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Quasi-Reflectionless Microstrip Bandpass Filters With Improved Passband Flatness and Out-of-Band Rejection

High-order quasi-reflectionless bandpass filters with improved passband flatness and good impedance matching both in-band and out-of-band are proposed in this work. The proposed design consists of conventional coupled-lines bandpass sections loaded with absorptive stubs at the input and output. Analysis shows that the absorptive stub is equivalent to a 2-pole bandstop filter. Compared to the prior art, the higher-order nature of the presented absorptive stub enables a flatter passband and better out-of-band rejection. The overall filter stopband attenuation can be readily improved by increasing the number of coupled-lines sections without altering the passband responses. Furthermore, cross-coupling between the two absorptive stubs can be used to improve the out-of-band rejection by introducing two transmission zeros without affecting the absorption characteristics. The proposed design concepts are experimentally validated by the design and fabrication of a set of 2.4-GHz 1-, 2-, and 3-pole microstrip quasi-reflectionless bandpass filters. Measured frequency responses of these filters closely match those of the simulation.


I. INTRODUCTION
As fundamental building blocks in radio frequency (RF) and microwave systems, filters are used to better define the system bandwidth or reject undesired signals [1]. In a conventional filter, rejection is achieved by presenting significantly mismatched impedance to the undesired (out of band) signals. In some systems, such reflection will deteriorate the performance of adjacent non-linear devices, such as analog-to-digital converters, mixers, and high gain amplifiers [2], [3]. A common solution to this problem is to insert nonreciprocal devices, such as isolators or circulators, to redirect the reflected signal to an absorptive load. This approach comes at a cost because the magnetic non-reciprocal devices are known to be bulky, expensive, bandwidth-limited, and difficult to integrate with the rest of the active electronics.
The associate editor coordinating the review of this manuscript and approving it for publication was Wenjie Feng.
Recently, there has been an increasing interest in reflectionless (absorptive) filters which effectively mitigate the reflection signals. In contrast to the conventional reflectivetype filters, the non-transmitted stopband signals in a reflectionless filter are dissipated within the filter itself. In other words, reflectionless filters have good impedance match not only in the passband but also in the stopband.
Several reflectionless bandstop filter designs have been reported based on lossy resonators [6]- [11], the bridge-T structure [12], and the complementary diplexer architecture [13]. Essentially, directional filter topologies are used to implement reflectionless bandpass filters [14]- [16]. The penalty paid, however, are the limited bandwidth, the large circuit size, and the high insertion loss. Asymmetrical quasireflectionless bandapass filter with tunable passband is presented in [17] by using lossy resonator close to one of its termination ports. Reflectionless bandpass filters, particularly those that are well matched at both the input and output ports, still remain a challenge to design.   [4], (b) [5], and (c) this work.
Based on the even-odd-duality theory, lumped-element reflectionless prototypes having low-pass, high-pass, bandpass, or bandstop characteristics are presented in [18] with theoretically all-band fully reflectionless characteristics at both the input and output ports. Based on the reflectionless lowpass prototypes, distributed first-order and higher-order reflectionless filters are derived in [19], [20] and [4], respectively. In these designs, however, only the order of the absorptive section circuit is increased whereas a single bandpass resonator is used in the main channel. Despite sharper passband roll-off at the transmission zero frequencies, the far-out roll-off of the filter remains the same, as shown in Fig. 1 (a). Lumped implementations of symmetrical lumped reflectionless filters with arbitrary prescribed transmission responses are recently presented in [21]- [23].
