M.N.S. Swamy

BIBO stability in the presence of nonessential singularities of the second kind in 2-D digital filters

In this paper, the open problem regarding the BIBO stability of two-dimensional linear shift invariant filters, in the presence of nonessential singularities of the second kind, is considered. Necessary and sufficient conditions for boundedness, and l_2 and l_1 stabilities of a function G(z_1, z_2)= P(z_1, z_2)/[Q(z_1, z_2)]^n , where P/Q has simple nonessential singularities of the second kind on T^2 , are obtained. These conditions are expressed in a very simple way in terms of the multiplicity of the zeros of certain resultants of two-variable polynomials. Many illustrative examples are also given.

A new type of wave digital filter

Abstract This paper proposes a new wave digital filter derived from doubly terminated LC-ladder networks by replacing each series or shunt arm element of the ladder by its equivalent digital two-port. It is shown that such two-ports may be cascaded without the use of adapters defined by Fettweis ( 1 ). A number of realizations of the wave digital two-ports, which are canonic with respect to both multipliers and delays, have been obtained. Also a realization which is canonic with respect to multipliers only is given and an example considered using this realization. The sensitivity of this filter with respect to the multiplier coefficient changes due to finite word length is compared with the conventional cascaded digital filter and also the one proposed by Renner and Gupta. It is found that the proposed filter appears to be a more desirable form of implementation than the conventional cascade form and comparable to that of Renner and Gupta ( 2 ).

A bound for the zeros of polynomials

A new kind of circular bound on the zeros of polynomials is derived by determining Cauchys bound on zeros of its transformed pair first. The transformation is based on a nonlinear transformation of the variable, which conceptually should give a better upper bound. The advantage of such a transformation is illustrated through several examples that show the improvement over the existing bounds. Convergence of the bound after iterative transformations of the original polynomial is also examined. >

Two-dimensional wave digital filters using doubly terminated two-variable LC-Ladder configurations

Abstract This paper proposes a method of obtaining a two-dimensional wave digital filter of the recursive type from a doubly terminated LC-ladder network in two variables by replacing each series or shunt arm element of the ladder by its equivalent digital two-port. A number of realizations of the wave digital two-ports, which are canonic in multipliers, have been obtained. An example of a circularly symmetric low-pass two-dimensional digital filter is considered using these realizations. The sensitivity of this filter with respect to the multiplier coefficient changes due to finite word length is compared with that of the direct realization. It is found that the wave digital filter appears to be a more desirable form of implementation than the conventional cascade form.

Approximation of two-variable filter specifications in analog domain

This paper describes a technique for approximating 2-variable filter specifications in the continuous or analog domain. It is shown that the design of 2-variable filter functions using this approach reduces to the problem of identifying a suitable 2-variable reactance function g(s_1,s_2) and the realization of a stable single-variable transfer function T(s) . Then T(g(s_1,s_2)) is the desired 2-variable stable transfer function which is guaranteed to have at least one realization whenever T(s) and g(s_1,s_2) are realizable. Applications of the theory developed in this paper are presented in the design of lumped-distributed filters and 2-dimensional digital filters.

An efficient numerical scheme to compute 2-D stability thresholds

Stability thresholds (margins) of two-dimensional (2-D) digital filters were recently defined in terms of the singularities of the transfer function. Stability thresholds hold a close relationship to the settling time of the 2-D impulse response and can therefore serve as a measure of stability of 2-D digital filters. In this paper, a simple, unified procedure for the computation of stability thresholds (margins) for 2-D digital filters based on the z_1 and z_2 resultants of 2-D polynomials is presented. To illustrate the process, simple examples are also provided.

Generalized dual transposition and its applications

Abstract A theorem, similar to Tellegens theorem, but for networks having dual topologies, is developed. This theorem is used in the frequency domain to define a new network operation called “generalized dual transposition”, whose applications in network synthesis are discussed. Finally, it is used in the time domain to define a network called “dual adjoint” and its relation to dual and adjoint networks established.

Sufficiency conditions for the stability of a class of 3-D functions with nonessential singularities of the second kind

In this correspondence, we state sufficiency conditions for boundedness, and 1_{2^-} and 1_{1^-} stabilities of a 3-D function G(z_1, z_2, z_3) = P(z_1, z_2, z_3)/Q(z_1, z_2, z_3) , where Q is a polynomial which is linear in all its three variables, and P/Q has some second kind nonessential singularities in \overline{U}^3 - U^3 . Some illustrative examples are also given.

Ladder realizations of multivariable positive real functions

Necessary and sufficient conditions are obtained for the realization of an m-variable positive real function (PRF) as the impedence function of a resistively-terminated ladder network of m lossless two-ports connected in cascade. Each two-port is a single-variable lossless ladder with all of its transmission zeros either at the origin or at finity. Conditions are also obtained when each of the two-ports is a Fujisawa-type lowpass ladder.

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

Abstract 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 T2, 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 l2- and l1-stabilities and for theboundedness for a function of the type G(z 1 ,z 2 ) = P(z 1 ,z 2 ) {Π n 1 Q ν i i (z 1 , 1 ,z 2 )} , where P Q i has simple nonessential singularities of the second kind on T2. 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.