Degrees of Freedom of a Communication Channel and Kolmogorov numbers

TL;DR

This paper explores the relationship between Kolmogorov numbers and the degrees of freedom in communication channels modeled by compact operators, providing insights into their connection and methods for numerical evaluation.

Contribution

It establishes a close relationship between Kolmogorov numbers and degrees of freedom in channels, and offers a numerical method for computing Kolmogorov numbers.

Findings

Kolmogorov numbers correspond to jump points in degrees of freedom versus noise level.

A numerical method for evaluating Kolmogorov numbers is developed.

Theoretical link between operator theory and communication channel capacity is demonstrated.

Abstract

In this note, we show that the operator theoretic concept of Kolmogorov numbers and the number of degrees of freedom at level $\epsilon$ of a communication channel are closely related. Linear communication channels may be modeled using linear compact operators on Banach or Hilbert spaces and the number of degrees of freedom of such channels is defined to be the number of linearly independent signals that may be communicated over this channel, where the channel is restricted by a threshold noise level. Kolmogorov numbers are a particular example of $s$-numbers, which are defined over the class of bounded operators between Banach spaces. We demonstrate that these two concepts are closely related, namely that the Kolmogorov numbers correspond to the "jump points" in the function relating numbers of degrees of freedom with the noise level $\epsilon$. We also establish a useful numerical computation result for evaluating Kolmogorov numbers of compact operators.