On Necessary and Sufficient Conditions for Perfect Reconstruction Multidimensional Delay Chain Systems

In this correspondence, we present some necessary and sufficient conditions for a perfect reconstruction (PR) modified multidimensional (MD) delay chain system. With these results, one is able to systematically check if a D-dimensional delay chain system with a D × D sampling matrix M and a D × D delay matrix L has PR property in a simpler way than before, where the matrix modulo operations are avoided. Moreover, given a D × D sampling matrix M, in many cases one can determine all possible D × D delay matrices L so that the delay chain systems with the sampling matrix M have PR property. Several examples are provided. We also present a method to generate D × D sampling and delay matrices M and L such that their corresponding traditional MD delay chain systems are PR.