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A new 4times1 scheme of transmitting two real orthogonal space-time block codes (O-STBC) on the I- and Q-channels with pulse-shaped offset QAM is proposed. This scheme achieves full rate and employs simple real-symbol-wise decoding. Two families of pulse shapes for the offset QAM signal are considered: i) pulse shapes with zero cross-channel interference (CCI) between I/Q-channels and ii) pulse shapes with controlled CCI. In the zero CCI case, the real-symbol-wise decoding is optimum and the bit error probability (BEP) performance is better than that of conventional full-rate, full-diversity quasi-orthogonal space-time block codes (QO-STBC), as well as rate-1/2 and rate-3/4 complex O-STBCs. To achieve good spectral confinement while maintaining decoding simplicity, we investigate pulse shapes that introduce a small amount of controlled CCI at the sampling instants so that some performance gains can be traded off. We derive a BEP upperbound for the case of nonzero CCI and formulate the design criteria on how to construct the two real O-STBCs on the I/Q channels in order to minimize the performance degradation caused by the controlled CCI. The optimum real O-STBCs are then identified and their BEP performance is compared with that of the benchmark schemes.