On the capacity of the dither-quantized Gaussian channel

TL;DR

This paper analyzes the capacity of a Gaussian channel with dithered uniform quantization, showing how it approaches the unquantized capacity as quantization becomes finer and examining low-SNR behavior under various power constraints.

Contribution

It provides a detailed analysis of the channel capacity with dithered quantization, including asymptotic behaviors and the impact of quantization resolution and power constraints.

Findings

Capacity approaches unquantized Gaussian channel as quantization step size decreases.

Capacity tends to zero as quantization step size increases.

Low-SNR asymptotic capacity matches unquantized channel when peak-power constraint is absent.

Abstract

This paper studies the capacity of the peak-and-average-power-limited Gaussian channel when its output is quantized using a dithered, infinite-level, uniform quantizer of step size $\Delta$. It is shown that the capacity of this channel tends to that of the unquantized Gaussian channel when $\Delta$ tends to zero, and it tends to zero when $\Delta$ tends to infinity. In the low signal-to-noise ratio (SNR) regime, it is shown that, when the peak-power constraint is absent, the low-SNR asymptotic capacity is equal to that of the unquantized channel irrespective of $\Delta$. Furthermore, an expression for the low-SNR asymptotic capacity for finite peak-to-average-power ratios is given and evaluated in the low- and high-resolution limit. It is demonstrated that, in this case, the low-SNR asymptotic capacity converges to that of the unquantized channel when $\Delta$ tends to zero, and it tends to zero when $\Delta$ tends to infinity. Comparing these results with achievability results for (undithered) 1-bit quantization, it is observed that the dither reduces capacity in the low-precision limit, and it reduces the low-SNR asymptotic capacity unless the peak-to-average-power ratio is unbounded.