The Basic Geometric Structures of Electromagnetic Digital Information: Statistical characterization of the digital measurement of spatio-Doppler and polarimetric fluctuations of the radar electromagnetic wave

TL;DR

This paper introduces geometric and statistical methods to characterize electromagnetic wave measurements, using Fréchet barycenters and information geometry to analyze radar data in Doppler, spatio-temporal, and polarimetric contexts.

Contribution

It develops new geometric approaches and models, including a maximum entropy extension, for analyzing electromagnetic wave fluctuations in radar measurements.

Findings

Introduces a metric based on Fisher's information geometry for covariance matrices.

Defines an 'average' electromagnetic state as a Fréchet barycenter.

Provides examples of radar applications using these new tools.

Abstract

The aim is to describe new geometric approaches to define the statistics of spatio-temporal and polarimetric measurements of the states of an electromagnetic wave, using the works of Maurice Fr{\'e}chet, Jean-Louis Koszul and Jean-Marie Souriau, with in particular the notion of 'average' state of this digital measurement as a Fr{\'e}chet barycentre in a metric space and a model derived from statistical mechanics to define and calculate a maximum density of entropy (extension of the notion of Gaussian) to describe the fluctuations of the electromagnetic wave. The article will illustrate these new tools with examples of radar application for Doppler, spatio-temporal and polarimetric measurement of the electromagnetic wave by introducing a distance on the covariance matrices of the electromagnetic digital signal, based on Fisher's metric from Information Geometry.