Research

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Introduction
High and excellent wireless communications and information transfer have huge demand, creating advanced device technology systems [1].So implementing microwave devices, such as stopband or bandpass filters (BPF) and power divider circuits, requires approaches from simple to complex structures but is easy to integrate [2].Thereby, the filter component plays a key role in these new challenges [3,4].They are several kinds of filters found in the literature for electronics applications [5][6][7], and they may be grouped into three leading families: reflecting (constituted of lumped elements), notch (dissipate internally unneeded signals), and dissipation (produce losses through the use of material losses) filters.Filters can be designed using lumped components [8] or transmission lines [9,10].Initially, transfer functions were used, especially when the microwave electronics domain was underdeveloped.But researchers have adapted that principle to the radiofrequency area [11].A large panel of studies focused on the line technique in the microwave area, terminated by open-circuit or short-circuited configurations, exist [12][13][14].In addition, the filter structures can have multipath [12], use coupling lines [5,15], have symmetrical shapes [16], etc.Those possibilities often offer the right to control [17] the bandwidth (BW) and their implementation with the rest of the integration systems [18].Hence, the odd and even modes are the approach basis [19].Based on the even-and-odd modes, the principle gave a vast opportunity to develop new and great BPF [20,21].Due to multiple controllable resonant modes, designers often use the multimode resonators (MMRs) technique.The stub-loaded resonators (SLRs) and the stepped-impedance resonators (SIRs) [22] are most commonly used in multiband filter designs [7].The literature is rich enough in some techniques, such as SIR [23,24], stepped-impedance ring-loaded resonator (SIRLR) [21], stub loaded with uniform impedance resonators (UIR) [6], and substrate-integrated waveguide (SIW) [25,26].Sometimes, the mix of techniques improves physical dimensions or electric performance [27].Another new approach used nowadays is the defected ground structure (DGS) [28] or/ and defected stub [29] technique.It finds several shapes, among them circular resonators [30].Metamaterials are becoming an up-and-coming option, especially with high frequencies [31][32][33].At the same time, some designers are developing new filter technologies to cover the terahertz area through the graphene principle as well illustrated in [34][35][36].All these designs are studied and prototyped to have at least two BWs and several applications [37,38].
Above all, the T-shaped resonators, mixing identical open-and-short-circuit configurations, have not gotten enough attention nowadays.The BPF can be obtained using stubs terminated by short-circuited configurations according to the considered wavelength.Based on the mathematical formulation's simplicity and the designed circuit facility to ease the study [39], this paper suggests a new approach, focusing on the Q-factor [40] determination when stub sections are identical and alternate the short-and-open ended configurations to split into two subbands a specific BPF bandwidth.This technique is based on the voluntary creation of mismatched bands around the rejection band to create subbands for particular applications.The excellent final goals are achieved when a transmission zero is obtained via a stopband.The design dimensions are based on the choice of the Q-factor [15], which fixes and controls the bandwidth of both main bands and four subbands.This Q-factor is closely related to the resonator's admittance through the rejected frequency and a specific value to be found.On a wafer surface 22:45 × 5:72 mm 2 , approximately 0:32 λ 0 × 0:082 λ 0 mm 2 , where λ 0 is the free-space wavelength of the first BW center frequency, a dual ultrawide BPF has been designed and prototyped.The proposed BPF occupies a sur-face area of 56.567 mm 2 .The BPF is measured using an Anritsu MS4642B network analyzer.We assume that the BPF BW is considered at 3 dB of the insertion loss maximum level [41] but not necessarily according to cases [42][43][44].Also, we assume that the system is matched from the 10 dB of the return loss [45].

