On the Low SNR Capacity of Peak-Limited Non-Coherent Fading Channels with Memory

TL;DR

This paper investigates the capacity limits of non-coherent Gaussian fading channels with memory under peak-power constraints in low SNR conditions, revealing a link between capacity and prediction error of the fading process.

Contribution

It establishes a novel connection between low SNR capacity behavior and the prediction error of fading processes, distinguishing between different fading law categories and optimal input strategies.

Findings

For slowly forgetting channels, IID inputs achieve low SNR capacity.

For quickly forgetting channels, block-constant magnitude inputs are near-optimal.

The capacity behavior relates to the prediction error of the fading process.

Abstract

The capacity of non-coherent stationary Gaussian fading channels with memory under a peak-power constraint is studied in the asymptotic weak-signal regime. It is assumed that the fading law is known to both transmitter and receiver but that neither is cognizant of the fading realization. A connection is demonstrated between the asymptotic behavior of channel capacity in this regime and the asymptotic behavior of the prediction error incurred in predicting the fading process from very noisy observations of its past. This connection can be viewed as the low signal-to-noise ratio (SNR) analog of recent results by Lapidoth & Moser and by Lapidoth demonstrating connections between the high SNR capacity growth and the noiseless or almost-noiseless prediction error. We distinguish between two families of fading laws: the ``slowly forgetting'' and the ``quickly forgetting''. For channels in the former category the low SNR capacity is achieved by IID inputs, whereas in the latter such inputs are typically sub-optimal. Instead, the asymptotic capacity can be approached by inputs with IID phase but block-constant magnitude.