Waterfilling Theorems for Linear Time-Varying Channels and Related Nonstationary Sources

TL;DR

This paper establishes waterfilling theorems for the capacity of linear time-varying channels and the rate distortion of related nonstationary sources using time-frequency analysis and Szego theorems, extending classical results.

Contribution

It introduces waterfilling theorems for LTV channels and nonstationary sources based on the spread Weyl symbol and Szego theorem, providing a new theoretical framework.

Findings

Waterfilling characterizes LTV channel capacity in the time-frequency plane.

Reverse waterfilling describes the rate distortion of nonstationary sources.

The approach aligns with classical results and extends them to nonstationary scenarios.

Abstract

The capacity of the linear time-varying (LTV) channel, a continuous-time LTV filter with additive white Gaussian noise, is characterized by waterfilling in the time-frequency plane. Similarly, the rate distortion function for a related nonstationary source is characterized by reverse waterfilling in the time-frequency plane. Constraints on the average energy or on the squared-error distortion, respectively, are used. The source is formed by the white Gaussian noise response of the same LTV filter as before. The proofs of both waterfilling theorems rely on a Szego theorem for a class of operators associated with the filter. A self-contained proof of the Szego theorem is given. The waterfilling theorems compare well with the classical results of Gallager and Berger. In the case of a nonstationary source, it is observed that the part of the classical power spectral density is taken by the Wigner-Ville spectrum. The present approach is based on the spread Weyl symbol of the LTV filter, and is asymptotic in nature. For the spreading factor, a lower bound is suggested by means of an uncertainty inequality.