L. Baratchart, P.Enqvist,

Given a square Schur function S, we consider the problem of constructing a symmetric Darlington synthesis of minimal size. This amounts essentially to finding a stable all-pass square extension of S of minimal size. The characterization is done in terms of the multiplicities of the zeros. As a special case we obtain conditions for symmetric Darlington synthesis to be possible without increasing the McMillan degree for a symmetric rational contractive matrix which is strictly contractive in the right half-plane. This technique immediately extends to the case where, allowing for a higher dimension of the extension, we require no increase in the McMillan degree. Both the complex and the real case are examined (the last one being that of interest for applications to circuits). Also in this case we obtain sharper results than those existing in the literature.