In this article we address linear anti-windup design for linear discrete-time control systems guaranteeing regional and global stability and performance. The techniques that we develop are the discrete-time counterpart of existing techniques for anti-windup augmentation which lead to convex constructions by way of linear matrix inequalities (LMIs) when adopting static and plant order anti-windup augmentation. Interesting system theoretic interpretations of the performance bounds for the non-linear closed loop can also be given. We show here that parallel results apply to the discrete-time case. We derive the corresponding conditions and prove their effectiveness by adapting the continuous-time approaches to the discrete-time case.