribution of electric current along the conducting elements of an antenna. This distribution plays a major role in determining the radiation pattern of the antenna, which affects its ability to send or receive electromagnetic signals⁴². The single-element surface current distribution peaks at 21130.8 A/m at 5 THz, as shown in Fig. 19a. The feedline and the frame in the patch’s center are where the surface currents are strongest. The anticipated surface current distribution for the MIMO antenna with one excited port and corresponding loads terminating at the other ports is displayed in Fig. 19b. No antenna in Fig. 19c shows a surface current unless we take out Ant.1, which is activated by a surface current. Aside from Antenna 2, none of the other antennas generate any surface current, as seen in Fig. 19c.

Fig. 19

Surface Current (a) Single Element Antenna; (b) MIMO Antenna (port 1); (c) MIMO Antenna (port 2).

Radiation pattern

An antenna’s radiation pattern is a crucial component of its performance in a wireless communication system, and various key factors must be considered while analyzing this pattern⁴³. The electric field vector and the direction of maximal radiation are contained in the E-plane, also called the electric field plane. The vector of the magnetic field and the direction of maximal radiation are contained in the H-plane, also called the magnetic field plane⁴⁴. At 5.03 THz, the E-field along ϕ = 0° exhibits a main lobe magnitude of 17.8 dB V/m with an HPBW of 51.7°, while along θ = 90°, the HPBW is 38.2°. The H-field along ϕ = 90° has a broader HPBW of 75.7°. At 6.2 THz, the E-field along ϕ = 0° achieves a main lobe magnitude of -1.68 dB V/m, while the H-field along θ = 90° has an HPBW of 37.1°, reflecting a focused radiation pattern with minimal sidelobes. At 7.77 THz, the E-field along ϕ = 0° has a main lobe magnitude of 13.5 dB V/m, while along θ = 90°, the main lobe magnitude increases to 16.2 dB V/m with an HPBW of 37.3°. The H-field along ϕ = 90° shows an HPBW of 51.6°, representing a balanced spread of energy in the plane orthogonal to the E-field. At 9.07 THz, the E-field along ϕ = 0° achieves an HPBW of 36.8°, ensuring sharp directivity. The H-field along ϕ = 0° has a main lobe magnitude of -45 dB A/m with an HPBW of 34.6°, indicating concentrated energy in a narrow angular range. Along θ = 90°, the H-field’s main lobe magnitude is -35.4 dB A/m, showcasing consistent performance in orthogonal orientations, as shown in Fig. 20 .

Fig. 20

Simulated Radiation Pattern of recommended MIMO Antenna.

RLC equivalent circuit

We created an R-L-C (Resistance-Inductance-Capacitance) circuit model after thoroughly analyzing the electromagnetic properties of the system in order to construct an enhanced antenna system for future wireless communication⁴⁵. An effective tool for illustrating the electrical characteristics of the antenna is this RLC circuit. The capacitance (C) in this model denotes the antenna’s ability to store electrical energy, whereas the inductance (L) indicates its ability to store magnetic energy. Energy wasted as radiation and heat is taken into account by the resistance (R)⁴⁶. A crucial component for efficient signal transmission, the resonance frequency of the antenna is ascertained with the use of this framework. Using Advanced Design System (ADS) software, the RLC equivalent circuit model was created, and its response was confirmed. The MIMO antenna’s final equivalent circuit is shown in Fig. 21.

Fig. 21

Equivalent circuit of the proposed MIMO Antenna.

The resonance frequency, essential for the antenna’s performance, is represented by a parallel circuit configuration consisting of resistance (R1), capacitance (C1), and inductance (L1). The first resonance frequency is defined by R1, C1, and L1; the second by R2, C2, and L2; the third by R3, C3, and L3; and the fourth by R4, C4, and L4. The square shape at the center of the patch is represented by inductances L5, L6, L7, and L8, while the plus-shaped patch sections are characterized by capacitances C5, C6, C7, and C8. Additionally, the “Lattice Diamond Patch” design is represented by the pairs (C9, L9), (C10, L10), (C11, L11), and (C12, L12). To incorporate the feedline into the circuit model, elements such as resistance (R5), capacitance (C13), and inductance (L13) were added to accurately capture its electrical characteristics. These parameters significantly influence the overall impedance and transmission properties of the antenna system. By carefully combining these individual elements, we developed a comprehensive model that accurately represents the behavior of the single-element antenna.

