A new representation of power spectral density and correlation function by means of fractional spectral moments

Abstract

In this paper, a new perspective for the representation of both the power spectral density and the correlation function by a unique class of function is introduced. We define the moments of order $\gamma$ ($\gamma \in \mathbb{C}$) of the one sided power spectral density and we call them Fractional Spectral Moments (FSM). These complex quantities remain finite also in the case in which the ordinary spectral moments diverge, and are able to represent the whole Power Spectral Density and the corresponding correlation function.

Introduction

The Spectral Moments (SMs) introduced by Vanmarcke [1] are the moments of the one-sided Power Spectral Density (PSD) function. These quantities have been introduced because of their relevance in many cases of engineering interest such as the prediction of the first excursion failure, fatigue failure, and so on. In some previous works [2], [3], [4] it has been shown that in time domains the SMs are covariances of a complex process constructed in such a way that its real part is the given process, while the imaginary part is its Hilbert transform. Such a complex process is usually referred to in literature as analytic process. In particular the zero-th and second order SMs are the common variances of the process and of its derivative, respectively, while the first SM is the cross covariance of the analytic process and its first derivative. Higher order SMs have been also introduced for statistical distribution of peaks and ranges [5]. Other relevant information on the SM may be found in [6], [7], [8], [9], [10], [11], [12], [13].

In any cases, we can say that spectral moments up to the fourth order give some information on the process but higher order spectral moments are not in general useful. Moreover, higher SMs might be divergent quantities [3] and, in any case the knowledge of the complete set of SMs cannot be used to restore the whole PSD.

In this paper, we introduce a different perspective on the spectral moments problem, that allows us to formulate a representation formula for both the entire PSD and the correlation function, in all the ranges $0<\omega <\infty$ and $0<\tau <\infty$, respectively. To this aim we define the moments of order $\gamma (\gamma \in \mathbb{C})$ of the one sided PSD and we call them as Fractional Spectral Moments (FSM). These quantities remain finite quantities also in the case in which the ordinary SMs diverge, by simply selecting a proper value of the real part of $\gamma$.

Then, a new generalized Taylor expansion in integral form in terms of Riesz fractional derivatives and integrals, involving FSMs, is introduced. By using such a Taylor integral form both the PSD and the correlation function are expressed in terms of the FSM function. Discretization of such a Taylor integral form give quite satisfactory results also in some pathological cases like the exponential type correlation function and Pierson–Moskowitz spectrum.