Transmit diversity using rotated constellations with Hadamard transform

We introduce a general framework to build block space-time (ST) codes by the combination of rotated constellations and the Hadamard transform. We prove that these codes have a maximum transmit diversity over a quasi-static and fast fading channels. Furthermore, we prove that the conjunction of the rotated constellations and the Hadamard transform is quasi-optimum in the sense of maximizing the coding gain. The main modulated data stream is split into n substreams which, by the aid of a rotation and the corresponding Hadamard transform, are encoded into a ST n×n matrix. Given m receive antennas, the received signal at each one is the superposition of n symbols affected by independent fades and disturbed by noise. The generalized algebraic space-time (GAST) codes transmit at a rate of 1 symbol/sec for any number n such that n is 1, 2 or a multiple of 4. They maintain their rate, diversity and coding gains whether the used constellation is real or complex. Using the GAST codes with complex constellations largely outperforms the codes from orthogonal design for high spectral efficiency since the latter transmit at 1/2 symbol/sec, and the GAST codes are available for a much larger spectrum of dimensions. Maximum likelihood (ML) decoding is performed by the sphere decoder (SD) and has a moderate complexity (up to n=32)