Stopband Performance Research Articles

Compact IPD bandpass filter with wide upper stopband based on modified topology for 5G applications

AbstractA compact bandpass filter (BPF) with wide stopband performance is proposed and implemented on Si‐based integrated passive device (IPD) technology. A modified topology consisting of a novel Pi‐section and two capacitors are introduced in this BPF. The Pi‐section topology can achieve a band‐pass performance and generate four transmission zeros out‐of‐band. These two capacitors can improve the in‐band matching and also widen the stopband. The proposed BPF with a compact size of 1.0 mm × 0.5 mm × 0.3 mm is fabricated using Si IPD technology and measured by on‐wafer probing. It shows that the insertion loss is less than 1.75 dB, the return loss is better than 18 dB in the 5G band, and the upper stopband suppression level is better than 17.5 dB up to 18 GHz. The simulated and measured results of the proposed BPF are in reasonably good agreement.

Design of miniaturized 2.5D frequency selective surface for X‐ and Ku‐band applications

AbstractThis paper presents the design of 2.5D frequency selective surface (FSS) for X and Ku band applications. Proposed FSS's unit cell is constructed with two independent slitted rings on both side of the FR4 substrate and is connected with metallic vias. The FSS offers a wide stop band performance from 6.26 to 18 GHz, with a fractional bandwidth of 96.7% measured at S21 < −10 dB and has a low insertion loss (IL) of 0.018 dB. The footprint of the FSS is only about 0.11 with the shielding effectiveness of 83.365 dB, and the transverse electric and transverse magnetic mode performances are stable up to 70 degrees. Prototype FSS is fabricated and measured for its performance validation. The performance validation is carried out using applied wave research and the equivalent circuit model tools.

A high-selectivity ceramic bandpass filter with controllable transmission zeros

Abstract This paper proposes a microstrip filter with controllable transmission zeros based on the λ/4 stepped impedance resonator. This filter utilizes hybrid electric-magnetic coupling to generate transmission zeros, which can improve selectivity and stop-band performance. A fifth-order filter is designed and manufactured on a ceramic substrate with a center frequency of 26 GHz, bandwidth of 2 GHz and the signal suppression level is better than 42.51 dB from 10 to 22.8 GHz and 48.23 dB from 29.1 to 40 GHz. Furthermore, the proposed filter has a compact size of 2.074 × 1.071 mm (0.566λ g × 0.292λ g ).

Miniaturized electronically controlled notched band filter based on spoof surface plasmon polaritons

In this paper, a novel miniaturized electronical controlled notch band filter based on spoof surface plasmon polaritons (SSPPs) with inverted “山”-shaped unit is designed and experimentally demonstrated. The notch band filter is mainly composed of four parts: microstrip transmission line, transition structure, inverted “山”-shaped SSPPs, and split ring resonator (SRR) structure, and a varactor diode is embedded in the slit notch of the SRR structure to realize electronic control. Comparing with the traditional SSPP unit with the same lateral size, the dispersion curve of the proposed inverted “山”-shaped unit shows better slow wave characteristics, and the asymptotic frequency is reduced to 55%. The frequency of the notch band can be dynamically controlled by adjusting the external bias voltage at both ends of the varactor diode. As the external bias voltage increases from 0.5 V to 30 V, the notch band frequency can be changed from 2.1 GHz to 2.3 GHz and achieve easily electronic regulation. The simulation results show that the notched band filter achieves low insertion loss (<i>S</i><sub>21</sub> < –1 dB) and great return loss (<i>S</i><sub>11</sub> > –10 dB) in the pass band, which has the advantage of miniaturization with the size only 0.78<i>λ</i><sub>g</sub> × 0.35<i>λ</i><sub>g</sub>. It is worth noting that when the equivalent capacitance of the slit notch is changed, the transmission coefficient of the notched band is always less than –15 dB, showing superior band-stop performance. At the same time, by comparing and analyzing the electric field distribution of notch band filter, the transmission mechanism of microwave signal is further verified. In order to verify the its effectiveness, the traditional printed circuit board technology is used to fabricate notch band filter. The measurement results are in good agreement with the simulation ones, verifying the feasibility of the design. The electronically controlled notch band filter has higher integration and can effectively suppress the interference frequency band.