The Filter Topology and Methodology Approach
2.1.The Prototype Topology.The literature presents various basic concepts for creating filters.At the same time, some have proposed specific bandpass filter (monoband or multiband) topologies [46] but not some details for many bandpass filters like the one developed in this paper.This paper presented a new simple, easy topology to achieve great results for specific applications.This technique is based on the voluntary creation of mismatched bands around the rejection band to create subbands for particular applications.Hence, T-shaped stubs in short-and-open terminated configurations are put together through an alternation of these to achieve the goals.Also, all stubs and the mainline section have identical electric lengths except for their width.The design dimensions are based on the choice of Q-factor, which must be approximated when a frequency is chosen.This Q-factor fixes and controls the bandwidth of both main bands and four subbands.The excellent final goals are achieved when a transmission zero is obtained via a stopband.The bandpass filter's structure comprises five identical resonators and four transmission line sections.All resonators are connected by a line section, as presented in Figure 1.The short circuit terminates three resonators.The metallic vias symbolize shortcircuit configurations.Two resonators, ended by the open circuit configurations, are inserted between two short-circuited stubs.As described in [46], each design section can be transformed as a combination of the lumped elements.But the lumped element topology has attracted the researcher's attention.The inductance and capacitance values are determined using the Chebyshev and/or Butterworth's table or special calculations [11,[47][48][49].
The line section (constituted of θ the electric length and Z 0 the characteristic impedance) is perpendicular to stubs whose admittance is Y 1 and long of θ).That gives the BPF's shape.The normalized admittance y 1 is provided as follows, where Y 0 is the line section admittance.The electric length can be defined as a frequency variation parameter and given by the following equation, with ε ef f ′ and μ ef f ′ are the effective permittivity and permeability, respectively, c the vacuum's light velocity, l 0 the transmission line section length, and f 0 the operating 2 International Journal of RF and Microwave Computer-Aided Engineering frequency.At the same time, the filter's electric length θ T is given below, The total PBF length is L, and f is the frequency covering the entire scanned frequency.Let us consider w, the stub's width.In that case, the BPF global length is given by Equation ( 4), As shown in Figure 1, stubs are placed in a T-shaped and cascading position.

The Methodology Approach.
In such a configuration, the first step consists of choosing the resonator's number and length.This work uses two open-circuited and three shortcircuited.That makes the design a five-order filter.Next, a line section is settled between two stubs that differ from their terminated configuration.It means that the short-and-open terminated stubs are mixed through their alternation position.All section lines and stubs are quarter-wavelength ðλ/ 4Þ and well described in [46,50].The third step focuses on the Q − factor of each stub and the global Q g (also called initial quality factor Q i to fix) ones, where the subscript letter "g " indicates the global computation.It appears that using Q g and λ/4 becomes a great option [51].The open-circuited λ/4 line leads to the stopband filter [52].In the methodology presented in this paper, the stubs and the mainline transmission sections have identical lengths but differ in width.It means that their impedances (or admittances) are equal.Let us consider Q p and Q s , the normalized Q of parallel and series stubs as described in [46], respectively.These are given in the following equations for five resonators: and In the proposed approach, three short-circuited stubs are interspersed by two open-circuited stubs, as shown in Figure 1.As defined in [40], the Q ap is given as follows, So, Equation ( 7) becomes as given below, This equation establishes a link between the Q ap and the admittance of the stubs and the line sections.The fourth step needs to define the normalized admittance and apply an empirical formula to determine a value that lets the designer approximate the Q i needed through the computer-aided software.
where Z 0 = 50 Ω, the connector impedance that is welded to the prototype input and output accesses.Finally, the designer chooses the operating frequency f 0 and Q i to approximate ðQ ap ≅ Q i Þ.Both parameters define the stopband bandwidth around f 0 , creating two bandwidths where the high-frequency band is the harmonic of the lowfrequency band.Consequently, mixing both configurations leads to the transmission zero (TZ) and a dual-band BPF with subbands in each bandwidth.The approach developed in this paper focuses on the study's simplicity.The system must be designed with perfect symmetry to avoid input and output mismatch and those of the subbands, while the main dual-wideband is mismatched.
Because of the stubs and the transmission mainline section dimensions, the rejection band shifts or slides by 1 GHz from the theoretical f 0 .
The BPF analysis is made from the scattering Sparameters matrix.The characteristics (return loss Γ, insertion loss IL, bandwidth Δ, cut-off frequency f c , group delay φ gd ) of the BPF are read from those S-parameters.The return loss (RL) is the signal's power loss reflected by a discontinuity in a transmission line.This symbolizes the ratio of the reflected power to the incident power [53], as given The cross-section of the prototype passband filter.
3 International Journal of RF and Microwave Computer-Aided Engineering below, where S 11 is the reflection coefficient.Furthermore, insertion loss [8] is the amount of energy a signal loses as it travels through a component or a system [54,55], such as the BPF device.
where S 21 is the transmission coefficient.Whatever the BPF is, it presents a bandwidth Δ, defined from two cutoff frequencies, f h and f l , where ðf h > f f Þ.In that case, the fractional bandwidth (FBW) is often called to indicate how large the bandwidth is from the cut-off frequencies.That parameter varies between 0% and 2%.The FBW is given as follows, where Reading Equation ( 12), the FBW depends on the bandwidth.The wider the bandwidth, the bigger the FBW.The Q is the inverse of the fractional resonator bandwidth and is determined as follows, Therefore, the time delay is given as shown in the following equation: where v p is the phase velocity.This parameter, also named the phase delay ϕ pd , expressed in second, can also be defined as given [56,57] below, where the electric length θð f Þ is expressed in radians.The phase delay can be expressed in nanoseconds (ns) if the frequency is in gigahertz (GHz).At the same time, the group delay φ gd is obtained from the phase delay [58] through the following equation: The substitution of Equation (2) in Equation ( 16) gives Equation (18), By subtracting θð f Þ in Equation ( 2) and replacing it in Equation ( 17), the filter's group delay becomes as follows: which is expressed in seconds.Usually, this characteristic quantity is given in nanoseconds.Then, Equation ( 19) is rewritten as, Another important parameter that helps analyze the filter in high frequencies is the phase velocity in meters per second.From Equations ( 3) and ( 15), the substitution Equation ( 21) talks about the phase velocity, which depends on the electric length behavior.That velocity decreases, stabilizes, or increases as the electric length increases or decreases.
The methodology adapted can be summed up as follows: first, the short-and-open terminated stubs are mixed through their alternation position, creating the subbands.Second, all stubs and mainline sections are quarter-wavelength and calculated at the same operating frequency and the stub's characteristic impedance.Third, at that frequency, the mainline impedance is chosen as 50 Ω and differs from the stub ones.This creates the low-frequency and harmonic (high-frequency) bandwidths mismatch, depending on the Q.Fourth, an initial Q i that is fixed (chosen according to the designer's objectives) and found must be recovered.This new Q ap called approximated quality factor is linked to the structure's impedance.Finally, the computer-aided software finds the unknown parameter leading to satisfying the condition Q ap = Q i .In that case, the operating frequency belongs to the stopband, creating the transmission zero.

