Level 2: Spectral Estimation

In the next level, the mathematical model incorporates little a priori information and is used to analyse the information content (spectrum, time-frequency, etc...) of the raw measurement data, in an attempt to draw some rough conclusions about the nature of the signals under investigation (Fig. (6)). The gaol of the spectral estimation is to estimate the spectral density of a random signal from a sequence of time samples of the signal. The spectral density characterizes the frequency content of the signal. The technique of spectral estimation is common used for monitoring of rotating machineries such as for example gear- or bearing applications and the application on rotating machinery has made the transition from research topic to actual practice [1], [19].

x(t)

Spectral Es/mator

Frequency

Phase angle ω

θ ω₀

θ₀

Figure 6: Typical example of a spectral estimation process: analysis of two sine waves.

Fourier Transform

If a signal is discrete and periodic then it can be represented by a Fourier series [20]. The discrete Fourier series representation of a signal is given by the pair

x(t) =X

where am are the discrete Fourier coefficients and e^−jx= cos(x) − jsin(x). These relations imply that the discrete signal under investigation can be decomposed into a unique set of harmonics. In other words, the representation of the signal as a Fourier series enables us to extract spectral content from the signal, and the corresponding coefficients can be considered as an equivalent representation of the information in the signal under investigation. However, the information about the harmonic base functions has global support. For example, in decomposing a signal where a discontinuity in time is present, all the weights of the basis function will be affected; the phenomenon of discontinuity is diluted. Therefore, FT is usually used for stationary signals [21].

Short-Time Fourier Transform

The classical Fourier transform is a very strong signal analysis tool for stationary signals. However, for non-stationary signals it fails to describe how the frequency evolves with time. For these types of signals the short-time Fourier transform (STFT) is used, as it is able to extract the frequency content of the signal and its corresponding time value. The STFT breaks up the non-stationary signal into small segments, so-called windows, and then the FT is applied to each segment to ascertain the frequencies that exist in that segment. These spectra, taken together, indicate how the spectrum varies in time [21, 5]. However, the STFT windowing process results in a trade-off between the time and the frequency resolutions, and therefore accuracy cannot be obtained simultaneously in both time and frequency domains.

The method of STFT is very commonly used for analysing measured response

signals from structures, since the STFT gives a very brief overview of the frequency spectra of the signal. Here, only the most cited publications are named, e.g., Ihn and Chang [22] or Giurgiutu and Cuc [23]. Droubi et al. [24] used in their work for the processing of measured data from acoustic emission data the Fast-Fourier Transform in order to extract the spectral information.

Wavelet Transform

The wavelet transform (WT) breaks up a signal into a series of wavelets that are shifted and scaled. The functional basis consists of dilated and shifted versions of a single basis function called a mother wavelet, which can be seen as a wave packet. The WT has the advantage that it can adjust the window length according to the needs of the real signal. Therefore, detailed information (high-frequency components) can be obtained with a narrow window and general information (low-frequency components) with a wide window. A good introduction to WT is given in the book by Staszewski et al. [10], where the two main types of WT, the continuous wavelet transform and the discrete wavelet transform, are described. However, the WT has difficulty in picking up the correct wavelet for a specific target signal, and therefore the application of WT requires considerable knowledge and experience.

A good overview about wavelet transform for SHM applications is given in the compendium provided by Reda Taha et al. [25] although the work was done in 2006. More actual, Shaopeng et al. [26] used the so called empirical wavelet transform (EWT) for analysing multi-mode signals gained from acoustic emission data. The idea of EWT is decompose a signal accordingly to its contained infor-mation. Sarrafi et al. [27] combined the wavelet transform with a probabilistic model in order to overcome the problems with uncertainties coming from opera-tional/environmental variability. A novelty approach for the design of a mother wavelet for the processing of measured signals from guided waves is presented in the work from Chen et al. [28]. They used a emitted tone-burst signal as mother wavelet for the detection of damage features caused by corrosion.

Summary: Spectral Estimation

In summary, all damage feature extraction methods which are based on transforms are powerful tools when the information about possible damage in the structure is hidden in the frequency content of the signal. Examples of such applications are primarily in the field of civil infrastructures, where the frequency behaviour of the structure under investigation is analysed. Due to the outdated infrastructure in many countries, SHM applications for civil structures have become very popular in the last few decades, leading to many publications. For example, Amezequita et al. [29] and Noel [30] used the WT for extracting information about vibration from the measured signal. For other damage detection methods, e.g., guided ultrasonic waves, the application of transfer methods is limited to signals where the damage-related features are clearly separated from other features in the signal. Staszewski et al. [10] used the WT for analysing measured signals obtained from guided ultrasonic wave propagation. They filtered out the information about the different wave modes within the measured signal.