This article consists of a brief discussion of the energy density over time or frequency that is obtained with the wavelet transform. Also an efficient algorithm is suggested to calculate the continuous transform with the Morlet wavelet. The energy values of the Wavelet transform are compared with the power spectrum of the Fourier transform. Useful definitions for power spectra are given. The focus of the work is on simple measures to evaluate the transform with the Morlet wavelet in an efficient way. The use of the transform and the defined values is shown in some examples.
The wavelet transform is a method for time-frequency analysis. The application for acoustic signals can be found in several publications. These publications deal for example with the analysis of dispersive waves [1,2], source or damage localization [3,4], investigation of system parameters [5,6] or active control [7]. A comparison of the short time Fourier transform and the wavelet transform is done by Kim et.al. [8]. It is found that the continuous wavelet transform CWT of acoustic signals is a promising method to obtain the time -frequency energy distribution of a signal. Another sort are wavelets for spatial transforms, that are an effective tool for damage localisation [9,10]. The aim of this publication is to present an algorithm for the continuous Email address: richard.buessow@tu-berlin.de (Richard Büssow).
Preprint submitted to Elsevier wavelet transform CWT with the Morlet wavelet. Caprioli et al. state in [11] that “The high computational complexity (of the CWT) is not a serious worry for today’s computer power; still it might be an obstacle for an “on line” application.” Alternatives to the CWT are characterised by low computational complexity, but the best results are obtained by the CWT. The presented algorithm implements simple measures to reduce the computational complexity of the CWT. An implementation in Java is written and is open source and freely available online 1 . Nevertheless it should be mentioned that the Wigner approximation is also very useful in the given context. A recent publication is for example [12]. Other methods, that are adapted to the analysis of dispersive waves are discussed in [13,14]. This publication starts with a discussion of the energy values that are obtained by the wavelet transform. It follows a description of an efficient algorithm to evaluate the transform.
The theoretical background of the wavelet transform can be found in textbooks [15,16]. Only the used definitions are stated. The wavelet transform with the wavelet ψ of a signal y(t) is defined as
where a is called dilatation and b translation parameter. The Morlet wavelet ψ which is sometimes called Gabor wavelet is given by
and c ψ = π/β. The values β = ω 2 0 and ω 0 are defined in a particular application so that the admissibility condition is valid [16]. The function (2) is called mother wavelet. The Morlet wavelet is common for the time frequency analysis of acoustic signals. For a comparison of different so called mother wavelets see Schukin et. al. [17].
When using the Fourier transform one usually develops the spectrogram. The wavelet transform has scaling factors. The analog to the spectrogram is the scalogram defined as
(3) The scalogram is a measure of the energy distribution over time shift b and scaling factor a of the signal. It holds that the energy E y of a signal y is
If instead of the scaling factor a the frequency value f = 1/a is used, the value f is only the real frequency if ω 0 = 2π. It follows with da
It is possible to divide this total energy into an energy density over time and over frequency. This is achieved by one integration over frequency or time.
The energy density over time is defined by
The energy density over frequency, or the energy density spectrum is defined by
The value |W y ψ (f, b)| 2 is sometimes called Morlet power spectrum, but here the term energy density is used. A Comparision of the values obtained with the Fourier transform and the wavelet transform is, for example, given in [18]. A power signal is characterized by
If the wavelet transform is applied to a power signal one recognises that for higher frequencies the energy density is lower, as shown in the examples in section 4. To achieve a better match, Shyu [19] proposes a modified equal amplitude wavelet power spectrum that is given by
The above definition corresponds more closely to power spectra obtained with the discrete Fourier transform. This can be explained since P = 1 T E and c ψ a is the effective length of the wavelet so T ef f = c ψ /f is the value to scale the energy density. Since one is usually familiar with power spectra, the above definition ( 9) is useful but a problem is that
This fact can be compared with the windowed Fourier transformation. If the window like for example, the flat top is defined so that the peak value is constant2 , it follows that the effective noise band width is altered by the windowing function. The Morlet wavelet transform can be interpreted as a windowed Fourier transformation with a frequency dependent window size.
2.1.1.1 Alternative definition of the power spectrum In the following a new definition of the power spectrum is given. It is defined analogous to the discrete Fourier transform, so that the power can be calculated by summation of the power values. The discrete Fourier transform is a filter with the bandwidth ∆f , so each value of the power spectrum is the integral over ∆f . The power spectrum that is defined with the energy values of th
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