State space representations of 2-D FIR lossless transfer matrices

This paper examines the properties of state-space realizations of square 2-D FIR lossless transfer matrices. Such matrices have found applications for the design of 2-D orthogonal perfect reconstruction filter banks. It is shown that 2-D FIR lossless matrices admit a minimal first-level realization with respect to one transform variable, where the dynamics are FIR and lossless in the other variable. This result is then employed to prove that 2-D FIR lossless transfer matrices admit a minimal 2-D Roesser realization whose dynamics are parametrized by a single constant orthonormal matrix, which satisfies constraints ensuring that the resulting transfer matrix is FIR. This parametrization is used to compute the number of degrees of freedom available for the synthesis of a 2-D FIR lossless transfer matrix of fixed McMillan degree in each variable. It is also shown that 2-D lossless FIR transfer matrices cannot necessarily be represented as a product of lossless FIR factors of lower degree, and conditions for the existence of such factorizations are given