Wide-Stopband Substrate Integrated Waveguide Filters Based on Single-Mode Cavities With Multiple-Cross Slots

A novel design method for wide-stopband substrate integrated waveguide (SIW) bandpass filters (BPF) is proposed, which is realized by combining TE102-and TE101-based single-mode cavities with multiple-cross slots (MCSs). To demonstrate the proposed method, a planar SIW BPF with MCSs is proposed firstly. The ingenious MCSs have no effect on resonant modes, but have a great impact on non-resonant modes resulting in staggered their resonances. Thus, the stopband performance can be improved significantly and unload quality factors are not deteriorated simultaneously. To realize size-reduced wide-stopband SIW filters, a multiple-layer SIW filter is proposed on the basis of the planar filter. Finally, two SIW filters are fabricated and measured to verify the performance of the proposed SIW filters and the design method.

Wideband and Highly Selective BandPass Filter Using Substrate Integrated Waveguide Technology for Ku-Band Applications

A new compact design of bandpass filter using substrate integrated waveguide (SIW) is proposed and analyzed in this paper. It offers good in-band return loss, low insertion loss, wide passband, and high stopband attenuation. The filtering response is achieved by loading a couple of inverted tuning fork-shaped defected ground structures (DGS) on the bottom of the SIW cavity. DGS is utilized to enhance the upper stopband performance. To validate, the filter is fabricated using Rogers RT/Duroid 5880 with 0.508 mm thickness and tested. The filter has a center frequency of 15.44 GHz with 19% fractional bandwidth (FBW). The return loss is more than 20 dB and the insertion loss is about 1.2 dB. It finds its application in Ku band satellite communication.

Compact hybrid eighth‐mode substrate integrated waveguide and microstrip line filters with mixed electric and magnetic coupling

AbstractThis paper proposes a new type of hybrid eighth‐mode substrate integrated waveguide (EMSIW) and microstrip line (MSL) bandpass filter (BPF) with mixed electric and magnetic coupling. Firstly, two second‐order EMSIW BPFs with mixed coupling and wide stopband performance are investigated. Then, in order to realise high‐order BPF with decent performance, half‐wavelength (λ/2) open‐circuited MSL resonators are employed in the hybrid BPFs design. In the end, two third‐order BPFs and a fourth‐order BPF are developed by combining the EMSIW and MSL resonators together. Extra transmission zeros can be introduced due to the cross couplings in the trisection and quadruplet topologies. For demonstration, the proposed prototypes are designed, fabricated and measured. Reasonable agreements can be achieved between the measurement and simulation. The proposed third‐order BPFs compete against the state of the art, exhibiting the competitive characteristics of the operating bandwidth, physical footprint and upper‐stopband rejection with less order. The proposed fourth‐order BPF features a compact size of 0.34 × 0.34λ0, a low insertion loss of 1 dB, a broad 3‐dB fractional bandwidth of 22.9% and an excellent lower‐stopband suppression.

Design of superconducting ultra‐wideband filter with wide stopband based on quintuple‐mode resonator

AbstractIn this paper, a new ultra‐wideband (UWB) bandpass filter (BPF) based on QMR (quintuple‐mode resonator) loaded with quasi‐lumped elements is proposed for wide stopband application. The method of connecting five folded U‐shaped resonators with rectangular patches to construct a QMR was proposed. By adjusting the relative position of the rectangular patches, the frequency band width of the filter can be freely adjusted. Further, quasi‐lumped elements were loaded to the QMR to generate three transmission zeros to enhance stopband performance. The filter designed in this paper can independently adjust both the stopband and the passband, providing greater design flexibility. The passband of the final filter is 3.18–9.43 GHz, and the FBW (fractional relative bandwidth) is 99%. The out‐of‐band rejection is better than −15 dB in the frequency range of 9.6–28.42 GHz. The measurement results show good performance without any tuning, and are in good agreement with the simulation results.

