The necessity of relay selection for beamforming in amplify-and-forward networks

We determine necessary conditions on the structure of symbol error rate optimal quantizers for limited feedback beamforming in wireless networks with one transmitter-receiver pair and R parallel amplify-and-forward relays. It was previously shown that any quantizer codebook that contains the single-relay selection (SRS) codebook achieves full-diversity. We prove the converse: Every full-diversity codebook contains the SRS codebook. We also determine the structure and the achievable diversity of codebooks with cardinality less than R.