## 2 Materials and Methods
The array factor (AF) of a uniform linear array with N elements quantifies the combined contribution of all elements to the array's overall radiation pattern and it is given by [32]  
\[\Delta L=0.412h\frac{\Big{(}\epsilon_{reff}+0.3\Big{)}\Big{(}\frac{W}{h}+0.264 \Big{)}}{\Big{(}\epsilon_{reff}-0.254\Big{)}\Big{(}\frac{W}{h}+0.8\Big{)}} \tag{2}\]1  
This equation describes the combined contribution of each element to the overall radiation pattern. Designing an antenna array requires the careful selection of both N and d to achieve optimal radiation characteristics, including gain, directivity, and beamwidth. Increasing N proportionally enhances gain and directivity and narrows the major lobe beamwidth, thereby improving angular resolution, which is required for specific wireless applications such as radar. However, a larger N can lead to increased sidelobe levels, which may cause interference, necessitating amplitude tapering techniques to mitigate these effects. Practical constraints such as physical size, cost, and feeding network losses further limit the feasible number of elements in compact array designs. Element spacing is equally critical, with an optimal spacing of d = \(\lambda/2\) minimizing mutual coupling, preventing grating lobes, and ensuring uniform phase progression across the array. Using smaller spacing (d < \(\lambda/2\)) yields a compact array with a slightly enhanced major lobe gain due to constructive interference, but it also increases mutual coupling and the risk of pattern distortion. Conversely, larger spacing (d > \(\lambda/2\)) can reduce coupling issues but often introduces undesirable grating lobes, which degrade overall array performance. In special cases, such as d = \(\lambda\) or greater, the formation of significant grating lobes makes the design impractical for high-performance applications. Practical applications in 5G/MIMO systems typically require a balanced approach, with a spacing of d = \(\lambda/2\) (or slightly less) and a moderate number of elements to achieve a compact design. In this study, the simulation results indicate that an array with N = 8 elements and spacing slightly less than half of the guided wavelength (\(\lambda_{\text{g}}/2\)) can achieve a high-gain, high-efficiency MIMO antenna. An important step for an efficient antenna array is to attain good impedance matching that is critical for maximizing power transfer efficiency within the array. Ensuring that each antenna element and the feed network are impedance-matched to the system impedance (often 50 \(\Omega\)) minimizes reflections and enhances overall performance. MIMO antennas are assessed using several parameters that measure their performance. Among these metrics, the envelope correlation coefficient (ECC) stands out as a key measure of the correlation between signal envelopes received from different antennas. In this regards, an ECC calculation helps in evaluating the correlation among signals and their impact on MIMO performance, which can be determined using  
\[\Delta L=0.412h\frac{\Big{(}\epsilon_{reff}+0.3\Big{)}\Big{(}\frac{W}{h}+0.264 \Big{)}}{\Big{(}\epsilon_{reff}-0.254\Big{)}\Big{(}\frac{W}{h}+0.8\Big{)}} \tag{2}\]2  
where the terms Si, Si, Si\({}_{\text{i}}\), Si\({}_{\text{i}}\)*, and Si* denote to the complex and conjugate complex components of the scattering matrix, representing the received signals by antennas i and j, respectively. A lower ECC value indicates strong diversity performance, suggesting an improved reception of uncorrelated signals and enhanced diversity gain. Another important parameter is the diversity gain (DG), which quantifies the received signal quality achieved by employing multiple numbers of antennas to reduce signal fading through spatial diversity.