A new robust controller design method that satisfies the H-infinity criterion is developed for linear time-invariant single-input single-output (SISO) systems. A data-driven approach is implemented in order to avoid the unmodeled dynamics associated with parametric models. This data-driven method uses fixed order controllers to satisfy the H-infinity criterion in the frequency domain. The necessary and sufficient conditions for the existence of such controllers are presented by a set of convex constraints. These conditions are also extended to systems with frequency-domain uncertainties in polytopic form. It is shown that the upper bound on the weighted infinity norm of the sensitivity function converges monotonically to the optimal value, when the controller order increases. Additionally, the practical issues involved in computing fixed-order rational H infinity controllers in discrete- or continuous-time by convex optimization techniques are addressed.