The Necessity of Relay Selection

We determine necessary conditions on the structure of symbol error rate (SER) optimal quantizers for limited feedback beamforming in wireless networks with one transmitter-receiver pair and R parallel amplify-and-forward relays. We call a quantizer codebook "small" if its cardinality is less than R, and "large" otherwise. A "d-codebook" depends on the power constraints and can be optimized accordingly, while an "i-codebook" remains fixed. It was previously shown that any i-codebook that contains the single-relay selection (SRS) codebook achieves the full-diversity order, R. We prove the following: Every full-diversity i-codebook contains the SRS codebook, and thus is necessarily large. In general, as the power constraints grow to infinity, the limit of an optimal large d-codebook contains an SRS codebook, provided that it exists. For small codebooks, the maximal diversity is equal to the codebook cardinality. Every diversity-optimal small i-codebook is an orthogonal multiple-relay selection (OMRS) codebook. Moreover, the limit of an optimal small d-codebook is an OMRS codebook. We observe that SRS is nothing but a special case of OMRS for codebooks with cardinality equal to R. As a result, we call OMRS as "the universal necessary condition" for codebook optimality. Finally, we confirm our analytical findings through simulations.