2-D non-separable paraunitary matrices and Grobner bases

A 2-dimensional FIR transfer matrix H(z, z) is paraunitary if the underlying FIR system is a lossless system. Venkataraman-Levy [1994] and Basu-Choi-Chiang [1993] have constructed 2-D FIR paraunitary matrices of McMillan degrees (2,2) that are not factorable into simpler paraunitary matrices. This compares to the factorizability of 1-D FIR lossless transfer matrices in terms of Givens rotations, which produces the parameters that can be used for an optimal design of lossless FIR filter banks with prespecified filtering characteristics. Because of the state-space realization used in the construction, these 2-D counter-examples are floating-point approximations. In this paper, we formulate the lossless condition and nonseparability condition using multivariate polynomials in the coefficients. By studying the polynomial system, we obtain a continuous one parameter family of 2-D 2 2 non-separable paraunitary matrices