Complex Chebyshev Approximation Research Articles

Interpolatory [formula omitted] model reduction

We introduce an approach to H∞ model reduction that is founded on ideas originating in realization theory, interpolatory H2-optimal model reduction, and complex Chebyshev approximation. Within this new framework, we are able to formulate a method that remains effective in large-scale settings with the main cost dominated by sparse linear solves. By employing Loewner “data-driven” partial realizations within each optimization cycle, computationally demanding H∞ norm calculations can be completely avoided. Several numerical examples illustrate that our approach will produce high fidelity reduced models consistently exhibiting better H∞ performance than those produced by balanced truncation; these models often are as good as (and occasionally better than) those models produced by optimal Hankel norm approximation. In all cases, reduced models are produced at far lower cost than is possible either with balanced truncation or with optimal Hankel norm approximation.

Optimal design of IIR filters using linear semi‐indefinite programming method

AbstractAn optimal design method for stable IIR (Infinite Impulse Response) filters under the min‐max criterion is proposed. The design problem considered is the complex Chebyshev approximation of a rational function including the stability constraint. We formulate this problem as a problem of real linear semi‐infinite programming problem using the real rotation theorem. The problem is solved by the three‐phase method; one of the methods is used for solving semi‐infinite programming problems. The three‐phase method includes three operations. In the first operation, some active constraint candidates are selected by the iterative simplex method. Next, the second operation integrates some degenerate constraints. In the third operation, the approximate solution obtained up to the second operation is adjusted so as to satisfy the optimality condition. The filters designed by the method are found to be more precise than those designed by the conventional method. Several design examples are presented to demonstrate the effectiveness of the proposed method. © 2011 Wiley Periodicals, Inc. Electron Comm Jpn, 94(6): 17–23, 2011; Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/ecj.10330

Optimal Design of IIR Filters Using Linear Semi-Infinite Programming Method

In this paper, an optimal design method for stable IIR(Infinite Impulse Response) filters in a criterion of min-max sense is proposed. The design problem is considered one of the complex Chebyshev approximation for rational function including the stability constraint, we formulated such the problem as a real linear semi-infinite programming using the real rotation theorem. Then, the problem is solved by the three phase method that is one of the methods solving semi-infinite programming problem. The three phase method is composed of three operations. In the first operation, some candidates of active constraints are selected by the iterative simplex method. Next, the second operation integrates some degenerate constraints. In the third operation, the approximation solution obtained until second operation is adjusted so as to satisfy the optimality condition. As a result, the filters designed by the method are more precise than one designed by conventional method. Several design examples are shown to present effectiveness of the proposed method.

Algorithm for Spatial Straightness Evaluation Using Theories of Linear Complex Chebyshev Approximation and Semi-infinite Linear Programming

Abstract This paper presents a novel methodology for evaluating spatial straightness error based on the minimum zone criterion. Spatial straightness evaluation is formulated as a linear complex Chebyshev approximation problem, and then reformulated as a semi-infinite linear programming problem. Both models for the primal and dual programs are developed. An efficient simplex-based algorithm is employed to solve the dual linear program to yield the straightness value. Also a general algebraic criterion for checking the optimality of the solution is proposed. Numerical experiments are given to verify the effectiveness and efficiency of the presented algorithm.

Complex Chebyshev approximation for FIR filters using linear semi‐infinite programming method

AbstractIn the optimum design of an FIR filter by the complex Chebyshev method, it is difficult to limit the increase of computational effort and to guarantee convergence to an optimum solution due to the nonlinearity of the problem and the instability of the frequency points for the optima. In this paper, the complex Chebyshev approximation problem is formulated as a linear semi‐infinite programming problem by means of the real rotation theorem. An optimum design method is proposed using the three‐phase method, a linear semi‐infinite programming method. In the present method, the design problem is first reduced to a linear programming problem with constraints and a provisional solution is derived by the simplex method. After the constraints corresponding to the degenerated optimum frequencies contained in the provisional solution are combined, the solutions are adjusted to satisfy the optimum condition so as to obtain an optimum solution. Since low accuracy is sufficient for the provisional solution, memory requirements and computational effort can be reduced. By means of a design example, it is shown that the proposed method is superior in terms of solution accuracy and computational effort to the technique based on the simplex method. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 87(3): 49–57, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10104

Complex Chebyshev approximation for IIR digital filters based on eigenvalue problem

This paper presents an efficient method for designing complex infinite impulse response digital filters in the complex Chebyshev sense. The proposed method is based on the formulation of a generalized eigenvalue problem by using the Remez multiple exchange algorithm. Hence, the filter coefficients can be easily obtained by solving the eigenvalue problem to find the absolute minimum eigenvalue, and then the complex Chebyshev approximation is attained through a few iterations starting from a given initial guess. The proposed algorithm is computationally efficient because it not only retains the speed inherent in the Remez exchange algorithm but also simplifies the interpolation step. Some design examples are presented and compared with the conventional methods. It is shown that the results obtained by using the method proposed in this paper are better than those obtained by the conventional methods.

