Path generating functions and continued fractions

The denominator polynomial of a given causal stable z-domain transfer function is modified so that the magnitude of the frequency response remains the same. This simple modification permits an infinite number of decompositions of the modified denominator into a mirror-image polynomial (MIP) and an anti-mirror-image polynomial (AMIP). Two types of Discrete Reactance Functions (DRF) are constructed. From these DRFs, continued fraction expansions (CFE) are considered and some properties are obtained. These properties indicate whether the original denominator polynomial has all its roots within the unit circle (is minimum phase) or not.

Contiguous relations, basic hypergeometric functions, and orthogonal polynomials, I

The goal of this work is to show a software package, based in Matlab, that simplifies the process of designing analog filters. This package is menu-oriented using the Matlab capabilities for doing so. The package is able to design lowpass, highpass, bandpass and bandreject filters. The approximation magnitude responses can be Butterworth (maximally flat magnitude), Chebyshev (equal-ripple), inverse Chebyshevi and elliptic. In the case of phase response, there is also the Thomson (maximally flat delay) also known as the Bessel response. The output for the approximation stage is a set of data that includes the transfer function poles and zeros, quadratic factors for both numerator and denominator polynomials, and the numerator and denominator polynomials. The network realizations can be either passive or active realizations. For the passive realizations, low sensitivity ladder realization is used. This realization is available for any approximation type. In the case of the active realizations, the authors have implemented the cascade of biquad stages but they are working on the active ladder realization. For the biquad stages, the Sallen-Key realization, multiple feedback, state-variable KHN, state variable Tow-Thomas and the GIC based biquad are available.

It is well-known that a continued fraction may be regarded as a sequence of Mobius maps. In this partly expository paper we consider continued fractions by examining the action of Mobius maps in hyperbolic space, and in all dimensions. We obtain a general version of the Stern-Stolz theorem valid in all dimensions, and we draw comparisons between the theory of continued fractions and the theories of discrete groups of Mobius maps, and complex dynamics.

The ordinary and matrix continued fractions in the theoretical analysis of Hermitian and relaxation operators

Combinatorial proofs of identities of Calkin and Hirschhorn

Throughout the present Chapter, we are essentially concerned with the following problem: for which functions Ψ : ℝ ≥1 → : ℝ ≥0 is it true that, for a given real number ξ, or for all real numbers ξ in a given class, the equation |ξ – p/q | q ) has infinitely many solutions in rational numbers p/q ? We begin by stating the results on rational approximation obtained by Dirichlet and Liouville in the middle of the nineteenth century. In Section 1.2, we define the continued fraction algorithm and recall the main properties of continued fractions expansions. These are used in Section 1.3 to give a full proof of a metric theorem of Khintchine. The next two Sections are devoted to the Duffin–Schaeffer Conjecture and to some complementary results on continued fractions. Dirichlet and Liouville Every real number ξ can be expressed in infinitely many ways as the limit of a sequence of rational numbers. Furthermore, for any positive integer b , there exists an integer a with |ξ – a/b | ≤ 1/(2 b ), and one may hope that there are infinitely many integers b for which |ξ – a/b | is in fact much smaller than 1/(2 b ). For instance, this is true when ξ is irrational, as follows from the theory of continued fractions.

A connection between the symbolic description of the geodesic flows on certain modular surfaces and the theory of continued fractions is developed. The well-known properties of these dynamical systems then lead to some new results about continued fractions.

The subject of this paper is the current state of art in theory of continued fractions, intermediate fractions and their relation to the best rational approximations of the first and second kind. The paper provides an overview of the some well known and even some new properties of continued fractions, and the various terms associated with them. In addition to intermediate fractions, paper considers the fine intermediate fractions and gave some statements to position these fractions in the continued fraction representation of numbers.

A method for calculation of maximum allowable gain or closed Ioop systems having plants with structured uncertainties is given. The method accommodates plant transfer functions with linearly dependent coefficient perturbations in the denominator polynomial and first order multiplicative coefficient perturbations in the numerator polynomial. Examples are given which show the versatility of the calculation.

It is shown that magnitude hyperspherically symmetric transfer functions of multidimensional (MD) causal recursive digital filters must have numerator and denominator polynomials that are separately magnitude hyperspherically symmetric. Further, the exact reference-domain magnitude hyperspherically symmetric denominator polynomial is of infinite order, possessing only one free parameter, and the magnitude hyperspherically symmetric numerator polynomial itself has to be a radial even function. The corresponding MD design problem is shown to be essentially a one-dimensional design problem. Filter transfer functions having good symmetry and moderate degree can be designed by using the presented procedure. >

Michael Somos conjectured a relation between Hankel determinants whose entries ${1\over 2n+1}{3n\choose n}$ count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss's continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Hankel determinants. We also describe a simple method for transforming determinants using the generating function for their entries. In this way we transform Somos's Hankel determinants to known determinants, and we obtain, up to a power of $3$, a Hankel determinant for the number of alternating sign matrices. We obtain a combinatorial proof, in terms of nonintersecting paths, of determinant identities involving the number of ternary trees and more general determinant identities involving the number of $r$-ary trees.

The paper deals with the problem of robust absolute stability of systems with parametric uncertainties. By using a 2q-convex parpolygonal value set of a family of polynomials of the form P(s, q) = a <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> (q) + a <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> (q)s +………+ a <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> (q)s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> whose coefficients depend linearly on q = [q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> ,…, q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</inf> ] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> and the uncertainty box is Q = {q : q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> ∊ [q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> , q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> ], i = 1,2,….q}, an extremal boundary result for a transfer function, whose numerator and denominator polynomials are in the form of P(s, q), is first given. Then, using this boundary result, the robust versions of the classical absolute stability criteria of Lur'e, Popov and the Circle criterion are obtained. An example is included to illustrate the benefits of the method presented.

Provides author's reply to "Comments on 'Optimizing simultaneously over the numerator and denominator polynomials in the Youla-Kucera parameterization" (ibid., vol. 50, no. 9, pp. 1369-1374, Sep 05).

Another approach for generating reduced order discrete time system using composite method is being proposed. The proposed method is validated by considering a Linear Time Invariant Discrete System (LTIDS) having single input single output(SISO) of higher order being represented in frequency domain. Eigen Spectrum Analysis and Least Squares moment matching methods form the proposed composite method to obtain the reduced numerator and denominator polynomials. The suitability of the proposed method is judged by comparing the step, impulse responses of the original and reduced systems. The performances of the systems are measured in terms of Steady State Error (SSE). It is seen from the numerical examples that the results are quite competitive as compared to other methods thus justifying the applicability of the proposed method.