The equivalent circuit for the MIMO antenna was subsequently created by combining two identical single-antenna circuits, as the MIMO configuration comprises two antenna elements. The mutual inductance and capacitance between these two antenna elements in the MIMO system are represented by C14 and L14, which are connected in parallel between the antennas.

To confirm the accuracy of the Agilent ADS simulation, we executed a comparative study by matching the results of the CST simulation with those from a parallel circuit simulation, focusing particularly on the S11 parameter. Figure 22 shows the comparison between the circuit simulation and CST simulation results, offering a thorough evaluation of our R-L-C circuit model’s accuracy and reliability in representing the antenna system’s behavior. The specific parameter values are specified in Table 3.

Fig. 22

Reflection Coefficient Analysis: CST and Equivalent Circuits in ADS.

Table 3 Values of the parameter of RLC equivalent circuit.

Antenna optimization through ML techniques

Antenna optimization through machine learning involves using data-driven models to predict and enhance antenna performance by fine-tuning design parameters. Traditional antenna design often requires extensive parameter sweeps and costly simulations. Scientists adjust frequency, size, and materials to achieve optimal performance in gain, efficiency, and resonance frequency metrics. Machine learning allows for analyzing large datasets of simulated designs to develop predictive models that reveal the relationships between design factors and performance outcomes. This approach reduces the need for repetitive simulations and enables rapid exploration of the design space to identify optimal combinations⁴⁷.

Numerous machine-learning techniques, including Random Forest, XGBoost, Neural Networks, and Stacking Ensembles, are frequently employed. These models are capable of accurately predicting antenna performance by capturing intricate, non-linear interactions between design elements. Engineers can rapidly predict the performance of new designs by training these models on simulation data, which enables them to improve parameters at a cheap computational cost. The design can then be iteratively refined using the best-performing model, which will result in improved performance metrics and a significant reduction in the time and resources required for antenna development.

The flowchart shown in Fig. 23 provides an overview of a thorough procedure for designing antennas and using machine learning to enhance their performance. Each factor is described in depth below:

• Antenna design: Determine important parameters including application, frequency, and performance goals. The operational frequency and necessary performance parameters of the antenna are specified in this step.

• Parameter sweep simulation: Run simulations by radically changing the antenna’s parameters. This process helps to improve important variables and evaluates the performance effects of various design choices.

• Data collection and preparation: Gather simulation output data, including resonance frequency, efficiency, and gain. Compile this data into a dataset, which will serve as input for the machine-learning models.

• Data partitioning: Training Dataset (80%)-The dataset is distributed, with 80% designated for training machine learning models. Testing Dataset (20%) - The remaining 20% is set aside for testing, evaluating the model’s performance on unseen data.

• Machine learning models: Several machine learning algorithms are employed, including: Random Forest, CatBoost XGBoost, Neural Network and Stacking Ensemble. A method that merges multiple models to improve prediction accuracy. These models are trained using the training dataset to predict antenna performance based on design parameters.

• Model assessment and selection: The models are meticulously tested using the testing dataset to determine which is the most effective. This evaluation, based on metrics such as accuracy, precision, and other relevant criteria, is crucial. It leads to the selection of the optimal model for predicting antenna performance, ensuring the accuracy of the prediction.

Fig. 23

Diagram illustrating the sequential process involved in the development of machine learning.

This flowchart provides an organized approach for developing and refining an antenna through simulations and machine-learning methods. It ensures the selection of the most efficient model for accurate performance prediction.

Neural network architecture

To predict the geometrical characteristics of the antenna, we used a neural network architecture. The complex, non-linear connections between the input features and the output geometrical parameters were precisely captured by this network. Five fully connected (dense) layers with Rectified Linear Unit (ReLU) activations made up the architectural design. Dropout layers were added to help with regularization and avoid overfitting⁴⁸. Figure 24 provides a thorough illustration of the network structure.

