Nonessential Singularities Research Articles

Routh Zero Location Tests Unhampered by Nonessential Singularities

The generalization of the Routh stability test to the corresponding zero location problem of determining how many of the zeros of a polynomial have negative real parts, positive real parts, or are purely imaginary encountered intensive and controversial activity about overcoming singular cases. This paper revisits this problem and presents two (MaxQ and LinQ) ZL methods for an arbitrary complex polynomial. The first method produces for an investigated polynomial <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P(s)$</tex-math> </inline-formula> of degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$</tex-math> </inline-formula> a sequence of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\leq $</tex-math> </inline-formula> <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n+1$</tex-math> </inline-formula> of para-even or para-odd (para-paritic) polynomials of descending degrees obtained by a polynomial remainder routine with quotients of maximal degrees. The second method uses successively linear quotients for the reduction of degrees and thus assign to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P(s)$</tex-math> </inline-formula> always a sequence of exactly <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n+1$</tex-math> </inline-formula> para-paritic polynomials. Previous so-called first-type singularities become “non-essential singularities” in the sense that they are now absorbed into modified forms of the polynomial recursions. The distribution of zeros with respect to the imaginary axis is extracted in both methods by sign variation rules posed on the leading coefficients (that stay real when testing a complex polynomial as well) of the polynomials in the sequence.

Piecewise holomorphic systems

Holomorphic systems are well known in the literature due to their important properties: reversibility, integrability, non-existence of limit cycles and complete knowledge of the phase portraits around their non-essential singularities. In this paper we are concerned about studying the piecewise holomorphic systems (PWHS). Specifically, we study the sewing, sliding, and tangential regions of the PWHS and we classify the typical singularities of these systems. Also, we are interested in understanding how the trajectories of the regularized system associated with the PWHS transit through the region of regularization. In addition, we know that holomorphic systems have no limit cycles, but piecewise holomorphic systems do, so we provide conditions to ensure the existence of limit cycles of these systems. Additional conditions are provided to guarantee the stability and uniqueness of such limit cycles.

Design of Stable Circularly Symmetric Two-Dimensional GIC Digital Filters Using PLSI Polynomials

A method for designing stable circularly symmetric two-dimensional digital filters is presented. Two-dimensional discrete transfer functions of the rotated filters are obtained from stable one-dimensional analog-filter transfer functions by performing rotation and then applying the double bilinear transformation. The resulting filters which may be unstable due to the presence of nonessential singularities of the second kind are stabilized by using planar least-square inverse polynomials. The stabilized rotated filters are then realized by using the concept of generalized immittance converter. The proposed method is simple and straight forward and it yields stable digital filter structures possessing many salient features such as low noise, low sensitivity, regularity, and modularity which are attractive for VLSI implementation.

Stability Testing of 2-D Digital Filters Including Those Having Non-essential Singularities of the Second Kind

Stability Testing of 2-D Digital Filters Including Those Having Non-essential Singularities of the Second Kind

Bibo stability in 3-d digital filters

In this brief, we consider the problem of binary input binary output stability of three-dimensional linear shift-invariant filters, in the presence of nonessential singularities of the second kind. We derive necessary and sufficient conditions for l/sub 1/ and l/sub 2/ stabilities of a function G=P/Q/sup n/, where P and Q are polynomials in the complex variable z.

Zero location of polynomials with respect to the unit-circle unhampered by nonessential singularities

A method to determine the distribution of the zeros of a polynomial with respect to the unit-circle, proposed by this author in the past, is revisited and refined. The revised procedure remains recursive and nonsingular for polynomials whose Schur-Cohn matrix is not singular. Other nonessential singularities that previously caused interruption of the recursion are assimilated into a more general regular form of the three-term recursion of symmetric polynomials that underlies the method. The new form of the procedure does not compromise the simplicity of the rules to extract the information on the distribution that are proved using a different and more direct proof, based on the evaluation of the Cauchy index along the unit-circle. The low count of operations of the original procedure (recognized as the least cost solution for the problem) is maintained and actually gets better by the elimination of nonessential singularities. The improved features make the revised procedure a better all-around unit-circle zero location method for any real or complex polynomial. Its wider range of regularity should also benefit a variety of related signal processing and algebraic problems including some that were already affected by the original formulation.

Zero Location of Polynomials with Respect to the Unit-Circle without Nonessential Singularities

The author's method to determine the distribution of the zeros of a (real or complex) polynomials with respect to the unit-circle is revisited and refined. The three-term recursion of symmetric polynomials that the method uses is generalized such that it remains regular for all polynomials whose Schur-Cohn matrix is not singular. The refined version eliminates nonessential singularities without compromising the low computational cost and the simplicity of the zero location rules of the original procedure.