Reflectionless filters can also be realized by using complementary diplexer architectures. A rigorous susceptancecancellation methodology is obtained in [24] to design arbitrary odd-order input-port reflectionless lowpass filters. Reflectionless filter prototype and its distributed implementation are presented in [25] with only input-port quasi-reflectionless characteristics in a limited frequency range. Bandpass-bandstop-diplexer topologies are presented in [26] using coupling matrix analysis to design one-port reflectionless bandpass filters. The idea is then extended to one-port reflectionless bandpass filters using a mixture of distributed-and lumped-elements [27], split-type input-reflectionless multiband filters [28], filtering power dividers [29], dual-band microstrip BPF [30], and multiplexers [31], [32]. In [33], the main bandpass channel, which consists of a bandpass filter and spur-lines, together with the bandstop auxiliary channel lead to a wideband bandpass filter with broad stopband and ultra-wideband reflectionless characteristics at the input port. Similarly, a wideband reflectionless filtering coupler is presented in [34].
To achieve symmetrical two-port reflectionless properties using the complementary diplexer architecture, auxiliary absorptive circuits need to be loaded at both the input and output ports [5], [35]. The theoretical analysis demonstrates symmetrical quasi-reflectionless properties and provide general guidance in terms of implementations using acoustic, lumped-element, transmission-line, etc. However, the resulting overall circuit size is nearly three times that of its bandpass filter circuit, as shown in [5]. Besides, the demonstrated microstrip filters show absorptive behavior only in a limited frequency range and they degenerate back to reflectivetype responses at the stopband center frequency, as shown in Fig. 1 (b).
An absorptive coupled-line consisting of a conventional coupled-line [ Fig. 2 (a)] and a loaded absorptive stub, as shown in Fig. 2 (b), is presented in [36] as the feed line for coupled-lines bandpass filters. To enhance the all-band matching performance, the singly-loaded absorptive stub is split to two identical ones. A series of 1-, 2-, and 3-pole quasi-reflectionless microstrip bandpass filters with all-band reflectionless responses are presented. However, because the absorptive stub has a single resonance, the passband transmission in [36] deteriorates significantly, particularly at the edge of the passband.
In this work, high-order quasi-reflectionless filters with improved passband flatness and out-of-band rejection, as shown in Fig. 1 (c), are presented using the proposed absorptive stub section [ Fig. 2 (c)]. Analysis shows that this structure absorbs not only the stopband reflection but also the close-to-passband signals, thus improving the passband roll-off and close-in rejection. The higher order nature of the absorptive circuit significantly improves the passband transmission flatness, particularly at the edge of the passband. In addition, higher-order reflectionless passband can be easily realized by increasing the order of the bandpass sections, without changing the matching performance.
To further improve the out-of-band rejection, cross-coupling can be introduced between the absorptive stubs. The proposed designs advance the state-of-the-art by demonstrating highorder distributed quasi-reflectionless filters with a flat passband, enhanced out-of-band rejection, and good impedance matching both in-band and out-of-band.
The rest of this paper is organized as follows. Section II introduces the design and analysis of the proposed absorptive coupled-lines. In Section III, the design of quasi-reflectionless filters is studied and the possibility of improving out-of-band rejection by introducing crosscoupling is discussed. The design and optimization guidelines are given in Section IV. Section V details the comparison between this work and reported distributed filter works to clearly show the advantages of the proposed designs. Section VI presents the design and measurement of several microstrip absorptive filters to validate the proposed design methodology. Finally, conclusions are provided in Section VII.

