Envelope correlation coefficient (${\rho _{ECC}}$)

The diversity measure that shows how the neighboring MIMO antenna components correlate is $({{{\varepsilon }}_r}$. It may be estimated using the S-parameters or radiation patterns. It’s essential to evaluate its value using the far-range radiation pattern, as ${\rho _{ECC}}$ explains the unique radiation patterns of different radiating parts in MIMO systems. Additionally, it is evident that the majority of planar antennas experience loss. It’s best to avoid using the S-parameters to calculate ${\rho _{ECC}}$. The allowable limit of envelope correlation coefficient (ECC) in a realistic situation must be less than 0.5. Equations (1) and (2) provide the formulas for utilizing information about the radiation pattern of a MIMO system. In contrast, Equation (3) provides the formula for ${\rho _{ECC}}$ utilizing information about the S-parameters [Reference Muttair, Shareef, Mosleh, Ghazi Zahid, Shakir and Qasim18, Reference Tiwari, Singh, Kumar and Kanaujia19].

(1)\begin{align}{\rho _{ECC}} = \frac{{{{\left| a \right|}^2}}}{{a \times a}}\,\end{align}

(2)\begin{align}a = \int\limits_0^{2\Pi } \int \limits_0^\Pi \left( {{\varrho ^*}_{{{\theta }}p}{\varrho _{{{\theta }}q}}{P_{{\theta }}}{X_{{\text{PR}}}} + {\varrho ^*}_{\varphi p}{\varrho _{\varphi q}}{P_\varphi }} \right)d\Omega \end{align}

(3)\begin{align}{\rho _{ECC}} = \frac{{\left( {{{\left| {{S_{\left( {ii} \right)}}\,{S_{\left( {ij} \right)}} + {S_{\left( {ji} \right)}}\,{S_{\left( {jj} \right)}}} \right|}^2}} \right)}}{{\left( {1 - \left( {{{\left| {{S_{\left( {ii} \right)}}} \right|}^2} + {{\left| {{S_{\left( {ji} \right)}}} \right|}^2}} \right)} \right)\left( {1 - \left( {{{\left| {{S_{\left( {jj} \right)}}} \right|}^2} + {{\left| {{S_{\left( {ij} \right)}}} \right|}^2}} \right)} \right)}}\end{align}

Where ${X_{{\text{PR}}}}$ is the cross-polarization level, it can be a percentage of the mean power across the φ & θ directions. $S$ is the S-parameter (reflection coefficient) for each port at different frequencies, while $i$ and $j$ represent the number of ports in the MIMO array.

Diversity gain (DG) (G_Diversity)

In wireless networks, ${G_{Diversity}}$ represents the quality and dependability of MIMO antennas. As a result, the ${G_{Diversity}}$ for the MIMO antenna within the permitted frequency spectrum has to be high (10 dB). Equation (4) can be utilized to calculate the ${G_{Diversity}}\,$used the value of ${\rho _{ECC}}$ [Reference Muttair, Shareef and Taher20].

(4)\begin{align}{G_{Diversity}} = 10*\,\sqrt {1 - {{\left| {{\rho _{ECC}}} \right|}^2}} \end{align}

Channel capacity loss (CCL)

It denotes the volume of data that can be conveyed across a communication link, accounting for potential channel loss. A specified MIMO system’s predetermined channel capacity loss (CCL) value is 0.4 bits/s/Hz. Equation (5) provides the formula for CCL through S-parameters [Reference Khalid, Syeda, Niamat, Rahman, Fawad, Mirjavadi, Khan and Amin21].

(5)\begin{align}& CCL =\nonumber\\ & \quad - {\log _2}\left[ det\left[ \begin{array}{@{}cc@{}} {1 - \left[ {{{\left| {{S_{11}}} \right|}^2} + {{\left| {{S_{12}}} \right|}^2}} \right]}& { - \left[ {{S^*}_{11}{S_{12}} + {S^*}_{21}{S_{12}}} \right]} \\ { - \left[ {{S^*}_{22}{S_{21}} + {S^*}_{12}{S_{21}}} \right]}& {1 - \left[ {{{\left| {{S_{22}}} \right|}^2} + {{\left| {{S_{21}}} \right|}^2}} \right]} \end{array} \right] \right]\end{align}

Average effective gain (AEG)

It is a crucial diversity parameter for MIMO antennas, indicating the additional power received compared to an isotropic antenna. The average effective gain (AEG) ratio has to be <3 dB for a MIMO antenna to function better at equal power output. Using Eq. (6), the evaluation value of AEG can be determined [Reference Khalid, Syeda, Niamat, Rahman, Fawad, Mirjavadi, Khan and Amin21].

