Spectral representation and asymptotic properties of certain deterministic fields with innovation components

In this paper we derive the spectral and ergodic properties of a special class of homogeneous random fields, which includes an important family of evanescent random fields. Based on a derivation of the resolution of the identity for the operators generating the homogeneous field, and using the properties of measurable transformations, the spectral representation of both the field and its covariance sequence are derived. A necessary and sufficient condition for the existence of such representation is introduced. Using an analysis approach that employs the solution to the linear Diophantine equations, further characterization and modeling of the spectral properties of evanescent fields are provided by considering their spectral pseudo-density function, defined in this paper. The geometric properties of the spectral pseudo-density of the evanescent field are investigated. Finally, necessary and sufficient conditions for mean and strong ergodicity of the first and second order moments of these fields are derived. The analysis, initially carried out for complex valued random fields, is later extended to include the case of real valued fields.