A new approach to the realization of two-dimensional (2- D) FIR and IIR digital filters is introduced based on the so-called "lower-upper triangular (LU) decomposition" of matrix coefficients of their two-dimensional polynomials. It is demonstrated that the LU realization scheme enjoys a number of attributes for VLSI implementation including high parallelism, modularity, and regularity. This paper also shows that the computational requirements of the LU realization structure are much smaller than those of the Jordan (JD), singular value (SV), and canonical (direct) realizations. Furthermore, it is shown that if a canonical 2-D IIR filter is realizable, then a realizable LU decomposition form can be always obtained, a property that is not shared by the JD and SV decomposition forms. Finally, the extension of the LU decomposition approach to the realization of m-D digital filters is discussed.

The LU decomposition theorem and its implications to the realization of two-dimensional digital filters

It is shown that an arbitrary rational 2-D transfer function can be expanded in first order terms, each one of which is a function of only one of the two variables. This method leads naturally to reconfigurable filters with great modularity and parallelism, which can realize any rational transfer function up to a given order.

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A decomposition theorem and its implications to the design and realization of two-dimensional filters

The paper presents the troubles and trends of research in multidimensional system theory, with special emphasis in the areas of multivariable network analysis and synthesis, and multidimensional digital filters. The possibilities as well as perils of extending established single-dimensional techniques to multidimensional situations is discussed, and the nature and effect of certain fundamental problems present when applying the mathematics of several variables is given cognizance. Open problems are identified at various stages and some recommendations for future research are made.

Problems and progress in multidimensional systems theory

A possible extension of a well-known stabilization technique for one-dimensional recursive digital filters to the two-dimensional case was proposed by Shanks via a conjecture, stating that the planar least squares inverse of a two-dimensional filter polynomial is minimum phase and hence stable. In the present work, the conjecture has been verified first for a class of polynomials which are linear in one variable and quadratic in the other and then extended to a class of polynomials of higher degrees in the same variables. Though the conjecture is known to be false, in general, some conditions under which the conjecture is valid are explored.

Stabilization of certain two-dimensional recursive digital filters

A unified approach for the analysis and synthesis of a class of nonlinear systems is discussed. The analytical approach is based on the Volterra series representation of the input/output relationship of nonlinear systems. A canonic structure is presented for a class of nonlinear systems. The necessary and sufficient condition for the realizability of a given nonlinear system in terms of the canonic structure is presented. This synthesis procedure is based on a simple algorithm for separating a multivariable polynomial using the zeroes of a single variable polynomial. The synthesis procedure is also shown to yield computationally simpler models for a class of nonlinear systems.

Analysis and synthesis of a class of nonlinear systems

Necessary and sufficient conditions for reducibility of certain classes of multivariable polynomials are considered. The special case when a polynomial in that class can be factored into a finite product of single variable polynomials is investigated. A set of necessary and sufficient conditions for decomposing a multivariable p.r.f. into a sum of single variable p.r.f.'s is given. Examples to illustrate applications of sum and product separabilities of functions are given.

Sum and product separabilities of multivariable functions and applications

This paper presents a general theory of decoupling a system of nonuniform lossless coupled lines with the aid of linear transformations. The lines have a common return and support only TEM waves. It is shown that the solution of the decoupling problem is equivalent to that of the simultaneous diagonalization of a set of real constant matrices through specially related pairs of congruent and similarity transformations. The solution of the equivalent problem yields the necessary and sufficient conditions for decoupling the system of coupled lines. A systematic procedure for decoupling is then presented. A special case is studied for which the decoupling problem becomes simpler. Finally, we have derived a set of equivalent but considerably simpler necessary and sufficient conditions for decoupling the frequently used system of a pair of lossless coupled lines.

S. Chakrabarti

Papers

Sum and product separabilities of multivariable functions and applications

A general theory of decoupling a system of lossless coupled lines using linear transformations