Multiple Zeros Research Articles

A new family of Sakurai–Torii–Sugiura type iterative methods with high order of convergence

In this paper, we introduce a new family of iterative methods for finding simultaneously all zeros (multiple or simple) of a polynomial. The proposed family is constructed by combining the known Sakurai–Torii–Sugiura iteration function with an arbitrary iteration function. We provide a detailed convergence analysis in the following two directions: local convergence if the polynomial has multiple zeros with known multiplicity and semilocal convergence if the polynomial has only simple zeros. As an application, we study the convergence of several particular iterative methods with high order of convergence.

Absorptive/transmissive frequency selective surface with multiple transmission zeros for reduced transition bandwidth

An absorptive/transmissive frequency selective surface (ATFSS) with multiple transmission zeros is proposed, which consists of one lossy layer and two lossless layers separated by air space. As the key features, the slot-line loading and interlayer coupling technologies are adopted in the lossy layer and lossless layers, respectively, to introduce the infinite impedance points and four transmission zeros, thereby simultaneously achieving low insertion loss and high selectivity of the transmission band. Benefiting from the ohmic loss of the lumped resistance in the lossy layer, effective absorption can be readily implemented. A detailed qualitative analysis of the proposed structure is presented through the equivalent circuit theory, surface current, and power loss density distributions. Under normal incidence, the overall structure exhibits two discrete absorption bands over the frequency range of 6.34 to 13.40 GHz, whereas a transmission behavior is obtained from 9.34 to 11.40 GHz with a minimum insertion loss of only 0.14 dB. In particular, there are only 3.9% and 3% transition bandwidths between absorption and transmission functions. Then, a sample prototype is fabricated for demonstration, and the measured results are well consistent with the simulated one. As expected, the proposed ATFSS has the advantage of multiple transmission zeros, loss insertion loss, and narrow transition band, which is of great potential in the application of radar stealth radome.

Design and Modelling of a Compact Triband Passband Filter for GPS, WiMAX, and Satellite Applications with Multiple Transmission Zero’s

Designing microwave filters with high selectivity and sharp roll-off between the stop and pass bands can be challenging due to the complex nature of the R.F. signals and the requirements for achieving high performance in a limited physical space. To achieve a high selectivity and sharp roll-off rate, this paper presents a compact filter with a triple passband response. The two different passbands at 1.57 GHz and 3.5 GHz are achieved using a step impedance resonator (SIR) with metallic slots perturbation added to the lower corner of the high impedance section of the SIRs, which helps to enhance the filter’s selectivity and size reduction greatly. The embedded L-shaped structure originates a third passband at 4.23 GHz, resulting in a triband response with eight transmission zeros below and above the passbands at 1.22/1.42/1.98/3.18/3.82/3.98/4.38/4.53 GHz, respectively. The prototype has low signal attenuation of <1.2 dB and high signal reflection of >25 dB for the three passbands. The fractional bandwidths achieved are 2.54%, 4.2%, and 1.65% at 1.57/3.57/4.23 GHz, respectively, with rejection levels in the stopband greater than 15 dB. Lastly, the structure is fabricated on RO-4350B PCB and observed good matching between experimental and measured results. This demonstrates that the prototype can be successfully implemented in real-world applications such as GPS, WiMAX, and Satellite systems. The area occupied by the filter on a substrate or in a circuit is 0.31 λg × 0.24 λg, where λg is the guided wavelength of the material calculated at the lowest frequency.

Inequalities for the polar derivative of a polynomial with restricted zeros

Let [Formula: see text] be a polynomial of degree at most [Formula: see text] with real or complex coefficients, then by a prize winning result of Bernstein [Formula: see text] In this paper, we assume that [Formula: see text] has a zero of multiplicity [Formula: see text] at a point inside the unit disc and the remaining zeros outside or inside the disc of radius [Formula: see text] and prove some Bernstein-type inequalities. We also draw the attention of readers to the wrong conclusion of a result due to [Mir and Wani, A note on two recent results about polynomials with restricted zeros, J. Math. Inequalities, 14 (2020) 45–50].