II. ABSORPTIVE COUPLED-LINES
In this section, we provide a detailed analysis of the proposed absorptive coupled-line which consists of a conventional coupled-line section and an absorptive stub loaded at one of the open ends of the coupled-line.

A. THE PROPOSED ABSORPTIVE STUB
To help understand the working principle of the absorptive stub, we liken it to a simple 2-pole bandstop filter.  The impedance parameters of the bandstop filter circuit can be extracted from its lowpass filter prototype as follows, where γ is given by Here, f 1 is the cutoff frequency of the lowpass prototype filter (f 1 = 1 Hz), g i (i = 0, 1, 2, and 3) are the element values of the prototype lowpass filters [37], f 0 is the stopband center frequency, and f 1 is the cutoff frequency of the stopband filter.
In the following analysis, Z A is set to 50 and the stopband center frequency f 0 is normalized to 1.0 GHz. where The real part and imaginary part of Z in1 can be derived as As defined in (1), Z 1 and Z 12 have the following relationship with respect to the termination impedance Z A Applying (6) to (5), we have On the other hand, the input impedance Z in2 of the circuit in Fig. 3 By setting Z in2 can be made equal to Z in1 , implying that the circuit in Fig. 3 (b) is equivalent to the one in Fig. 3 (a) in terms of input impedance, but with a much more compact size due to the reduced transmission line length. By inserting quarter-wavelength transformer with impedance Z c into Fig. 3 (b), the proposed absorptive stub circuit can be obtained, as shown in Fig. 3 (c). The required Z c is given by and For a lossless two-port network, such as the one in Fig. 3 (a), conservation of power requires [1] For the absorptive single-port circuit in Fig. 3 (c), the power is either reflected at Port 1 or absorbed by the resistor R. Defining S 11,AS and S a,AS as the reflection and absorption coefficients of the absorptive stub, we have Comparing (13) and (12), it is clear that S a,AS behaves just like the transmission coefficient S 21 in the sense that signal power is transmitted to the resistor R to be absorbed. With this definition in mind, we compare the reflection and transmission characteristics of the three circuits of Fig. 3 in Fig. 4. Here, a bandstop filter at 1.0 GHz with 0.1-dB passband ripple and 60% stopband bandwidth is used as an example. The lowpass prototype element values are found to be g 0 = 1.0, g 1 = 0.8431, g 2 = 0.6220, g 3 = 1.3554. According to (1), (2), and (9) 2) It is clear from Fig. 4 that the three circuits have the same transmission (absorption) coefficient in both the passband and stopband. The transmission coefficients of the three circuits merge together because the discrepancies due to reflection differences can be neglected.
Here, BW r and BW 3dB correspond to the ripple bandwidth and 3-dB bandwidth, respectively. In this specific example, BW 3 = 0.54BW r [37]. 3) Due to the dispersion of the quarter-wavelength transformer, circuit Fig. 3 (c) is equivalent to circuit Fig. 3 (a) and (b) only in the vicinity of the center frequency. For example, the three circuits have the same BW r (60%) and BW 3dB (32%). An interesting and unique property of circuit Fig. 3 (c) is that it is well matched at the even harmonics, e.g., 0 GHz and 2.0 GHz, in contrast to the circuits of Fig. 3 (a) and Fig. 3 (b). As will be seen in Section III, this property helps the stopband attenuation and enables wideband absorption characteristics. VOLUME 8, 2020

B. ABSORPTIVE COUPLED-LINES
Similar to [36], absorptive coupled-lines can be realized by loading the proposed absorptive stub to the open-end of a conventional coupled-line section, as shown in Fig. 2 (c). Fig. 5 compares the frequency responses between the absorptive coupled-line and a conventional coupled-line section at a nominal center frequency of 1.0 GHz. Since the absorptive stub is only loaded to the open-end of Port 1, the two-port absorptive coupled-line of Fig. 2 (c) shows asymmetrical matching characteristics, i.e., Port 1 is all-band reflectionless while Port 2 is only matched at the center frequency. At the center frequency, the conventional coupled-line and the absorptive coupled-line exhibit the same transmission and reflection responses. As we move away from the center frequency, the absorptive coupled-line exhibits much better impedance matching than the conventional coupled-lines at Port 1. For example, at 0 and 2.0 GHz, the conventional coupled-line is fully reflective whereas the absorptive coupled-line is strongly absorptive at the input port, with both |S 11 | and |S 22| below −40 dB. In addition, the absorptive stub absorbs not only the out-of-band reflections but also the close-to-passband signals, leading to an enhanced passband selectivity.
In [36], the absorptive stub [ Fig. 2 (b)] is split into two identical stubs to improve the absorption performance and ease the implementation. In this work, however, this method is not feasible due to the high transmission line impedance, e.g., Z b is 157.8 in the above example ( Fig. 4 and Fig. 5).

