Necessary and sufficient conditions for full diversity order in correlated Rayleigh fading beamforming and combining systems

Transmit beamforming and receive combining are low complexity, linear techniques that make use of the spatial diversity advantage provided by transmitters and/or receivers employing multiple antennas. There has been a growing interest in designing beamforming schemes for frequency division duplexing systems that use a limited amount of feedback from the receiver to the transmitter. This limited feedback conveys a beamforming vector chosen from a finite set known to both the transmitter and receiver. These techniques often use a set of beamforming vectors where the probability of error expression can not be easily formulated or bounded. It is of utmost importance to guarantee that the sets of beamforming and combining vectors are chosen such that full diversity order is achieved. For this reason, necessary and sufficient conditions on the sets of possible beamformers and combiners are derived that guarantee full diversity order in correlated Rayleigh fading.