Chebyshev Approximation Research Articles

Microstrip Lowpass Filter Realization Using Monic Chebyshev Polynomial

This paper presents a novel approach to microwave lowpass filter design that addresses several challenges in modern microwave systems, including size, performance, and cost. With that in mind, we propose using monic Chebyshev polynomials as the characteristic function for the filter to achieve a low passband insertion loss and maximum cutoff slope. The half-power cutoff frequency characterizes these filters, and the ripple factor is not considered a design parameter. To demonstrate the effectiveness of the monic Chebyshev approximation, a comparison is made with chained function (CF) filters designed for microwave applications. It has been shown that the monic Chebyshev filter outperforms CF filters in all applications, including microwaves.

ChebNet: Efficient and Stable Constructions of Deep Neural Networks with Rectified Power Units via Chebyshev Approximation

ChebNet: Efficient and Stable Constructions of Deep Neural Networks with Rectified Power Units via Chebyshev Approximation

Estimation of the domain of attraction for continuous-time saturated positive polynomial fuzzy systems based on novel analysis and convexification strategies

In this paper, the domain of attraction (DOA) of the continuous-time positive polynomial fuzzy systems subject to input saturation is estimated by using the level set of the linear copositive Lyapunov function. To relax the estimation of DOA, the restriction on the level set is removed by embedding the expression of the level set into the stability conditions and positivity conditions. Referring to the nonconvex terms caused by above novel analysis strategy, some polynomial inequality lemmas are proposed to handle them; the nonconvex terms caused by imperfect premise matching (IPM) nonlinear membership functions are dealt with by sector nonlinear methods and advanced Chebyshev membership-function-dependent (MFD) methods. In this advanced MFD method, the state space segmentation and polynomial order selection of the Chebyshev approximation method are improved based on breakpoints of the first derivative and curvature, respectively, which is helpful to reduce the conservatism and computational burden of the result. Thus, this advanced Chebyshev MFD method not only optimizes the convexification strategy, but also can further be extended to estimate the DOA when it is used to introduce the membership functions information for convex stability and positivity conditions. Finally, a numerical example and the lipoprotein metabolism and potassium ion transfer nonlinear model are presented to validate the effectiveness and feasibility of the aforementioned analysis and convexification strategies in the expansion of DOA estimation.

Randomized semi-quantum matrix processing

We present a hybrid quantum-classical framework for simulating generic matrix functions more amenable to early fault-tolerant quantum hardware than standard quantum singular-value transformations. The method is based on randomization over the Chebyshev approximation of the target function while keeping the matrix oracle quantum, and is assisted by a variant of the Hadamard test that removes the need for post-selection. The resulting statistical overhead is similar to the fully quantum case and does not incur any circuit depth degradation. On the contrary, the average circuit depth is shown to get smaller, yielding equivalent reductions in noise sensitivity, as explicitly shown for depolarizing noise and coherent errors. We apply our technique to partition-function estimation, linear system solvers, and ground-state energy estimation. For these cases, we prove advantages on average depths, including quadratic speed-ups on costly parameters and even the removal of the approximation-error dependence.

Theoretical design and implementation of equal and unequal split ultra-wide band Wilkinson power divider with Chebyshev impedance transform

Abstract In this study, we designed multi-section UWB Wilkinson power dividers for both equal and unequal power divider. The key innovation is successfully matching these power dividers using the Chebyshev method. The proposed method achieves very low insertion losses and high transmission coefficients across the ports. For the three-section equal power divider, Chebyshev polynomials were used to model the reflection coefficient, and the characteristic impedance for each section was calculated symmetrically. The design achieved a fractional bandwidth of 106% for a 20 dB return loss, with an insertion loss of 0.3 dB. For unequal multi-section dividers, power division ratios of 1.5:1 and 2:1 were chosen. Theoretical microwave calculations were used to determine the input and output transmission line impedances. Chebyshev approximation was then applied to design the three-section unequal power dividers. The return losses for these unequal dividers were measured at 17 dB and 15 dB, with fractional bandwidths of 120% and 126%. Both circuits had insertion losses of 0.25 dB. The consistent results from analytical calculations, simulations, and measurements confirm the effectiveness of the Chebyshev method for both equal and unequal power division.

Design of distributed impedance‐matching networks for ultra‐wideband low‐noise amplifiers

AbstractAn ultra‐wideband low‐noise amplifier (LNA) design was developed based on distributed network synthesis to achieve impedance matching of an active device by parasitic absorption. The equivalent device models of a low‐noise device were generated using negative‐image device modeling and traveling‐wave matching, and matching circuits were produced using the Chebyshev approximation. The minimum insertion loss and ripple of the matching networks were adjusted correctly to allow the parasitic absorption to match the device models. This paper describes the design methodology for distributed matching circuits and its application to the design of an ultra‐wideband LNA.