III. QUASI-RELFECTIONLESS BANDPASS FILTER DESIGN A. 1-POLE QUASI-REFLECTIONLESS FILTER EXAMPLE
By connecting two absorptive coupled-lines [ Fig.2 (c)], one can realize a ''1-pole'' bandpass filter of symmetrical dual-port reflectionless performance. Fig. 6 shows the transmission-line circuit model of such a circuit. Here, Z c1 and k c1 are the coupled characteristic impedance and coefficient, respectively. The coupled-line even-mode and oddmode impedance Z ec1 and Z oc1 are given by Z c1 √ k c1 and Z c1 / √ k c1 , respectively. To optimize the transmission and absorption performance, the impedance of the lines are slightly adjusted by three scaling parameters r a , r b , and r c , as shown in Fig. 6. The transmission and reflection characteristics of the quasi-reflectionless filter are studied parametrically in Fig. 7-10. The nominal values of the various parameters follow the same example given in Section II-A.
The bandwidth of the quasi-reflectionless filter is mainly determined by the bandwidth of the absorptive stub circuit because the coupled-lines section usually provides a much larger bandwidth [36]. This is evident from Fig. 7 in which we can see that the frequency response of the filter is not strongly dependent on the coupling coefficient of the coupled-lines section.
Admittedly, the reflection at the passband-stopband transition region, e.g., around 0.8 GHz and 1.2 GHz, is high. The r a , r b , and r c are used to improve the absorption performance.
As shown in Fig. 8, the overall reflection changes significantly with respect to r c . Larger r c benefits lower reflection not only in the passband but also in the transition band, although at the cost of the deteriorated stopband matching. A slight increase in BW 3dB with increased r c can also be observed. For a well-balanced overall reflection performance, an optimal value of r c = 1.3 is chosen, resulting in the passband reflection below −20 dB, the transition  band reflection below −12 dB, and the stopband reflection below −14 dB. Fig. 9 shows that the frequency responses of the quasireflectionless BPF are nearly the same as r a increases, except for an improvement in the reflection at the passband-stopband transition region. In contrast, the overall reflection remains almost the same but an obvious improvement in the passband reflection, as well as a decrease in BW 3dB , can be observed as r b increases (Fig. 10).
Following the above analysis, the transmission and reflection responses of the optimized 1-pole quasi-reflectionless bandpass filter are given in Fig. 11. In this figure, the responses of its bandpass section are also included for comparison. Obviously, the bandpass section exhibits a conventional 1-pole bandpass responses with its passband  centered at 1.0 GHz and stopband located at 2.0 GHz. It is well-matched at the passband center frequency with one reflection zero and is fully reflective in the stopband.
The quasi-reflectionless filter has a spectral period of [0, 2f 0 ) due to the periodic nature of the transmission line sections. The reflection of the 1-pole quasi-reflectionless filter is below −14 dB across all the frequency range. Owing to the absorptive stub's higher-order response, the filter exhibits a ''flat'' passband with reduced insertion loss variations. In addition, the absorptive stub also helps to eliminate out-of-band signals near the passband and significantly improves the close-in rejection. In this example, the closein roll-off approaches that of a quasi-elliptic filter, although the far out-of-band roll-off is still that of a 1-pole filter, i.e., −20 dB/dec.  Fig. 12 compares the transmission coefficient of the 1-pole quasi-reflectionless filter |S 21,ABF | and the reflection coefficient of its absorptive stub |S 11,AS |. Here, the subscript ABF and AS represent absorptive bandpass filter and absorptive stub, respectively. From Fig. 12, we can see that |S 21,ABF | is almost the same as |S 11,AS | in the vicinity of the passband. A qualitative explanation can be offered as follows. In a resistor-embedded circuit such as the quasireflectionless filter of Fig. 6, the input signal power is either reflected, transmitted, or absorbed. Conservation of power requires

B. A DISCUSSION ON THE FILTER BANDWIDTH
where S a,ABF is the absorption coefficient of the filter. Since |S 11 | ≈ 0 for the quasi-reflectionless filter, (14) reduces to Comparing (13) and (15) leads to the conclusion That is, the transmission response of the quasi-reflectionless filter is close to the reflection response of its absorptive stub. Note that this relationship is approximate because (13) and (15) are derived under slightly different impedance conditions. As a consequence, the quasi-reflectionless filter's passband BW 3dB is almost identical to the 3-dB reflection bandwidth of its absorptive stub. This observation serves as a starting point in the filter design procedure in terms of synthesizing the absorptive stub circuit from a prescribed bandwidth specification.

