Chebychev Approximation Research Articles

Microstrip diplexer design based on two square open loop bandpass filters for RFID applications

AbstractIn this work, a simple and efficient approach to design a microstrip diplexer is proposed. It is based on the combination of two compact Square Open Loop Resonator band pass filters. These filters are designed for Radio Frequency Identification application at 2.45 GHz frequency. Using the Chebychev approximation, the two filters and diplexer are designed, simulated, fabricated, and measured. The results show that the diplexer presents a high selectivity, a good insertion loss and high isolation (>40 dB) with the compact size. The diplexer also shows a good performance when associated with the patch antenna operating at 2.45 GHz.

Characterization of best Chebychev approximation using the frequency response of IIR digital filters with convex stability

This paper deals with the application of Chebychevʼs approximation theory to IIR digital filter frequency response (FR) approximation. It explores the properties of the frequency response of IIR digital filters as a nonlinear complex approximating function; IIR digital filter frequency response is used to approximate a prescribed magnitude and phase responses. The approximation problem is closely related to optimization. If the set of approximating functions is non-convex, the optimization problem is difficult and may converge to a local minimum. The main results presented in the paper are proposing a convex stability domain by introducing a condition termed “sign condition” and characterization of the best approximation by the Global Kolmogorovʼs Criterion (GKC). The Global Kolmogorovʼs Criteria is shown to be also a necessary condition for the approximation problem. Finally, it is proved that the best approximation is a global minimum. The sign condition can be incorporated as a constraint in an optimization algorithm.

Local reduction based SQP-type method for semi-infinite programs with an infinite number of second-order cone constraints

The second-order cone program (SOCP) is an optimization problem with second-order cone (SOC) constraints and has achieved notable developments in the last decade. The classical semi-infinite program (SIP) is represented with infinitely many inequality constraints, and has been studied extensively so far. In this paper, we consider the SIP with infinitely many SOC constraints, called the SISOCP for short. Compared with the standard SIP and SOCP, the studies on the SISOCP are scarce, even though it has important applications such as Chebychev approximation for vector-valued functions. For solving the SISOCP, we develop an algorithm that combines a local reduction method with an SQP-type method. In this method, we reduce the SISOCP to an SOCP with finitely many SOC constraints by means of implicit functions and apply an SQP-type method to the latter problem. We study the global and local convergence properties of the proposed algorithm. Finally, we observe the effectiveness of the algorithm through some numerical experiments.

A survey on OR and mathematical methods applied on gene-environment networks

In this paper, we survey the recent advances and mathematical foundations of gene-environment networks. We explain their interdisciplinary implications with special regard to human and life sciences as well as financial sciences. Special attention is paid to applications in Operational Research and environmental protection. Originally developed in the context of modeling and prediction of gene-expression patterns, gene-environment networks have proved to provide a conceptual framework for the modeling of dynamical systems with respect to errors and uncertainty as well as the influence of certain environmental items. Given the noise-prone measurement data we extract nonlinear differential equations to describe and analyze the interactions and regulating effects between the data items of interest and the environmental items. In particular, these equations reflect data uncertainty by the use of interval arithmetics and comprise unknown parameters resulting in a wide variety of the model. For an identification of these parameters Chebychev approximation and generalized semi-infinite optimization are applied. In addition, the time-discrete counterparts of the nonlinear equations are introduced and their parametrical stability is investigated by a combinatorial algorithm which detects the region of parameter stability. Finally, we analyze the topological landscape of the gene-environment networks in terms of structural stability. We conclude with an application of our analysis and introduce the eco-finance networks.

A review on data mining and continuous optimization applications in computational biology and medicine

An emerging research area in computational biology and biotechnology is devoted to mathematical modeling and prediction of gene-expression patterns; it nowadays requests mathematics to deeply understand its foundations. This article surveys data mining and machine learning methods for an analysis of complex systems in computational biology. It mathematically deepens recent advances in modeling and prediction by rigorously introducing the environment and aspects of errors and uncertainty into the genetic context within the framework of matrix and interval arithmetics. Given the data from DNA microarray experiments and environmental measurements, we extract nonlinear ordinary differential equations which contain parameters that are to be determined. This is done by a generalized Chebychev approximation and generalized semi-infinite optimization. Then, time-discretized dynamical systems are studied. By a combinatorial algorithm which constructs and follows polyhedra sequences, the region of parametric stability is detected. In addition, we analyze the topological landscape of gene-environment networks in terms of structural stability. As a second strategy, we will review recent model selection and kernel learning methods for binary classification which can be used to classify microarray data for cancerous cells or for discrimination of other kind of diseases. This review is practically motivated and theoretically elaborated; it is devoted to a contribution to better health care, progress in medicine, a better education, and more healthy living conditions.

On optimization, dynamics and uncertainty: A tutorial for gene-environment networks

An emerging research area in computational biology and biotechnology is devoted to mathematical modeling and prediction of gene-expression patterns; to fully understand its foundations requires a mathematical study. This paper surveys and mathematically expands recent advances in modeling and prediction by rigorously introducing the environment and aspects of errors and uncertainty into the genetic context within the framework of matrix and interval arithmetic. Given the data from DNA microarray experiments and environmental measurements we extract nonlinear ordinary differential equations which contain parameters that are to be determined. This is done by a generalized Chebychev approximation and generalized semi-infinite optimization. Then, time-discretized dynamical systems are studied. By a combinatorial algorithm which constructs and follows polyhedra sequences, the region of parametric stability is detected. Finally, we analyze the topological landscape of gene-environment networks in terms of structural stability. This pioneering work is practically motivated and theoretically elaborated; it is directed towards contributing to applications concerning better health care, progress in medicine, a better education and more healthy living conditions.

PSpice for Filters and Transmission Lines

Abstract In this book, PSpice for Filters and Transmission Lines, we examine a range of active and passive filters where each design is simulated using the latest Cadence Orcad V10.5 PSpice capture software. These filters cannot match the very high order digital signal processing (DSP) filters considered in PSpice for Digital Signal Processing, but nevertheless these filters have many uses. The active filters considered were designed using Butterworth and Chebychev approximation loss functions rather than using the ‘cookbook approach’ so that the final design will meet a given specification in an exacting manner. Switched-capacitor filter circuits are examined and here we see how useful PSpice/Probe is in demonstrating how these filters, filter, as it were. Two-port networks are discussed as an introduction to transmission lines and, using a series of problems, we demonstrate quarter-wave and single-stub matching. The concept of time domain reflectrometry as a fault location tool on transmission lines is ...

Nonlinear Approximation with Dictionaries I. Direct Estimates

We study various approximation classes associated with m-term approximation by elements from a (possibly redundant) dictionary in a Banach space. The standard approximation class associated with the best m-term approximation is compared to new classes defined by considering m-term approximation with algorithmic constraints: thresholding and Chebychev approximation classes are studied, respectively. We consider embeddings of the Jackson type (direct estimates) of sparsity spaces into the mentioned approximation classes. General direct estimates are based on the geometry of the Banach space, and we prove that assuming a certain structure of the dictionary is sufficient and (almost) necessary to obtain stronger results. We give examples of classical dictionaries in Lp spaces and modulation spaces where our results recover some known Jackson type estimates, and discuss some new estimates they provide.

Acoustic infinite elements for non-separable geometries

We present new infinite element formulations for solving acoustic scattering and radiation problems in the exterior of long, slender bodies. The new infinite elements are geometrically constructed from a prolate spheroid inscribed by the scatterer. These elements need not begin on a level surface of the prolate spheroidal coordinate system. Instead, they may be attached to any convex surface, including that of the scatterer itself. This scheme reduces, or even completely eliminates, finite element modeling of the exterior medium. The formulations may easily be extended to the cases of an interior oblate spheroid or ellipsoid. We present both conjugated and unconjugated formulations without any weighting factors, although it would be simple to include them. We describe a fast numerical scheme for computing the element integrals based on Chebychev approximation. We include numerical results for scattering from spheres and capped cylinders. These results demonstrate the accuracy and the dramatic reduction in computational expense of our new formulations compared to other coupled finite element/infinite element methods.

