Capacity Pre-Log of Noncoherent SIMO Channels via Hironaka's Theorem

TL;DR

This paper demonstrates that adding a second receive antenna in a noncoherent SIMO channel significantly reduces the capacity pre-log penalty caused by channel uncertainty, approaching a near-coherent performance as block length increases.

Contribution

It introduces a novel approach using algebraic geometry to analyze the capacity pre-log of SIMO channels, showing the benefit of multiple antennas in resolving channel uncertainty.

Findings

Adding one receive antenna reduces the pre-log penalty from 1-Q/L to 1/L.

The second antenna enables the system to effectively resolve channel uncertainty.

The approach leverages Hironaka's Theorem from algebraic geometry.

Abstract

We find the capacity pre-log of a temporally correlated Rayleigh block-fading SIMO channel in the noncoherent setting. It is well known that for block-length L and rank of the channel covariance matrix equal to Q, the capacity pre-log in the SISO case is given by 1-Q/L. Here, Q/L can be interpreted as the pre-log penalty incurred by channel uncertainty. Our main result reveals that, by adding only one receive antenna, this penalty can be reduced to 1/L and can, hence, be made to vanish in the large-L limit, even if Q/L remains constant as L goes to infinity. Intuitively, even though the SISO channels between the transmit antenna and the two receive antennas are statistically independent, the transmit signal induces enough statistical dependence between the corresponding receive signals for the second receive antenna to be able to resolve the uncertainty associated with the first receive antenna's channel and thereby make the overall system appear coherent. The proof of our main theorem is based on a deep result from algebraic geometry known as Hironaka's Theorem on the Resolution of Singularities.