Finite-Time Capacity: Making Exceed-Shannon Possible?

TL;DR

This paper investigates the finite-time capacity of communication channels, revealing that transmitted bits per second can surpass Shannon capacity within finite observation windows, and provides new formulas and proofs for this phenomenon.

Contribution

It introduces a novel finite-time capacity formula, proves the Exceed-Shannon phenomenon, and links the problem to operator theory for correlated Gaussian processes.

Findings

Finite-time capacity can exceed Shannon capacity.

Derived a finite-time capacity formula using Mercer expansion.

Proved the existence and achievability of the Exceed-Shannon phenomenon.

Abstract

Shannon-Hartley theorem can accurately calculate the channel capacity when the signal observation time is infinite. However, the calculation of finite-time capacity, which remains unknown, is essential for guiding the design of practical communication systems. In this paper, we investigate the capacity between two correlated Gaussian processes within a finite-time observation window. We first derive the finite-time capacity by providing a limit expression. Then we numerically compute the maximum transmission rate within a single finite-time window. We reveal that the number of bits transmitted per second within the finite-time window can exceed the classical Shannon capacity, which is called as the Exceed-Shannon phenomenon. Furthermore, we derive a finite-time capacity formula under a typical signal autocorrelation case by utilizing the Mercer expansion of trace class operators, and reveal the connection between the finite-time capacity problem and the operator theory. Finally, we analytically prove the existence of the Exceed-Shannon phenomenon in this typical case, and demonstrate the achievability of the finite-time capacity and its compatibility with the classical Shannon capacity.