## Visual Representation of Results of Spectral Analysis of Experimental Data by Different Techniques

### Reporter: Mironov D.

Most of experimental data correspond to time dependence of measuring value.  Realizations itself are not very representative especially in case when obtained signals are result of fluctuation processes. In order to obtain qualitative and quantitative information about phenomenon from experimental data spectral methods of data analysis are employed. For the most commonly used Fourier transform it is suggested that original signal is infinite and stationary. Whereas in practice many physical phenomena are not steady and experimental data never have infinite duration. To eliminate influence of mentioned properties time-frequency methods can be employed, such as wavelet transform and Hilbert-Huang transform.
Wavelet transform is actually convolution product of the signal with so called mother wavelet function which has to have finite energy and be localized in time [1]. Time-frequency wavelet spectrum is obtained by convolution with differently dilated and translated versions of the mother wavelet function.
Hilbert-Huang transform is a method of unsteady data analysis proposed by Huang and others [2]. It consists of two main steps: empirical mode decomposition (EMD) and Hilbert transform. EMD is based on the direct extraction of the energy associated with various intrinsic time scales, the most important parameters of the system. Using this decomposition any complicated data set can be decomposed into a finite and often small number of “Intrinsic Mode Functions” (IMF) that admit well-behaved Hilbert transforms which is given by:

With this definition,   and   form the complex conjugate pair, so we can have an analytic signal,  as

where

– directly gives instantaneous magnitude of a signal, and instantaneous frequency can be calculated by differentiation:

Using magnitude and frequency distribution time-frequency spectra can be plotted.
Examples of obtained spectra are presented in Fig. 1. and correspond to fluctuations generated by flow over open shallow cavity at M = 0.75 .
a)  b)
Fig. 1. Examples of a) wavelet and b) Hilbert-Huang spectra

Peaks at the frequency of resonant fluctuations are obvious at the wavelet spectrum. From such a representation of data we can see that higher frequencies have much less magnitude, than lower ones. It can be mentioned that resonant fluctuations have an intermittent behavior and magnitude of most of peaks is approximately the same except one short upsurge at time moment 0.02 s.
Spectrum in Fig. 1 b) is plotted using Hilbert transform of several intrinsic mode functions one of which corresponds to resonant fluctuations. Surge of amplitude is also well detected here. Additionally we can see that frequency of dominant mode in the flow is not constant and varies with time. Large dispersion of frequency on Hilbert-Huang spectra is caused by unsteadiness of the flow phenomenon. It becomes larger for lower amplitudes due to digital differentiation when instantaneous frequency is calculated.
Because of smoothness of wavelet transform it is impossible to distinguish variation of amplitude of a signal shorter then several periods of fluctuation process. Hilbert-Huang can eliminate this disadvantage since magnitude is calculated for every sample of the signal and therefore has better temporal resolution.
REFERENCES
1. Bratteli O. Jorgensen P. Wavelets Through a Looking Glass. The World of Spectrum, Applied and Numerical Harmonic Analysis. — Berlin, 2002.
2. Huang N.E., Shen Z., Long S.R. and others. The Empirical Mode Decomposition and The Hilbert Spectrum for Nonlinear and Non-stationary Time Series Analysis// Proc. R. Soc. Lond. A. 1998. Vol. 454.  — pp. 903-995.
3. Mironov D.S. An Experimental Study of pressure Fluctuations Generated by an open Shallow Cavity Performed using joint Time-Frequency Techniques of Data Analisys//Thermophysics and Aeromechanics. 2011. - Vol. 18, No. 3. - pp. 369-379.

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