In this paper, convergence analysis of the extended Kalman filter (EKF), when used as an observer for nonlinear deterministic discrete-time systems, is presented. Based on a new formulation of the first-order linearization technique, sufficient conditions to ensure local asymptotic convergence are established, Furthermore, it is shown that the design of the arbitrary matrix, namely R(k) in the paper, plays an important role in enlarging the domain of attraction and then improving the convergence of the modified EKF significantly, The efficiency of this approach, compared to the classical version of the EKF, is shown through a nonlinear identification problem as well as a state and parameter estimation of nonlinear discrete-time systems.