Vertical line time domain green function and its applications in numerical simulation of ship seakeeping performance

In present study, a vertical line source time domain Green function is proposed and the subsequent numerical evaluation methods are described in detail. For which, Taylor expansion and Chebyshev approximation algorithms are originally extended from point source to line source for small parameters, as well as asymptotic expansions are extended for large parameters. Meanwhile, polynomials are originally proposed to replace the asymptotic expansions to improve the computational efficiency. The accuracy and stability of the proposed method are validated through several cases and good agreement can be obtained.Furthermore, a hybrid Green function method based on the vertical line source Green function and Rankine source is developed for time domain simulation of seakeeping performances of ships. With respect to the hybrid method, the flow domain is divided into inner part and outer part by an artificial control surface. The first-order Taylor Expansion Boundary Element Method (TEBEM) based on the Rankine panel distribution is applied in the inner part, while three-dimensional panel method based on the vertical line source Green function distribution is adopted in the outer part. Under this circumstance, motions of the single-ship (Slender Wigley and Blunt Wigley) and two-ship (two side-by-side ships) in head waves are numerically investigated. The numerical results agree well with the experimental data, which proved the feasibility and the accuracy of present hybrid Green function method for motion prediction of both single-ship and two-ship.

The $$L_q$$-weighted dual programming of the linear Chebyshev approximation and an interior-point method

The $$L_q$$-weighted dual programming of the linear Chebyshev approximation and an interior-point method

Chebyshev Approximation of Multivariable Functions by a Power Expression

Chebyshev Approximation of Multivariable Functions by a Power Expression

Chebyshev approximation technique: analysis and applications

Chebyshev approximation technique: analysis and applications

Iterative Chebyshev approximation method for optimal control problems

We present a novel numerical approach for solving nonlinear constrained optimal control problems (NCOCPs). Instead of directly solving the NCOCPs, we start by linearizing the constraints and dynamic system, which results in a sequence of sub-problems. For each sub-problem, we use finite number of Chebyshev polynomials to estimate the control and state vectors. To eliminate the errors at non-collocation points caused by conventional collocation methods, we additionally estimate the coefficient functions involved in the linear constraints and dynamic system by Chebyshev polynomials. By leveraging the characteristics of Chebyshev polynomials, the approximate sub-problem is changed into an equivalent nonlinear optimization problem with linear equality constraints. Consequently, any feasible point of the approximate sub-problem will satisfy the constraints and dynamic system throughout the entire time scale. To validate the efficacy of the new method, we solve three examples and assess the accuracy of the method through the computation of its approximation error. Numerical results obtained show that our approach achieves lower approximation error when compared to the Chebyshev pseudo-spectral method. The proposed method is particularly suitable for scenarios that require high-precision approximation, such as aerospace and precision instrument production.

On the handling of inconsistent systems of max-min fuzzy relational equations

In this article, we study the inconsistency of systems of max-min fuzzy relational equations. We first compute by an explicit analytical formula the Chebyshev distance (defined by the L-infinity norm) between the second member of an inconsistent system and the set of second members of consistent systems defined with the same matrix: that of the inconsistent system. Then we study the set of Chebyshev approximations of the second member of the inconsistent system, which are the second members of the consistent systems defined with the same matrix as the inconsistent system, whose distance to the second member of the inconsistent system is equal to the computed Chebyshev distance. This set is non-empty and associated to the approximate solutions set of the inconsistent system in the following sense: an approximate solution is a solution vector of a consistent system defined with the same matrix as the inconsistent system, whose second member is a Chebyshev approximation of the second member of the inconsistent system. As main results, we describe both the structure of the approximate solutions set and that of the set of the Chebyshev approximations of the second member of the inconsistent system. As an application of our results, we show that our work can help to find approximate values for the rule parameters of a possibilistic rule-based system.

LiFi-enabled UAV swarm networks.

Unmanned aerial vehicle (UAV) swarm communication is a powerful component of aerial relays; however, conventional radio frequency (RF)-based UAV swarm networks struggle to ensure timely and reliable communication. To this end, a light spectrum-based wireless system, LiFi, is presented to supplement in this work thanks to its distinctive benefits. We present the analytical derivation of the average block error probability (ABEP) as Chebyshev approximation, lower and upper bounds. Then, the key performance metrics of reliability, throughput, and latency expressions are provided as a function of the ABEP. The results show that the severe requirements of ultra-reliable (99.99%) and low-latency (sub-millisecond) communication (URLLC) are satisfied at even low signal-to-noise ratio (SNR) values. Additionally, in the numerical results, the impact of the blocklength, packet size, different distances among UAVs, SNR value, and light-emitting diode (LED) semi-angle are explored.

