Optimal Chebyshev Research Articles

Optimal approximation of the 1/ x function using Chebyshev polynomials and magic constants

In this paper we analyze low-cost accurate approximation of the function \(1/x\) using Chebyshev polynomials of the first kind and minimizing number of elementary operations in computer codes (in particular, by using the so called magic constants). It is shown that Newton-Raphson iterative method is not optimal and a new approach is proposed. We prove that optimal Chebyshev polynomials can be factorized in terms of Chebyshev polynomials of lower order which leads to new -optimal- iteration schemes. We also construct a family of new algorithms by dividing the considered interval into sub-intervals where different magic constants and multiplicative factors are used (in order to increase the accuracy). Theoretical considerations and proofs are completed with numerical tests on three types of microcontroller processors.

Performance analysis of optimum Chebyshev filters at microwave frequencies

The paper suggests that optimum Chebyshev filters present an excellent choice for microwave filter design, as this approximation offers minimum return loss in the half-power passband, provides a maximum cutoff slope, and maintains low sensitivity in the passband. A comparative analysis is conducted with the chained-function filters to demonstrate the efficacy of this approximation, since chained-function filters also utilize Chebyshev polynomials to shape the filter’s characteristic function. The optimum Chebyshev filter, along with two chained-function filters, each of the sixth-degree, are realized, simulated, and compared as lowpass stepped impedance filters with a half-power passband of 1 GHz. After a detailed comparison, it is concluded that the optimum Chebyshev filter outperforms, or at least matches, the performance of the chained-function filters. This suggests that the optimum Chebyshev filter can effectively replace chained-function filters in all applications, offering a superior solution.

A parametric family of spline-based finite impulse response filters and a method for finding the optimal parameter

A new parametric family of filters with finite impulse response is proposed based on convolutions of splines with a rectangular impulse. An algorithm for searching for a parameter that optimizes the deviation of the amplitude-frequency characteristic in passbands and suppression bands. A numerical experiment was carried out, which involved comparing research of new filters with window filters and optimal Chebyshev filters.

Optimum Chebyshev filter with an equalised group delay response

ABSTRACT The first part of the paper discusses designing of the lowpass filter that gives both a constant group delay over the passband and a sharp cut-off slope, which is one of the most difficult challenges in continuous-time signal processing. Furthermore, equalisation of the group delay distortion of the Chebyshev filter has been performed through a cascade connection of the allpass filters. The Newton–Raphson method is applied for solving the system of nonlinear equations that calculate the allpass filter parameters for equalising the constant group delay in an equiripple sense. This procedure gives nearly symmetric impulse response of the resulting filtering system, as well as a longer duration. In the second part of the paper, two solutions for synthesising polynomial filter transfer functions are discussed: filter with monotonic amplitude characteristic (LSM) and filter using Chebyshev polynomial. For the purpose of comparison, the passband ripple factor of the Chebyshev filter is necessary to be adjusted so that both filters have the same stopband insertion loss, group delay and transient response. It is shown that Chebyshev filter is preferable since the passband magnitude distortions and sensitivity to the element deviation tolerance are significantly superior.

Optimum allpole filters with Chebyshev passband magnitude response

A new kind of the allpole lowpass filter functions with Chebyshev behaviour in the passband is derived by minimizing the passband insertion loss. The ripple parameter ε, commonly used to control the attenuation at the cutoff frequency, in proposed approximation is used as a degree of freedom by which the passband attenuation can be controlled, while keeping the attenuation at the cutoff frequency equal to 3 dB. The approximation method relies on solving the nonlinear equation such that the area under the filter characteristic function in passband is minimized. Obtained filters, called Optimum Chebyshev filters, exhibit minimum insertion loss and maximum cutoff slope. Filter polynomial coefficients and the elements values for the LC ladder implementation are tabulated for filter degrees from 3 to 11.Comparison of the Optimum Chebyshev and Least Square Monotonic filters is also discussed in the paper, since both filters are developed subject to the minimum reflected power in the passband; the first with the equiripple and the second with the staircase (monotonic) behavior in the passband. It is also revealed that the Optimum Chebyshev approximation can be considered as the optimal solution among the class of filters whose passband response is bound to be a staircase.

Weyl Spreading Sequence Optimizing CDMA

This paper shows an optimal spreading sequence in the Weyl sequence class, which is similar to the set of the Oppermann sequences for asynchronous CDMA systems. Sequences in Weyl sequence class have the desired property that the order of cross-correlation is low. Therefore, sequences in the Weyl sequence class are expected to minimize the inter-symbol interference. We evaluate the upper bound of cross-correlation and odd cross-correlation of spreading sequences in the Weyl sequence class and construct the optimization problem: minimize the upper bound of the absolute values of cross-correlation and odd cross-correlation. Since our optimization problem is convex, we can derive the optimal spreading sequences as the global solution of the problem. We show their signal to interference plus noise ratio (SINR) in a special case. From this result, we propose how the initial elements are assigned, that is, how spreading sequences are assigned to each users. In an asynchronous CDMA system, we also numerically compare our spreading sequences with other ones, the Gold codes, the Oppermann sequences, the optimal Chebyshev spreading sequences and the SP sequences in Bit Error Rate. Our spreading sequence, which yields the global solution, has the highest performance among the other spreading sequences tested.

Algorithms for constructing optimal n-networks in metric spaces

We study optimal approximations of sets in various metric spaces with sets of balls of equal radius. We consider an Euclidean plane, a sphere, and a plane with a special non-uniform metric. The main component in our constructions of coverings are optimal Chebyshev n-networks and their generalizations. We propose algorithms for constructing optimal coverings based on partitioning a given set into subsets and finding their Chebyshev centers in the Euclidean metric and their counterparts in non-Euclidean ones. Our results have both theoretical and practical value and can be used to solve problems arising in security, communication, and infrastructural logistics.

Optimal Gegenbauer quadrature over arbitrary integration nodes

This paper treats definite integrations numerically using Gegenbauer quadratures. The novel numerical scheme introduces the idea of exploiting the strengths of the Chebyshev, Legendre, and Gegenbauer polynomials through a unified approach, and using a unique numerical quadrature. In particular, the developed numerical scheme employs the Gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms. For a large-scale number of expansion terms, the numerical quadrature has the advantage of converging to the optimal Chebyshev and Legendre quadratures in the L∞-norm and L2-norm, respectively. The key idea is to construct the Gegenbauer quadrature through discretizations at some optimal sets of points of the Gegenbauer–Gauss (GG) type in a certain optimality sense. We show that the Gegenbauer polynomial expansions can produce higher-order approximations to the definite integrals ∫−1xif(x)dx of a smooth function f(x)∈C∞[−1,1] for the small range by minimizing the quadrature error at each integration point xi through a pointwise approach. The developed Gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes. Exact integrations are obtained for polynomials of any arbitrary degree n if the number of columns in the developed Gegenbauer integration matrix (GIM) is greater than or equal to n. The error formula for the Gegenbauer quadrature is derived. Moreover, a study on the error bounds and the convergence rate shows that the optimal Gegenbauer quadrature exhibits very rapid convergence rates, faster than any finite power of the number of Gegenbauer expansion terms. Two efficient computational algorithms are presented for optimally constructing the Gegenbauer quadrature. We illustrate the high-order approximations of the optimal Gegenbauer quadrature through extensive numerical experiments, including comparisons with conventional Chebyshev, Legendre, and Gegenbauer polynomial expansion methods. The present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods.

Iterative WLS Design of SAW Bandpass Filters

It has been demonstrated by several authors that the well-known weighted least squares (WLS) approximation can be equiripple if a suitable weighting function is applied. In the present paper, the WLS algorithm is generalized to SAW filter synthesis with prescribed magnitude and phase specifications. Several weighting techniques producing quasi-equiripple designs are presented. The frequency sampling technique is used for SAW filter frequency response approximation to reduce the number of optimized variables. The WLS algorithm rapidly converges both for linear and nonlinear phase SAW filters. Typically, no more than 5-10 iterations are required to obtain the WLS solution to accuracy better than 0.5-1 dB in the stopband when compared with the optimum Chebyshev approximation. Moreover, it is shown that the WLS technique can be effectively applied for second-order effect compensation.

A condition for the superiority of the (2, 2)-step methods over the related Chebyshev method

The (2, 2)-step iterative methods related to an optimal Chebyshev method for solving a real and nonsymmetric linear system A x = b are studied. A condition under which the asymptotic rate of convergence of the optimal Chebyshev method can be improved by a related (2, 2)-step method is derived. The condition depends not only on the location of the extreme eigenvalues of T but also on whether the ratio of the minor axis to the major axis of the optimal ellipse is greater than the golden ratio. Two numerical examples are given to illustrate our results.

A personal history of the Parks-McClellan algorithm

This article describes the work that led to what is now known as the Parks-McClellan algorithm. Within the bigger picture of filter design methods, this paper recount events that had an impact on the inspiration to develop the Parks-McClellan algorithm, i.e., the Remez exchange algorithm with optimal Chebyshev approximation for FIR filter design.

The superiority of a new type (2,2)-step iterative method over the related Chebyshev method

A new type (2,2)-step iterative method related to an optimal Chebyshev method is developed for solving real and nonsymmetric linear systems of the form Ax= b. It is an extension of the (2,2)-step iterative method introduced in [Numer. Linear Algebra Appl. 7 (2000) 169]. The superiority of the new type (2,2)-step iterative method over the optimal Chebyshev method is derived in the case where the known (2,2)-step iterative method may not improve the asymptotic rate of convergence. Two numerical examples are given to illustrate the results.

Bessel planar arrays

In this paper, we introduce a new class of planar arrays that we call the Bessel planar arrays. A formula for the current distribution in the elements of these arrays is presented, which is related to Bessel functions. For the Bessel planar arrays, the maximal sidelobe level is controllable, the directivity is very high, and the half‐power beam width is slightly larger compared to the optimal Chebyshev planar arrays. Methods to set the maximal sidelobe level and compute the directivity and the half‐power beam width are described, and numerical examples are given to illustrate the features of the proposed arrays.

Optimal Chebyshev Preemphasis and Filtering

A solution is presented to the problem of designing filters which minimize the error in transmitting a signal along an analog channel with additive interference. A uniform Chebyshev criterion is used in accordance with which the maximum value of the error is minimized without changing the energy of the transformed signal. An estimate is made of the potential interference immunity of Chebyshev filtering, a condition is derived for its effectiveness, and an optimal Chebyshev filter is synthesized. The results are given of a computer modeling of the processing of the signal and interference in accordance with the procedure proposed by the authors.

Is a Chebyshev method optimal for an elliptic region also optimal for a nearly elliptic region?

The asymptotic rate of convergence of an optimal Chebyshev semiiterative method for solving a real and nonsymmetric linear system x =T x + c can be improved by the related (2,2)-step iterative methods under certain conditions. The condition for which a Chebyshev method asymptotically optimal for an elliptic region is also asymptotically optimal for a nearly elliptic region is presented. Thus a (2,2)-step method is asymptotically superior to the Chebyshev method asymtotically optimal for a nearly elliptic region under certain conditions. A numerical example illustrates our results.

Comparison between the convergence rates of the Chebyshev method and the related (2,2)-step methods

An optimal Chebyshev method for solving Ax = b, where all the eigenvalues of the real and non-symmetric matrix A are located in the open right half plane, is dependent on an optimal ellips∂Ω* such that the spectrum of A is contrained in Ω*, the closed interior of the ellipse. The relationship between the convergence rates of the Chebyshev method and the closely related (2,2)-step iterative methods are studied. (2,2)-step iterative methods are faster than an optimal Chebyshev method under certain conditions. A numerical example illustrates such an improvement of a (2,2)-step iterative method. Copyright © 2000 John Wiley & Sons, Ltd.

