Sankar Basu

New results on stable multidimensional polynomials-Part I: Continuous case

New classes of multivariable polynomials arising in studies of passive multidimensional systems have been identified and their properties have been studied. In particular, polynomials occurring as the numerators and denominators of multivariable reactance functions and positive functions are characterized. Related properties of these and other classes of multivariable Hurwitz polynomials are also studied. Finally, a nontrivial test for the property of positivity of rational functions, holomorphic in a domain, in terms of their behavior on the distinguished boundary is formulated.

On boundary implications of stability and positivity properties of multidimensional systems

Multidimensional generalizations of various 1-D results on the robustness of Hurwitz, Schur, and positivity properties of polynomials and rational functions are considered. More specifically, the convexity property of the stable region in the coefficient space of multivariable polynomials is studied. Multidimensional generalizations of Kharitonov-type results are reviewed, and further extensions, including that of the 1-D edge theorem, are discussed. Interval positivity properties of multivariate rational functions are characterized in terms of ratios of a finite number of Kharitonov-type polynomials constructed from the extreme values of the intervals of perturbation. >

On the multidimensional generalization of robustness of scattering Hurwitz property of complex polynomials

Recent stability results on the scattering and the immitance description of passive multidimensional systems are used to characterize the robustness of the scattering Hurwitz property of a given multidimensional (complex) polynomial in terms of the scattering Hurwitz property of a finite number of multidimensional (complex) polynomials. The result is a complete proof of a recent conjecture extending V.L. Kharitonovs (Differential Equations, vol.14, p.1483-5, 1979) theorem on the characterization of the interval (strict sense) Hurwitz property of real as well as complex polynomials to multidimensions. The multidimensional versions of the weak and strong forms of Kharitonovs one-dimensional results are presented along with their proofs. >

New results on stable multidimensional polynomials- Part II: Discrete case

Properties of various multidimensional polynomials arising in studies on discrete multidimensional systems are investigated. Reactance Schur polynomials and immittance Schur polynomials occurring, respectively, as the denominators (and numerators) of discrete reactance functions and discrete positive functions are introduced and their properties studied. The role of these polynomials in scattering or immittance descriptions of passive discrete-time domain multiports are brought out. The interrelations between various classes of multidimensional polynomials arising in studies on discrete systems and the corresponding classes of polynomials in the context of continuous systems are also studied via the bilinear transformation.

New results on stable multidimensional polynomials. III. State-space interpretations

Lyapunov characterization of scattering Schur, reactance Schur, and immittance Schur properties of multidimensional polynomials are given. Since these polynomials arise in the description of passive systems, energy-like functions can be conveniently defined for such systems, thus making Lyapunov-type characterization readily available. It is shown that the notion of modal observability and its further variants for multidimensional systems are necessary ingredients of such considerations. Two approaches are followed. One is essentially passive synthesis based, whereas the other is that of viewing multidimensional systems as a parametric family of one-dimensional systems. A nontrivial example is included to illustrate the former method. >

On the factorization of scattering transfer matrices of multidimensional lossless two-ports

The problem of cascade decomposition of arbitrary bivariate lossless two-ports is addressed via factorization of the associated scattering transfer matrix. Necessary and sufficient conditions for the existence of solution to the factorization problem so formulated, are obtained. It is shown that the criterion for factorability can be expressed in terms of solvability of a set of linear simultaneous equations, which is overdetermined in general. Since no restrictions are imposed on the type of transmission zeros of the given scattering transfer matrix, earlier results in the literature can, in principle, be viewed as special cases of the results obtained here.

Tests for polynomial zeros on a polydisc distinguished boundary

That several applications require the testing for zeros on the distinguished boundary of an unit polydisc of a multivariable polynomial has been noted. Here, procedures to. test whether or not a polynomial B(z_1 , z_2,cdots,z_n) has zeros on |z_1| = |z_2| = cdots = |z_n|= 1 are developed. Two indirect tests are fornulated and the details of a direct test are given for the n = 2 case with possible extensions for the n > 2 case noted. Singular cases are given adequate attention.

Hermite-like reduction method for linear phase perfect reconstruction filter bank design

Complete parametrization of multiband linear phase biorthogonal filter banks are given. The method uses matrix reduction methods similar to the Hermite reduction method of linear system theory. Computational algorithms are derived for design, and examples are worked out.

On a generalized factorization problem for structurally passive synthesis of digital filters

The problem of structurally passive synthesis of multidimensional digital filters of the quarter-plane causal type as an interconnection of more elementary building blocks directly in the discrete domain is addressed by means of the factorization of the chain matrix, the hybrid matrix and the transfer function matrix associated with a prescribed multidimensional lossless two-port. By using recent results on the discrete domain representation of such matrices, a generalized lossless two-port matrix is introduced to present all three factorizations in a unified setting. Necessary and sufficient conditions for factorability as well as an algorithm for computing these factors when they exist are obtained. In particular, it is shown that, in one dimension the factorizations can always be performed. Thus, in one dimension, a discrete domain algorithm for synthesizing previously unpublished internally passive structures and alternative methods of synthesis for more conventional structures such as the cascade structure are also obtained. >

On multidimensional linear phase perfect reconstruction filter banks

We consider the multidimensional version of the problem of linear phase perfect reconstruction filter bank design. We give conditions for linear phase property of the filter bank, enumerate the number of symmetric and antisymmetric filters in the bank and give results on the possibility of construction of the entire filter bank if all filters in the bank except one are specified without any restriction other than the linear phase property.<<ETX>>