C. EXTENSION TO HIGHER-ORDER DESIGNS
Building upon the 1-pole quasi-reflectionless filter of Fig.6, higher-order quasi-reflectionless filters can be readily realized by cascading more bandpass coupled-line sections. Fig. 13 (a) and (b) show examples of a 2-pole and a 3-pole quasi-reflectionless filter, respectively. Z c2 and k c2 are the coupled-lines impedance and the coupling coefficient, respectively. Note that the added coupled-line sections have the same parameters as the filter order increases. Higher-order filter can be created by cascading more coupled-lines sections. Fig. 14 (a) and (b) show the respective frequency responses of the 2-pole and 3-pole quasi-reflectionless filters with respect to k c2 when Z c2 is set to 60 . The transfer functions of both filters are almost unchanged as k c2 varies, which again verifies that the bandwidth of this kind of absorptive filters is mainly determined by their absorptive stubs. Changing k c2 also has almost no effect on the stopband reflection since the latter is directly related to the absorption. However, the reflection in the vicinity of the passband depends strongly on k c2 . A good compromise between passband reflection (|S 11 | < −25 dB) and stopband reflection (|S 11 | < −14 dB) is achieved at k c2 = 1.8 for both the 2-pole and the 3-pole quasi-reflectionless filters. Fig. 15 compares the simulated transmission and reflection coefficients of the 1-, 2-, and 3-pole quasi-reflectionless filters with the same absorptive stub. We can see that the passband bandwidth is almost identical for the three filters. Increasing the filter order only improves the far-out-of-band rejection. Consequently, the passband bandwidth and stopband attenuation specifications can be individually fulfilled in the design process.

D. CROSS-COUPLING FOR IMPROVED OUT-OF-BAND REJECTION
The performance of the proposed quasi-reflectionless filters may be further improved by introducing cross-coupling between the absorptive stubs. Cross-coupling introduces additional signal paths between the input and output ports of a filter to create transmission zeros at certain frequencies [38]. The transmission zeros result in sharper rejection without having to use additional resonators. As such, cross-coupling techniques are widely used in the design of conventional filters.
Source-load cross-coupling is incorporated into the quasireflectionless BPF scheme to generate two transmission zeros in [5]. In [18] and [19], cross-coupling is implemented in the form of two-path signal-cancellation. In this work, crosscoupling is introduced between the absorptive stubs to realize additional transmission zeros. To illustrate this, three examples are shown in this section.
Cross-coupling will be introduced into the proposed absorptive BPFs, which are already systematically studied in Section III. Fig. 16 (a) shows a cross-coupled 1-pole quasi-reflectionless filter similar to that of Fig. 6. Crosscoupling is introduced by connecting together the last quarterwavelength resonator in each absorptive stub. As will be shown later in Section VI-B, this essentially merges the two quarter-wavelength stubs into one. As seen in Fig. 16 (b), two transmission zeros are generated at 0.6 GHz and 1.4 GHz, with a slight penalty in close-in rejection. Interestingly, the cross-coupling also helps to improved the passband return loss.
Although the filter of Fig. 16 (a) looks similar to the filter in [19], they are in fact fundamentally different because they are designed based on the reflectionless feed line and even-odd duality, respectively. Such difference is much more obvious for their high-order forms, as will be seen in the comparison studies in Section V. Fig. 17 (a) shows a 2-pole cross-coupled quasireflectionless filter. In this case, cross-coupling is achieved through a short-ended coupled-lines section with coupled impedance Z cr2 and coupling coefficient k cr2 . Fig. 17 (b) shows a parametric study of the frequency response of this VOLUME 8, 2020 filter with respect to k cr2 when Z cr2 is 70 . As k cr2 increases from 1.6 to 2.8, the two transmission zeros at 0.8 GHz and 1.2 GHz move closer to the imaginary axis but the return loss at the transition band (e.g., 0.83 GHz and 1.17 GHz) deteriorates with an increased transmission insertion loss.
Following a similar method as the above, cross-coupling can be introduced in a 3-pole quasi-reflectionless filter as shown in Fig. 18 (a). Two short-circuit quarter-wavelength coupled-lines connect the last quarter-wavelength resonators in each absorptive stub. Two transmission zeros at 0.6 GHz and 1.4 GHz can be generated with almost unaltered filter reflection responses when the coupled coefficient k cr2 is properly chosen. Unlike the 2-pole filter of Fig. 17 (b) where the insertion loss is degraded by the cross-coupling, the 3pole quasi-reflectionless BPF with cross-coupling develops an equi-ripple passband with the help of additional transmission zeros created by the cross-coupling.