A Hybrid EI-EDM-PO Method Integrated with Two-Dimensional Chebyshev Approximation for Efficiency RCS Analysis

A Hybrid EI-EDM-PO Method Integrated with Two-Dimensional Chebyshev Approximation for Efficiency RCS Analysis

Об интегральном подходе при использовании метода коллокации

The article describes a matrix method of polynomial Chebyshev approximation using an integral approach to construct a solution to a nonhomogeneous fourth-order differential equation with mixed boundary conditions of the first kind. The proposed method is based on the expansion of the fourth-order derivative of the desired function into a series in terms of Chebyshev polynomials of the first kind and the representation of the partial sum of this series as a product of matrices whose elements are, respectively, the Chebyshev polynomials and the coefficients in this expansion. In this paper, using analytical formulas for calculating integrals of Chebyshev polynomials, we obtain a representation of the desired function in terms of the product of the matrices defined above. The use of points of extrema and zeros of Chebyshev polynomials of the first kind as nodes, as well as the properties of the sums of products of Chebyshev polynomials at these points, made it possible to reduce the boundary value problem by the collocation method to a system of inhomogeneous linear algebraic equations with a sparse matrix of this system. It is shown that the solution constructed in this way satisfies the differential equation at all nodes, including the boundary ones, in contrast to the approximate solution obtained by approximating the exact solution in the form of a finite sum of the Chebyshev series. The effectiveness of the proposed method is demonstrated by considering a boundary value problem with a known analytical solution. The convergence analysis of the constructed solution is carried out.

Optimizing the technological modes of lamination of digital prints

The intensive use of lamination technology for the decoration of printing prints made by various printing techniques, as well as a wide range of films, require solving an urgent problem – optimization of influencing factors to ensure the quality of laminated products. The objects of research were prints of inkjet and laser printing on paper, photo paper, and cardboard. A glossy film based on bio-oriented polypropylene BOPP was used for lamination. Studies of densitometric indicators of prints after lamination confirmed an increase in optical density by 0.28 units. This is especially true for inkjet prints. However, after lamination, all digital prints showed a tendency to improve colorimetric characteristics, namely an increase in the brightness of CMYK colors. The relationship between the technological regimes of lamination (speed and temperature) and the strength of the laminates, estimated by breaking and pressing forces, was established. To describe the dependence of the laminate strength on the temperature and speed of lamination, polynomial models were built that reproduce the values of the breaking and pushing forces of the laminate with satisfactory accuracy. The selected algorithm for calculating the parameter values of these models is based on the application of the Chebyshev approximation of the functions of many variables. The optimization models built will allow, if necessary, for making corrections in the technological process of laminating printing prints to ensure the predicted required quality and strength of laminates.

Galerkin–Chebyshev approximation of Gaussian random fields on compact Riemannian manifolds

A new numerical approximation method for a class of Gaussian random fields on compact connected oriented Riemannian manifolds is introduced. This class of random fields is characterized by the Laplace–Beltrami operator on the manifold. A Galerkin approximation is combined with a polynomial approximation using Chebyshev series. This so-called Galerkin–Chebyshev approximation scheme yields efficient and generic sampling algorithms for Gaussian random fields on manifolds. Strong and weak orders of convergence for the Galerkin approximation and strong convergence orders for the Galerkin–Chebyshev approximation are shown and confirmed through numerical experiments.

Various-Order Low-Pass Filter with the Electronic Change of Its Approximation.

The design of a low-pass-frequency filter with the electronic change of the approximation characteristics of resulting responses is presented. The filter also offers the reconnection-less reconfiguration of the order (1st-, 2nd-, 3rd- and 4th-order functions are available). Furthermore, the filter offers the electronic control of the cut-off frequency of the output response. The feature of the electronic change in the approximation characteristics is investigated for the Butterworth, Bessel, Elliptic, Chebyshev and Inverse Chebyshev approximations. The design is verified by PSpice simulations and experimental measurements. The results are also supported by the transient domain response (response to the square waveform), comparison of the group delay, sensitivity analysis and implementation feasibility based on given approximation. The benefit of the proposed electronic change in the approximation characteristics feature (in general signal processing or for sensors in particular) is presented and discussed for an exemplary scenario.

A Parallel Algorithm for Computing Partial Spectral Factorizations of Matrix Pencils via Chebyshev Approximation

A Parallel Algorithm for Computing Partial Spectral Factorizations of Matrix Pencils via Chebyshev Approximation

Algebraic Solution of Tropical Best Approximation Problems

We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions.