Oversampling Increases the Pre-Log of Noncoherent Rayleigh Fading Channels

TL;DR

This paper demonstrates that oversampling the output of a noncoherent Rayleigh fading channel at twice the symbol rate increases the capacity pre-log at high SNR, surpassing the traditional symbol matched filtering approach.

Contribution

It proves that oversampling improves the capacity pre-log, showing symbol matched filtering is not optimal for continuous-time channel capacity at high SNR.

Findings

Oversampling doubles the capacity pre-log compared to symbol matched filtering.

Symbol matched filtering is suboptimal for maximizing capacity in noncoherent Rayleigh channels.

Oversampling achieves a capacity pre-log of at least 1-1/N, exceeding the 1-Q/N from symbol sampling.

Abstract

We analyze the capacity of a continuous-time, time-selective, Rayleigh block-fading channel in the high signal-to-noise ratio (SNR) regime. The fading process is assumed stationary within each block and to change independently from block to block; furthermore, its realizations are not known a priori to the transmitter and the receiver (noncoherent setting). A common approach to analyzing the capacity of this channel is to assume that the receiver performs matched filtering followed by sampling at symbol rate (symbol matched filtering). This yields a discrete-time channel in which each transmitted symbol corresponds to one output sample. Liang & Veeravalli (2004) showed that the capacity of this discrete-time channel grows logarithmically with the SNR, with a capacity pre-log equal to $1-{Q}/{N}$. Here, $N$ is the number of symbols transmitted within one fading block, and $Q$ is the rank of the covariance matrix of the discrete-time channel gains within each fading block. In this paper, we show that symbol matched filtering is not a capacity-achieving strategy for the underlying continuous-time channel. Specifically, we analyze the capacity pre-log of the discrete-time channel obtained by oversampling the continuous-time channel output, i.e., by sampling it faster than at symbol rate. We prove that by oversampling by a factor two one gets a capacity pre-log that is at least as large as $1-1/N$. Since the capacity pre-log corresponding to symbol-rate sampling is $1-Q/N$, our result implies indeed that symbol matched filtering is not capacity achieving at high SNR.