Delocalisation of one-dimensional marginals of product measures and the capacity of LTI discrete channels

TL;DR

This paper proves that discrete LTI channels with phase independence have positive capacity by applying a delocalisation theorem for product measures, extending existing results in measure theory.

Contribution

It demonstrates the positivity of capacity for a class of LTI channels under phase independence using advanced measure delocalisation techniques.

Findings

Capacity of LTI channels is positive under PI assumption

Extended delocalisation results for product measures

Application of measure theory to information theory

Abstract

We consider discrete linear time invariant (LTI) channels satisfying the phase independence (PI) assumption. We show that under the PI assumption the capacity of LTI channels is positive. The main technical tool that we use to establish the positivity of the capacity is the delocalisation theorem for one-dimensional marginals of the product measure due to Ball and Nazarov. We also prove two delocalisation results that can be seen as extensions of Ball-Nazarov Theorem.