On necessary and sufficient conditions for perfect reconstruction multidimensional delay chain systems

This paper solves an open problem and presents both a necessary and sufficient condition for a perfect reconstruction (PR) modified multidimensional (MD) delay chain system. In fact, the determination of the PR property of the MD delay chain system is equivalent to the determination of the complete solutions of integer linear equations. In this paper, we develop a systematic technique for solving the linear equations. With these results, one can separate the large delay chain system with a complicated sampling matrix into a series of the small delay chain systems with the simple sampling matrices, intrinsically, each being a 1D delay chain system, the PR property of which is easily checked by using a recurring algorithm