Effective Bounds on Strong Unicity in L1-Approximation

<p>In this paper we present another case study in the general project of Proof Mining which means the logical analysis of prima facie non-effective proofs with the aim of extracting new computationally relevant data. We use techniques based on monotone functional interpretation (developed in [17]) to analyze Cheney's simplification [6] of Jackson's original proof [9] from 1921 of the uniqueness of the best L1-approximation of continuous functions f in C[0, 1] by polynomials p in Pn of degree <= n. Cheney's proof is non-effective in the sense that it is based on classical logic and on the non-computational principle WKL (binary K¨onig lemma). The result of our analysis provides the first effective (in all parameters f, n and epsilon) uniform modulus of uniqueness (a concept which<br />generalizes `strong uniqueness' studied extensively in approximation theory). Moreover, the extracted modulus has the optimal epsilon-dependence as follows from Kroo [20]. The paper also describes how the uniform modulus of uniqueness can be used to compute the best L1-approximations of a fixed f in C[0, 1] with arbitrary precision, and includes some remarks on the case of best Chebychev approximation.</p>

Recursive Halfband-Filters

The paper describes the properties and the design of recursive halfband-filters.The two possibilities of being complementary are introduced. The lowpass with the transfer function HLp(z)and the corresponding highpass, described by HHp(z) = HLp(-z)can either be strictly complementary or power complementary. According to the respective symmetry, the impulse responses, transfer functions and frequency responses possess certain characteristic properties, which are described in section 2. It turns out that these resulting symmetries of the frequency response reduce the number of the choosable design parameters. We can only prescribe the cutoff frequency and the tolerated deviation either for the passband or the stopband. In the third section we treat the design of halfband-filters with approximately linear phase. By coupling an appropriately designed allpass of even degree nA with a delay of order m=nA±1 we obtain the desired solution by solving a corresponding approximation problem for the phase of the allpass. The resulting lowpass and highpass are strictly as well as power complementary!The kind of approximation will be done in the sense of maximal flatness, where a closed form solution exists [8], or in the sense of Chebychev, where the solution is obtained iteratively [13]. The design of systems with minimum phase is presented in section 4. The resulting lowpass and highpass are power complementary. Closed form solutions yield Butterworthand Cauer filters, if a maximal flat or a Chebychev approximation is desired. In all cases a fixed relation exists between the passband frequency ΩP and the tolerated deviation δP in the passband when the degree n has been chosen.

Frequency response identification of the dynamics of a Tokamak plasma

We describe the application of an H/sup /spl infin// system identification procedure to a Tokamak. The work is motivated by the need to create linear models which are suitable for controller design and which may be used to validate different models derived from physics principles. The paper develops an H/sup /spl infin// system identification algorithm and demonstrates its successful application to the Tokamak configuration variable (TCV). Each of the required steps is detailed, from the design of identification experiments to the creation of low-order models from a combination of Hankel model reduction and Chebychev approximation. The method described is a worst-case identification technique, in that it aims to minimize the H/sup /spl infin// error between the identified model and the true plant. Such a model is particularly well suited for robust controller design. The identified model of TCV is compared with various physics-based models.

Reliable Location and Regression Estimates with Application to Range Image Segmentation

Range images provide important sources of information in many three-dimensional robot vision problems such as navigation and object recognition. Many physical factors, however, introduce noise to the discrete measurements in range images, identifying the need to reassess the error distribution in samples taken from real range images. This paper suggests the use of the L norms to yield reliable estimates of location and regression coefficients. This particular approach is compared against two commonly used approaches: Equally Weighted Least Squares, which minimizes the L_2 norms and the Chebychev approximation, which minimizes the L_1 norm. The problem is a weighted least squares case where the weights are derived from the chosen parameter, p, and its ability to yield a variety of location estimates spanning from the sample mean to the sample median. These two estimates have a wide application in image processing that includes noise removal. This paper will show the problems associated with these two techniques, and suggest experimental solutions to minimize them. A specific operating range is determined in which the L norms perform well and a regression module is used in conjunction with a region-growing segmentation algorithm to provide a reliable partition of range images.

