Compact multi-band bandpass filters using quad-mode stub-loaded resonators are proposed in this letter. Firstly, a novel approach about the mode-splitting characteristics of the quadruple-mode resonator is investigated, which can provide dual-band behavior. Secondly, a quad-band filter is proposed and designed by cascading two quadruple-mode resonators; the upper one operates at 1.8/2.4 GHz (GSM- and WiMax-band) and the lower one at 1.57/2.1 GHz (GPS- and WCDMA-band). Finally, the proposed filters have been fabricated. Respectable agreement between simulation and measurement verifies the validity of this design methodology. Multi-band bandpass filters (BPFs) have attracted much attention due to their wireless communication applications, such as in wireless local area networks (WLANs), Worldwide Interoperability for Microwave Access (WiMAX), and Wideband Code-Division Multiple Access (WCDMA). The dual-mode BPFs, based on DWG resonators and hexagonal resonators [1] , have been presented before, but they can not meet the requirements of multiple passbands at the same time. To meet various application requirements, many methods have been proposed for multi-band BPFs. In general, the reported quad-band BPF design methods can be classified into three typical categories. The first category is to introduce transmission zeroes inside passbands of the dual-band BPF to split both passbands to obtain four frequency responses [2] . The second category is to cascade two or more types of BPF in parallel to form a quad-band filter [3] . The third category is based on the quad-mode resonator [4] . Among these design methods, the third method is simple and only uses a single type of resonator. But, the number of resonators required for a given filter degree is minimum. Hence, the quad-band BPF (based on the quad-mode resonators), which results in a compact configuration with a high degree of design freedom is worthy of study. In this letter, a microstrip quad-mode stub-loaded resonator is used to design multi-band BPFs. A novel approach to the mode-splitting characteristics of the quadruple-mode resonator is investigated. Note that the dual- or quad-band passbands of the proposed filter can be tuned independently with a high degree of design freedom. Based on this theory, a compact quad-band filter is designed to provide experimental verification and theoretical prediction. Figure 1 illustrates the proposed quad-mode resonator from [5] , which was originally used for a single wideband filter design. In our work, a novel approach about the mode-splitting characteristics of the quadruple-mode resonator is investigated, which can provide multi-band behaviors. The proposed quadmode resonator is formed by adding two identical opencircuited stubs, denoted by length L ₂ and width w ₂ , at both sides and another open-circuited stub ( L ₁ , w ₁ ) at the center plane along a high impedance microstrip line with length of (2 L ₀ + 2 s ) and width of w ₀ . Because the proposed quad-mode resonator is a symmetrical structure its operating mechanism can be justified by an even/odd mode analysis [6] . Under an even-mode excitation, its equivalent circuit is shown in Fig. 2(a) . The characteristic admittance of the widths w ₀ , w ₁ /2, and w ₂ is Y ₀ , and the electrical lengths of the four sections with lengths L ₀ , L ₁ , L ₂ and s are θ ₀ , θ ₁ , θ ₂ , and θ_s , respectively. The input admittance Y _ineven of the even-mode equivalent circuit is expressed as (1) $$Y_{\text{ineven}}=j{Y}_0\frac{\tan {\theta }_{\text{0}}+\tan {\theta }_2\text{+}\tan ({\theta }_{\text{1}}+{\theta }_s)}{1-\tan {\theta }_0[\tan {\theta }_2+\tan ({\theta }_{\text{1}}+{\theta }_s)]}$$ Similarly, the input admittance Y _inodd of the odd-mode equivalent circuit in Fig. 2(b) can be expressed as (2) $$Y_{\text{inodd}}=j{Y}_0\frac{\tan {\theta }_{\text{0}}+\tan {\theta }_2-\cot {\theta }_s}{1-\tan {\theta }_0\left(\tan {\theta }_2-\cot {\theta }_s\right)}$$ From the resonant conditions Y _ineven = 0 and Y _inodd = 0, the resonant frequencies can be expressed as (3) $$\tan {\theta }_0+\tan {\theta }_2+\tan ({\theta }_1+{\theta }_s)=\mathrm{0,}$$ (4) $$\tan {\theta }_0+\tan {\theta }_2-\cot {\theta }_s=0.$$ According to (3) and (4), it is found that the even-mode resonant frequencies are determined by L ₀ , L ₁ , L ₂ , and L_s , whereas the odd-mode resonant frequencies can be controlled by tuning L ₀ , L ₂ , and L_s . That is to say, the resonator can provide multi-band performances using the above-mentioned physical parameters; thus, it offers a high degree of design freedom. For demonstration purposes, the resonant characteristics of a resonator with different L ₁ and different L ₂ are shown in Figs. 3(a) and 3(b) , respectively. In Fig. 3(a) , by changing the length L ₁ of the center-loaded open stub, the resonator frequency f _even2 can be shifted within a wide range, whereas the other resonator frequencies f _even1 , f _odd1 , and f _odd2 remain stationary. In addition, the location of one transmission zero, TZ ₁ , clearly moves toward the lower frequency as L ₂ increases. Figure 3(b) shows the resonant characteristics of the proposed resonator for cases of different length L ₂ . It is observed that the stub length L ₂ brings the resonator frequencies f _odd1 and f _even1 closer