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A full-rate distributed space-time (ST) code design is proposed for amplify-and-forward cooperative wireless channels. By employing a signal space diversity technique at the source node and a unique signature vector at each relay node, the proposed distributed ST codes can achieve full cooperative diversity and full rate. The achievable diversity gain is shown to be M + 1, where M is the number of relay nodes. Optimal power allocation is also proposed to maximize the coding gain under a total power constraint.