(6)\begin{align}AEG = \frac{{AE{G_i}}}{{AE{G_j}}} = \frac{{0.5 \times \left[ {1 - {{\left| {{S_{ii}}} \right|}^2} - {{\left| {{S_{ij}}} \right|}^2}} \right]}}{{0.5 \times \left[ {1 - {{\left| {{S_{ij}}} \right|}^2} - {{\left| {{S_{jj}}} \right|}^2}} \right]}}\end{align}

Overall active reflection coefficient (OARC)

The overall active reflection coefficient (OARC) of a MIMO system is calculated by dividing the total reflected power by the total incident power. The ratio of the total power incident upon the patch is compared to the total reflecting power resulting from the radiating components. Equation (7) provides the equation for the expanded OARC for multi-port MIMO antennas. If we want to calculate the OARC for two ports, it will be according to the formula shown in Eq. (8) [Reference Pei, Zhu, Wang and Wen22].

(7)\begin{align}OARC = \frac{{\sqrt {\mathop \sum \nolimits_{i = 1}^N {{\left| {{b_i}} \right|}^2}} }}{{\sqrt {\mathop \sum \nolimits_{i = 1}^N {{\left| {{a_i}} \right|}^2}} }}\end{align}

(8)\begin{align}OARC = \frac{{\sqrt {{{\left| {{S_{11}} + {S_{12}}{e^{j{{\theta }}}}} \right|}^2} + {{\left| {{S_{21}} + {S_{22}}{e^{j{{\theta }}}}} \right|}^2}} }}{{\sqrt 2 }}\end{align}

Where ${a_i}$ and ${b_i}$ represent the coefficients of the scattering matrices, and is the number of ports in the MIMO configuration.

Dual-port MIMO antenna designs for various applications

In a recent study [Reference Kiani, Savci, Munir, Sedik and Mostafa23], researchers proposed the design of a two-port compact antenna with its middle layer (substrate) made of RO5880, which has a thickness of 0.787 mm. The overall dimensions of this antenna are 52 × 28 $m{m^2}$, as shown in Figs. 3(a) and (b). This antenna is designed for operation in a resonant frequency range of 2.3–11.5 GHz, as shown in Figs. 4(a), (b) and (c), making it suitable for UWB applications. The suggested antenna shows significant isolation improvement, reaching up to −16 dB, as shown in Fig. 4(a), achieved using shared radiators with small rectangular slots. This feature reduces interference, boosts overall performance, and demonstrates its capabilities through a detailed analysis of MIMO performance factors in Table 1. A good range of results was identified. The simulations and measurements indicate that the antenna design is feasible and successful. Its broad bandwidth, small size, and improved isolation qualities make it an attractive option for UWB imaging with microwave systems of the future.

Figure 3. Antenna structural shapes on both sides: (a) simulation design; (b) realistic manufacturing design [Reference Kiani, Savci, Munir, Sedik and Mostafa23].

Figure 4. The reflection coefficient parameter of the antenna in simulation and realistic measurements. (a) The simulation side; (b) The simulation and practical side of S11; (c) The simulation and practical side of S22 [Reference Kiani, Savci, Munir, Sedik and Mostafa23].

Table 1. An overview of the latest research into the advancement of wideband two-port antennas

In addition, the authors presented a two-element MIMO antenna that operates at mmWave frequencies (25.2–29.5 GHz). This antenna is characterized by its rectangular outer shape, as shown in Figs. 5(a) and (b), with geometric dimensions (W = 50 mm, L = 12 mm, H = 0.787 mm). The material used for its substrate layer is Rogers 5880, which has a permittivity (${{{\varepsilon }}_r}\,$ = 2.2) and loss tangent (δ = 0.02). The authors clarified that the antenna resonates at 28.4 GHz and that the isolation ratio between the first and second ports is lower than −32 dB, as demonstrated in Fig. 6(a). They added that because the antenna offers acceptable performance parameters, it is appropriate for communications systems based on mmWave frequencies. For example, the gain rate reached 11.4 dBi, the ECC is <0.00025, and the diversity gain (DG) is >9.996 dB, as shown in Figs. 6(b) and (c), respectively [Reference Arshad, Ahmad, Amin, Babar Abbasi and Choi24].

