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The aim of this work is to provide a unified observer design method for nonlinear Lipschitz discrete-time systems with extension to H infin synchronization and input recovery used for communication systems. The stability analysis is performed using a suitable Lyapunov function that leads to the solvability of linear matrix inequalities. One of the main challenges is to establish nonconservative stability conditions, particularly with respect to large Lipschitz constants. Simulations and experimental results have been provided in order to show the performances of the proposed method.