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In this paper, we consider a noncoherent amplify-and-forward (AF) half-duplex cooperative relay system, where the channel gains are completely unknown at the destination. For such a system, the currently available design of noncoherent unitary space-time block codes for the multiple-input-multiple-output (MIMO) systems cannot be applied because of the signal matrix having a certain triangular structure. In this paper, we propose the design of full-diversity nonunitary and unitary diagonal distributed space-time block codes for the noncoherent AF half-duplex protocol. We prove that both the nonunitary and unitary codes enable the unique identification of the channel and transmitted signals in the noise-free case for any given nonzero received signals, as well as full noncoherent diversity in the noise case for the least square error (LSE) receiver. Computer simulations show that our unitary code obtains about 5-dB signal-to-noise-ratio (SNR) gains at the symbol error rate of 10-5 over the precoded training scheme and the differential coding scheme.