## 2 HT Distribution and Simulation Results  
### Elements and Linear Array  
Figure 3 displays two randomly generated elements using F4 BM with \(\epsilon_{r}=2.2\) and \(\tan\delta=0.0007\). The element employing the edge feed is a vertical (we can also call it 0\({}^{\circ}\)) polarization structure composed of a copper ground at the bottom of the F4 BM and an etched copper patch on top of the F4 BM, with a height of 1.57 mm. The specific dimensions of element A are listed in Figure 3a as follows: a = 16.4 mm, b = 9.7 mm, c = 26.1 mm, and L = 50 mm. Element B, which uses the aperture-coupled feed, is a 45\({}^{\circ}\) polarization antenna that consists of a square copper patch with edge length d = 2.9 mm on the top of the upper F4 BM1 (i.e., the upper F4 BM) with a height of 1.016 mm, a copper ground with an etched slot between the upper and the lower dielectric (F4 BM1 and 2; F4 BM2's height is 0.203 mm), and a feed stripe at the bottom of F4 BM2 (i.e., the lower F4 BM). Figure 3b lists the material parameters of element B: i.e., e = 2.2 mm, f = 3.4 mm, L\({}_{2}\) = 13 mm. \[\Gamma=\frac{Z_{in}-Z_{0}}{Z_{in}+Z_{0}} \tag{1}\]  
\[|S_{11}|=20\lg|\Gamma| \tag{2}\]  
Figure 2: Flow chart of the HT method. \(\mid S_{11}\mid\) can be obtained using Equations (1) and (2), where \(\Gamma\) is the return loss of the antenna, \(Z_{in}\) is the impedance of the antenna, and \(Z_{0}\) is the impedance of the feed (50 \(\Omega\) in the work). Generally, people think antennas can operate well when its \(\mid S_{11}\mid\) is less than \(-\)10 dB. These elements are analyzed using commercial computer simulation technology 2019 (CST2019) software. We can enhance their electromagnetic characteristics by modifying the structure factors. Figure 4 shows the final S-parameter results. Figure 4a,b show that element A is a narrow-band antenna whose center resonator frequency is 3.5 GHz and frequency band is about 0.25 GHz, and element B is a broad-band antenna that can work at 22-27 GHz. Figure 5 shows their radiation patterns, based on which we can determine that their maximum gains are 7.2 dBi and 6.1 dBi, respectively, when they are operating independently at 3.5 GHz and 26 GHz. First, we determine the unit number N of the linear array. Following that, we must generate 2N Taylor random numbers, choose the first N items, and apply sidelobe depression (\(dp\)) weighting to them, as shown in Equation (7), where \(nl\) is the sampling times, \(N\) is twice the number of elements, and \(qh\) is the desired weighted Tyler excitations. \[at=10^{-\frac{dp}{20}} \tag{3}\]  
\[a=\log(at+\sqrt{at^{2}-1}) \tag{4}\]  
\[\text{sgm}=\text{nl}/\sqrt{a^{2}+\left(nl-5\right)^{2}} \tag{5}\]  
\[\text{S}=\prod_{m=1}^{nl-1}\left(1-n^{2}/(\text{nl}/\sqrt{a^{2}+\left(nl-5 \right)^{2}}\times\sqrt{a^{2}+\left(m-5\right)^{2}})\right) \tag{6}\]  
\[qh=1+2\times\frac{\sum_{n=1}^{nl-1}\frac{\Gamma(nl)^{2}}{\Gamma(nl+n)}\times s \times\cos(\frac{A}{N}-1)}{N-1}\times n\times\pi \tag{7}\]  
At this stage, we can adjust the suppression degree as needed. To satisfy the demand of higher gain and specific anti-power fluctuations, it can use a higher \(dp\), such as \(-\)30 dB. For higher anti-power fluctuations, it can use a lower \(dp\), such as \(-\)60 dB. We will describe carefully the differences observed with different \(dp\) values in this paper. We can obtain the ONF beam by using the N-weighted data as the excitations of an N-unit array. We validate these using two such antennas with different frequencies, polarizations, feed techniques, and bandwidths. Additionally, we can obtain the steering ONF beam by modifying the phase difference of each unit. As seen in Figure 6, we construct an N-unit linear array using the elements A and B described above, respectively, where N is an arbitrary number. Additionally, there are half wavelengths, approximately, between elements d1 and d2.