Fabrication of C-Band High-Temperature Superconducting Microstrip Triplexer with High Accuracy

This article presents a miniaturized C-band triplexer fabricated with high-temperature superconducting (HTS) microstrip lines and to be applied to quantum computers. The triplexer consists of three bandpass filters and branch lines. In the design of the bandpass filters, folded step impedance resonators (SIRs) are firstly used to achieve good stopband performances by harmonic suppression. Secondly, cross-coupling structures are also utilized to further improve the passband edge steepness of the bandpass filters. Finally, the parasitic transmission zeros originating from the assembling box sizes are employed to achieve a deeper out-of-band rejection. Three bandpass filters are designed and connected to the common input port of the triplexer. After that, the triplexer is coarsely optimized by advanced design system (ADS) for time-saving and is finely optimized by Sonnet for better accuracy. The optimized layout of the triplexer is fabricated on a 2-inch diameter and 0.5 mm thickness MgO wafer; here, the MgO wafer is double-sided coated with HTS YBa2Cu3O7 (YBCO) thin films. The measured passband bandwidths (return loss greater than 12 dB) of the triplexer are 6.420-6.680 GHz, 6.780-7.030 GHz, and 7.110-7.380 GHz, respectively, which satisfy the required design specifications.

Millimeter-Wave Filtering Metasurface Antenna Array With Printed RGW Technology

A millimeter-wave (MMW) printed ridge gap waveguide (PRGW) fed filtering metasurface antenna (MSA) array is presented for 5G applications. The MSA element consists of a parasitic loop and a stacked patch. The elaborately designed parasitic loop around MSA not only provides satisfactory lower stopband performance but also brings an in-band resonance. Meanwhile, another upper edge radiation null is generated by introducing two square-ring slots on the MSA, improving the suppression level. Therefore, with no extra filtering circuits, desirable filtering characteristics are achieved. A 1×4 filtering MSA array is fabricated and tested to demonstrate the concept. Based on measurement results, the array prototype exhibits an overall bandwidth of 27.2% within |S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</sub> |<-10 dB. Besides, the average tested gain of about 11.1 dBi is realized, and the gain suppression outside the passband exceeds 27 dB on both sides.

Design of compact and broadband filtering balun based on N-shaped spoof surface plasmon polaritons

This paper presents a novel compact and broadband filtering balun using N-shaped spoof surface plasmon polaritons (SSPPs). The N-shaped SSPP unit cell’s lateral size has a 48% reduction compared to the rectangle-shaped counterpart. The proposed filtering balun is constructed by periodically etching N-shaped and mode-matching SSPP unit cells into the slotline balun consisting of the microstrip-to-slotline transition (MST) structures. The combination of the N-shaped SSPP unit cells and slotline balun realize the compactness and the broadband bandpass filtering response. Moreover, we can individually adjust the passband’s lower and upper cut-off frequencies. Due to the larger phase constant of the N-shaped SSPP unit cell, the field-confined ability of the proposed filtering balun is enhanced. The proposed design is fabricated and measured. It has advantages in bandwidth, stopband performance, size, and degree of design freedom compared with existing SSPP filtering baluns.

