## 2 Design and Analysis  
### Single Antenna Element  
Before investigating the proposed MIMO design, it is essential first to explore the design and operational principles of a single antenna element. The geometry of the visualized tri-band antenna is illustrated in Figure 1. This antenna, printed on a low-loss Rogers RT-5880 substrate with a height (h) of 0.787 mm and a dielectric constant (\(\varepsilon_{\mathrm{r}}\)) of 2.2, and a \(9\times 10^{-4}\) loss-tangent, encompasses a modified rectangular patch fed by a microstrip line. The antenna is a single-patch antenna with dimensions \(\mathrm{W_{P}}\times\mathrm{L_{P}}\), with two inset feeders with dimensions \(\mathrm{W_{inset}}\times\mathrm{L_{inset}}\) to enhance the impedance matching of the antenna. The design features the rectangular slot 1 with dimensions of \(\mathrm{W_{slot1}}\times\mathrm{L_{slot1}}\), established at a distance \(\mathrm{S}\) from the right edge of the patch. This is followed by slot 2, which is a half-circle with a radius of \(\mathrm{R_{slot2}}\). Another rectangular slot is created to enhance the antenna's bandwidth for the three bands. Feeding the slotted patch is a 50 \(\Omega\) microstrip feedline with \(\mathrm{W_{f}}\) and \(\mathrm{L_{f}}\) dimensions. The substrate dimensions are \(\mathrm{W_{sub}}\) and \(\mathrm{L_{sub}}\), and the ground plate dimensions are also \(\mathrm{W_{g}}\) and \(\mathrm{L_{g}}\). Impedance matching at resonant frequencies was attained using optimization for slot dimensions and the inset feeder (\(\mathrm{W_{inset}}\) and \(\mathrm{L_{inset}}\)). Optimal performance was achieved with dimensions being 0.2 mm wide and 2.5 mm long for the inset feeder. All the design parameters are listed as shown in Table 1. The design incorporates an MTM unit cell with the dimensions shown in Figure 1c. The reflection coefficient (\(\mathrm{S_{11}}\)) of the antenna, shown in Figure 1d, reveals resonant frequencies at 28, 34.95, and 38.16 GHz, with reflection coefficients below \(-25\) dB, \(-15.5\) dB, and \(-20.5\) dB at their corresponding frequencies of interest. The bandwidths around these resonance frequencies are 1.0 GHz, 0.83 GHz and 1.9 GHz as the bands of interest, respectively. These simulated results provide critical insights for performance analysis and optimization during the pre-fabrication phase. Figure 1: (**a**) Front, (**b**) back view of single element antenna, (**c**) MTM unit cell dimensions, and (**d**) reflection coefficient parameter, \(\mathrm{S_{11}}\), versus frequency. After conducting a parametric sweep, it became evident that the inset length (L\({}_{\text{inset}}\)) holds paramount importance among the various dimensions listed in Table 1. Hence, altering the L\({}_{\text{inset}}\) from 2.4 to 3.5 would have a substantial impact on each band resonance frequency, as well as the reflection coefficient values and the achieved bandwidth, as demonstrated in Figure 2. The reason behind this lies in the intricate relationship between the lengths of the two insets and the input impedance of the antenna. Altering length leads to a consequential modulation of the impedance characteristics, which in turn plays a pivotal role in determining the reflection properties, resonance behavior, and bandwidth efficiency of the antenna. The best reflection characteristics of the tri-band were obtained at an L\({}_{\text{inset}}\) value of 2.5 mm. The radiation patterns in Figure 3 characterize the single-element antenna's performance across frequencies in both E-plane and H-plane. These patterns reveal key radiation characteristics including the gain, main lobe direction, and beamwidth: