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Automatic modulation classification (AMC) is a very important process for any receiver that has no, or limited, knowledge of the modulation scheme of the received signal. In addition to military systems, AMC has been receiving an increasing amount of interest in the context of spectrum sharing cognitive radio systems. In this paper, we present an AMC system based upon the distributed processing of local classification decisions made by multiple radios. These radios consist of two stages: a cyclic spectrum feature-based AMC stage and a decision making (DM) stage that makes a local decision that is sent to a fusion center. This fusion center then makes a global decision based on its own AMC stage and on the local decisions made by the radios. A nonlinear Gauss-Seidel iterative algorithm is used to find the person-by-person optimum decision rules for the fusion center and DMs. It is shown that the proposed distributed approach results in a significant increase in the probability of signal detection and correct classification, at the expense of requiring messages to be transmitted among the fusion center and the radios in the system.