Equiripple design in complex domain for FIR digital filters by transforming desired response

In this paper, a new design method is proposed for complex equiripple approximation of an FIR digital filter. In this method, the desired characteristics are varied and the least-squares method is applied iteratively until convergence to an equiripple solution is reached. Although the obtained solution is quasi-optimal, the necessary computational complexity is extremely small in each iteration and the computation error can be kept small. Therefore, the method has the advantage that the design of higher-order filters, which is difficult in the conventional methods, can now be executed at high speed. By computer simulation, the method is compared with the conventional complex Chebyshev approximation method and the effectiveness of the present method is demonstrated. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 84(2): 30–37, 2001

Least-squares approach to IIR digital filters and an application to equiripple solutions

This paper formulates the modified least-squares method presented by Mullis in a general form so that weighting can be applied. Then, all algorithm is proposed with which a least-squares approximation can be derived for an IIR digital filter, by solving iteratively a system of equations. The method is extended to a complex Chebyshev approximation that minimizes the maximum magnitude of the complex error, by multiplying the weight, which is newly derived in this paper, with the modified least-squares error. The method is compared with a conventional complex Chebyshev approximation by means of a computer simulation and the effectiveness of the proposed method is demonstrated. © 1998 Scripta Technica. Electron Comm Jpn Pt 3, 81(7): 28–36, 1998

THE SIMULTANEOUS APPROXIMATION OF MAGNITUDE AND PHASE BY FIR DIGITAL FILTERS. I: A NEW APPROACH

The non-linear problem of simultaneous approximation of the magnitude and phase by an FIR digital filter is studied. Usually this problem is solved indirectly by solution of the complex Chebyshev approximation problem of the frequency response of the filter, which can be described as a linear or non-linear convex semi-infinite programming problem. In the present first part of the paper the relationship between the magnitude/phase problem and the Chebyshev problem is investigated and an alternative approximate representation in the form of a convex semi-infinite programming problem is derived. The latter problem can be solved by the same methods for such problems which were used previously by the authors for the solution of (constrained) complex Chebyshev approximation problems. In part II of the paper some earlier approaches to FIR filter design are reconsidered and discussed with regard to the magnitude/phase problem. It is further demonstrated there by several examples of filter designs that the new approach of this paper leads to very good results.

A Multiple-Exchange Algorithm for Complex Chebyshev Approximation by Polynomials on the Unit Circle

This paper presents a generalization of the Remez multiple-exchange (ME) algorithm for solving complex Chebyshev approximation by polynomials on the unit circle. The difficulties of implementing the two fundamental steps of the Remez algorithm in the complex case are pointed out and methods for overcoming these difficulties are proposed. It is shown that generalization of the first step, which involves solving a complex Chebyshev approximation subproblem over a set of guessed extremal points, can be implemented by solving a maximization problem with simple bound constraints using Newton’s method. Furthermore, generalization of the second step, which involves finding a new set of guessed extremal points, can be realized by use of an effective exchange rule derived from the concept of extremal signatures. Numerical results suggest that the proposed ME algorithm has fast convergence rate and good numerical properties even for large-scale problems.

Complex chebyshev approximation for infinite impulse response digital filters

AbstractIn this paper, a new Chebyshev approximation (amplitude and phase simulataneous approximation) method for IIR digital filters is proposed. the proposed algorithm is based on the Remez algorithm. By solving the eigenvalue problem at each iteration, the frequency characteristic of the filter is interpolated so that the error function in the complex domain is made with equal ripples. Since the previously proposed amplitude and phase simulataneous approximation methods for the IIR digital filters are based on the nonlinear equation, they all require substantial computation. On the other hand, the present method is extremely simple and the amount of compuation can be reduced singnificantly. Nevertheless, the performance equal to or more than the conventional one can be obtained. Further, the present method can be applied not only for the real coefficient filters but also for the complex coefficient filtes. It is possible to carry out characteristic approximation limited to the positive frequency range. Hence, a further reduction of the filter order needed for signal processing can be attempted.

The design of FIR filters in the complex plane by convex optimization

The design of FIR filters in the complex domain is performed by complex Chebyshev approximation where the continuous complex approximation problem is considered as a convex semi-infinite programming problem. This approach permits the filter design under (in)finitely many additional convex constraints on the system function of the filter. For the solution of the semi-infinite programming problem a new method is presented which can be interpreted as a further development of the well-known Kelley-Cheney-Goldstein cutting plane method for finite convex programming. This method is simpler and as reliable as the authors' method in [30, 31], the only other method until now which likewise has proved convergence and can solve continuous design problems with constraints. Filters designed by the method are presented, in particular one with 1000 coefficients. For a number of test examples the method is compared with that in [30, 31].