On a class of conformal metrics arising in the work of Seiberg and Witten

We examine a class of conformal metrics arising in the N = 2 supersymmetric Yang-Mills theory of Seiberg and Witten. We provide several alternate characterizations of this class of metrics and proceed to examine issues of existence and boundary behavior and to parameterize the collection of Seiberg-Witten metrics with isolated non-essential singularities on a fixed compact Riemann surface. In consequence of these results, the Riemann sphere b does not admit a Seiberg-Witten metric, but for all >0 there is a conformal metric on b C of regularityC 2 which is Seiberg-Witten off of a finite set.

A new transform for the stabilization and stability testing of multidimensional recursive digital filters

We present a new transform which, when applied to the denominator polynomial of the transfer function of an unstable multidimensional recursive digital filter (of a special class) will yield a stable polynomial with good preservation of the magnitude spectrum. In fact, the discrete Hilbert transform (DHT), used to stabilize 2-D and 1-D recursive digital filters, is a special case of the general multidimensional transform we present here. We also address the problem of stability testing of a multidimensional recursive digital filter and show that the new transform may be used to implement a straightforward test for stability of any causal multidimensional recursive digital filter, having no nonessential singularities of the second kind.

Necessary conditions for the stability of N-D digital filters with nonessential singularities of the second kind

In this brief we consider the open problem concerning the BIBO stability of N-D digital filters with nonessential singularities of the second kind in the region U~/sup N/-U/sup N/. We apply transformations to reduce the transfer function to the two-dimensional (2-D) case. Illustrative examples are included.

On the stability preserving property of the double bilinear transformation on a class of 2-D transfer functions

This paper addresses the BIBO (bounded-input bounded-output) stability of a class of discrete 2-D quarter-plane filters in the presence of nonessential singularities of the second kind (NSSK's) on the unit bidisk. Conditions under which the double bilinear transformation (DBT) preserves stability are derived. The results presented here also extend the class of systems whose stability can be predicted. Use of the inverse DBT to produce a continuous equivalent of the discrete 2-D transfer function allows easy application of a continuous-domain equivalent of a criterion developed by Dautov. The necessary and sufficient condition for stability derived in this work provides a simple check for the class of systems under consideration. From this class of systems, it is also possible to construct stable pairs of mutually inverse transfer functions.

A computationally efficient technique for deriving 2-variable VSHP

In this paper, a method is presented for the generation of 2-variable Very Strictly Hurwitz Polynomial (VSHP) free of any non-essential singularities of the second kind. The proposed technique utilizes the properties of the positive definite and semi-definite matrices along with resistive matrices to avoid calculation of higher order derivatives which are cumbersome.

Stability of mutually inverse rational 2-D digital transfer functions

This brief considers the open problem concerning the BIBO stability of mutually inverse 2-D digital filters in the presence of nonessential singularities of the second kind on T/sup 2/. Necessary and sufficient conditions are obtained for the BIBO stability of mutually inverse pairs of 2-D digital filters having such singularities on T/sup 2/. Several illustrative examples are considered. >

A new algorithm to test the stability of 2-D digital recursive filters

Abstract Any algorithm for testing the stability of 2-D digital recursive filters has to process a 2-D polynomial of degree n and m with respect to each variable. The number of computations needed by the algorithm can be expressed as a polynomial of the variables n and m . The total degree of this polynomial defines the complexity order of the algorithm. In this paper we establish that the matrix associated with Bezout's resultant appearing in the stability test of causal or semicausal recursive filters has 2 as its displacement rank. This permits us to apply the generalized Levinson-Szego algorithm to derive a new algorithm for testing the 2-D digital filters' stability. This algorithm is proposed for quarter-plane or nonsymmetric half-plane recursive filters with real coefficients and without nonessential singularities of the second kind. Its complexity order is equal to 4. Note that the complexity order of the fastest existing algorithms is equal to 5.

Nonessential singularities of the second kind and stability for multidimensional digital filters

In this paper, certain problems concerning nonessential singularities of the second kind (NSSKs), especially continuous NSSKs for 3-D digital filters, are discussed. It is shown that continuous NSSKs, both inT 2 U and onT 3, may exist in stable systems. Based on the Weierstrass theorem of multidimensional complex analysis, the existence criteria of continuous NSSKs inT 2 U and onT 3 are obtained. Certain basic properties of this type of singularities are given. A stability condition for a transfer functionH(Z) with continuous NSSKs and the inverse ofH(Z) is derived. Furthermore, for isolated NSSKs, a new necessary and sufficient condition for stability is developed for a class of systems.

