In this paper, noncoherent cooperative amplify-and-forward (AF) half-duplex relay systems are considered. For such systems, an asymptotic formula of pairwise error probability (PEP) for the least square error (LSE) detector is derived by using the perturbation theory on the eigenvalues. The result shows that the full-diversity-gain function mimics coherent cooperative AF half-duplex relay systems, whereas the coding gain function mimics noncoherent multiple-input-multiple-output (MIMO) systems. In addition, it is proved that, for any given nonzero received signal, the unique blind identification of both the equivalent channel and the transmitted signals in a noise-free case is equivalent to full diversity with the LSE detector in a Gaussian noisy environment. In particular, for the noncoherent AF half-duplex protocol with three nodes, a full-diversity unitary diagonal distributed space-time block code (STBC) is designed by utilizing the full-diversity criterion established in this paper and recently developed uniquely factorable constellation. Furthermore, the generalized likelihood ratio test receiver is derived for this specific coded protocol. Simulation results show that the code presented in this paper substantially outperforms the differential code and the optimally precoded training scheme in the current literature.