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In this paper, the systematic design of a space-time block code is considered for a wireless communication system with multiple transmitter-receiver antennas and flat fading, in which channel state information is completely unknown. From the viewpoint of blind signal processing, a necessary and sufficient condition is given for the unique identification of the multi-input multi-output flat fading channel and transmitted signal. Then, some novel unique factorizations for a pair of coprime P-ary and Q-ary phase shift keying (PSK) constellations are established. With this and currently available coherent space-time block code designs, a method is developed to systematically construct full diversity blind nonunitary space-time block codes as well as unitary codes by just performing the QR decomposition of the nonunitary codes. It is proved that under a noise-free environment, the code design proposed in this paper can guarantee that the transmitted signals and the channel coefficients are uniquely identified, and that under a complex Gaussian noise environment in which Pth-order and Q th-order statistics (P and Q are co-prime) of the received signals are available, the channel coefficients can be still uniquely identified. In addition, a closed-form solution to determine the channel coefficients is obtained.