The authors describe in this consistent paper a new fast recursive algorithm for the solution of an unconstrained rational interpolation problem by exploiting the displacement structure concept. Divided in 9 parts (Introduction, Triangular factorization and the generalized Schur algorithm, Cascade systems: blocking and interpolation properties, The unconstrained interpolation problem, Physical interpretation, State-space descriptions, Connection to the Lagrange polynomial, Column reduced transfer matrices and Concluding remarks) this paper shows how to construct recursively a convenient feedforward cascade that parametrizes all solutions.