Chebychev approximation methods for evaluating conicity

This paper addresses the problem of determining the right circular cone that best fits a finite set of data points when the vertex of the cone has been specified. The problem of determining the best-fit cone when only the axis is specified is also discussed. In both problems, we have determined the best-fit (substitute) cone by minimizing the maximum normal deviation of the data points from the substitute cone. In other words, we have the Chebychev or the minimum zone solution. Finally, we develop a method to fit a cone, without a specified vertex point, to a set of data points. This is achieved with a simplex search of the vertex space to converge to the optimal vertex location where, at each step of the search, a vertex location is chosen, and the envelope width associated with the Chebychev cone for that vertex location is determined.

Design of FIR Hilbert transformers and differentiators in the complex domain

This paper presents a method for the design of FIR Hilbert transformers and differentiators in the complex domain. The method can be used to obtain conjugate-symmetric designs with smaller group delay compared to linear-phase designs. Non-conjugate symmetric Hilbert transformers are also designed. This paper is an extension of our previous work, which presented the algorithm for the design of standard frequency selective filters. The minimax criterion is used and the Chebychev approximation is posed as a linear optimization problem. The primal problem is converted to its dual and is solved using an efficient quadratically convergent algorithm developed by Tang (1988). When a constant group delay is specified, the filter designs have almost linear phase in the passbands. When the specified group delay is half the filter length, the algorithm results in exactly linear-phase designs.

Design of FIR filters with complex desired frequency response using a generalized Remez algorithm

Complex approximation with a generalized Remez algorithm is used to design FIR digital filters with nonconjugate symmetric frequency responses. The minimax criterion is used and the Chebychev approximation is posed as a linear optimization problem. The primal problem is converted to its dual and is solved using an efficient, quadratically convergent algorithm developed by Tang. Optimal Chebychev real-coefficient FIR filters with group delay smaller than half the filter length can be designed with slightly better magnitude responses compared to linear-phase filters. Linear-phase filters can also be designed when the group delay is specified to be half the filter length. Most importantly, the design method is capable of producing filters with complex coefficients that approximate nonconjugate symmetric frequency responses.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Residuation in fuzzy algebra and some applications

The system (〈0, 1 〉, max, min) has a number of well-known applications, many of which involve the inversion of linear relations. We show the advantage of residuation theory for this process and apply it to the analysis of the problem of Chebychev approximation.

A transformation on Chebychev approximation resulting in improved delay characteristics

An analytic technique involving a transformation on the Chebychev transfer function is presented in this paper. This method will enable the designer to reduce the value of the spike in the group-delay profile at the expense of the stopband roll-off rate, while maintaining a pre-specified equi-ripple characteristic in the passband. Since the parameters of the transformation are continuous-valued, an infinite family of Chebychev designs is possible. the approach is presented for the analog case, but is easily extended to digital filters via bilinear transformation.

A numerical evaluation of the hankel transform of bandlimited functions

The numerical evaluation of the Hankel transform of bandlimited functions arises in many different fields, particularly in image reconstruction from projections. We present a new numerical evaluation based on Chebychev approximation and asymptotic expansions that reduces the problem to a finite integration.

Stability of best rational Chebyshev approximation

In this paper we consider best Chebyshev approximation to continuous functions by generalized rational functions using an optimization theoretical approach introduced in [ B. Brosowski and C. Guerreiro, On the characterization of a set of optimal points and some applications, in “Approximation and Optimization in Mathematical Physics” (B. Brosowski and E. Martensen, Eds.), pp. 141–174, Verlag Peter Lang, Frankfurt a.M./Bern, 1983 ]. This general approach includes, in a unified way, usual, weighted, one-sided, unsymmetric, and also more general rational Chebychev approximation problems with side-conditions. We derive various continuity conditions for the optimal value, for the feasible set, and the optimal set of the corresponding optimization problem. From these results we derive conditions for the upper semicontinuity of the metric projection, which include some of the results of Werner [On the rational Tschebyscheff operator, Math. Z. 86 (1964), 317–326] and Cheney and Loeb [On the continuity of rational approximation operators, Arch. Rational Mech. Anal. 21 (1966), 391–401].