Filter design under magnitude constraints is a finite dimensional convex optimization problem

We consider the design of filters satisfying upper and lower bounds on the frequency response magnitude. The paper contribution is to prove that such a problem is equivalent to a finite dimensional convex optimization program involving linear matrix inequality constraints. Note that this filter design problem is usually reduced to a semi infinite dimensional linear programming optimization problem under the additional assumption that the filter poles are fixed (for instance, FIR design). In fact, weighting function design in the standard H_∞ approach to control is our motivating application. Unavailable systematic design method precludes a wider use of the H_∞ approach