Figure 5. The fabrication geometry of the proposed antenna is (a) front view and (b) back view [Reference Arshad, Ahmad, Amin, Babar Abbasi and Choi24].

Figure 6. The antenna performance curves for (a) reflection coefficient (S11 and S22) and isolation (S12 and S21), (b) ECC, and (c) DG [Reference Arshad, Ahmad, Amin, Babar Abbasi and Choi24].

Other scholars have recently published numerous studies on the design of two-port MIMO antennas, which are well detailed in Table 1. Table 1 showcases the comparative data of diverse antenna designs tailored for various purposes. Because mutual coupling is crucial in MIMO antennas, researchers in reference [Reference Elabd25] found that MIMO elements have a notable isolation ratio of −65 dB. Although the remaining designs were satisfactory and produced similar results, one stood out above the others.

The remaining recent works have been compiled in Table 2 using the same methodology. Researchers present a dual-port MIMO antenna operating in mmWave bands. It has been discovered that a significant number of the designs utilized similar materials to fabricate the antennas. In addition, there is a discrepancy in the results: the researchers in reference [Reference Megahed, Abdelhay, Abdelazim and Soliman26] achieved good results for isolation ratio, efficiency, ECC, DG, CCL, and gain. While the other proposed designs also produced good results, they showed varied performance due to the different dimensions and geometries suggested by each author for the antenna design.

Table 2. A detailed comparison and summary of recent research papers introducing dual-port antennas in the mmWave bands

Four-port MIMO antenna designs for various applications

Recently, researchers presented in reference [Reference Govindan, Palaniswamy, Kanagasabai and Kumar66] a foldable MIMO antenna for smart apparel applications with UWB capability. The MIMO antenna operated in the 2.9–12 GHz band, as shown in Fig. 7, and consisted of four octagonal radiators with empty holes built into them, as shown in Figs. 8(a), (b) and (c). The dimensions and thickness of this antenna are 50 × 50 × 1.6 mm³. The antenna’s radiation and diversification performances are analyzed, and the metrics obtained include ECC < 0.045, DG > 9.9 dB, OARC < − 14 dB, and CCL < 0.13 bits/s/Hz. Additionally, the suggested antenna has a 20 mm bend radius, making it appropriate for applications in wearable devices. This antenna is suitable for applications involving patient monitoring.

Figure 7. The S-parameter curves versus the different frequencies [Reference Govindan, Palaniswamy, Kanagasabai and Kumar66].

Figure 8. (a) Prototype antenna with four ports on the front; (b) antenna bending model at 20 mm; and (c) antenna performance measurement using a vector network analyzer device [Reference Govindan, Palaniswamy, Kanagasabai and Kumar66].

In another recent work in reference [Reference Ibrahim and Abo Sree67], a compact UWB four-element MIMO antenna design with band rejection is presented, as shown in Figs. 9(a) and (b). The recommended antenna can function at 3–12 GHz and has S11 ≤ −10 dB thanks to the four components’ rectangle radiators with curving sides and partially grounded planes with an engraved slot. The antennas were arranged orthogonally without decoupling features, simplifying the engineering process and ensuring high isolation between the components. On a cheap FR4 substrate, the recommended design dimension and thickness are 47 × 47 × 1.6mm³. The antenna operates within the operational bands with a maximum gain of 4.8 dBi, as shown in Fig. 10(a). It has an ECC of less than 0.005, as shown in Fig. 10(b). The antenna has a DG of 9.98 dB, as shown in Fig. 10(c). The CCL of the antenna is less than 0.4 bit/s/Hz, as shown in Fig. 10(d). The results were consistently positive, which allowed the recommended antenna to be employed in UWB MIMO communications systems.

Figure 9. An antenna manufacturing prototype (a) on the front side and (b) on the back side [Reference Ibrahim and Abo Sree67].

Figure 10. The basic parameters to determine the efficiency of the proposed antenna in reference [Reference Ibrahim and Abo Sree67] are (a) gain curves, (b) ECC curves, (c) DG curves, and (d) CCL curves.

