A framework for reformulating input design problems in prediction error identification as convex optimization problems is presented. For linear time-invariant single input/single output systems, this framework unifies and extends existing results on open-loop input design that are based on the finite dimensional asymptotic covariance matrix of the parameter estimates. Basic methods for parametrizing the input spectrum are provided and conditions on these parametrizations that guarantee that all possible covariance matrices for the asymptotic distribution of the parameter estimates can be generated are provided. A wide range of model quality constraints can be handled. In particular, different frequency-by-frequency constraints can be used. This opens up new applications of input design in areas such as robust control. Furthermore, quality specifications can be imposed on all models in a confidence region. Thus, allowing for statements such as "with at 99% probability the model quality specifications will be satisfied."