New Sufficient Conditions of N-D DTLBR Matrices and Some Properties of 2-D First Order DTLBR Functions-Multidimensional State Space Approach

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Recently discrete-time lossless bounded real functions and matrices have been receiving a great deal of attention in the various areas of one-dimensional (1-D) and N-dimensional (N-D) digital signal processing. In this paper a new sharper sufficient condition for state space realizations of structurally stable N-D discrete time lossless bounded real (DTLBR) matrices is given, which is an improvement of the authors' previous work. This new condition covers more wide class of structurally stable N-D DTLBR matrices compared with the authors' previous condition. The properties of minimal state space realizations of structurally stable 2-D first order DTLBR functions are studied in detail and a necessary and sufficient condition for a minimal state space realization of a structurally stable 2-D first order DTLBR function is obtained. Finally it is shown that both in cases of Roesser model and Fornasini-Marchesini second model a structurally stable 2-D first order DTLBR function can have two distinct minimal state space realizations for which no similarity transformation exists. This fact implies that minimal realizations of a given N-D system can be classified into several groups and if two minimal realizations belong to distinct groups, they cannot be connected by a constant similarity transformation matrix.
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