Fig. 24

Multi-Layer Neural Network Architecture for Predicting Antenna Geometry Using Performance Metrics.

Here, X ∈ R^n×d where n is the batch size, and d is the number of input features; W₁∈R^d×128 and b₁∈R¹²⁸; W₂∈R^{128 × 256} and b₂∈R²⁵⁶; W₃∈R^{256 × 128} and b₃∈R¹²⁸; W₄∈R^{128 × 64} and b₄∈R⁶⁴; W₅∈^{R64 × 10} and b₅∈R¹⁰ and y is the output vector which contains the geometric parameters.

Machine learning models and hyperparameter tuning

The neural network model has many weights. We used different machine learning algorithms to estimate antenna geometry characteristics due to data restrictions. Since these perform at non-linear regression, Random Forest, Gradient Boosting, XGBoost, and CatBoost were chosen.

CatBoost

Using minimal preprocessing, CatBoost efficiently handles categorical and numerical regression features. Ordered boosting and target-based encoding reduce target leakage and improve prediction accuracy⁴⁹. CatBoost accurately predicts complicated geometrical parameters like patch dimensions, substrate qualities, and feed characteristics by applying it to resonance features like gain, efficiency, and return loss (RL). This targeted encoding captures antenna design’s complex interactions.

Random forest regression

The Random Forest regression method uses an ensemble of decision trees trained on random data subsets to improve prediction robustness and variance⁵⁰. Gain, efficiency, and return loss at resonance frequencies are used to forecast complicated characteristics like patch dimensions and substrate qualities while reducing overfitting. Random Forest’s ensemble nature captures input-output correlations, predicting antenna design accurately.

XGBoost regression

Instead of a single model, XGBoost regression uses an ensemble of gradient-boosted decision trees. Regularization and parallel processing make it efficient at managing enormous datasets and accurate at predicting⁵¹. Input factors like gain, efficiency, and return loss at resonance frequencies let XGBoost anticipate complex geometrical aspects like patch dimensions and substrate qualities. Its overfitting resistance and complicated relationship modeling improve antenna design forecasts.

We hyperparameter tuned each model and cross-validated to ensure generalization to maximize performance. We utilized grid search and randomized search to discover the optimal hyperparameters for each model. By testing multiple hyperparameter combinations, these searches find the best-performing set based on cross-validated performance measures. For instance, Random Forest used a grid search over the number of trees (n_estimators), maximum depth (max_depth), and minimum number of samples for splitting nodes. We also used 5-Fold Cross-Validation to ensure model generalizability. This divides the dataset fivefold. Five times, each model is trained on four folds and validated on five. The fold-average performance is used to assess the model’s performance. This approach offers a reliable approximation of the model’s capacity to generalize to unknown inputs.

Leveraging stacking-based ensemble learning for enhanced prediction accuracy and robustness

Stacking is an ensemble method that integrates numerous models. In contrast to bagging and boosting, stacking is generally employed to amalgamate various classifiers⁵². As illustrated in Fig. 25, stacking consists of two levels: the base learner and the stacking model learner (meta learner):

Fig. 25

Framework for the proposed Stacking Approach.

This method trained on input features using four base models: Random Forest (RF), XGBoost, CatBoost, and Neural Network (NN). Xtrain ∈Rn×p and estimate output characteristics ytrain ∈ Rn×k, where n, p, and k are the number of samples, features, and target geometric parameters, respectively. Based on the test data Xtest, the models generate base predictions⁵³.

$$\:{\widehat{y}}_{i}={{f}_{i}(X}_{test})$$

(13)

The models for i ∈{1,2,3,4} are Neural Network, CatBoost, XGBoost, and Random Forest. The general algorithm for the base learner is presented as follows:

Base Learner ().

Input: Training Data V (features m, instances p).

Output: Predictions P for test instances I.

For B = 1 to Bn // Base classifiers.

For F = 1 to Fm // Target features.

For I = 1 to N // Test instances.

Model_B ← Train(B, V) // Train base classifier.

P_BF(I) ← Predict(Model_B, I) // Predict for test instance I.

EndFor.

EndFor.

EndFor.

Output: Predictions P_BF for all test instances.