Chebyshev digital FIR filter design

A new multiple-exchange ascent algorithm is presented for designing optimal Chebyshev digital FIR filters with arbitrary magnitude and phase specifications. Compared to existing Chebyshev design techniques, the new design algorithm exhibits faster convergence while maintaining high accuracy, and is guaranteed to converge to the optimal solution. In addition, the proposed algorithm exactly reduces to the classic second Remez (Parks–McClellan) algorithm when real-only or imaginary-only filters are designed and is, therefore, a generalization of the classic Remez algorithm to the complex case. The described algorithm has been incorporated as part of the MATLAB signal processing toolbox. Design examples are presented to illustrate the performance of the proposed algorithm.

Bark and ERB bilinear transforms

Use of a bilinear conformal map to achieve a frequency warping nearly identical to that of the Bark frequency scale is described. Because the map takes the unit circle to itself, its form is that of the transfer function of a first-order allpass filter. Since it is a first-order map, it preserves the model order of rational systems, making it a valuable frequency warping technique for use in audio filter design. A closed-form weighted-equation-error method is derived that computes the optimal mapping coefficient as a function of sampling rate, and the solution is shown to be generally indistinguishable from the optimal least-squares solution. The optimal Chebyshev mapping is also found to be essentially identical to the optimal least-squares solution. The expression 0.8517[arctan(0.06583fs)]/sup 1/2/-0.916 is shown to accurately approximate the optimal allpass coefficient as a function of sampling rate f/sub s/ in kHz for sampling rates greater than 1 kHz. A filter design example is included that illustrates improvements due to carrying out the design over a Bark scale. Corresponding results are also given and compared for approximating the related "equivalent rectangular bandwidth (ERB) scale" of Moore and Glasberg (ACTA Acustica, vo.82, p.335-45, 1996) using a first-order allpass transformation. Due to the higher frequency resolution called for by the ERB scale, particularly at low frequencies, the first-order conformal map is less able to follow the desired mapping, and the error is two to three times greater than the Bark-scale case, depending on the sampling rate.

On the optimal spectral Chebyshev solution of a controlled nonlinear dynamical system

In a recent paper we considered the numerical solution of the controlled Duffing oscillator: minimize J = 1/2∫ -T 0 U 2 (τ)dτ, subject to X(τ) + w 2 X(τ) + ∈X 2 (τ) = U(τ) (- T ≤ τ ≤ 0), where T is known, with X(- T) = x 0 , X(0) = 0, by the pseudospectral Legendre method, which shows that in order to maintain spectral accuracy the grids on which a physical problem is to be solved must also be obtained by spectrally accurate techniques. This paper presents an alternative spectrally accurate computational method of solving the nonlinear controlled Duffing oscillator. The method is based upon constructing the Mth-degree interpolation polynomials, using Chebyshev nodes, to approximate the state and the control vectors. The differential and integral expressions which arise from the system dynamics and the performance index are converted into an algebraic nonlinear programming problem. The results of computer-simulation studies compare favourably with optimal solutions obtained by closed-form analysis and/or by other numerical schemes.

Complex Chebyshev approximation for FIR filter design

The alternation theorem is at the core of efficient real Chebyshev approximation algorithms. In this paper, the alternation theorem is extended from the real-only to the complex case. The complex FIR filter design problem is reformulated so that it clearly satisfies the Haar condition of Chebyshev approximation. An efficient exchange algorithm is derived for designing complex FIR filters in the Chebyshev sense. By transforming the complex error function, the Remez exchange algorithm can be used to compute the optimal complex Chebyshev approximation. The algorithm converges to the optimal solution whenever the complex Chebyshev error alternates; in all other cases, the algorithm converges to the optimal Chebyshev approximation over a subset of the desired bands. The new algorithm is a generalization of the Parks-McClellan algorithm, so that arbitrary magnitude and phase responses can be approximated. Both causal and noncausal filters with complex or real-valued impulse responses can be designed. Numerical examples are presented to illustrate the performance of the proposed algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>