Orthogonal designs with maximal rates

Orthogonal designs have been used as space-time block codes for wireless communications with multiple transmit antennas, which can achieve full transmit diversity and have a very simple decoupled maximum-likelihood decoding algorithm. The rate of an orthogonal design is defined as the ratio of the number of transmitted information symbols in a block of channel uses to the length of the given block, which reflects the bandwidth efficiency of the employed space-time block code constructed from the orthogonal design. This paper focuses on the analysis and synthesis of orthogonal designs with the maximum possible rates, which may be real or complex and square or rectangular matrices. We first provide several representations of orthogonal designs and their characterizations in terms of Hurwitz-Radon matrix equations. Next, we observe that the real orthogonal designs, square or rectangular, and the complex square orthogonal designs with maximal rates have been well understood from the existing results in the mathematics literature which can be dated back to 1890s. However, unfortunately, it is not the case for the complex rectangular orthogonal designs with rates as high as possible. We then construct a class of complex orthogonal designs for any number of transmit antennas. The proposed complex orthogonal designs for the number of transmit antennas n=2m-1 and 2m have the same rate m+1 and 2m, where m is any natural number. Finally, we demonstrate that, for n=2m-1 and 2m with any given natural number m, the value m+1 and 2m is the maximum possible rate that the complex orthogonal designs, square or rectangular, with n transmit antennas can achieve.