Noncoherent SIMO Pre-Log via Resolution of Singularities

TL;DR

This paper derives a lower bound on the noncoherent capacity pre-log of a SIMO channel, showing that adding a single receive antenna can significantly reduce the penalty caused by channel uncertainty, using algebraic geometry techniques.

Contribution

It introduces a novel lower bound on the SIMO channel capacity pre-log, leveraging resolution of singularities to improve understanding of channel capacity limits.

Findings

Adding one receive antenna reduces the penalty to 1/L under certain conditions.

The lower bound generalizes previous results for arbitrary rank and block-length.

Resolution of singularities is used as a key technical tool.

Abstract

We establish a lower bound on the noncoherent capacity pre-log of a temporally correlated Rayleigh block-fading single-input multiple-output (SIMO) channel. Our result holds for arbitrary rank Q of the channel correlation matrix, arbitrary block-length L > Q, and arbitrary number of receive antennas R, and includes the result in Morgenshtern et al. (2010) as a special case. It is well known that the capacity pre-log for this channel in the single-input single-output (SISO) case is given by 1-Q/L, where Q/L is the penalty incurred by channel uncertainty. Our result reveals that this penalty can be reduced to 1/L by adding only one receive antenna, provided that L \geq 2Q - 1 and the channel correlation matrix satisfies mild technical conditions. The main technical tool used to prove our result is Hironaka's celebrated theorem on resolution of singularities in algebraic geometry.