Sufficient Condition for State-Space Representation of N-D Discrete-Time Lossless Bounded Real Matrix and N-D Stability of Mansour Matrix

The definition of an N-D discrete time lossless bounded real (DTLBR) matrix and sufficient conditions on a state space representation to correspond to an Af-D DTLBR matrix are given. Based on these sufficient conditions, it is shown that the transfer function of the n-D lattice filter, which is obtained from a stable 1-D lattice filter by replacing each delay element z with zi (i = 1,2,⋯, n), becomes an n-D DTLBR function. From this fact an n × n Mansour matrix, which is an A -matrix of the state space representation of a 1-D lattice filter, is proved to be stable for any dimension up to n, if it is stable in the one-dimensional case.