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 N-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 z/sub i/ (i=1,2,.....,n), becomes a 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.<
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