# Design of a Compact Ultra-Wideband Microstrip Bandpass Filter
## 2 Filter Design
The curves (b) in Figure 3 show that the roll-off factor is greatly improved after loading stubs, and the stopband harmonic suppression performance is also optimized. Introduction of two transmission zeros optimizes the passband selectivity, but in-band reflection coefficient becomes slightly worse. Of course, the stopband rejection characteristics are also not good. In the subsequent design process, the stopband suppression characteristics would be optimized by adjusting parameters on the premise of maintaining the passband characteristics. Figure 4 is the current distribution of the top coupling patch after parameter adjustment. The positions of the transmission zeros have a slight frequency shift compared with the initial one. It can be seen that, at the frequency corresponding to the transmission zeros, two groups of open-circuit stubs have played their roles. To optimize the reflection characteristics in the passband of the filter, a central connecting microstrip is added between two symmetrical U-shaped microstrip lines, as shown in Figure 2c, so that part of the energy of the input is not only coupled down through the  
Figure 4: Current distribution of top coupling patch at the frequency of transmission zeros: (a) 9.666 GHz; (b) 16.925 GHz. U-shaped microstrip line, but also directly transmitted to the U-shaped microstrip line on the other side through the central connecting microstrip. At the same time, part of the energy reflected from the lower layers will be transmitted through the central connecting microstrip, thus reducing the return loss of signal energy, corresponding to the curves (c) in Figure 3. We can see that in-band reflection has been obviously improved, the reflection coefficient of passband is optimized by 5 dB to 8 dB, the roll-off coefficient of transmission characteristics increases, the low-frequency cut-off frequency of passband moves to a higher frequency and the bandwidth decreases slightly. In order to further optimize the in-band reflection characteristics, a DGS composed of three rectangular slots is loaded on the bottom ground plate as shown in Figure 2e. DGS can improve current distribution and return loss, adjust cut-off frequency and resonance frequency and optimize stopband characteristics [23; 24]. The curves of reflection and transmission characteristics after loading the DGS are shown in curves (d) in Figure 3. These three rectangular slots change the surface current path of the metal grounding layer. Two wider and longer rectangular slots on the outside introduce the low-frequency resonance point, while narrower and shorter rectangular slot introduces high-frequency resonance point. The reflection coefficient in the passband is optimized by changing the parameters to adjust the positions of the two resonance zeros. Finally, the reflection coefficients in passband are all less than \(-20\) dB, and the two resonance points reach \(-36.85\) dB and \(-30.44\) dB at 11.361 GHz and 14.943 GHz, respectively. In order to obtain good flat passband performance, excellent out-of-band roll-off and harmonic suppression, a series of parameters of the proposed filter are optimized. By changing different structural parameters one by one, the influences of these parameters on the filter performance are studied. Finally, the optimal parameters of the filter in Table 1 are obtained. This bandpass filter can also be approximated using a Chebyshev function with a transfer function to analyze [25]:  
\[\left|S_{21}(j\Omega)\right|^{2}=\frac{1}{1+\varepsilon^{2}T_{n}^{2}(\Omega)} \tag{4}\]  
where \(\varepsilon\) is the ripple constant. Low cutoff frequency of the bandpass filter \(f_{c1}=10.215\) GHz and high cutoff frequency \(f_{c2}=(\pi/\theta_{c}-1)\times f_{c1}\), where \(\theta_{c}\) is the electrical length of the distributed parameter element. If the electrical length of the element is \(71.7^{\circ}\), the high cutoff frequency is \(15.375\) GHz.