Synthesis Design of Multiband Bandpass Filters Employing Multimode Bandstop Resonators With Star-Like Topology

Multimode resonators (MMRs) can be utilized to build multiband bandpass filters (MBPFs). Each individual mode serves as the operating mode for one of the passbands in a well-known coupling topology. However, several difficulties are involved with the classical design method. In this article, we present a novel star-like topology to design MBPFs based on MMRs. Differently from the classical use, in the proposed novel topology, they are so regarded as bandstop resonators and provide multiple controllable transmission zeros. To explain the proposed concept and the implementation method, two bandstop MMRs – a classical dual-mode stub loaded resonator (DSLR) and a new tri-mode dual-stub loaded resonator (TDSR) – are investigated with detailed design formulas and diagrams. Then, a tri-band MBPF with equal individual sub-bands, a quad-band and two quint-band MBPFs with unequal individual sub-bands are synthesized and designed using the proposed bandstop MMRs. Additionally, simulations and experiments are used to verify the efficiency of all prototypes. Good agreements with the theoretical calculations are observed for all the MBPFs.

Grid-Connected Phase-Locked Loop Technology Based on a Cascade Second-Order IIR Filter

The moving average filter-based phase-locked loop (MAF-PLL) can obtain grid synchronization signals accurately under adverse grid conditions with a large amount of harmonics due to the high filtering capability of the MAF. However, MAF-PLL cannot achieve a fast dynamic response in the case of frequency drift, phase angle steps, and unbalanced voltage sag. MAF is essentially an FIR filter, and its filtering performance is hard to be adjusted. To address this issue, this paper proposes an alternative to MAF consisting of a set of cascading second-order IIR filters (CIIRF). Based on MAF, CIIRF introduces multiple zeros and poles from the zero–pole replacement perspective, and by changing the position of the poles, the filter performance can be adjusted. To improve the anti-interference ability of PLL based on CIIRF (CIIRF-PLL) in the presence of grid frequency drift, a frequency-adaptive scheme is also proposed. Simulation and experimental results show that CIIRF-PLL can accurately track the grid voltage phase in the case of frequency steps, phase angle jumps, harmonics injection, and unbalanced voltage sag and has good steady-state and dynamic performance.

Triple-Passband Channelized Passive Filter With Infinite Input-Quasi-Reflectionless Range

A transmission-line-based sharp-rejection channelized passive bandpass filter (BPF) with 1.87/2.5/3.21-GHz triple-band response and infinite input-quasi-reflectionless (IQR) range is reported. Two RF power input absorption mechanisms are combined into it, as follows: 1) a balanced-type/two-branch circuit with a 90 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$^{\circ}$</tex-math> </inline-formula> at 2.5-GHz transmission line input phase-delay stage in one channel to attain a 1.6–3.4-GHz 10-dB-referred IQR range and 2) an input lossy circuit that dissipates the RF power echoes in the 0–1.6-GHz and 3.4–5-GHz bands, respectively. In this manner, as the overall transfer function is frequency quasi-periodic with the 5-GHz spectral period, a theoretically infinite IQR bandwidth is attained. Moreover, by exploiting transversal signal-interference principles owing to the transmission-line-based phase-delay stages that are inserted at the outputs of its two channels, an increased selectivity triple-band BPF response with multiple transmission zeros (TZs) is obtained. The basic RF operational foundations of the engineered triple-passband IQR channelized filter are presented, along with the measured results of a manufactured proof-of-concept microstrip prototype.

