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This paper provides a nuclear norm minimization approach to an identification of linear systems with finite word-length data. Measurement data sampled from low resolution sensors are sequence of a few bit data and have large quantization errors, which deteriorate the identification accuracy. In this paper, the identification problem is formulated as a rank minimization problem, and the nuclear norm heuristic is introduced to estimate the model order and precise values of finite word-length data. An iterative algorithm is proposed based on the weighted nuclear norm minimization and its semidefinite programming formulation. Numerical examples demonstrate that we can estimate both the model order and parameters and show the effectiveness of the proposed method.