together, whereas the other two resonant frequencies remain almost unchanged. At the same time, the location of TZ ₁ is almost unchanged, and the location of TZ ₂ is greatly improved. Thus, the resonant frequency f _even2 and transmission zero TZ ₁ can be allocated in a desired location by reasonably choosing the length L ₁ and the resonant frequencies f _even1 and f _odd1 . The transmission zero TZ ₂ can be adjusted within the desired passband by changing the parameter L ₂ . Furthermore, the bandwidths of the first passband can be adjusted by the length L ₁ , while the bandwidth of the second passband can be changed by adjusting the length L ₂ ; therefore, a dual-band BPF is generated with the help of the quad-mode resonator. Based on the above discussion, we designed and implemented a compact dual-band filter using the proposed quad-mode resonator shown in Fig. 4 . The proposed filter consists of a quad-mode resonator and a pair of 50 Ω T-shape feed lines. It is meandered to realize compactness of the structure. The coupled-line structure is employed to design the input/output coupling structure, which enhances degrees of freedom. The filter was designed, analyzed, and simulated by a commercial simulator — namely, Sonnet — and fabricated on a substrate with a relative dielectric constant of 3.5 and a thickness of 0.76 mm. The dimensions of the filter are optimized as follows: L _f = 10.7, L ₀ = 14.75, L ₁ = 18.3, L ₂ = 14.05, w ₀ = 0.5, w ₁ = 0.6, g = 0.2, and s ₀ = 2.1 (all in millimeters). Simulated and measured frequency responses of the designed dual-band BPF are shown in Fig. 5 . The two passbands of the filter are centered at 2.4 GHz (WLAN-band) and 3.5 GHz (WiMax-band) with fractional bandwidths of 3.8% and 3.1%, respectively. Maximum insertion loss within the passband is less than 1.0 dB, which would be mainly attributed to the conductor and dielectric loss. The configuration also displays extra transmission zeroes at 2.46 GHz and 3.83 GHz, which are caused by virtual ground due to stubs L ₁ and L ₂ . Following the above analysis, a quad-band bandpass response can be achieved by cascading two quad-mode stubloaded resonators in parallel to form a quad-band performance. As shown in Fig. 6 , the I/O feed lines are located between the upper quad-mode resonator (in pink) and the lower quad-mode resonator (in blue) so that the resonant frequencies of the two resonators can be adjusted independently. That is, there is no mutual coupling between the two resonators. And the coupling between the external circuit and the quad-mode resonators (external Q) can be characterized by the equation in [7] . To reduce the overall circuit area, the quad-mode resonators are modified with inward-folded and outward-folded curve stubs, respectively The schematic view of the proposed quad-band BPF is shown in Fig. 6 . The upper and lower quad-mode stub-loaded resonators are chosen and designed to operate at 1.8/2.4 GHz and 1.57/2.1 GHz, respectively. According to the design equations (3) and (4) in section II, the physical dimensions of the filter are chosen and optimized as follows: L _d = 14.15, L ₃ = 21.75, L ₄ = 22.4, L ₅ = 19.5, L ₆ = 24.15, L ₇ = 29.7, L ₈ = 21.25, w ₀ = 0.5, w ₃ = w ₄ =1, s ₁ = s ₂ = 2.55, and g ₁ = g ₂ = 0.2 (all in millimeters). The proposed filter is fabricated on the same substrate as mentioned above. The proposed fabricated filter occupies only about 28.5 mm × 24.75 mm (about 0.15 λ ₀ × 0.13 λ ₀ , where λ ₀ is the guided wavelength in the free space at the central frequency of the first passband). A comparison of the simulated and measured frequency responses are shown in Fig. 7 . Results show that the designed quad-band filter is centered at 1.57 GHz (GPS-band), 1.8 GHz (GSM-band), 2.1 GHz (WCDMA-band) and 2.4 GHz (WiMax-band) with a 3 dB fractional bandwidth of 5.9%, 4.4%, 4.4%, and 4.6%, respectively. The measured maximum insertion loss among the passbands is less than 0.9 dB. Note that the upper quad-mode resonator produces four resonant modes at 1.79 GHz, 1.81 GHz, 2.39 GHz, and 2.41 GHz and that the lower one exhibits four resonant modes at 1.56 GHz, 1.585 GHz, 2.095 GHz, and 2.11 GHz. In addition, the transmission zero TZ ₂ at 1.73 GHz, generated by the transversal interference between the two signal paths from one port to another, is shown in Fig. 7 . The four transmission zeroes (TZ ₁ , TZ ₃ , TZ ₄ , and TZ ₅ at 1.53 GHz, 1.86 GHz, 2.2 GHz, and 2.61 GHz, respectively) are due to open stubs, explained in section II, which greatly improve the selectivity and stopband suppression. Respectable agreement between the simulated and measured results demonstrates the proposed structure. A compact quad-band BPF using two quad-mode resonators was introduced in this letter. With the help of even- and odd-mode analysis, the synthesis method was developed to explain quad-mode characteristics that can provide dual-band behavior with a high degree of design freedom. Furthermore, a quadband BPF implemented by cascading two quad-mode resonators was designed and fabricated, from which several resonant modes and transmission zeroes were created to greatly improve selectivity and stopband suppression. For a system integrating multi-band applications, this design could be applied to filter signals among multiple commercial bands.