Minimal seven‐element series‐parallel realizability of a certain positive‐real biquadratic impedance

SummaryThe minimal complexity realization of low‐order positive‐real immittances as resistor–inductor–capacitor (RLC) (or damper‐spring‐inerter) networks is an essential unsolved problem in the theory of network synthesis, and the results can be directly applied to the inerter‐based mechanical systems control. This paper solves the realizability problem of a certain class of biquadratic impedances as any seven‐element series‐parallel RLC network, where the impedance contains a repeated zero and pole of multiplicity two. In contrast to the previous results of this problem, the work in this paper considers the general case and removes the restrictions on the network structures and the number of reactive elements. The final results present a necessary and sufficient condition for the realizability of such an impedance as any seven‐element series‐parallel RLC network. A network synthesis example in suspension control design is presented to illustrate the results of this paper and the applications in inerter‐based mechanical control.

Filtering linear polarization converters based on multi-mode unit for harmonic suppression

In this article, a class of filtering linear polarization converters based on a single quasi-single-layer triangular resonator unit is proposed. By loading the stub-line and slot lines, the degenerate modes of the single fundamental resonant mode of an isosceles right-angle patch are separated into two for a second-order filtering characteristic, and two additional resonant modes are then generated for the second conversion band. Attributed to the mixed electric and magnetic coupling in the coupling section of the loaded stub-line, several conversion zeros are introduced at the same time. As the contribution of this work, multiple resonant modes and conversion zeros can be independently controlled by the corresponding parameters, enabling second-order single- and dual-band response with multi-mode high selectivity side-band and harmonic rejection characteristics out-of-band. With the detailed analysis of the surface current and field distributions, the polarization conversion mechanisms are revealed. Finally, the single-band and dual-band linear polarization converters are processed, and the measured and simulated results are in good agreement. As expected, the two polarization converters show excellent filtering features, which is beneficial for improving modulation accuracy and avoiding unwanted electromagnetic interference in various applications of multi-passband and multi-functional metasurfaces.

Dual-Band and Triple-Band Filtering Power Dividers Using Coupled Lines

In this brief, new dual-band and triple-band filtering power dividers (FPDs) with sharp frequency selectivity are proposed. Firstly, the dual-band FPD is developed and mainly composed of a quarter-wavelength three-line parallel coupled line structure, five transmission lines, and a pair of short-ended parallel coupled lines. Based on this specific structure, with two extra transmission lines imported, a triple-band FPD is further attained. Multiple transmission zeros are generated successfully in the above two designs, thus leading to a sharp skirt selectivity. For the above FPD designs, only a single isolated resistor is arranged between the two side lines of three-line parallel coupled line for nice port-to-port isolation within wide passbands. For demonstration, a prototype dual-band FPD operating at 2.38 and 3.80 GHz with a size of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.67{\lambda }\text{g}\,\,{\times }\,\,0.68{\lambda }\text{g}$ </tex-math></inline-formula> , and a triple-band FPD operating at 2.38 GHz, 3.42 GHz, and 3.91 GHz with a size of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.66{\lambda }\text{g}\,\,{\times }\,\,0.68{\lambda }\text{g}$ </tex-math></inline-formula> are designed, fabricated, and tested, respectively.

Novel Synthesis Method for Reconfigurable Filtering Phase Shifter With Frequency-Dependent Coupling Using Optimization Technique

A novel optimization technique is proposed for reconfigurable filtering phase shifter with frequency-dependent coupling in this brief. It incorporates both the coupling matrix entries and the circuit elements into optimization process. With the proposed technique, it can not only obtain desired insertion phase at the center frequency, align the phase slope between different phase states to be same, but also maintain satisfactory return loss in the passband and have multiple transmission zeros in the stopband. The proposed method well suits the reconfiguration application scenario where frequency-dependent coupling is adopted between adjacent resonators. To validate the proposed idea, a microstrip prototype is fabricated and measured. Experimental results show that the operating center frequency can be tuned from 0.6 GHz to 0.71GHz, with a continuously tunable phase shifting range of larger than 175° at each frequency state.

