On space-time block codes from complex orthogonal designs

Space-time block codes from orthogonal designs recently proposed by Alamouti, and Tarokh-Jafarkhani-Calderbank have attracted considerable attention due to the fast maximum-likelihood (ML) decoding and the full diversity. There are two classes of space-time block codes from orthogonal designs. One class consists of those from real orthogonal designs for real signal constellations which have been well developed in the mathematics literature. The other class consists of those from complex orthogonal designs for complex constellations for high data rates, which are not well developed as the real orthogonal designs. Since orthogonal designs can be traced back to decades, if not centuries, ago and have recently invoked considerable interests in multi-antenna wireless communications, one of the goals of this paper is to provide a tutorial on both historical and most recent results on complex orthogonal designs. For space-time block codes from both real and (generalized) complex orthogonal designs (GCODs) with or without linear processing, Tarokh, Jafarkhani and Calderbank showed that their rates cannot be greater than 1. While the maximum rate 1 can be reached for real orthogonal designs for any number of transmit antennas from the Hurwitz-Radon constructive theory, Liang and Xia recently showed that rate 1 for the GCODs (square or nons-square size) with linear processing is not reachable for more than two transmit antennas. For GCODs of square size, the designs with the maximum rates have been known, which are related to the Hurwitz theorem. In this paper, We briefly review these results and give a simple and intuitive interpretation of the realization. For GCODs without linear processing (square or non-square size), we prove that the rates cannot be greater than 3/4 for more than two transmit antennas.