On the irreducible cascade synthesis of a system with a real rational transfer matrix

In this paper, the problem of the irreducible synthesis of a real rational transfer matrix as a cascade (tandem connection) of first and second degree real systems is considered. A number of theoretical and practical results concerning the possibility of such a realization are deduced. In the process, the following results are obtained: 1) general forms for first and second degree real systems, 2) necessary and sufficient conditions for the extraction of a real first or second degree matrix, 3) a synthesis algorithm for a real rational transfer matrix as a cascade of first and second degree real systems, and 4) examples of situations where the real matrix cannot be so factored, even when it can be complex factored. The case where no cascade realization is available is rare, but possible. These results are of interest in simulation, in active and digital filter design of multiple input multiple output systems.