Design and Theoretical Analysis of Highly Negative Dispersion-Compensating Photonic Crystal Fibers with Multiple Zero-Dispersion Wavelengths

This paper presents a highly negative dispersion-compensating photonic crystal fiber (DC-PCF) with multiple zero dispersion wavelengths (ZDWs) within the telecommunication bands. The multiple ZDWs of the PCF may lead to high spectral densities than those of other PCFs with few ZDWs. The full-vectorial finite element method with a perfectly matched layer (PML) is used to investigate the optical properties of the PCFs. The numerical analysis shows that the proposed PCF, i.e., PCF (b), exhibits multiple ZDWS and also achieves a high negative chromatic dispersion of −15089.0 ps/nm·km at 1.55 μ m wavelength, with the multiple ZDWs occurring within the range from 0.8 to 2.0 μ m range. Other optical properties such as the confinement loss of 0.059 dB/km, the birefringence of 4.11 × 10 − 1 , the nonlinearity of 18.92 W − 1 k m − 1 , and a normalized frequency of 2.633 was also achieved at 1.55 μ m wavelength. These characteristics make the PCF suitable for high-speed, long-distance optical communication systems, optical sensing, soliton pulse transmission, and polarization-maintaining applications.

Covering almost all the layers of the hypercube with multiplicities

Given a hypercube Qn:={0,1}n in Rn and k∈{0,…,n}, the k-th layer Qkn of Qn denotes the set of all points in Qn whose coordinates contain exactly k many ones. For a fixed t∈N and k∈{0,…,n}, let P∈R[x1,…,xn] be a polynomial that has zeroes of multiplicity at least t at all points of Qn∖Qkn, and P has zeros of multiplicity exactly t−1 at all points of Qkn. In this short note, we show thatdeg(P)≥max⁡{k,n−k}+2t−2. Matching the above lower bound we give an explicit construction of a family of hyperplanes H1,…,Hm in Rn, where m=max⁡{k,n−k}+2t−2, such that every point of Qkn will be covered exactly t−1 times, and every other point of Qn will be covered at least t times. Note that putting k=0 and t=1, we recover the much celebrated covering result of Alon and Füredi (1993) [1]. Using the above family of hyperplanes we disprove a conjecture of Venkitesh (2022) [23] on exactly covering symmetric subsets of hypercube Qn with hyperplanes. To prove the above results we have introduced a new measure of complexity of a subset of the hypercube called index complexity which we believe will be of independent interest.We also study a new interesting variant of the restricted sumset problem motivated by the ideas behind the proof of the above result.

A compact diamond-shaped dual-band bandpass filter with multiple transmission zeros

In this paper, a compact diamond-shaped dual-band bandpass filter with seven transmission zeros in the stopband and two poles in each passband is designed, analyzed, and implemented. The proposed filter consists of coupled lines and stepped impedance open stubs. Theoretical analysis based on lossless transmission line theory is carried out to verify the transmission zero frequencies. A filter prototype with center frequencies of 1.17 GHz and 2.85 GHz is fabricated to exhibit the feasibility of the design. The prototype is printed on a 0.79 mm Rogers RT Duroid 5880 with a dielectric constant of (ϵr = 2.2) and occupies an area of 0.29λg×0.17λg, where λg is the guided wavelength at the first center frequency. The experimental results show a 3-dB fractional bandwidth of 13.65% and 3.5% at low and high center frequencies, respectively. The simulated and experimented frequency responses are consistent with each other.