IV. DESIGN PROCEDURES
According to the above studies, the design and optimization procedures of the proposed quasi-reflectionless bandpass filters can be summarized as follows.
1) Specify the center frequency f 0 , passband bandwidth BW 3dB , and stopband attenuation. 2) The absorptive stub is synthesized from a low-pass prototype filter [ Fig. 3 (a)]. The ripple bandwidth BW r and hence the cut-off frequency f 1 of the prototype filter can be calculated from BW 3dB [37]. Initial parameters of the absorptive stub circuit (Z a , Z b , Z c ) are then determined by (1), (9), (10), and (11). For simplicity, the resistor R can be set to be the same with the termination reference impedance Z 0 . 3) A 1-pole quasi-reflectionless bandpass circuit is first constructed using back-to-back connection of two absorptive coupled-lines (Fig. 6). Z c1 and k c1 of the absorptive coupled-lines are used to cover the required transmission bandwidth. The values of the impedance scaling parameters r a , r b , and r c can be found by parametric analysis to realize a good compromise between a flat passband and good overall return loss. 4) To meet the stopband attenuation specification, the order of the filter can be increased by simply cascading coupled-line sections (Z c2 and k c2 ) to the above 1-pole absorptive filter without significantly changing the passband transmission and reflection. 5) Stopband attenuation can be further improved by introducing cross-coupling [ Fig. 16 (a), Fig. 17 (a), and Fig. 18 (a)]. For a given attenuation requirement, the choice between increasing the order and using cross-coupling should be carefully weighed.

V. COMPARISON WITH THE STATE-OF-THE-ART
At the time of this writing, only a few distributed dualport reflectionless bandpass filters have been reported in the literature, viz., the fully-reflectionless bandpass filter based on the theory of even-odd-duality [4], [19], [20], the quasireflectionless bandpass filter based on a complementary bandpass-bandstop-diplexer architecture [5], and the quasireflectionless coupled-line filters [36]. In this subsection, a detailed comparison between the proposed work and those reported works is presented bellow. Fig. 19 (a), (b), (c), and (d) reproduce exemplary designs of a 3-pole reflectionless bandpass filter using methodologies presented in [4], [5], [36], and this work, respectively. All the transmission line electrical length is quarter-wavelength at the passband center frequency of 1.0 GHz. The proposed work uses different high-order extension methodology as those of [4] and [5]. Besides, it has improved passband transmission flatness and out-of-band rejection when compared to these reported works.
In particular, Fig. 19 (a) shows the architecture of the distributed 3-pole fully-reflectionless bandpass filter presented in [4]. Its corresponding lowpass prototype form is reported in [18]. Due to the even-odd duality, only the order of absorptive section circuit is increased when the overall filter order increases and only one bandpass resonator is used. Besides, such kind of filters have different high-order characteristic from conventional filters: it exhibits lower stopband rejection due to the higher out-of-band peaks as the filter order increases [4], which has been reproduced in Fig. 1 (a). This can also be observed in the corresponding high-order lowpass prototype filters in [18].
In [5], quasi-reflectionless bandpass filters of complementary bandpass-bandstop-diplexer architectures are presented based on a coupling-matrix-level analysis. Theoretical results of the frequency-independent coupling-routing diagrams show all-band reflectionless properties at both the input and output ports. The two auxiliary bandstop channels have the same circuit order as the main bandpass channel. As seen in Fig. 19 (b), both the auxiliary and main channels are scaled to perform high-order responses, resulting in a relatively large circuit size especially for high-order filters. The overall circuit area is three times of its bandpass channel.
In contrast, a different high-order extension methodology is used in this work and [36]. Here, increasing the filter order is readily achieved by increasing the order of the bandpass section alone without having to change the absorptive stubs, as shown in Fig. 19 (c) and (d). As such, a much more compact circuit size than Fig. 19 (b) is obtained. Besides, detailed analysis in Section III shows that this kind of high-order filters allows independent designs of passband bandwidth and stopband attenuation.
In [36], two resonance-dependent absorptive stubs are loaded to the parallel coupled line, as shown in Fig. 19 (c). In contrast, the proposed absorptive stub is of different configuration and can be equivalent to a 2-pole bandstop filter when we take the embedded resistor as an internal port, as illustrated in Fig. 2. This high-order nature leads to the benefits of improved transmission flatness and outof-band rejection, as shown in the compared responses in Fig. 19 (e) and (f). The work in [36] relies on the resonancedependent absorptive stub of Fig. 2 (b). It has maximum transmission at the center frequency point and drops dramatically at other frequencies. Same limitations can be found in input-reflectionless filters [17], [25]- [27], [31]. For a clear performance-level comparison between this work and [36], the same filter topology as [36] is used here in Fig. 19 (d). It is clear that the proposed absorptive feed line is superior in terms of passband transmission flatness and close-in matching due to the additional reflection poles at 0.75 GHz and 1.25 GHz.
In addition, Fig. 19 (e) and (f) compare the performance between the proposed work and those of [4] and [5]. In Fig. 19 (e), the proposed work outperforms [4] and [5] in terms of out-of-band rejection. Although multiple transmission zeros are generated in [4], it has worse out-of-band rejection due to the higher stopband peaks. This is more obvious for higher-order responses, e.g., the out-of-band peak is as high as −5.2 dB for a 3-pole design [ Fig. 19 (f)].
The distributed design in [5] uses a branch-line bandpass filter as its main transmission path and a branch-line bandstop filter as its absorptive circuits. It can only achieve return loss better than 10 dB in a relatively narrow range of 0.58-1.42 GHz. Outside of this range, the filter degenerates back to a reflective filter. Moreover, strong spurious responses can be observed in the lower and upper stopband. In comparison, the quasi-reflectionless filters in this work achieve better than 15-dB return loss across all frequencies.
Besides, cross-coupling implementations for high-order quasi-reflectionless filters are presented in this work. Although the first-order filter of Fig. 16 (a) has similar cross-coupling structure as the filter presented in [19], the cross-coupling implementations of the two are fundamentally different in their high-order extension circuits [4]. We also note that the cross-coupling is discussed in the coupling diagram analysis in [5], although no further details are given whether it is possible to implement such filters in a distributed fashion.
In summary, compared with the reported works in [4], [19], [20] and [5], the arbitrary-order filters in this work are of different extension topologies, which benefit an enhanced roll-off and out-of-band rejection. The same filter topologies as [36] are used in this work to better clarify the priorities of this work, i.e., the improved passband flatness due to the high-order nature of the proposed absorptive stubs, and the cross-coupling in this work introduces an additional pair of transmission zeros. Eventually, the absorptive feeding coupled-line in this work can enrich the designs of quasireflectionless filters by using the various resonator configurations and coupling schemes, which have been widely studied in conventional filter designs [38].

