On the Capacity of the Discrete-Time Channel with Uniform Output Quantization

TL;DR

This paper analyzes the capacity of discrete-time channels with uniform output quantization, comparing saturation and wrapping schemes, and provides bounds and capacity-achieving input distributions for these quantization methods.

Contribution

It offers a detailed analysis of two quantization schemes, determining their capacities and the input distributions that achieve these capacities, extending prior theoretical work.

Findings

Capacity of wrapping quantization is explicitly determined.

When noise is small, saturation quantization's capacity is bounded below by wrapping.

As quantization levels increase, capacity bounds differ by only 0.26 bits.

Abstract

This paper provides new insight into the classical problem of determining both the capacity of the discrete-time channel with uniform output quantization and the capacity achieving input distribution. It builds on earlier work by Gallager and Witsenhausen to provide a detailed analysis of two particular quantization schemes. The first is saturation quantization where overflows are mapped to the nearest quantization bin, and the second is wrapping quantization where overflows are mapped to the nearest quantization bin after reduction by some modulus. Both the capacity of wrapping quantization and the capacity achieving input distribution are determined. When the additive noise is gaussian and relatively small, the capacity of saturation quantization is shown to be bounded below by that of wrapping quantization. In the limit of arbitrarily many uniform quantization levels, it is shown that the difference between the upper and lower bounds on capacity given by Ihara is only 0.26 bits.