Parametrized Family of 2-D Non-factorable FIR Lossless Systems and Gröbner Bases

Abstract

The factorability of one-dimensional (1-D) FIR lossless transfer matrices [1] in terms of Givens rotations produces the parameters that can be used for an optimal design of filter banks with prespecified filtering characteristics. Two dimensional (2-D) FIR lossless systems behave quite differently, however. Venkataraman-Levy [2] and Basu-Choi-Chiang [3] have constructed 2-D FIR paraunitary matrices of McMillan degrees (2,2) that are not factorable. Because of the state-space realization used in the construction, they are floating-point approximations, and they do not produce explicit parametrizations that can be used for optimal design process. In this paper, we formulate the lossless condition and nonfactorability condition of a 2-D FIR paraunitary matrix using multivariate polynomials in the coefficients. The resulting polynomial system can be explicitly solved with Gröbner bases. By studying the polynomial system, we obtain a continuous one parameter family of 2-D 2×2 non-factorable paraunitary matrices. As an example, we get a closed-form expression for a 2-D 2×2 paraunitary matrix that is not factorable into rotations and delays.