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Automatic modulation recognition is advantageous for wireless communication systems employing adaptive modulation, software-defined radio, and cognitive radio. In this paper, we consider a phase based maximum likelihood (ML) approach for identifying the modulation format of a linearly modulated signal. Since the optimal ML scheme is computationally intensive, we propose two approximate ML alternatives, which can offer close-to-optimal performance with reduced complexity. We then present a general performance analysis for classification of K types of modulation constellations. For K<;=5, probability of correct classification (Pcc) can be evaluated via simplified integration. In the case of K>;5, we obtain a set of upper bounds on Pcc, which provide a tradeoff between accuracy and complexity in calculating the Pcc. In addition, asymptotic behavior of phase based ML classification schemes is investigated.