An efficient implementation of Lawson's algorithm with application to complex Chebyshev FIR filter design

This paper presents an efficient implementation of Lawson's algorithm and illustrates its application in complex Chebyshev FIR filter design. It is shown that the update terms in Lawson's algorithm can be efficiently achieved by computing a proper subspace projection, provided that the number of points involved in Lawson's algorithm is sufficiently small (not much greater than n where n is the number of variables). An application of this particular implementation form for Lawson's algorithm is then demonstrated by using it to solve the subproblems involved in a multiple exchange algorithm. In particular, this exchange algorithm is based on generalizing Remez second exchange algorithm to the complex case which requires solving a sequence of subproblems where each subproblem is itself a complex Chebyshev approximation problem defined over a finite number of points; the subproblems are systematically defined using a simple exchange procedure. The effectiveness of this multiple exchange algorithm for designing complex FIR filters is illustrated through various design examples, including a long filter with length n=125. >

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>

Iterative reweighted least-squares design of FIR filters

Develops a new iterative reweighted least squares algorithm for the design of optimal L/sub p/ approximation FIR filters. The algorithm combines a variable p technique with a Newton's method to give excellent robust initial convergence and quadratic final convergence. Details of the convergence properties when applied to the L/sub p/ optimization problem are given. The primary purpose of L/sub p/ approximation for filter design is to allow design with different error criteria in pass and stopband and to design constrained L/sub 2/ approximation filters. The new method can also be applied to the complex Chebyshev approximation problem and to the design of 2D FIR filters. >

FIR filter design with additional constraints using complex Chebyshev approximation

A powerful single exchange algorithm for the solution of the complex Chebyshev approximation problem was introduced by Tang. It was picked up and modified by Alkhairy et al. for the design of digital FIR filters. In this paper we extend this algorithm to solve the approximation problem in conjunction with additional constraints, such as constraints on the filter coefficients, constraints on the magnitude response and / or its derivatives or constraints on the group delay. Since the algorithm deals with a linear optimization problem, all constraints have to be linear with respect to the filter coefficients. We show this linearization and the inclusion into the complex approximation problem and the algorithm as well. A final example will demonstrate results achieved with the modified algorithm.

Design of real FIR filters with arbitrary complex frequency responses by two real Chebyshev approximations

Abstract Since the real coefficients of a FIR filter with arbitrary complex-valued desired frequency responses are neither symmetric nor antisymmetric, the Remez exchange algorithm cannot be applied directly. The problem can be solved by dividing the original complex approximation into two real ones such that the Remez exchange algorithm can be applied by slightly modifying the Parks-McClellan program. This method is much easier than the currently existing methods using linear programming or complex Chebyshev approximation, and the performance is satisfactory. More importantly, the magnitudes of the resultant complex errors are also equiripple as the direct complex Chebyshev approximation designs. Several numerical examples including a low-pass filter, a full-band differentiator, a wide-band Hilbert transformer, and a chirp-delay and sine-delay FIR all-pass phase equalizer are given to show the effectiveness of this approach.

Complex Chebyshev approximation for FIR digital filters

An adaptive approach to FIR (finite impulse response) filter design is applied to a fully general complex specification, showing it to be capable of giving an optimum Chebyshev solution. It is thought to be one of the first practical approaches to achieve this in the case of a complex function. Design examples include a bandpass filter with nearly linear phase and all-pass filters with general shaping phase. Comparisons are drawn with equivalent results from other algorithms. >

Improvements of a complex FIR filter design algorithm

The iterative algorithm for designing FIR filters by complex Chebyshev approximation introduced by Preuss is considered again. This algorithm yields the desired solution and is slightly than another one, presented by Chen and Parks for an approximative method. In this paper we propose a modification, yielding a significant acceleration. Furthermore the handling of the extremal points by Preuss aimed to the use of the Newton interpolation formula is reviewed and a modification is suggested. Specifically, the Newton interpolation formula is substituted by a Gaussian relaxation algorithm. Finally examples are shown illustrating the improvements obtained.

Complex chebyshev approximation for IIR digital filters with arbitrary log magnitude and phase functions

AbstractThis paper proposes a technique for designing IIR digital filters to have prescribed magnitude and phase responses. The technique is based on an iterative weighted least‐squares approach in the frequency domain.A weight updating procedure is introduced to minimize the complex logarithmic error in the Chebyshev sense. The real and imaginary parts of the complex logarithmic error represents the log magnitude and phase errors, respectively. Both the log magnitude and phase errors of the resulting filter become nearly equiripple. Filter coefficients at each iteration are computed efficiently using FFT and a fast recursive algorithm for a set of linear equations derived from the weighted least‐squares problem. Several design examples are shown to demonstrate the effectiveness of the proposed technique.