Analytic interpolation problems and lattice filter design

The so-called analytic interpolation problem is addressed and solved. The objective is to find the family of rational interpolants which are analytic in a certain region of the complex plane. It turns out that the usual linear fractional map cannot be used to describe the solution set conveniently. Instead, an affine parametrization formula is proposed as the natural framework to impose analyticity constraint on the interpolants. All solutions of the interpolation problem are characterized in terms of a generating system, which can be obtained efficiently via a fast recursive algorithm. The recursive procedure can be used to update the solutions whenever a new interpolation constraint is added to the input data set. It is shown that the analytic interpolation problem is solvable if and only if the corresponding unconstrained problem is solvable, i.e., if and only if the interpolation data-set is consistent. The above results have many applications in different areas such as stable lattice filter design, channel identification, and Q-parametrization of stabilizing controllers