The Necessity of Relay Selection

TL;DR

This paper establishes the fundamental structure of optimal quantizers for relay selection in wireless networks, showing that the single-relay selection and orthogonal multiple-relay selection are necessary for optimal diversity performance.

Contribution

It proves that full-diversity optimal codebooks must include SRS and OMRS structures, providing a universal necessary condition for relay selection codebook optimality.

Findings

Full-diversity i-codebooks contain SRS codebooks.

Optimal large d-codebooks asymptotically contain SRS codebooks.

Small codebooks' maximal diversity equals their cardinality.

Abstract

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.