On the nonexistence of rate-one generalized complex orthogonal designs

Orthogonal space–time block coding proposed recently by Alamouti [1] and Tarokh, Jafarkhani, and Calderbank [4] is a promising scheme for information transmission over Rayleigh-fading channels using multiple transmit antennas due to its favorable characteristics of having full transmit diversity and a decoupled maximum-likelihood (ML) decoding algorithm. Tarokh, Jafarkhani, and Calderbank extended the theory of classical orthogonal designs to the theory of generalized, real, or complex, linear processing orthogonal designs and then applied the theory of generalized orthogonal designs to construct space–time block codes (STBCs) with the maximum possible diversity order while having a simple decoding algorithm for any given number of transmit and receive antennas. It has been known that the STBCs constructed in this way can achieve the maximum possible rate of one for every number of transmit antennas using any arbitrary real constellation and for two transmit antennas using any arbitrary complex constellation. Contrary to this, in this correspondence we prove that there does not exist rate-one STBC from generalized complex linear processing orthogonal designs for more than two transmit antennas using any arbitrary complex constellation.