Propagation of the Cross-Spectral Density of Stochastic Electromagnetic Fields

In the case of EMI problems we have to deal with stochastic electromagnetic fields. If a stochastic electromagnetic field originates from a sufficiently large number of statistically independent processes, the field amplitudes will exhibit a Gaussian probability distribution due to the central limit theorem. A Gaussian process can be described completely by its mean value and its second order moments. These second order moments are represented by the respective auto- and cross correlation functions. Stochastic electromagnetic fields with Gaussian probability distribution can be described completely by the autocorrelation spectrum of each field variable and the cross correlation spectra of field variables at distinct points of observation [1, 2]. Due to the equivalence principle an equivalent source distribution determined by amplitude and phase scanning of the tangential electric or magnetic field on a surface enclosing the radiating structure is equivalent to the internal sources and allows to model the environmental field. Characterization of a stochastic electromagnetic field requires the sampling of the EM field in pairs of observation points and the determination of the cross correlation functions for all pairs of field samples [2]. For full characterization of a stochastic electromagnetic field in the near-field region two-point sampling allows high resolution far below the wavelength limit. On the other hand, due to the increase of transverse coherence with increasing distance from the source [3], with increasing distance the required density of reference scanning points decreases. It has been shown, that in the far-field the correlation matrices summarizing the sampled E-field correlation spectra exhibit Toeplitz character [4]. Therefore in the far-field a single reference point is required. On the other hand, scanning in the far-field region limits the reconstruction of the spatial source distribution to a resolution in the order of half the wavelength. In this paper we introduce the Helmholtz equation for the correlation dyadics of stochastic electromagnetic fields and investigate the propagation of the cross-spectral density of stochastic electromagnetic fields to show the trade-off between achievable resolution in source reconstruction and the required density of two-point sampling points.     «

In the case of EMI problems we have to deal with stochastic electromagnetic fields. If a stochastic electromagnetic field originates from a sufficiently large number of statistically independent processes, the field amplitudes will exhibit a Gaussian probability distribution due to the central limit theorem. A Gaussian process can be described completely by its mean value and its second order moments. These second order moments are represented by the respective auto- and cross correlation functi...     »