Signals Featuring Harmonics with Random Frequencies -- Spectral, Distributional and Ergodic Properties

Abstract

It has been observed that an interesting class of non-Gaussian stationary processes is obtained when in the harmonics of a signal with random amplitudes and phases, frequencies can also vary randomly. In the resulting models, the statistical distribution of frequencies determines the process spectrum while the distribution of amplitudes governs the process distributional properties. Since decoupling the distributional and spectral properties can be advantageous in applications, we thoroughly investigate a variety of properties exhibited by these models. We extend previous work that represented processes as finite sum of harmonics, by conveniently embedding them into the class of harmonizable processes. Harmonics are integrated with respect to independently scattered second order non-Gaussian random measures. The proposed approach provides with a proper mathematical framework that allows to study spectral, distributional, and ergodic properties. The mathematical elegance of these representations avoids serious conceptual and technical difficulties with limiting behavior of the models while at the same time facilitates derivation of their fundamental properties. In particular, the multivariate distributions are obtained and the asymptotic behavior of time averages is formally derived through the strong ergodic theorem. Several deficiencies following from the previous approaches are resolved and some of the results appearing in the literature are corrected and extended. It is shown that due to the lack of ergodicity processes exhibit an interesting property of non-trivial randomness remaining in the limit of time averages. This feature maybe utilized to modelling signals observed in the presence of influential and variable random factors.