A recent research article [Reference Hasan, Islam, Abdul Rahim, Alam, Rmili, Alzamil, Islam and Soliman68] presented a quad-port broadband metamaterial (MM) antenna to achieve a high gain in new radio communications operating at sub-6 GHz, as shown in Figs. 11(a) and (b). The suggested four MIMO antennas are arranged orthogonally to the neighboring antennas. It achieves the compact size and properties of 55.2% bandwidth with a low interelement edge-to-edge length of 0.19 ${{-}\mkern-10mu\lambda _{min}}$ at 3.25 GHz. The intended MIMO system is implemented using an inexpensive FR-4 printed material, measuring just 0.65 ${{-}\mkern-10mu\lambda _{min}}$×0.65 ${{-}\mkern-10mu\lambda _{min}}$×0.14 ${{-}\mkern-10mu\lambda _{min}}$, as shown in Figs. 12(a), (b) and (c). A high peak output gain of about 7.1 dBi between −9 and −50 dB isolation is displayed by the developed miniature MIMO system with an MM reflector, as shown in Fig. 11(a). Furthermore, the proposed broad-spectrum MM increases MIMO’s various perspectives and radiation properties, with an average overall efficiency of 68% throughout the target bands. The specified antenna for the MIMO system has good multiplex efficiency, with a value of more than −1.4 dB, an acceptable CCL of less than 0.35b/s/Hz, an exemplary DG of 9.96 dB, and an exceptional ECC of less than 0.045. The values of the rest of the parameters that determine antenna performance are listed in Table 3. In the end, the researchers explained that the proposed antenna is a potential approach to the 5G system.

Figure 11. The return loss curves versus the various frequencies for (a) simulation side curves and (b) comparisons between simulation side curves and manufacturing measurements [Reference Hasan, Islam, Abdul Rahim, Alam, Rmili, Alzamil, Islam and Soliman68].

Figure 12. The design of the proposed antenna shapes (a) CST simulation, (b) practical design on the front side, and (c) practical design on the back side [Reference Hasan, Islam, Abdul Rahim, Alam, Rmili, Alzamil, Islam and Soliman68].

Table 3. A summary of recent studies on the design of a four-port antenna that operates at frequencies below 27 GHz

In another recent study [Reference Alharbi, Rafique, Ullah, Khan, Abbas, Ali, Alibakhshikenari and Dalarsson69], researchers proposed a four-port MIMO antenna with dimensions of 90 × 90 × 1.6 mm³, as shown in Figs. 13(a), (b), (c) and (d). The researchers explained that the antenna is proposed for wide-band communications systems. Based on the results achieved by the antenna, it was good in most parameters to determine performance and efficiency. It worked in a wide frequency range that reached 9.33 GHz because the antenna operates at frequencies from 2.67 to 12 GHz, as shown in the reflection coefficient (return losses) curves in Fig. 14(a). In addition, it achieves an isolation ratio between ports of less than −15 dB, as shown in Fig. 14(b), an ECC of less than 0.1, and a DG of 9.97 dB. While the gain and efficiency for the antenna reached 5 dBi and 75%, respectively, the researchers concluded that the antenna is well-suited for UWB applications.

Figure 13. Antenna designs for simulation and manufacturing: (a) simulation front side, (b) simulation back side, (c) fabrication front side, and (d) fabrication back side [Reference Alharbi, Rafique, Ullah, Khan, Abbas, Ali, Alibakhshikenari and Dalarsson69].

Figure 14. Curves of S-parameters versus different frequencies from 2 to 12 GHz (a) return loss curves and (b) isolation curves between ports [Reference Alharbi, Rafique, Ullah, Khan, Abbas, Ali, Alibakhshikenari and Dalarsson69].

In addition, in a recent article [Reference Sharaf, Zaki, Hamad and Omar70], academics proposed manufacturing a four-port MIMO antenna, as shown in Figs. 15(a) and (b). This antenna is made of copper for both the patch and ground layers. While its substrate layer is made of FR4, the design dimensions of the antenna are 50 × 20. This antenna operates at two resonant frequencies: 38 and 60 GHz, as shown in Fig. 16(a). The proposers confirmed that the antenna is suitable for 5G system applications. The antenna provides satisfactory results for the isolation ratio between the four ports, which reached −42 dB at 38 GHz and −47 dB at 60 GHz (Fig. 16(b)). Moreover, the value of ECC is <0.05, and the DG is >9.98. The gain values reach 6.5 at 38 GHz and 5.5 dBi at 60 GHz.

Figure 15. Practical aspects of the proposed antenna include (a) manufacturing the antenna and (b) measuring the antenna’s performance using an analysis device (Rohde & Schwarz) [Reference Sharaf, Zaki, Hamad and Omar70].

