Wavelets Transform

Wavelet transform (WT) is a signal processing tool which employs what is called wavelets. A wavelet is a waveform of effectively limited duration that has an average value of zero (Mathworks, 1997).

The mathematical formulation of the wavelet transform was first proposed in the form of a new orthonormal function by Alfred Haar in 1909. This then led to the invention of the simplest orthogonal wavelet, which was later named the Haar wavelet ((Graps,1995), (Li et al. 2007)).

Chapter 2 – Review of Vibration Signal Analysis in Fault Diagnosis 25 The idea of using a wavelet transform in signal analysis is to meet the need for analysis of signals using different sized scaling factors (window or scale). A large scale refers to a big frame or window for viewing or analysing a signal, while a small scale refers to a small frameor window used for viewing the details of a signal. In this context, changing from a large scale (window or frame) to a small is equal to the zooming process and vice versa.

In Fourier analysis, a signal is decomposed into sine waves of various frequencies. Similarly, in wavelet analysis, a signal is decomposed into shifted and scaled versions of what is called the original (or mother) wavelet. The aim of the wavelet transform is to overcome the shortcomings of the Fourier transform such as signal- cutting (windowing) problems, along with the analysing of non-stationary signals.

Wavelet transform uses a fully scaleable modulated window to solve signal- cuttingproblems. The window is shifted along the signal and for every position, the spectrum is calculated, as shown in Figure 2.1. The process can be repeated many times with a slightly shorter (or longer) window or scalein every new cycle. The end result will be a collection of time-frequency representations of the signal, all with different resolutions.

Figure 2.1 Wavelet transform principle (Saravanan and Ramachandran, 2010)

The mother wavelet with a defined scale size is translated from the beginning to the end of the signal to be analysed, as shown diagrammatically in Figure 2.1. The

process is iterated by determining a new scale of the wavelet function and the process is cycled again from the beginning to the end of the signal.

The process produces segments of signal which are comprised of “Approximated versions” and “Detailed versions” of the processed signal. This process is called the wavelet transform of the signal which provides the method for synthesising or disassembling the signal into two parts which are labelled approximation (a) and detail (d) parts.

The approximated version has a low frequency content which approximates the original of the processed signal, while the detailed version contains high frequency information on the processed signal.

The wavelet transform (WT) was proposed to address the limitations of the Fourier transform since the sine and cosine functions used in the FT are continuous and therefore not suited to particular needs. This is due to the fact that they are not localised functions and that they stretch out to infinity. In this context, if sine and cosine functions are used in the approximation of non-stationary signals the result is not satisfactory. In the WT, better results are achieved since the approximating functions used are limited to a finite time period (Graps, 1995).

The difference between the FT and the WT is at the ability of the WT to decompose signals using multi-scale analysis with dilation and translation in order to extract the time-frequency features or characteristics of the signal.

The wavelet analysis uses the wavelet function as the basic function to produce localised features of the original signal in a scaled domain (Paya et al. 1997). The basic functions comprise a family of functions which are derived from a single wavelet function called the mother wavelet.

The basic functions of the wavelet are useful in cases where there is a need to apply more suitable functions, other than sines and cosines, to approximate irregular signals or data with sharp discontinuities (Crandall, 1994). The basic function gives

Chapter 2 – Review of Vibration Signal Analysis in Fault Diagnosis 27 the wavelet the capability of analysing the signal in localised time (or space) along with the frequency (or scale) domains (Al-Badour, et al. 2011).

An advantage of the wavelet transform is that it can be used to analyse signals in different frequency bands and to study each band with a resolution based on the wavelet scaling factor, as shown in Figure 2.1. If a longer scale of wavelet is used, the analysis produces low frequency information regarding a signal. The high frequency information is produced when a shorter wavelet scale is used (Graps, 1995).

There are many possible different types of wavelets; each of them is specified by its own coefficients. A common orthogonal set wavelet defined by Inggrid Daubechies (Daubechies, 1990) called the Daubechies n wavelet (abbreviated as db-n wavelet) is commonly used in various applications. An example of the practical implementation of wavelet transform using Daubechies wavelet is presented in Appendix 1.

The wavelet transform provides a time-scaled result of a signal which is different to the classical representation in the time-frequency used by FT.

In the process of wavelet transformation, the basic function (mother wavelet) is translated and dilated to provide improved time resolution for high-frequency information and simultaneously it provides limited time resolution for low frequency information.

The detail of time-frequency information of wavelet transform is depicted in Figure 2.2.

Figure 2.2 Daubechies Wavelet basis function, time frequency tiles and coverage of the time-frequency plane (Graps, 1995)

Figure 2.2 shows the coverage areas in the time-frequency plane with a Daubechies wavelet function. There are three different time scales of Daubechies wavelet, shown in the upper part of Figure 2.2. The shape of wavelet varies with the size of time- frequency coverage windows.

The Daubechies wavelet belongs to the family of orthogonal wavelets. An orthogonal wavelet is a discrete wavelet transform and it is defined by a maximal number of vanishing moments/points within a given support range. There is a scaling function for each wavelet type in this class or family which generates an orthogonal multi-resolution analysis. The functions of the Daubechies wavelet () are shown in Figure 2.3 and the scaling () functions are shown in Figure 2.4.

Chapter 2 – Review of

The scale factor is an important parameter in the

processing vibration signals. As shown in the top section of Figure 2.1, there are three different Daubechies (db) wavelets with different scaling. Each of the wavelets has a particular time scale which produces differently

Larger scale wavelets produce a “wider” time window but with a correspondingly low frequency coverage span. Smaller time scale wavelets produce a “narrower” time window but with a higher frequency coverage area. The number of transformation levels determines the frequency and time resolution segmentation. A higher transformation level will improve frequency resolution at the expense of decreased time resolution.

Figure 2.

Review of Vibration Signal Analysis in Fault Diagnosis

The scale factor is an important parameter in the wavelet transform when used for processing vibration signals. As shown in the top section of Figure 2.1, there are three different Daubechies (db) wavelets with different scaling. Each of the wavelets has a particular time scale which produces differently sized time-frequency windows. Larger scale wavelets produce a “wider” time window but with a correspondingly low frequency coverage span. Smaller time scale wavelets produce a “narrower” time window but with a higher frequency coverage area. The number of transformation levels determines the frequency and time resolution segmentation. A higher transformation level will improve frequency resolution at the expense of decreased time resolution.

Figure 2.3 Daubechies wavelet (Ψ) functions

Vibration Signal Analysis in Fault Diagnosis 29 wavelet transform when used for processing vibration signals. As shown in the top section of Figure 2.1, there are three different Daubechies (db) wavelets with different scaling. Each of the wavelets frequency windows. Larger scale wavelets produce a “wider” time window but with a correspondingly low frequency coverage span. Smaller time scale wavelets produce a “narrower” time window but with a higher frequency coverage area. The number of transformation levels determines the frequency and time resolution segmentation. A higher transformation level will improve frequency resolution at the expense of

Figure 2.