VI. EXPERIMENTAL VALIDATION
To validate the proposed designs, a set of 1-, 2-, and 3-pole quasi-reflectionless bandpass filters are designed, simulated, and measured. A common set of target performances for the filters are a center of frequency of 2.4 GHz, a 3-dB fractional bandwidth of 32%, and all-band return loss better than 10 dB. The schematic and EM designs are carried out in the Keysight Advanced Design System software (ADS). The microstrip filters are implemented on 0.813-mm thick Rogers 4003C substrates with a relative permittivity of 3.55 and a loss tangent of 0.0027.

A. WITHOUT CROSS-COUPLING
A conventional second-order 0.1-dB-ripple Chebyshev bandstop filter is considered as an example. The 3-dB reflection bandwidth of the absorption circuit is 32%, which is also the 3-dB absorption bandwidth BW 3dB . The 0.1-dB ripple bandwidth BW r is BW 3 ÷0.54 = 60% [37]. Correspondingly, f 1 /f 0 in (2) is 0.7 and γ is 0.51.
Following the design procedures, the critical circuit parameters for the absorptive stubs are: Z a = 21.5 , r a = 0.9, The absorptive coupled-lines (Z c1 = 50 and k c1 = 3.2) loaded with the absorptive stubs are connected back-to-back to form the 1-pole quasi-reflectionless filter. Coupled-lines sections (Z c2 = 60 and k c2 = 1.8) are inserted to realize the 2-and 3-pole filters.
In this design, the strong coupling (Z c1 = 50 and k c1 = 3.2) and the low-impedance stub (Z a r a = 19 ) present challenges to the circuit layout. For k c1 , capacitors loaded across the coupled-lines are used to dramatically decrease the odd-mode impedance, thus increasing the coupling strength [1]. Three identical 0.2-pF capacitors (PPI 0603N0R2AW251) are uniformly loaded over the coupledlines to provide the needed coupling. The low-impedance stub (Z a r a ) is realized by a parallel combination of two identical stubs each with doubled impedance (2Z a r a ).  For the 1-pole quasi-reflectionless filter, the measured return loss is better than 14 dB across the passband (2.0-2.75 GHz) and better than 11 dB across all the measured frequency range (0.5-4.5 GHz). The measured minimum insertion loss is 0.5 dB. The measured 3-dB bandwidth is 2.0-2.75 GHz (FBW 3dB = 31.6%).
The simulated and measured responses of the 2-pole quasireflectionless filter are shown in Fig. 20 (d). The measured return loss is better than 18 dB in the passband and better than 11 dB across all the measured frequency range (0.5-4.5 GHz). The measured minimum passband insertion loss is 0.67 dB and the 3-dB bandwidth is 2.1-2.8 GHz (FBW 3dB = 29%).

