Sampling Distributions of Random Electromagnetic Fields in Mesoscopic or Dynamical Systems

TL;DR

This paper derives the sampling probability density functions for localized random electromagnetic fields in various environments, enabling improved statistical estimation and confidence interval calculation for field measurements.

Contribution

It introduces new sampling pdfs for electromagnetic field quantities, accounting for dimensioned and standardized forms, and compares their behaviors to classical distributions.

Findings

Dimensioned electric field quantities follow Bessel K sampling pdfs.

Standardized quantities follow Student t, F, and root-F distributions with heavier tails.

Classical small-sample theory overestimates uncertainties for dimensionless quantities.

Abstract

We derive the sampling probability density function (pdf) of an ideal localized random electromagnetic field, its amplitude and intensity in an electromagnetic environment that is quasi-statically time-varying statistically homogeneous or static statistically inhomogeneous. The results allow for the estimation of field statistics and confidence intervals when a single spatial or temporal stochastic process produces randomization of the field. Results for both coherent and incoherent detection techniques are derived, for Cartesian, planar and full-vectorial fields. We show that the functional form of the sampling pdf depends on whether the random variable is dimensioned (e.g., the sampled electric field proper) or is expressed in dimensionless standardized or normalized form (e.g., the sampled electric field divided by its sampled standard deviation). For dimensioned quantities, the electric field, its amplitude and intensity exhibit different types of Bessel $K$ sampling pdfs, which differ significantly from the asymptotic Gauss normal and $\chi^{(2)}_{2p}$ ensemble pdfs when $\nu$ is relatively small. By contrast, for the corresponding standardized quantities, Student $t$, Fisher-Snedecor $F$ and root-$F$ sampling pdfs are obtained that exhibit heavier tails than comparable Bessel $K$ pdfs. Statistical uncertainties obtained from classical small-sample theory for dimensionless quantities are shown to be overestimated compared to dimensioned quantities. Differences in the sampling pdfs arising from de-normalization versus de-standardization are obtained.