The meta-learner uses the basic models’ predictions as input features. A matrix Pbase∈Rm×4k, where m is the number of test samples and 4Kcore represents the predictions for each geometric parameter from the four base models, can be used to represent these predictions⁵⁴.

$$\:{P}_{base}=\:\:\left[{{f}_{1}(X}_{test}\right)\:\:\:{{f}_{2}(X}_{test}\left)\:\:\:{{f}_{3}(X}_{test}\right)\:\:\:{{f}_{4}(X}_{test}\left)\right]$$

(14)

Each P_base column represents model predictions for a geometric parameter. These predictions are transformed into meta-learner training features. The meta-learner maps base predictions (P_base) to true test outputs (y_test) while minimizing prediction error using a linear regression model.

$$\:{\widehat{y}}_{final}={g(P}_{base})$$

(15)

where g(⋅) is the linear regression model, solving for the optimal weight matix W and bias b:

$$\:{\:{\widehat{y}}_{final}={WP}_{base}+b}^{\:}$$

(16)

This meta-learner combines base model predictions to ensure lower variance and higher accuracy than any single base model. The meta learner’s general algorithm is shown as follows:

Meta-Learner ().

Input: Predictions P from Base Classifiers, S.

Output: Final Predictions Pf from Meta-Learner E.

For D = 1 to Fm // Target features.

For M = 1 to Bn // Base learners.

If D ≠ M Then.

T(D, M) ← Concatenate predictions from Base Learner M (excluding D).

Meta_Model ← Train(E, T(D, M)) // Train Meta-Learner.

Pf(D, :) ← Predict(Meta_Model, Ltest) // Test and predict.

EndIf.

EndFor.

EndFor.

Output: Final Predictions Pf.

The stacked model combined base models’ strengths to reduce prediction errors and improve antenna geometric parameter predictions.

Results and analysis

We developed and evaluated four predictive models: Random Forest, CatBoost, XGBoost, and a Neural Network. We trained each model on our dataset to predict geometric parameters using gain and efficiency features.

MSE, RMSE, MAE, R2 Score, and Explained Variance Score are used to evaluate our model.

Mean squared error (MSE) measures the average of the squares of the errors, indicating the average squared deviation between the expected output features and their actual values. The Mean Squared Error (MSE) is a reliable metric for evaluating model performance, where diminished values signify enhanced accuracy in predicting geometrical parameters⁵⁵.

$$\:MSE=\frac{1}{n}{\sum\:}_{i=1}^{n}\:\:{({y}_{i}-\:{\widehat{y}}_{i})}^{2}$$

(17)

Where y_i is the actual value, $\:{\widehat{y}}_{i}$​ is the predicted value, and n is the number of observations.

Root mean squared error (RMSE) is the square root of the Mean Squared Error (MSE), yielding a comprehensible measure in the same units as the expected output variables. RMSE facilitates direct comparisons with the scale of geometrical parameters by expressing the average magnitude of prediction errors. Reduced RMSE values indicate improved predicted accuracy⁵⁶.

$$\:RMSE=\sqrt{\frac{1}{n}\:{\sum\:}_{i=1}^{n}\:\:{{(y}_{i}-\:{\widehat{y}}_{i})}^{2}}$$

(18)

Mean absolute error (MAE) determines the average absolute discrepancies between expected and actual output features. These metric measures accuracy and is less sensitive to outliers than MSE, making it a reliable choice for geometrical parameter model evaluation⁵⁷.

$$\:MAE=\frac{1}{n}\:{\sum\:}_{i=1}^{n}\:\:\left|{y}_{i}-\:{\widehat{y}}_{i}\right|$$

(19)

R² score, or coefficient of determination measures how much variance in output features can be explained by input variables. Values near 1 imply that the model captures a lot of geometrical parameter variance, whereas values near 0 indicate poor explanatory ability⁵⁸.

$$\:{R}^{2}=1-\frac{{\sum\:}_{i=1}^{N}\:\:{({y}_{i}-{\widehat{y}}_{i})}^{2}}{{\sum\:}_{i=1}^{N}\:\:{({y}_{i}-{\widehat{y}}_{i})}^{2}}$$

(20)

Where yˉ​ is the mean of the actual values.