Finite word-length effects in m-D digital filters with singularities on the stability boundary

The author addresses stability robustness with respect to finite word-length conditions for multidimensional (m-D) digital filters with singularities on the distinguished boundary. In particular, it is shown that all purely first-order digital filters with nonessential singularities of the second kind (NSSKs) are asymptotically stable under certain types of nonlinearities. Even bounded-input, bounded-output (BIBO)-unstable m-D digital filters with singularities on the distinguished boundary proved to be asymptotically stable in the first-order case. The results are then extended to certain classes of higher-order m-D digital filters. It is demonstrated that, although linear m-D digital filters with NSSKs on the distinguished boundary have no margin of stability in the l/sub 1/, l/sub 2/ or asymptotic sense, they might exhibit a rather robust asymptotic behavior with respect to nonideal effects such as finite word-length conditions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nonlinear and Stochastic Phenomena: The Grand Challenge for Partial Differential Equations

The major problems for partial differential equations are either nonlinear or stochastic or both. Considerable progress has occurred recently in these areas, much of which has resulted from a striking interplay among theory, computation, and applications. The nonlinear structures emphasized here are the nonlinear hyperbolic waves, which occur in the solutions of nonlinear hyperbolic conservation laws and associated dissipative equations. These equations are those most commonly used to model physical and chemical processes in continuum systems. In one space dimension these nonlinear waves form traveling waves, for which the methods of dynamical systems and bifurcation theory are very useful. Established points of view toward mathematical theories have been challenged by recent work (e.g., uniqueness of solutions, entropy conditions, and elliptic regions in hyperbolic equations). A type of geometry is defined in the state space of the conservation law by the nonlinear waves. This geometry is generically singular in the case of nonlinear resonance. A regularizing space, called the Fundamental Wave Manifold, has been constructed which removes nonessential singularities in this geometry; the remaining, essential, sin- gularities serve as organizing centers for the bifurcation of the nonlinear hyperbolic waves. Stochastic solutions of partial differential equations may result from stochastic data, for example, in the form of random initial conditions or equation coefficients. This phenomenon can be called forced, driven, or external chaos. Stochastic solutions can also result from internal considerations, i.e., from instabilities inherent in the equation itself, if the equation is nonlinear. The instabilities which give rise to chaotic solutions are generally unstable on a very large range of length scales, or in some approximation, on all length scales. Fractal geometries provide one model for the typically intermittent, or blotchy, chaos which can occur, and the renormalization group is an analytic tool which in some cases has given a quantitatively correct theory of chaotic solutions of partial differential equations.

Two observations regarding first-quadrant causal BIBO-stable digital filters

An n-dimensional digital filter is said to be normal if it is linear shift-invariant, first-quadrant causal, and bounded-input-bounded output (BIBO) stable. The transfer function of such a filter is holomorphic in the open polydisc U/sup n/ and continuous in its closure, making it a member of W. Rudin's polydisc algebra A(U/sup n/) (see W. Rudin) (Function Theory in Polydiscs, New York: W.A. Benjamin, 1969). It has been shown by Rudin that, nevertheless, every all-pass function in A(U/sup n/) is rational, free of nonessential singularities of the second kind, and of finite norm. This result appears to have been completely overlooked in the engineering literature. The material covered here is mainly tutorial, comprising a leisurely expanded account of the key ideas behind the Fejer and Rudin constructions. >

General results concerning the BIBO stability of inverse 2-D digital filters

Consideration is given to the open problem concerning the BIBO (bounded-input, bonded-output) stability of inverse 2-D digital filters in the presence of nonessential singularities of the second kind on T/sup 2/, the distinguished boundary of the unit bidisk. Necessary and sufficient conditions are obtained for the BIBO stability as well as the boundedness of a large class of mutually inverse 2-D digital filters having such singularities on T/sup 2/. Some illustrative examples are included. >

General results concerning the stability of 2-D digital filters with nonessential singularities of the second kind

Previous results concerning the stability of 2-D digital filters in the presence of simple nonessential singularities of the second kind are extended to a very broad class of functions having second kind singularities on T 2 , the distinguished boundary of the unit bidisk, where these singularities may be simple or of multiple order. Necessary and sufficient conditions are given for the ℓ 2- and ℓ 1-stabilities and for theboundedness for a function of the type G(z 1,z 2) = P(z 1,z 2) {Π n 1Q ν i i(z 1, 1,z 2 )} , where P Q i has simple nonessential singularities of the second kind on T 2 . These conditions are computationally simple and are expressed in terms of multiplicities of the zeros of certain resultants of two-variable polynomials. Several examples are considered to illustrate the results.