Wideband-Filtering Switches With Ultra-Wide Stopband Using P-I-N Diodes Loaded on Slotline

This paper presents a new design method for the microwave switch, which uses p-i-n diodes loaded on slotline. A single-pole single-throw (SPST) switch based on the microstrip-slotline-microstrip (M-S-M) transition with non-uniform loading structures is first designed to validate our design approach, which controls the on and off states of the switch by applying different voltages to the diodes. Then a single-pole double-throw (SPDT) switch using p-i-n diodes loaded on slotline T-junction is proposed, which achieves great high-pass response of ON-state channel and 20 dB OFF-state suppression (OSS) within dc to 4.64 GHz. To overcome the poor selectivity of two switches, transmission zeros need to be introduced to achieve band-pass response. Therefore, the slotline of the M-S-M transition is loaded with gradient length short stubs to bring in multiple transmission zeros and achieve SPST/SPDT wideband-filtering switches with ultra-wide stopband. Finally, the proposed filtering switches are fabricated and measured. The measurement results show that the SPDT filtering switch not only provides the expected band-pass response and wide stopband from 4 to 43.5 GHz of ON-state channel, but also has 22.8 dB rejection of OFF-state channel, and the 20.8 dB port-to-port isolation from dc to 43.5 GHz.

Design of dual-band BPFs with high selectivity

Abstract In this article, two dual-band microstrip bandpass filters (BPFs) with high selectivity and independently controllable passbands are designed based on three parallel-coupled lines (TPCLs). The first BPF proposed in this paper is achieved by employing a pair of folded F-type resonators (FFRs) and conventional TPCL structure. By adjusting the above parameters of each FFR, the center frequencies of the two passbands can be controlled easily. In the meantime, TPCL can enhance the coupling between the source and load, and introduce multiple transmission zeros (TZs), which can conduce to a higher selectivity. To obtain superior selectivity, the second BPF is designed based on a novel TPCL. The novel TPCL is based on the traditional one, which further enhances selectivity since transmission paths are doubled. In order to achieve miniaturization and enhance coupling, FFRs are folded. In addition, four transmission zeros (TZs) are generated to improve the selectivity of the two passbands. The measured results of the fabricated BPFs centered at 2.45/3.5 GHz agree well with the simulated ones, which validates the proposed design method.

A compact 5G band N77 bandpass filter with multiple zeros based on GaAs integrated passive device process

AbstractIn this paper, a compact 5G band N77 bandpass filter (BPF) with multiple zeros is designed based on GaAs integrated passive device (IPD) process. The BPF consists of lumped capacitors and inductors. Hybrid resonator technology which consists of a combination of shunt and series resonant tanks are proposed to produce six transmission zeros in the stopband and significantly improve the filter selectivity. In addition, a series resonator at the output port of the filter was introduced to produce another transmission zero at the upper stopband and further improve the stopband suppression. Meanwhile, to reduce the area and insertion loss of the filter, a stacked inductor is designed by using IPD technology, which can greatly reduce the area and size of the filter. The minimum insertion loss is 1.782 dB. The size of the entire BPF is 1.3 mm × 0.7 mm.

Inequalities for the derivative of rational functions with prescribed poles and restricted zeros

In this paper, instead of assuming that a rational function r(z) with prescribed poles has a zero of order s at origin, we suppose that it has a zero of multiplicity s at any point inside the unit circle, whereas the remaining zeros are within or outside a circle of radius k and prove some results which besides generalizing some inequalities for rational functions include refinements of some polynomial inequalities as special cases.

Robustness of disturbance observer based servo systems*

This paper gives a theoretical analysis of the robustness of servo systems with disturbance observers. The disturbance observers have been used as an easy way to enhance the robustness of servo systems because of their simple structure. However, the effectiveness has yet to be theoretically proved. Therefore, this paper aims to clarify the advantage of the disturbance observer. This paper considers the servo system consisting of a servo compensator and a disturbance observer. The servo system has robustness in suppressing the response variations against the system uncertainties. It is theoretically shown that the robustness comes from the systems having zeros of multiplicity two at s = 0, while the systems are type-1 servo systems. A numerical example is provided to confirm the theoretical result and to demonstrate the effectiveness of disturbance observers.

Design of a Planar Compact Dual-band Bandpass Filter with Multiple Transmission Zeros Using a Stub-loaded Structure

Design of a Planar Compact Dual-band Bandpass Filter with Multiple Transmission Zeros Using a Stub-loaded Structure