Least order coprime factorizations of rational matrices: the canonical case

For a given rational matrix G with complex coefficients and a given domain /spl Gamma/ in the closed complex plane, both arbitrary, we develop a complete theory of coprime factorizations of G over /spl Gamma/, with denominators of McMillan degree as small as possible. We consider both the cases in which the denominator is arbitrary and in which it has a certain symmetry, namely it is J all-pass, either with respect to the imaginary axis or to the unit circle. All the developments are carried out in terms of descriptor realizations associated with rational matrices, leading to explicit and computationally efficient formulas.