A novel filtering monopole antenna for 5G communications

PurposeThis study aims to design a compact filtering monopole antenna for 5G communication. The design is most suited for various applications within the frequency range of 2.2–3.8 GHz. It offers enhanced bandwidth and reasonable gain with wide-stopband performance.Design/methodology/approachA low-pass filter (LPF) of complementary split ring resonator (CSRR) with short-circuited stub lines is integrated with a compact defected coplanar waveguide fed truncated circular monopole ultrawideband (UWB) antenna. The reference UWB antenna etched on an FR4 substrate was coupled to the designed LPF to transform the UWB antenna into a wideband antenna. The effect of coupling is analyzed based on the real and imaginary responses of the terminal impedance (ZT) curve. Three short-circuited stub lines of asymmetric lengths are added to the CSRR LPF to suppress harmonics, thereby enhancing the stopband performance and impedance matching between the elements. The proposed filtering antenna is fabricated using a photolithography process, and the corresponding results are measured using a network analyzer (N9951A). The radiation parameters of the proposed filtering monopole antenna are tested in the anechoic chamber. The simulated/measured results are compared and are found in agreement with each other.FindingsThe proposed design suppresses 6.5f0 harmonics, resulting in wide stopband performance and increased gain selectivity at the transition edge. A peak suppression of −41 dB and an average suppression of −18 dB were attained throughout the stopband. An operating fractional bandwidth of 54.5%/143% with a peak gain of 3 dBi/5 dBi was obtained. The proposed filtering antenna supports 5G applications such as WiMAX, WLAN, n7, n38 IMT-E, n30 WCS, n40 TDD, n41 TDD, n48 TDD, n78 TDD and n90 TDD.Originality/valueThe proposed design is novel and compact and has a wide application in 5G communication. With the filter, the antenna operates in wideband, and without the filter, it operates in UWB. Besides, it offers enhanced stopband performance with high gain selectivity at the transition edge. Comparatively, a 50% improvement in bandwidth, 52% improvement in size reduction and 33% improvement in harmonic suppression are attained.

A four-port wideband filtering monopole MIMO array for sub-6 GHz 5G communications

AbstractThis communication presents the significance of a four-port 2×2 multiple input multiple output antenna operating in mid-band frequency for 5G handheld device application. A microstrip-fed reference truncated decagonal monopole antenna operating at 3.9 GHz and a modified split-ring resonator (SRR)-based low-pass filter that excites at a cut-off frequency of 6.5 GHz are designed. On introducing a stub line in the SRR structure, the proposed filter shows a good out-of-band rejection beyond 6.5 GHz. By integrating the filter into the reference monopole, a wideband filtering monopole with improved stopband performance is obtained. A four-port MIMO array is constructed by orthogonally positioning the antenna with a center-to-center separation of 0.25λ0. To improve isolation, a pair of inverted L-stubs of varying lengths is introduced. The closed loop formed by the interconnection of stubs enhances isolation within the frequency range of 3–3.5 GHz. The fabricated prototype fosters an impedance bandwidth of 66.67%, isolation &gt;18.5 dB, an average suppression of 15 dB, an average peak gain &gt;3.7 dB, envelope correlation coefficient &lt; 0.015, Actual Diversity Gain (ADG) &gt; 9.9 dB, Effective Diversity Gain (EDG) &gt; 8.8 dB, and efficiency &gt;88%. The proposed design is suitable for 5G sub-6 GHz applications such as n48, n77, n78, n79 bands, WiMAX, and LTE bands 42, 43, 46, 47, 48.

Modular Design of Waveguide Bandpass Filters With Improved Stopband and High Selectivity

A class of waveguide bandpass filters (BPFs) with parallel coupling topology is proposed by stacking rectangular cavities. Compared with transversal filters, each layer comprises more than one cavity, and these cavities are cascaded with electric or magnetic coupling. The stopband performance could be improved by properly adjusting the width-length ratio of rectangular cavity to push up the higher-order modes in the cavities, while the attractive property of transversal filter, i.e., modular design, is still valid and useful in simplifying the filter design process and producing transmission zeros (TZs) out of the passband. A fourth -, sixth -, and seventh-order BPF is successively designed, and frequency of the first spurious mode is increased from 1.12 to 1.48 times the operating frequency. Due to the parallel coupling topology, multiple TZs are generated at both sides of the passband, making the suppression more than 20 dB out of passband. Among these filters, the fourth- and sixth-order BPFs are fabricated, and the measured results demonstrate the effectiveness of the proposed method.