Fig
. 20 (f) shows the simulated and measured responses of the 3-pole quasi-reflectionless filter. The measured return loss is better than 13.5 dB in the passband and better than 10.0 dB across all the measured frequency range. The measured minimum passband insertion loss is 0.75 dB and the 3-dB bandwidth is 2.1-2.8 GHz (FBW 3dB = 29%).

B. WITH CROSS-COUPLING
Following the discussions in Section III-D, quasireflectionless filters with cross-coupling are also designed and fabricated. Photos of the 1-, 2-, and 3-pole quasireflectionless filters with cross-coupling are shown in Fig. 21 (a), (c), and (e), respectively.
Similar to the filters of Section VI-A, three 0.2-pF capacitors are loaded across the coupled-lines to provide the needed coupling coefficients. As shown in Fig. 21 (a), cross-coupling for 1-pole quasi-reflectionless BPF is introduced by using a common short-ended transmission-line stub for the two absorptive stubs. For the 2-and 3-pole quasireflectionless filters, coupled-lines between the two absorptive stubs introduce the cross-coupling, as shown in Fig. 21 (c) and Fig. 21 (e), respectively. Note that a short-ended resonator is formed in the configuration of the cross coupling of the 3-pole filter [ Fig. 18 (a)] and it has a resonance at f 0 . However, its resonant length is bounded by the layout arrangement of the bottom bandpass sections, as seen in Fig. 21 (e). Accordingly, an open-circuit stub is centrally loaded to act as a compensation factor without altering the passband performances.
It can be seen that two transmission zeros can be generated at the lower and upper stopband frequencies for each of the three filters. All the measured results maintain the same allband reflectionless properties, viz., |S 11 | lower than −12 dB, −11 dB, and −10 dB for the 1-, 2-, and 3-pole examples, respectively. Future works can be done to further improve the matching level of such kind of reflectionless filters.
Due to the manufacturing tolerances and parasitic effect of the lumped components, some discrepancies between the simulated and measured reflection responses can be observed, particularly for locations of the reflection poles and zeros at the higher frequency range. However, reasonably close agreements between simulation and measurement results are achieved. Flat passband transmission with good return loss has been achieved for all the demonstrated filters.
Note that although the cross-coupled first-order filter in Fig. 21 (a) looks the same to the first-order filter in [4], the design strategies and results in this work are fundamentally different from those in [4], [18], [19], especially for their high-order extensions, as have been compared in Fig. 1 and Fig. 19. High-order filters in this work are achieved by increasing the order of the bandpass coupled-lines to enhance the stopband rejection. Quasi-reflectionless and flat passband transmission of arbitrary-order filters are retained by using the proposed absorptive coupled-lines as the feeding lines. Eventually, the proposed absorptive feeding coupledline can find more applications in the designs of conventional filters [38].

VII. CONCLUSION
High-order distributed bandpass filters with all-band quasireflectionless and flat passband transmission have been demonstrated in this paper. The proposed absorptive filter architecture consists of absorptive coupled-line sections as the input and output coupling structures and conventional bandpass coupled-line sections to provide out-ofband rejection. Compared to the previous state-of-the-art, the proposed filters exhibit several advantages: 1) All-band quasi-reflectionless can be achieved at both input and output ports; 2) Flat passband transmission with reduced deterioration at the band edge frequencies is obtained owing to the higher-order nature of the absorptive stub; 3) The passband roll-off and close-in rejection are significantly improved by the proposed absorptive stubs; 4) cross-coupling between the absorptive stubs can be used to further improve the out-ofband rejection. Several microstrip filter examples are provided to validate the proposed designs. The work presented in this paper opens doors to further improvement of reflectionless filter designs to meet the demand of high performance RF and microwave systems.