Communicating One Bit over a Delay Constrained Gaussian MIMO Channel with Feedback

TL;DR

This paper derives the energy-optimal strategy for transmitting a single bit over a Gaussian MIMO channel with feedback within a limited number of uses, showing significant performance improvements especially with minimal feedback.

Contribution

It introduces a dynamic programming-based optimal coding/decoding scheme for one-bit communication over Gaussian channels with feedback, highlighting the benefits of feedback and its discontinuous nature.

Findings

Feedback significantly improves performance.

Most gains achieved with just one-bit feedback.

Optimal strategies are discontinuous.

Abstract

The energy-optimal scheme is found for communicating one bit over a memoryless Gaussian channel with an ideal feedback channel. It is assumed that the channel is allowed to be used at most N times before decoding. The optimal coding/decoding strategy is derived by dynamic programming. It is found that feedback gives a significant performance gain and that the optimal strategies are discontinuous. It is also shown that most of the performance increase can be obtained even with a one-bit feedback channel. The optimal scheme is compared with the strategy by Kailath-Schalkwijk and is found to be significantly more effective. For the case of a diagonal MIMO channel where measurement noise variances are equal along the sub channels we also show that the problem can be reduced to the previous case of transmitting one bit over a scalar feedback channel.