## 2 Antennas Design and Working Principle Analysis
### Stable Radiation Patterns and Titled Beams  
In Section 2.2, it is observed by Figure 6 that the shapes of the radiation patterns of the antenna during the working band differ obviously. In order to keep the shapes of the radiation patterns stable, the radiation stubs of the antenna are tapered. A network at one end is removed to minimize the antenna length and the end is open. The lengths of the tapered stubs are \(h_{1}\), \(h_{2}\), \(h_{3}\), \(h_{4}\), \(h_{5}\), and \(h_{6}\). Additionally, the influence of a metal plate on the antenna is also discussed in this part. The metal plate with length \(l_{g}\) and width \(w_{g}\) is arranged below the tapered-stub antenna with distance \(h_{g}\). The profile of the modified antenna is shown in Figure 8.  
By tapering the stubs, the radiation property of the proposed antenna obtains an obvious improvement, as shown in Figure 9. The radiation patterns become very stable from 5.7 GHz to 7.0 GHz. The realized gains are around 10 dBi, and the SLLs are less than \(-14\) dB. Figure 9: Normalized radiation patterns of the proposed antenna without metal plate in H-plane (YOZ plane). When the metal plate is set below the antenna, the backward endfire beams become tilted beams. The influence of the metal plate on the tilted angles of the radiation beams is illustrated in Figure 10. With the metal plate, the beams are tilted upward and scans from \(-30^{\circ}\) to \(-60^{\circ}\) with the increase in hg from 4 mm to 24 mm. The beam shape is distorted as the metal plate is too close to the antenna with \(h_{g}\) = 4 mm. As the metal plate keeps going away from the antenna, the radiation beam approaches the backward endfire. Therefore, \(h_{g}\) = 8 mm is chosen here to realize the fixed tilted-beam radiation, considering the low profile and the distortion of beam shape. Figure 11 gives the electric field distribution of the proposed antenna at frequency points of 5.7 GHz, 6.4 GHz, and 7.0 GHz. Without the metal plate, the electric fields at the three points all concentrate at the backward endfire direction, so the antenna radiates endfire beams, as shown in Figure 9. With the metal plate, the electric fields of the proposed antenna at the three points all concentrate at the upward tilted direction because of the reflection of the big metal plate. The strongest field is around \(-40^{\circ}\), which presents the radiation beam direction. Figure 11: Electric field distribution of the proposed antenna in YOZ plane. (**a**) without the metal plate, (**b**) with the metal plate (\(h_{g}\) = 8 mm). Figure 10: Radiation patterns of the proposed antenna with metal plate at 6.4 GHz as \(h_{g}\) varies. The theory of beam tilted by the metal plate is analyzed below. Based on the mirror image principle, the normalized directivity function \(F(\theta)\) of the proposed antenna is formed by unit factor and array factor given as:  
\[F(\theta)=F_{1}(\theta)\times F_{a}(\theta) \tag{3}\]  
The unit factor \(F_{1}(\theta)\) is the directivity function of the endfire antenna simulated by electromagnetic simulation software HFSS. The array factor \(F_{a}(\theta)\) is calculated by function of \(sin(h_{g}\)-\(k_{0}cos\theta)\) with \(h_{g}\) = 8 mm, to assume the ground is infinite. According to the function (3), the beam is upward tilted, and the calculated angle is \(-50^{\circ}\), given in Figure 12. Actually, the tilted angle of the proposed antenna with \(h_{g}\) = 8 mm is \(-40^{\circ}\), presented in Figures 10 and 11. The difference between the angles is mainly due to the assumption of the infinite ground in the function. Figure 13 illustrates the reflective coefficients and gains of the proposed antenna with and without the metal plate. The proposed antenna with or without the metal plate both has good impedance matching from 5.7 GHz to 7.7 GHz with \(\,|\)S11\(\,|<-10\) dB. Without the metal plate, the proposed antenna radiates backward endfire beams with stable gain from 9.3 dBi to 10.7 dBi over the frequency band of 5.7-7.3 GHz.