Variations on a theme by Schalkwijk and Kailath

TL;DR

This paper explains and extends the Schalkwijk-Kailath feedback coding scheme for Gaussian channels, showing improved error decay rates and bounds, including in the infinite bandwidth limit, with implications for reliable communication.

Contribution

It provides a simple derivation of the Schalkwijk-Kailath scheme, introduces a modified scheme with linearly increasing error decay, and establishes bounds on error probability for finite and infinite bandwidths.

Findings

Error probability decreases exponentially with block length.

Modified scheme achieves zero error probability in infinite bandwidth.

Bounds on error probability are derived for finite bandwidth.

Abstract

Schalkwijk and Kailath (1966) developed a class of block codes for Gaussian channels with ideal feedback for which the probability of decoding error decreases as a second-order exponent in block length for rates below capacity. This well-known but surprising result is explained and simply derived here in terms of a result by Elias (1956) concerning the minimum mean-square distortion achievable in transmitting a single Gaussian random variable over multiple uses of the same Gaussian channel. A simple modification of the Schalkwijk-Kailath scheme is then shown to have an error probability that decreases with an exponential order which is linearly increasing with block length. In the infinite bandwidth limit, this scheme produces zero error probability using bounded expected energy at all rates below capacity. A lower bound on error probability for the finite bandwidth case is then derived in which the error probability decreases with an exponential order which is linearly increasing in block length at the same rate as the upper bound.