## 2 Proposed UWB Antenna with Quadruple Band Rejections
### Design of Band-Notched Structures  
As shown in Figure 4, four independent band-stop structures are used to achieve quadruple rejections. The required four notches are divided into two categories. One is WLAN bands with narrow notch width requirements, which are realized by etching interdigital inductance slots on the ground. The interdigital inductance slots can be regarded as improved C-shaped slots. The structure of the interdigital inductance slot consists of two parts: a C-shaped slot and an interdigitated structure. The required resonant frequency band can be achieved by a reasonable size design. Signals can be restrained in a narrow band around their resonant frequency. As the interdigital structure increases the equivalent inductance and has a higher quality factor (Q value) than the general C-slot, it can achieve a narrower resonant bandwidth [21]. The frequency \(f_{0}\) of the interdigital inductance slots is \(1/2\pi\sqrt{L_{0}C_{0}}\). For a parallel resonant circuit, the width of the band-gap is proportional to \(\sqrt{L_{0}/C_{0}}\), where \(L_{0}\) and \(C_{0}\) stand for the equivalent inductance and capacitance of the interdigital inductance slots [22].  
The other category is frequency bands with a wider notch width. A split-ring resonator and a C-shaped slot are used, and their length is one-half of the wavelength corresponding to the center frequency of the WiMAX and the downlinks of X-band satellite communication. The following formula can calculate the key parameters of the structures:  
\[l_{element}=\frac{\lambda_{g}}{2}=\frac{\lambda_{0}}{2\sqrt{\varepsilon_{ref} }}=\frac{c}{2f_{metch}\sqrt{\varepsilon_{ref}}} \tag{1}\]  
\[\varepsilon_{ref}=\frac{\varepsilon_{r}+1}{2}+\frac{\varepsilon_{r}-1}{2} \Bigg{(}1+\frac{12h}{\omega_{f}}\Bigg{)}^{-0.5} \tag{2}\]  
Figure 3: Simulated VSWR of the proposed antenna without notched bands in case of different g.  
Figure 2: Simulated VSWR of the proposed antenna under different values of corner cut (**a**) W4 (**b**) I4.  
\(\lambda_{0}\) is the wavelength of free space, \(c\) is the speed of light, \(f_{notch}\) is the center frequency of the notched band, \(\varepsilon_{reff}\) is the effective permittivity, \(\varepsilon_{r}\) denotes dielectric constant of the substrate, \(h\) denotes the height of the substrate, and \(\omega_{f}\) denotes the width of the feed line.  
An inwardly curved extension branch is added to the tail of the general C-shaped slot and split ring resonator to increase the degree of freedom of design. By adjusting the branch's electrical length and distance, the band-stop width's center frequency and width can be changed.  
As shown in Figure 5, the curled C-shaped slot is directly etched on the circular antenna itself, which changes the antenna structure and affects the impedance bandwidth of the antenna. It is worth noting that the structures formed by slots can realize the band notched of the antenna alone, while curled split-ring resonator cannot realize the band rejects by itself well. However, its band-stop capabilities are greatly improved when it interacts with other slots etched on the patch and ground, according to Figure 5a,b.  
Figure 4: The geometry of band-notched structures: (**a**) curled split-ring resonator, (**b**) curled C-shaped slot, (**c**) interdigital inductance slots 1, and (**d**) interdigital inductance slots 2.  
Figure 5: Simulated VSWR of UWB antennas with (**a**) single band-notched structure and (**b**) all structures.