Explained variance score (EVS) evaluates the fraction of expected output feature variance attributable to actual values. A higher EVS score, like the R² score, indicates the model provides accurate predictions by capturing geometrical parameter variability⁵⁹.

$$\:explained\:variance\:\left(y,\widehat{y}\right)=1-\frac{Var\left(y-\widehat{y}\right)}{Var\left(y\right)}$$

(21)

Qualitative analysis

We compare the performance metrics of our proposed architecture with those of state-of-the-art models as shown in Table 4. First, we evaluate our model against individual regression models such as Random Forest, Catboost, XGBoost, and a Neural Network. Next, we include hyperparameter tuning and cross-validation in our assessment.

Table 4 Performance prediction metrics for various regression model.

The bar chart as illustrated in Fig. 26 shows the error matrix of numerous machine learning models, including Random Forest, CatBoost, XGBoost, Neural Network, and Stacking Ensemble, evaluated against three metrics: Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Mean Absolute Error (MAE). Notably, the Stacking Ensemble model attains the lowest error rates across all metrics, indicated by the shortest bars in each category, reflecting its superior predictive accuracy. CatBoost shows low error rates among the individual models, followed closely by Random Forest. In contrast, the Neural Network shows the highest RMSE and MAE, indicating less accurate predictions.

Fig. 26

Error matrix bar chart for machine learning regression model.

The Stacking Ensemble model records the lowest values for all three error measures, with a Mean Squared Error (MSE) of 0.00842, a Root Mean Squared Error (RMSE) of 0.00453, and a Mean Absolute Error (MAE) of 0.00999, establishing it as the top-performing model. Among the individual models, CatBoost exhibits commendable results, with a Mean Squared Error (MSE) of 0.0144, a Root Mean Squared Error (RMSE) of 0.06693, and a Mean Absolute Error (MAE) of 0.0223, indicating its effectiveness compared to other individual models. The Neural Network, on the other hand, presents the highest RMSE (0.1182) and MAE (0.1015), highlighting the most substantial prediction errors among the models assessed. This analysis demonstrates that the Stacking Ensemble method significantly enhances forecasting accuracy.

The bar chart as illustrated in Fig. 27 presents the accuracy of various machine learning models assessed through two metrics: R² Score and Explained Variance Score (EVS). The Stacking Ensemble model stands out with outstanding performance, achieving an R² of approximately 0.96 and an EVS close to 1, indicating its ability to capture almost all variance in the data and demonstrating remarkable predictive accuracy. The Neural Network also shows commendable results among the individual models, with an R² of around 0.44 and an EVS of about 0.35. CatBoost performs well, with metrics near 0.36 and 0.38, respectively. In contrast, XGBoost and Random Forest exhibit lower R² and EVS values, reflecting their reduced accuracy in representing the data’s variance compared to the ensemble model. This analysis underscores the effectiveness of the Stacking Ensemble approach in achieving superior prediction accuracy.

Fig. 27

Accuracy comparative bar chart for machine learning regression model.

Table 5 provides a detailed comparison of our proposed stacking ensemble model with other machine learning models used in related works. The results demonstrate that our model outperforms the others in most metrics, showcasing its effectiveness.

Table 5 Comparison of machine learning models for antenna design based on performance metrics.

The proposed stacking ensemble model achieved the lowest MAE (0.0999%), significantly better than the Random Forest Regression models in²² (3.15%) and⁶⁰ (86.1%), as well as the Gaussian Process in²³ (0.26%). Similarly, it demonstrated the lowest MSE (0.0842%) compared to 0.3% in²², 38.16% in⁶⁰, and 0.18% in²³.

In terms of RMSE, our model achieved the best performance (0.0453%), significantly outperforming the 4.51% in²² and the 0.34% in²³. For the R-squared metric, the Gaussian Process in²³ achieved the highest value (99.3%), slightly outperforming the proposed model’s 96.3%. However, the proposed model still demonstrated better R-squared performance than²² (95.51%) and⁶⁰ (82.6%).

Finally, the Variance Score of the proposed model was the highest (99.8%), outperforming²² (96.62%), and²³ (98.84%). These results demonstrate that the stacking ensemble method effectively integrates predictions from multiple base learners, resulting in superior accuracy and generalization compared to existing models.