A Miniaturized Harmonic Reject Waveguide Filter for Ku-Band VSAT Transmitter

In this paper, a novel miniaturized low-pass waveguide filter with wide stop-band is proposed for the harmonic rejection of Ku-band Very Small Aperture Terminal (VSAT) transmitters. The proposed filter consists of a corrugated waveguide low-pass filter and a metal septum in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H$ </tex-math></inline-formula> -plane at the center of the narrow side wall. Because it does not require additional impedance transformers, the proposed filter achieves a compact size. To estimate the amplitude level of the higher-order modes at the input port of the harmonic reject filter in a RF transmitter, higher-order modes generated by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$E$ </tex-math></inline-formula> -plane probe-type microstrip-to-waveguide (M2W) transition are analyzed and calculated using commercial numerical tools. To analyze the response of the proposed filter in higher-order modes, the mode conversions in the structure are also analyzed and calculated. The analysis shows that a wide stop-band free of spurious response in all higher-order modes can be obtained by co-designing M2W transition and filter. To validate the design concept, a Ku-band harmonic reject filter is designed using WR-75 waveguide. Simulation results show that the reflection is less than −23dB between 13 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim $ </tex-math></inline-formula> 15 GHz, while the suppression stop-band performance in TE 10 mode is better than 35dB in 17.5 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim $ </tex-math></inline-formula> 29.6 GHz. The effective length of the filter is 29 mm. Simulation results also indicate that the spurious pass-band performance in TE 20, TE 21 and TM 21 modes is improved by the M2W transition. Finally, to validate the proposed idea, a Ku-band GaN power amplifier is fabricated and measured. The measurements show that the amplifier has a minimum output power of 72.5 W with better than 20 dBc <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{nd}$ </tex-math></inline-formula> harmonic rejection. By using the proposed filter, the measured <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{nd}$ </tex-math></inline-formula> harmonic rejection of the power amplifier is greater than 52 dBc.

Study on a Multi-Point Differential Feeding Strategy for Design of Filtering Patch Antennas With Stopband Enhancement

In this article, a novel multi-point differential feeding technique is introduced and applied in the design of filtering patch antennas with a wide upper stopband. Different from the traditional differential feeding structure, the proposed method has four physical feeding points on the patch. Our analysis results indicate that the proposed feeding approach can not only effectively excite the desired fundamental TM <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$_{\mathbf {01}}$ </tex-math></inline-formula> mode but also flexibly allow for the elimination of high-order modes by selecting suitable feeding positions. Just relying on this feeding method, unwanted harmonic radiations can be totally removed up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4{f} _{\mathbf {0}}$ </tex-math></inline-formula> without any additional elements. For demonstration, a prototype of a stacked patch antenna with filtering response has been then designed. With resorting to the proposed feeding scheme, both driven and stacked patches are indeed able to support good excitation of TM <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$_{\mathbf {01}}$ </tex-math></inline-formula> mode with other harmonics to be suppressed, leading to a wide upper stopband performance. In addition, several radiation nulls are generated in the stopband to improve the filtering selectivity. Measured results agree well with the simulated ones, indicating that the designed differential antenna possesses enhanced fractional impedance bandwidths, high in-band gains, low cross polarization, as well as intrinsic wide stopband rejection, further verifying the proposed feeding concept.

Ultra-compact half-mode substrate integrated waveguide bandpass filters with IRCSRR metamaterial on reinforced hydrocarbon polymer ceramic composites

