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## Open Mathematics

Saturday Night, and musical anoxic aspects. The wavelet shrinkage and thresholding methods allow recovering the signal from noisy data. De-noising is possibly the most popular practice of wavelets on data because after decomposing the signal the further process is rather intuitive. The basic methods used for signal decomposition are the discrete and the continuous analysis.

The use of discrete or continuous decomposition techniques depends on the complexity of the signal. In fact, de-noising is a powerful but not the only application of wavelets on time series. They can be used for statistical methods like density estimation and detection of self-similar behavior in time series.

Wavelets show an alternative and fast way of getting information from acoustic emission signals where classical interpretation is often difficult and time consuming. Key words: wavelets, acoustic emissions, de-noising, filtering, scalegram 1 Introduction The wavelet transform is a mathematical tool which is able to examine a signal simultaneously in time and frequency. The method itself came up in the mid - s where it was originated to investigate seismic signals Goupillaud et al.

### Daubechies Wavelets on Intervals with Application to BVPS

But in many aspects wavelets are a synthesis of older ideas with new elegant mathematical results and efficient computational algorithms Percival andWalden, Fields of application are e. Transforms often result in simplified problem solving analysis Brigham, One such transform analysis technique and furthermore really widespread is the Fourier transform. Another one is the wavelet transform described in the following in detail. The Fourier transform is able to reveal the frequencies present in a signal.

But it is not possible to say when they are present. An effort to correct this deficiency is to analyze only a small section of the signal at a time - a technique called windowing the signal, also called the Short-Time Fourier Transform STFT which maps a signal into a two-dimensional function of time and frequency. The STFT represents a sort of compromise between the time- and frequency-based views of a signal. It provides some information about both when and at what frequencies a signal event occurs.

However information is optained with limited precision, and that precision is determined by the size of the window Misiti et al. This is the starting point for the Wavelet transform. Using the Wavelet transform a timescale joint representation is possible where scale is proportional to frequency. By decomposing a time series into time-scale equal to frequency space, one is able to determine both, the dominant modes of variability and how those modes vary in time Torrence and Compo, In addition, this scale based analysis is often easier to interpret because the knowledge of the variance in time makes the understanding of process development simpler.

As mentioned above, the wavelet transform offers a variety of features which are interesting for time series analysis. Especially the application to acoustic emission data is very successful Motz et al. One severe problem in acoustic emission analysis is that huge data sets often more than acoustic emission events with a low signal to noise ratio are detected Figure 1.

Fig 1: Three dimensional localization of acoustic emissions from a pull out test left. Typical signal right with a low signal to noise ratio and an offset from the y-axis. Acoustic emission data, e. Due to the testing process itself a low frequent signal, caused by the testing device loading machine , often superimposes the acoustic emission signal additionally.

This makes the onset detection difficult. The wavelet transform is a fast and efficient way of filtering and especially de-noising data Motz et al. Due to the large number of events the use of an automatic onset detection algorithm is preferable. The use of such an algorithm requires high quality data.

That means filtering and de-noising is a pre-condition for automatic data processing. Furthermore, the wavelet transform offers a variety of other application forms on time series. One example is the use of so called scalegrams Scargle et al.

### Wavelets: Theorie und Anwendungen (Teubner Studienbücher Mathematik) (German Edition)

Another field of application is the detection of self similarities and long term evolution processes Misiti et al. This listing about using wavelets leads to a closer look at the mathematical concept of the wavelet transform. Wavelets are mathematical functions that have to be well localized other requirements are of technical matter. The mathematical steps used for the wavelet transform can be summarized as follows: i a fully scalable modulated window solves the signal cutting problem ii this window is shifted along the signal and for every position the spectrum is calculated iii the process is repeated many times with a slightly shorter or longer window for every new cycle iv this results in a collection of time-scale representations of the signal, all with a different solution These four points represent the principle of the wavelet transform.

General concepts known from Fourier analysis form the application base. Therefore, the basic concepts of convolution and filtering of finite sequences and some basics about orthonormal functions are presented first before the discrete DWT and the continuous wavelet transform CWT will be discussed. Due to the fact that our focus of application here is set on the discrete wavelet transform the mathematical concept presented in the following is adopted from Percival and Walden Concerning acoustic emission data this filter is applied to a finite sequence.

The process is a convolution of two sequences. Suppose two infinite sequences u t and x t where the first one is the filter and the second one is the signal. The convolution is defined as: 1 or in flow diagram: 2 Regarding a finite sequence x t and a periodized filter u t which is a circular filter of length N the convolution is defined as: 3 Here a sequence of N variables is transformed into a new sequence of length N.

In essence the filter is cut into blocks of coefficients and these are overlain and then added to form the periodized filter.

## Book Wavelets Theorie Und Anwendungen German

Using a flow diagram this can be expressed as: 4 2. The series and its transform can be considered to be two representations of the same mathematical entity. The orthonormality property can be restated in terms of the inner product as: 5 That means the columns and the rows of are an orthonormal set of vectors. Such a matrix can be used for an orthonormal transformation in both directions, as it is applied in the DWT.

The N X N matrix consists of these periodized filters. The third point indicates that the filter is rescaled for every new cycle. That means the matrix consists of the wavelet and the scaling filter. The wavelet filter is a high pass filter with a nominal pass-band while the scaling filter is a low pass filter with a different nominal pass-band. Applying them on the time series X , the Wavelet coefficients W are gained which are a collection of time-scale representations of the signal, all with a different solution.

In other words: the wavelet is scaled and shifted and then moved along the time series. The main difference between the discrete and the continuous wavelet transform is that the DWT can be thought of as a subsampling of the wavelet coefficients with dyadic scales. Each element is one wavelet coefficient. The rows of that produce the wavelet coefficients the wavelet filter for a particular scale are circularly shifted versions of each other.

Furthermore, changes concerning the wavelet scale the scaling filter are also of dyadic order in the same range as for the wavelet filter. The variables w , n are the new dimension, scale and translation, after the wavelet transform.

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The CWT is calculated by continuously shifting a continuously scalable function over a signal. De-noising and compression are possibly the most popular practices of wavelets on data. Due to the time-scale joint representation of the wavelet transform a lot of information from the time series can be found separated in the coefficients. This information can be analyzed using different statistical approaches e. Some of them with a relevance for the application on acoustic emission data are discussed in the following.

For the wavelet decomposition to give an example a biorthogonal wavelet of the order 4. The comparison of the original and the filtered signal on the right side of Fig.