Predictive stability and variability in machine learning models for antenna design

The predictive accuracy of our suggested model was assessed for every geometrical parameter by contrasting the true values across sample indices with the predicted values (Fig. 28). This detailed examination emphasizes model strengths and weaknesses with numerical observations.

Fig. 28

Graphical comparison of model predictions and true values for antenna parameters.

Substrate width (sw) predictions are close to correct, with some minor differences. At index 15, the predicted value peaks at 0.95 while the true value is closer to 0.8. The average variance among indices is modest, showing sw is well-predicted despite these fluctuations. Patch thickness (pt) is accurately predicted by the model across indices. The greatest deviation comes around index 10, where the projected value is 0.55 and the true value is 0.6. Pt is one of the best-predicted factors due to this small disparity. For feed length (fl.), model predictions match true data with an average deviation of 0.05. The projected value is 0.65 at index 18, while the true value is 0.6. However, the model predicts fl. well by capturing the general trend. Substrate length (sl) is one of the most correctly predicted metrics, with almost perfect alignment across all indices. No significant differences exist between the expected and true values of 0.5. The model’s consistency shows its precision in predicting sl. Our model accurately predicts feed width (fw), especially when aligned with genuine values. At index 10, the anticipated value is 0.45 and the true value is 0.4. The average deviation is under 0.05, indicating accurate fw prediction. The outer radius (r3) forecast has a good match with the true values until index 10, when it jumps to 0.6 from 0.5. The model occasionally overshoots genuine values, therefore this peak suggests refinement. The model typically follows the true values for patch length (pl) but peaks at index 10, where the projected value is 0.9, much higher than the true value of 0.7. Although the overall trend is reflected, spikes show where the model struggles to maintain stability. Inner radius (r4) predictions match true values rather well until index 10, when the predicted value drops to 0.2 while the true value remains near 0.3. This divergence implies that r4 is a difficult model parameter, identifying opportunities for prediction stability improvement. The model predicts patch width (pw) close to true values but spikes at index 10. The true value is 0.65, whereas the anticipated value is 0.85. This disparity suggests pw is another predictively unstable parameter. The model accurately predicts substrate thickness (st), especially peaks and troughs. The highest deviation is at index 15, where the projected value is 1.0 and the true value is 1.1. The tiny variance and general alignment suggest that st is one of the most reliably predicted factors.

With average errors less than 0.05, our suggested model thus shows strong predictive ability for metrics like sl (Substrate Length), pt (Patch Thickness), and st (Substrate Thickness), all of which show little variance from true values. On the other hand, as they may exhibit more deviations—with maximum variations of up to 0.2 at specific indices—r4 (Inner Radius) and pw (Patch Width) are considered as more difficult parameters. These findings suggest that although the model works well overall, more fine-tuning might increase accuracy for more variable parameters.

Conclusion

In conclusion, this study introduces a breakthrough in compact, high-performance MIMO antenna design for terahertz 6G applications, achieving dual-band resonance with substantial bandwidths of 3.34 THz and 1 THz, high gain of 15.8 dB, and remarkable isolation beyond − 32.09 dB. By harnessing an advanced meta learner-based stacked ensemble machine learning approach, antenna parameters were optimized to an unprecedented degree of precision, with predictive metrics reaching R² of 0.96, EVS of 0.998, and error values MSE, RMSE, and MAE of 0.0002, 0.004, and 0.0015, respectively. Rigorous validation through CST simulations, complemented by RLC circuit modeling in ADS, underscores the antenna’s exceptional efficiency and resilience, supporting stable, high-speed data transmission even in dense 6G communication environments. Future work will focus on fabricating the antenna to validate simulated results with empirical data, a critical step toward real-world deployment. Additionally, integrating metamaterials presents a promising avenue for further enhancements in bandwidth, gain, and efficiency, potentially introducing new functionalities for extended terahertz applications. This design not only establishes itself as a strong candidate for next-generation 6G technology but also sets a benchmark in the fusion of THz communication and machine learning-driven antenna design, laying the groundwork for future advancements in wireless communication and sensing technologies.