Measurement: Results and Discussion
The approach technique has been simulated, prototyped, and experimentally validated with a well-known material: FR4 HTG-175, which has 1 mm of thickness, a theoretical relative permittivity (ε r ), and a loss tangent (tan δ) of 4.4 and 0.02, respectively.The material intrinsic parameters are supposed to be constants during the dimensions and admittance determination.Still, according to the technique used to extract those electric parameters, the accuracy changes as long as those parameters [59,60].Therefore, at the operating frequency f 0 = 10:7 GHz, Q g = 0:9 is the target to manufacture a dual BW and split them into two subbands.As a result, three metallic vias have been made to connect the substrate top and bottom for short-circuit stubs,     2. Table 1 gives the design-specific parameters.By using X int = 252:75 × 10 −15 , the BPF dimensions obtained are depicted in Table 1.

Extraction of the Electric
Length.The high-frequency structure simulator (HFSS) software has been used for the study.The experimental results are computed after measure-ments with a vectorial network analyzer (VNA) Anritsu MS4642B in the frequency range of 3 GHz-15 GHz.
Figure 2(a) shows the entire prototype, while Figure 2(b) presents only the feedline as a transmission line.During the deembedding stage, that feedline will be removed.Finally, in Figure 3, the prototype is connected to the SMA, linked to the prototype, and the VNA through the two access ports.2, the accurate filter's length is 22.45 mm.
In Equation ( 4) and the specifications in Table 1, the design length error can be estimated at 0.0256 mm.The measures made are plotted in Figures 4-7.
The attenuation coefficient parameter (ACP), as defined in [61,62], depends on the frequency, the RL, and the IL (see Figures 5 and 7).At the same time, when the system is well matched, the ACP equals the IL.There is no fixed matched level, as it depends on the designer and the target application domain.But, as long as the bandwidth is identified from the 10 dB of the RL, this assumption finds its importance, and the IL becomes lower than 10 dB.The filter's physical dimensions have been calculated from a fixed relative permittivity and operating frequency.However, this parameter changes with frequency, and its accuracy depends on the material characterization technique used as developed in [63,64].Hence, the BW will move slightly in comparison to the target.Furthermore, physical dimensions (length and via diameters) have tolerance gaps that can change the expected results.
Both wideband are separated by a stopband from 5.7 GHz to 11.57GHz with a peak rejection at 9.373 GHz, while the transmission coefficient reaches -28.118 dB.Therefore, a return loss of 2.044 dB is achieved at that frequency.Table 2 shows that lower and upper bands have 53.282% and 21.456% as FBW.Several applications are possible with this prototype, like 5G.Each wideband has two subbands, read at -10 dB of the reflection coefficient, level of system adaptation definition.Results closely conform with the concept step despite some manufacturing impacts (see Figures 4, 6, and  7).The main reason for the mismatch is the weakness of the Q-factor that has been used.As seen in Figures 8 and 9, which present the simulation results from the prototype designed on the Agilent system software (ADS), the higher the Q-factor, the more significant the stub's width and the subbands shrink.At the same time, the isolation gets better, and the BPF becomes a dual-band or three-band bandwidths structure (according to the manufacturing constraints and accuracy) with two harmonics at its high-frequency band.The lower the quality factor, the better the adaptation of the main band.The structure is reduced to a system without a subband, and the bandwidth becomes ultrawide.
It is noticed that all simulation and experimental results have harmonic in their high-frequency band.This is because the Q-factor does not control these harmonic bands.Finally, using a high Q-factor leads stub section line behaving as capacitor elements and inductance when it becomes deficient (low).At this stage, a compromise is necessary according to the designer's objectives.The stub width is w = 3:1755 mm, representing a characteristic impedance Z ≈ 36:04 Ω for Q = 1:2.Each split band from the wideband is well matched to the device system, and these sidebands cover a minimum of 372 MHz.Additionally, there is a wideband and a narrowband as a subband.Therefore, when analyzing Equation (12), the biggest BW does not mean the greater FBW, as noticed in Tables 2 and 3.The electromagnetic signal propagates at a velocity that depends on the medium to cross over and the frequency range, as Equation ( 21) says.This gives much information about the group delay (GD), as presented in Equation (17).Figures 10 and 11 depict the phase velocity and group delay curves below.
The linear phase response is required to avoid signal waveform distortion in communication systems [39].Moreover, the linear phase causes a flat GD response, which is essential, especially in the microwave domain [11,65].Therefore, the fluctuation of the filter GD is often controlled when below 0.7 ns, and the waveform distortion can be avoided [65].For example, Figure 11 presents a GD below 0.21 ns in the entire frequency range, with a flat response into the upper At the same time, the phase velocity decreases slightly.Combining Equations ( 3) and (21), by substituting the phase velocity when using the dielectric material, the phase velocity is inversely proportional to the effective permittivity in the propagation medium.Equation Figure 12 shows that the open-terminated resonator's electric field distribution is high.It is noticed that the resonance frequency is better enough, as read in Figure 4.However, as the E-field distribution decreases, the filter is not matched enough in Figure 10(b), unlike Figure 10(a).
Table 4 presents a few domains where the suggested BPF can be used.This new approach helps select some application domains and rejects the others when fixing the BW and the RL to be accepted in that range.For example, the WLAN can be inserted in the first subband while only a sub-3.6GHz for the 5G.Finally, a study comparison is made in Table 5 with other works to highlight the great advantage of developing such a structure.
In Table 5, the proposed BPF is one of the smallest compared to the others.At the same time, it has the largest 3 dB FBW, while the IL reaches a minimum of 0.656 dB and 3.027 dB in the lower and upper bands.Also, it suggests a    simple technology in its feasibility during the fabrication.To improve the insertion loss, choose lossless insulator material (dielectric) with low thicknesses, reduce the filter's size (the fabrication technology will be vital as it is limited nowadays to 100 μm), and the dielectric material with thick conductor width.In addition, the use of materials with very high conductivity is ideal (gold, etc.).Therefore, this BPF is suitable for several radars and other communication systems, as presented in Table 4.

