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

We derive some upper bounds of the rates of (generalized) complex orthogonal spacetime 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.