Dilated wavelet

Figure 8 mother wavelet y/(t) (left) and wavelet built out of the mother wavelet by time shift b, and dilatation a. Both functions are represented in the time domain and the frequency domain. 

Strang. G., Wavelets and dilation equations A brief introduction. SIAM Rev. 31, 614 (1989). Ungar, L. H., Powell, B. A., and Kamens, S. N., Adaptive Networks for fault diagnosis and process control. Comput. Chem. Eng. 14, 561 (1990). 

A family of wavelets is a family of functions with all its members derived from the translations (e.g., in time) and dilations of a single, mother function. If iffit) is the mother wavelet, then all the members of the family are given by... 

In the time domain, a wavelet placed at a translation point, u, has a standard derivation, o- , around this point. Similarly, in the frequency domain a wavelet is centered at a given frequency, w (determined by the value of the dilation parameter) and has a standard deviation, cr, around the specific frequency. The values of a- and o- are given by... 

As the dilation parameter, 5, increases, the wavelets have significant value over a broader time segment and the resulting resolutions in the time anc frequency domains change as 5o- and cr /j, respectively. Thus, at largt... 

To ensure that no information is lost on Fio)) as the dilation is discretized, the scale factors 2 " for m g Z must cover the whole frequency axis. This can be accomplished by requiring the wavelets to satisfy the following... 

Equation (6a) implies that the scale (dilation) parameter, m, is required to vary from - ac to + =. In practice, though, a process variable is measured at a finite resolution (sampling time), and only a finite number of distinct scales are of interest for the solution of engineering problems. Let m = 0 signify the finest temporal scale (i.e., the sampling interval at which a variable is measured) and m = Lbe coarsest desired scale. To capture the information contained at scales m > L, we define a scaling function, (r), whose Fourier transform is related to that of the wavelet, tf/(t), by... 

As a result of the dyadic discretization in dilation and translation, the members of the wavelet family are given by... 

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

Strang, G., Wavelets and dilation equations a brief introduction. SIAM Review, 31(4), 614—... 

The continuous wavelet transform (WT) is a space-scale analysis that consists in expanding signals in terms of wavelets that are constructed from a single function, the analyzing wavelet /, by means of dilations and translations [13, 27-29]. When using the successive derivatives of the Gaussian function as analyzing wavelets, namely... 

A wavelet is a general function, usually, but by no means exclusively, of time, g(t), which can be modified by translation (b) or dilation (expansion/contraction) (a). The function should add up to 0, and can be symmetric around its mid-point. A very simple example the first half of which has the value +1 and the second half —1. Consider a small spectrum eight datapoints in width. A very simple basic wavelet function consists of four —1 s followed by four —Is. This covers the entire spectrum and is said to be a wavelet of level 0. It is completely expanded and there is no room to translate this function as it covers the entire spectrum. The function can be halved in size (a = 2), to give a wavelet of level 1. This can now be translated (changing b), so there are two possible wavelets of level 1. The wavelets may be denoted by [n, m] where n is the level and m the translation. 

The concept of wavelets is illustrated by example of continuous wavelets [Bar2]. Wavelets are obtained from a single function w(jt) by translation and dilatation of the time axis. 

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).

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. 

A wavelet function with dilation parameter s and translation parameter u... 

The basis functions, or wavelets,, are dilated and translated versions of a wavelet mother function. A set of wavelets is specified by a particular set of numbers, called wavelet filter coefficients. To see how a wavelet transform is performed, we will take a closer look at these coefficients that determine the shape of the wavelet mother function. 

The basic idea of the wavelet transform is to represent any arbitrary function as a superposition of basis functions, the wavelets. As mentioned already, the wavelets P(x) are dilated and translated versions of a mother wavelet Tg. Defining a dilation factor d and a translation factor t, the wavelet function F(x) can be written as... 

The scaling of the mother wavelets Fg is performed by the dilation equation, which is, in fact, a function that is a linear combination of dilated and translated versions of it ... 

Wavelets are dilated and translated versions of a mother wavelet in wavelet transforms. 

Wavelet transforms (WT) are classified into continuous wavelet transforms (CWTs) and discrete wavelet transforms (DWTs). Wavelet is defined as the dilation and translation of the basis function /(t), and the continuous wavelet transforms is defined as [Shao, