On beamforming with finite rate feedback in multiple-antenna systems

TLDR

The paper examines a multiple‑antenna system in which the transmitter has quantized knowledge of instantaneous channel realizations. The authors aim to derive a universal lower bound on outage probability for any finite set of beamformers when the transmitter uses quantized channel information for beamforming. They use a geometrical bounding technique that minimizes the maximum inner product between beamforming vectors, equivalent to designing unitary space‑time codes, to derive the bound and propose a design criterion for beamformers. The bound quantifies the gain per feedback bit and shows that finite‑feedback systems approach the perfect‑information case as the number of bits increases, with optimal beamformers corresponding to optimal Grassmannian packings of two‑dimensional subspaces.

Abstract

We study a multiple-antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations. Assuming that the transmitter uses the quantized information for beamforming, we derive a universal lower bound on the outage probability for any finite set of beamformers. The universal lower bound provides a concise characterization of the gain with each additional bit of feedback information regarding the channel. Using the bound, it is shown that finite information systems approach the perfect information case as (t-1)2/sup -B/t-1/, where B is the number of feedback bits and t is the number of transmit antennas. The geometrical bounding technique, used in the proof of the lower bound, also leads to a design criterion for good beamformers, whose outage performance approaches the lower bound. The design criterion minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook, and is equivalent to the problem of designing unitary space-time codes under certain conditions. Finally, we show that good beamformers are good packings of two-dimensional subspaces in a 2t-dimensional real Grassmannian manifold with chordal distance as the metric.