Conclusions
This paper described a new and straightforward approach to implementing several matched subbands from mismatched dual wideband bandpass filters.Identical resonators have created a rejection band, and two unmatched wide bandwidths (low-frequency and one harmonic) split into four subbands.The design is an alternation of quarter-wavelength resonators with short-and-open circuit terminations.There are positioned to each other symmetrically, and an approximation Q ap is determined from an operation frequency and compared to the initial chosen Q i .The unknown parameter allows the final results to be calculated from computer-aided software that satisfies a developed mathematical expression.The suggested technique has been validated.Furthermore, a five-order BPF with the FR4 HTG-175 (ε r = 4:4 and tan δ = 0:02) having 1 mm of thickness has been prototyped.
The BPF provides two wide unmatched BW at 2.285 GHz/2.858GHz with 53.282%/21.456%as FBW, while its subbands have four matched BWs, 382 MHz/606 MHz/ 564 MHz/372 MHz obtained from 10 dB of the RL.The prototype has an IL of 0.656 dB/3.027dB with three TZ.This device covers radar and other applications, particularly 5G, Wi-Fi, WAS, WLAN, WiMAX, FSS, DBS, etc.The newly developed approach uses a mathematical model to determine an unknown parameter value to reach the targeted Q i , which controls the BPF bandwidth (dual wideband and four subbands) and the device matching when a fixed operation frequency is chosen.It has been designed, modelized, and prototyped before experimental validation with the VNA Anritsu MS4642B two-port measurements in the scanned frequency range of 3 GHz-15 GHz.The measured results have exhibited good agreement with simulated ones; slight discrepancies may have been observed due to fabrica-tion errors.The proposed BPF has the advantages of low IL and group delay, high FBW, wide BW, small sizes, and an extended range of communication systems.A high Q provides an ample stub width and excellent isolation but shrinks the subbands.
In contrast, the low Q leads to small stub width, broadens bandwidth, and mismatches the band.Both cases create harmonic in the high-frequency band.Therefore, as previously said, this prototype is a good candidate for some applications compared to others in the literature.

Figure 2 :
Figure 2: (a) The bandpass filter prototype, with its feedline at each access; (b) the bandpass filter feedline.

Figure 3 :
Figure 3: The passband filter prototype with the SMA access ports.

Figure 4 :
Figure 4: Comparison of the simulated and measured frequency response of the dual-wideband bandpass filter.

Figure 5 :
Figure 5: Frequency response measured comparison of the attenuation coefficient (AC) and insertion loss (IL).

Figure 6 :
Figure 6: (a) Zoom of the frequency response (2 GHz-6 GHz) of the reflection and transmission coefficients of the BPF; (b) zoom of the frequency response (11 GHz-15 GHz) of the reflection and transmission coefficients of the BPF.

Figure 7 :Figure 8 : 9 S11Figure 9 :
Figure 7: (a) Zoom of the frequency response (3 GHz-6 GHz) comparison of the attenuation coefficient and insertion loss; (b) zoom of the frequency response (11 GHz-15 GHz) of the attenuation coefficient and insertion loss.

Figure 10 :
Figure 10: Experimental results of the phase velocity of the proposed bandpass filter.

Figure 11 :
Figure 11: Experimental results of the group delay of the proposed bandpass filter.

( 21 )Figure 10
Figure 10 deals well with Equation ((22)).Figure12shows that the open-terminated resonator's electric field distribution is high.It is noticed that the resonance frequency is better enough, as read in Figure4.However, as the E-field distribution decreases, the filter is not matched enough in Figure10(b), unlike Figure10(a).Table4presents a few domains where the suggested BPF can be used.This new approach helps select some application domains and rejects the others when fixing the BW and the RL to be accepted in that range.For example, the WLAN can be inserted in the first subband while only a sub-3.6GHz for the 5G.Finally, a study comparison is made in Table5with other works to highlight the great advantage of developing such a structure.In Table5, the proposed BPF is one of the smallest compared to the others.At the same time, it has the largest 3 dB FBW, while the IL reaches a minimum of 0.656 dB and 3.027 dB in the lower and upper bands.Also, it suggests a

Table 1 :
The manufactured prototype parameters.

Table 2 :
Synthesis of bandpass filter electrical performances manufactured on FR4 HTG-175, selected at 3 dB of IL.

Table 3 :
Frequency subband characteristics of the manufactured bandpass filter prototype, considered at 10 dB of RL.

Table 4 :
Application areas of the suggested BPF prototype.