Figure 16. (a) S-parameter curves for the simulation and measurement sides, and (b) isolation curves between ports [Reference Sharaf, Zaki, Hamad and Omar70].

Other recent articles summarize the designs of quad-port (2 × 2) MIMO antennas based on frequencies below 27 GHz in Table 3, while designs based on mmWave bands are listed in Table 4. It has been found that antennas operating at sub-6 GHz frequencies have larger dimensions and do not experience isolation problems between the ports. In contrast, mmWave antennas are small in dimensions and suffer from isolation problems between antenna elements in MIMO configuration.

Table 4. A summary of the most recent research on quad-port antennas designed for mmWave frequency bands

Six and more ports MIMO antenna designs for various applications

In the recent manuscript [Reference Jayanthi and Kalpana120], the researchers designed a multi-port antenna (six ports), as shown in Figs. 17(a) and (b). The researchers focused on providing this antenna to operate at mmWave frequencies, so the antenna worked in two bands, the first band from 27.7 to 28.1 GHz and the second band from 36.92 to 39.5 GHz, as shown in Figs. 18(a) and (b).

Figure 17. The structural manufacturing design of the proposed antenna for (a) the front face and (b) the back face [Reference Jayanthi and Kalpana120].

Figure 18. Antenna performance measurement curves for (a) and (b) reflection coefficient and (c) transmission coefficient [Reference Jayanthi and Kalpana120].

The researchers were able to obtain good results in light of the challenges in the field of antenna manufacturing. So, the antenna achieved satisfactory outputs for the isolation value that reached less than −20, as shown in Fig. 18(c). The highest gain value for the first band reached 13.3 dBi, and the second band reached 10.09 dBi, as shown in Fig. 19(a). While the ECC value is <0.01, the DG value is >9.988, and the CCL value was less than 0.4 bits/s/Hz. In the end, the antenna achieved an overall efficiency of 92% and 94% for the two bands, respectively, as shown in Fig. 19(b).

Figure 19. The complementary results achieved by the antenna are (a) gain curves for both sides of CST simulation and actual measurements, and (b) total efficiency curves for simulation and manufacturing [Reference Jayanthi and Kalpana120].

Moreover, in the manuscript [Reference Shao, Chen, Wang and Wang121], the authors presented a new geometry for an eight-port antenna with geometric dimensions (150 × 80 × 1.6 mm³), as shown in Fig. 20. The authors focused on presenting an antenna composed of four elements, each containing two ports. Note that the antenna was designed with both simulation aspects using the HFSS software (version 2020), as shown in Fig. 20, and actual fabrication using practical laboratories, as shown in Figs. 21(a) and (b). The materials for manufacturing the antenna are copper for the patch and ground layers and FR4 (permittivity of 4.4 and tangent loss of 0.02) for the substrate layer. This antenna operates at sub-6 GHz frequencies ranging from 3.4 to 3.6 GHz, as shown in Fig. 22(a). The antenna achieved the lowest isolation value between the two ports (first and second) of −14 dB and the highest value between the two ports (first and fourth) of −43 dB, as shown in Fig. 22(b). It also achieved good performance values for the ECC parameter <0.065, with the highest gain reaching 6.24 dB. The efficiency for the simulation side ranges between 75% and 85%, and the practical side ranges between 60% and 75%, as shown in Fig. 23.

Figure 20. A geometric design of the proposed octa-port MIMO antenna using the HFSS simulation program [Reference Shao, Chen, Wang and Wang121].

Figure 21. The fabrication geometry of the proposed MIMO antenna for (a) the front view and (b) the back view [Reference Shao, Chen, Wang and Wang121].

Figure 22. (a) The reflection coefficients for the simulation and fabrication aspects, and (b) the mutual coupling between all ports [Reference Shao, Chen, Wang and Wang121].

Figure 23. Overall antenna efficiency for all simulations and fabrication measurements [Reference Shao, Chen, Wang and Wang121].

Furthermore, several recent studies have introduced MIMO antennas with eight or more ports. Summaries of designs operating at sub-27 GHz and mmWave frequencies are provided in Tables 5 and 6, respectively. In these studies, increasing the number of ports while reducing the antenna size has increased mutual coupling between MIMO antenna elements and deteriorated the results in some works.

Table 5. A detailed comparison of the latest articles proposing eight-port antennas operates at frequencies sub-27 GHz

Table 6. A comparison of recent research on eight-port or more antennas relying on mmWave bands