In this work, metamaterial-enabled ultra-compact evanescent-mode half-mode substrate integrated waveguide (HMSIW) bandpass filters are developed by loading inner-rotary complementary split-ring resonator (IRCSRR) on reinforced hydrocarbon polymer ceramic composites substrate. In the proposed IRCSRR, inner-rotary slots are etched inside the conventional complementary split-ring resonator (CSRR) metamaterials to enhance the product of equivalent inductance and capacitance without occupying extra circuit area. Therefore, the proposed HMSIW-IRCSRR filters can obtain more size reduction with the same operation frequency as compared with the conventional CSRR-based ones. Similar as the conventional CSRR case, the evanescent-mode transmission below the dominant cutoff frequency of HMSIW can be produced by the negative permittivity effect of IRCSRR, and a transmission zero (TZ) at the upper stopband can be simultaneously generated by the mutual coupling between HMSIW and IRCSRR. Then, to improve the upper stopband performance, source-load coupling topology is utilized in the two-pole filters, which can produce multiple extra TZs. Meanwhile, multiple microstrip L/T-shaped open stubs are employed to introduce hybrid low-pass effect to further extend the upper stopband much wider. To demonstrate the availability of the aforementioned concepts, a series of HSMIW-IRCSRR filters are experimentally designed, fabricated and verified. Measured results shows that both ultra-compact sizes and excellent frequency-selecting performance are achieved simultaneously, as well as good agreement with simulations, all of which illustrating quite good promises for the practical radio frequency and microwave applications.

An effective method to suppress high-order modes of SIW filters with compact size

In this paper, an effective method for substrate integrated waveguide (SIW) filter with improved stop band performance is proposed. By configuring the position and the direction of the coupling window, the transmission of the parasitic mode such as TE102 and TE201 can be effectively suppressed without adding any extra structure and element. Under the above theory, a Ka-band SIW filter centered at 35.71 GHz based on quartz glass substrate with compact size and simple structure is designed and tested. The stop rejection of the design is increased from 1.03 times to 1.95 times the center frequency within −35 dB of the filter with the insertion loss of −2.5 dB and the return loss of −20 dB. Finally, the measurement results validated the proposed design.

Compact low-pass filter (LPF) with wide harmonic suppression using interdigital capacitor

Abstract In this article, a compact microstrip low-pass filter (LPF) with low in-band insertion loss and a wide attenuation band is recommended. The structure is modeled using two rectangular-shaped open resonators (ORs) loaded with multiple sections of the interdigital capacitor. The proposed LPF has a cut-off frequency of 1.5 GHz, a passband insertion loss of 0.35 dB, and an electrical circuit size of 0.02 λ g 2. In the ground plane, two dissimilar defected ground structures (DGSs) are introduced to generate additional attenuation poles for enriching the stopband performance of the LPF. The harmonics suppression is achieved up to 15 GHz with tenth harmonic suppression and relative stopband rejection of almost 165%. The EM simulated results show a well-matched behavior with the experimental ones. The proposed LPF can be used for global positioning systems (1500 MHz), telecommunications (1600 MHz), and audio broadcasting (1400 MHz) applications.

Design for a Self-Packaged All-PCB Wideband Filter With Good Stopband Performance

A new design of a self-packaged all-printed circuit board (PCB) wideband filter based on the cascade connection of high-pass filter (HPF) and low-pass filter (LPF) is presented. The novel low- and high-pass quasi-lumped filter structure differs from the known ones by the existence of cross coupling between elements producing transmission zeroes (TZs) at the desired frequency. It is shown that high-order structures make it possible to design a wideband filter having the number of TZs in the low-frequency stopband equal to the order of the applied HPF and the number of TZs in the high-frequency stopband equal to half the order of the LPF. An implementation of two-layer metallization of the capacitor plates made it possible to manufacture all three types of filters using the common technology of multilayer PCBs with a good coincidence of the characteristics of the model and the fabricated filters. The filter fabricated using materials with a rather low quality factor (F4BM/RO4450B/RO4003C) has demonstrated a good performance: the central frequency is 0.99 GHz, the 1-dB fractional bandwidth is 152%, the minimum in-band loss is 0.3 dB, the selectivity ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta f_{\mathrm {30~dB}}/\Delta f_{\mathrm {3~dB}}$ </tex-math></inline-formula> ) is 1.21, and the 40-dB stopband width is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$6.16f_{0}$ </tex-math></inline-formula> .