Upper bounds of rates of complex orthogonal space-time block code

We derive some upper bounds of the rates of (generalized) complex orthogonal space-time block codes. We first present some new properties of complex orthogonal designs and then show that the rates of complex orthogonal space-time block codes for more than two transmit antennas are upper-bounded by 3/4. We show that the rates of generalized complex orthogonal space-time block codes for more than two transmit antennas are upper-bounded by 4/5, where the norms of column vectors may not be necessarily the same. We also present another upper bound under a certain condition. For a (generalized) complex orthogonal design, its variables are not restricted to any alphabet sets but are on the whole complex plane. A (generalized) complex orthogonal design with variables over some alphabet sets on the complex plane is also considered. We obtain a condition on the alphabet sets such that a (generalized) complex orthogonal design with variables over these alphabet sets is also a conventional (generalized) complex orthogonal design and, therefore, the above upper bounds on its rate also hold. We show that commonly used quadrature amplitude modulation (QAM) constellations of sizes above 4 satisfy this condition.