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In this paper, we address the problem of observer design for a class of Lipschitz nonlinear discrete-time systems with time-delay. The main contribution lies in the use of a new structure of the proposed observer. Thanks to this new design, a new nonrestrictive synthesis condition, expressed in term of linear matrix inequality (LMI), is obtained. Indeed, the obtained LMI contains more degree of freedom than those established by the approaches available in the literature which consider a simple Luenberger observer. An extension of the presented result to Hinfin performance analysis is given in order to take into account the noise (if it exists) affecting the considered system.