On the Achievable Rate of Stationary Rayleigh Flat-Fading Channels with Gaussian Inputs

TL;DR

This paper investigates the maximum achievable data rate over stationary Rayleigh flat-fading channels with Gaussian inputs, deriving new bounds and analyzing the impact of channel characteristics and power constraints on capacity.

Contribution

It introduces two novel upper bounds on the achievable rate with Gaussian inputs, including one based on channel prediction error, and characterizes the high SNR behavior with respect to Doppler frequency.

Findings

Derived tight bounds on achievable rate with Gaussian inputs.

Established the high SNR pre-log as 1-2f_d, linking it to Doppler frequency.

Compared Gaussian input bounds with peak power constrained capacity and pilot-based schemes.

Abstract

In this work, we consider a discrete-time stationary Rayleigh flat-fading channel with unknown channel state information at transmitter and receiver. The law of the channel is presumed to be known to the receiver. In addition, we assume the power spectral density (PSD) of the fading process to be compactly supported. For i.i.d. zero-mean proper Gaussian input distributions, we investigate the achievable rate. One of the main contributions is the derivation of two new upper bounds on the achievable rate with zero-mean proper Gaussian input symbols. The first one holds only for the special case of a rectangular PSD and depends on the SNR and the spread of the PSD. Together with a lower bound on the achievable rate, which is achievable with i.i.d. zero-mean proper Gaussian input symbols, we have found a set of bounds which is tight in the sense that their difference is bounded. Furthermore, we show that the high SNR slope is characterized by a pre-log of 1-2f_d, where f_d is the normalized maximum Doppler frequency. This pre-log is equal to the high SNR pre-log of the peak power constrained capacity. Furthermore, we derive an alternative upper bound on the achievable rate with i.i.d. input symbols which is based on the one-step channel prediction error variance. The novelty lies in the fact that this bound is not restricted to peak power constrained input symbols like known bounds, e.g. in [1]. Therefore, the derived upper bound can also be used to evaluate the achievable rate with i.i.d. proper Gaussian input symbols. We compare the derived bounds on the achievable rate with i.i.d. zero-mean proper Gaussian input symbols with bounds on the peak power constrained capacity given in [1-3]. Finally, we compare the achievable rate with i.i.d. zero-mean proper Gaussian input symbols with the achievable rate using synchronized detection in combination with a solely pilot based channel estimation.