Wavelets Morlet

Finally, the band pass filters corresponding to the Morlet wavelet have a "quicker" decrease towards null frequencies than filters obtained with the first derivative of gaussian wavelet (fig. 9). As a result, they... 

Scalograms have been calculated with Morlet wavelet for all tube samples, and give... 

Fig. 40.42. A family of Morlet wavelets with various dilation values.

Under such conditions it seems reasonable to describe the extended part of a quantum particle by a generalized Gaussian Morlet wavelet [4]... 

Fig. 4.4.4 [Bar2] Morlet wavelets (top) and their Fourier spectra (bottom) according to eqns. (4.4.21 a and b). Time and frequency are scaled in arbitrary units. Dilatation parameter from left to right a = 0.5,1.0,2.0 Widths of WabO) = 3,6,12 (top). Widths of Wai,((u) = 3.0,1.5,0.75 and corresponding peak positions at 12,6,3 (bottom).

Depending on the type of application, different wavelets can be chosen. In spectroscopy and imaging, signals are often well localized in frequency. In this case the Morlet wavelet can be used [Bar2]. The wavelet and its Fourier transform are given by... 

The Morlet wavelet can be understood to be a linear bandpass filter, centred at frequency m = coo/a with a width of /(aoa). Some Morlet wavelets and their Fourier spectra are illustrated in Fig. 4.4.4. The translation parameter b has been chosen for the wavelet to be centred at time f = 0 (top). With increasing dilatation parameter a the wavelet covers larger durations in time (top), and the centre frequency of the filter and the filter bandwidths become smaller (bottom). Thus depending on the dilatation parameter different widths of the spectrum are preserved in the wavelet transform while other signals in other spectral regions are suppressed. 

By compressing this function in time, Morlet was able to obtain a higher frequency resolution and spread it out to obtain a lower frequency resolution. To localize time, he shifted these waves in time. He called his transform the wavelets of constant shape and today, after a substantial number of studies in its properties, the transform is simply referred to as the Wavelet transform. The Morlet wavelet is defined by two parameters the amount of compression, called the scale, and the location in time. 

Fig, 12.1. (Taken from [17]) Reproducing kernel of the Morlet wavelet for three different scales (a) s=8, (b) s=32 and (c) s=128. The width in time and in scale direction increases linearly with scale (i.e. in scale direction it appears constant on a logarithmic scale axis)... 

Fig. 12.2. (Taken from [17]) Stochastic chirp with e = 0.3. (a) The spectrum m(6, a)p. (b) A typical realization in the time domain, calculated with a Morlet Wavelet, cuq = 6.

Fig. 12.6. (Taken from [17]) Pointwise significance test of the wavelet sample spectrum of Gaussian white noise (Morlet wavelet, = 6, rus = 0) against a white noise background spectrum of equal variance. Spuriously significant patches appear.

The Morlet and Mexican has wavelets are interesting because for these wavelets, there is no corresponding scaling function. Each of these wavelets can be expressed in closed form. The Mexican wavelet is written as v)/(t) = - t )e and the Morlet wavelet is equal to... 

As suggested in reference [25], the traditional sigmoidal function can be replaced with the Morlet wavelet basis function Fqwt in neural network analysis (Fig. 4(b)). When a spectral data, X, is applied to this WNN system, a response or an output value Ydwt >s obtained as follows ... 

Morlet wavelet is the kind of wavelet most used (ZHENG, LI, CHEN, 2002 BAYDAR BALL, 2003 LIN ZUO, 2003 LIN ZUO, 2004 SMITH etal, 2007 JAFARIZADEH etal., 2008). Other kinds of wavelets were applied such as mexican hat (BAYDAR BALL, 2003), daubechies (SMITH, et al, 2007), harmonic (LIN ZUO, 2003), gabor-based (BAYDAR BALL, 2003), haar (SMITH, et al, 2007) and modulus maxima distribution (MIAO MAKIS, 2007). 

In order to obtain a diagnosis of the state of health of gearboxes, the Morlet wavelet transform was appUed. The output of this transform is the matrix W(s,n), which is a matrix of S xN dimension. In this case, 5 = 49 and n = 8192. Because the Morlet wavelet transform was used, the values of this matrix are complex. By taking the absolute value of each of the elements of the matrix W s,n), the matrix V(s,n) is produced, where all elements of V s, n) are real. 

Although it is not possible to draw effective conclusions about Morlet wavelet from the signal in the time domain, these graphs can be useful to check the consistency of data and to have a first understanding of the data. 

Figure 3. The Morlet Wavelet of horizontal vibration (2000 rpm - full load - data set 2).

The Fourier transform of one of the prototype wavelets (called a Morlet wavelet) is given by... 

Haar wavelet, Morlet wavelet and Daubechies wavelet... 

Figure 4.4 shows a signal f t) along with the Morlet wavelet at three scales and shifts. As a increases, the wavelet stretches by a factor of two, which analyses the signal at lower frequency as b increases, the wavelet shifts right, which analyses the signal at different localisations. The variations of a and b lead to the multiscale and localised analysis with wavelet. 

Morlet wavelet transform at different scales and locations... 

To identify seiche events, records of the surface elevations at ROZ were Fourier band-passed filtered, passing the seiche frequency band (0.1-2.0mHz). A wavelet analysis based on the Morlet wavelet " has been applied to the filtered surface elevation data and meteorological data. This technique is suited for the detection of fluctuations that come in bursts. It has been used to identify the seiche events inside the harbor and the corresponding periods of increased levels of low-frequency energy at sea. Following first identification from the wavelet spectra, the time intervals with seiche events were reviewed in more detail. 

Morlet wavelet Heat flux Self-similar bifurcation and trifurcation phenomena Ross and Pence (1997)... 

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