On the realization problem for stationary,homogeneous, discrete-time systems

For degree 2, stationary, homogeneous systems, we consider realizations in the form of state-affine difference equations, and in the form of feedback-free interconnections of stationary linear systems, adders, and multipliers. Realizability conditions are presented in terms of the symmetric transfer function (or kernel), and in terms of a new representation called the regular transfer function (or kernel). A procedure is given for constructing a minimal dimension state-affine realization, and the corresponding interconnection.