PSI - Issue 37

Arvid Trapp et al. / Procedia Structural Integrity 37 (2022) 622–631 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 Higher-order ( > 2 ) spectra ( ) ( 1 , … , −1 ) can be interpreted as an extension of the second-order PSD 2 ( ) ( ) but include the phase information of the Fourier transform. They are multidimensional spectra, determining cross correlation between their multiple frequency arguments. HOS extend Parseval´s theorem to higher-order central moments = ∫ ( ( ) − ) ∞ − ∞ = ∫ … ∫ ( ) ∞ −∞ ( 1 , … −1 ) 1 … −1 = ∫ ( − ) ( ) ∞ − ∞ ∞ − ∞ (7) Decomposing higher-order moments such as skewness and kurtosis in the frequency domain enables to draw additional and essential information from their spectral decomposition. The trispectrum 4 ( ) ( 1 , 2 , 3 ) decomposes the fourth-order central moment into a three-dimensional frequency space. Accordingly, integrating over the trispectral set yields the scalar value of the fourth-order central moment 4 which can be normalized to extract the kurtosis (Gl. (4)). Analogously, the cumulant trispectrum 4 ( ) ( 1 , 2 , 3 ) decomposes the fourth-order cumulant 4 , which becomes the excess kurtosis after standardization. Consequently, all estimates of a cumulant trispectrum embody deviations from Gaussianity. Despite the HOS´s high potential and various research efforts (Collis, 1996; Swami et al., 1997), little practical applications have been drawn from them so far. The high complexity of HOS surmounts the simplicity of PSDs by far. Not only the prevalent redundancies, but also complex-valued entries and the multidimensionality impede the interpretation, visualization and estimation of HOS. An additional issue comes up in the visual representation of the trispectrum, when navigating through the multidimensional frequency set (Sec. Error! Reference source not found. ). 2.3. Estimating higher-order spectra (HOS) 625 4

( )

( )

(inverse) Fourier transform

analytic signal ,

filtering ( )

convolution ∫ …∫ + ∞ −∞ ∞ −∞

1 … + −1 1 … −1

multiplication 1 … −1 ∗ ( 1 + + −1 ) -th order spectrum ( ) ( 1 , … , −1 ) direct method

time-dom. resolution (integration) ∫ , 1 … , −1 ∞ −∞

filtering averaging

-th order autocorrelation ( ) ( 1 , … , −1 )

(inverse) Fourier transform

indirect method

time domain frequency domain

Figure 1: Non-parametric (conventional) estimation methods for higher-order spectra (HOS)

A central challenge of HOS is its estimation process due to the large computational effort linked to the multi dimensionality. It may be drastically reduced by using a parametric estimation approach (Nikias and Petropulu, 1993), however such approaches are unpopular as they are based on narrow assumptions. Thus, the estimation is commonly carried out employing non-parametric (conventional) estimation methods, summarized in Figure 1. These include the direct and indirect estimation methods, which solely differentiate by the chosen estimation domain (Wiener-Khinchin theorem). For relevant amounts of data, they obligate a time-segmentation, which inherently incorporates smoothing of HOS. A recently published estimator ‘filtering - averaging’ (Trapp and Wolfsteiner, 2021a) circumvents the short time analysis and provides HOS-estimation by their full time resolution using frequency-domain smoothing. This provides a robust and simple estimation approach and additional interesting applications (Trapp and Wolfsteiner, 2021c).