## 1 Introduction  
At present, electronic products tend to operate at high data rates and complexities. Transmission at a GHz data rate is widely used in modern digital circuits, such as serial advanced technology attachment (SATA) III/6 Gbit/s, universal serial bus (USB) 3.2/10 Gbit/s, and peripheral component interconnect express (PCIe or PCI-e) 4.0/16 GT/s. Differential transmission is the most suited for high-speed data raterequirements. Differential pairs should be symmetrical to maintain good signal integrity (SI) and avoid electromagnetic interference (EMI). However, asymmetric routing is unavoidable practical situations, even when using smooth bend structures [1], as it results in common-mode (CM) noise because of an imbalanced amplitude or a timing skew on the differential signal paths. CM noise may generate unwanted radiation and cause severe EMI problems, which results in a decreased performance of high-speed circuits [2]. Therefore, we expect the common-mode filter to maintain a good SI and EMI to avoid interfering with the sensor signals in the system. The conventional method is to use CM choke coils with high-permeability ferrite cores [3] or compact CM filters that were built on multilayer, low-temperature, co-fired ceramics [4]. They suppress the CM noise in the megahertz frequency domain. However, CM chokes do not work well in the frequency domain above 2 GHz. Furthermore, they are not practical for compact circuit designs. The differential-mode bandpass filter (BPF) and CM noise bandstop filter are better suited for GHz digital circuit systems; several methods are discussed in [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. First,the stopband-extended balanced BPF [5] and balanced dual-band BPF [6] were proposed as design concepts of the resonator for differential signaling. They achieved narrow passbands using adequately designed resonators. Next, wideband or ultra-wideband differential-mode BPFs were proposed in [7; 8; 9]. The results showed that differential-mode BPFs were limited in high-speed digital circuit applications because the BPFs did not pass low-frequency terms. In addition, the authors of [10; 11; 12; 13; 14; 15; 16; 17; 18] proposed many CM suppression filter designs using defected ground structures (DGSs). The primary design concept differed from the differential-mode BPF introduced earlier; CM filter designs can pass both passband and low-frequency differential signals. Theoretically, the differential signals were propagated on two coupled microstrip lines in the odd mode. The signal current traveled through one trace and returned through the other trace. In other words, the DGS on the ground plane was negligible in the odd propagation mode because the return current passed through the opposite line of the coupled microstrip line. On the other hand, the CM noises were propagated in the even mode; the return current of the CM noises flowed through the ground plane. Therefore, we designed an inductance (L) and capacitance (C) resonant circuit using the DGS on the ground plane. It acted as an ideal open circuit and reduced the magnitude of CM noise because the DGS blocked the return current. First, a dumbbell-shaped DGS in [10] obtained a bandstop frequency range from 3.3 GHz to 5.7 GHz using a \(-\)20 dB definition. In [11], quarter-wavelength open-circuit stub resonators with 8.5 mm and 6.5 mm lengths achieved a broadband-frequency-range performance from 4.27 GHz to 8.2 GHz using a \(-\)20 dB definition. Next, DGSs in [12; 13; 14; 15; 16; 17] obtained a bandstop range using a \(-\)15 dB definition. The DGS in [12] used one H- and two C-shaped units to achieve a bandstop frequency range of 3.68 GHz to 8.43 GHz. Similarly, the DGS in [13] obtained a bandstop frequency range of 5.4 GHz to 11.4 GHz. In [14], a CM suppression filter design using two quarter-wavelength resonators in multiple sets of differential pairs achieved a bandstop frequency range of 3.75 GHz to 6.95 GHz.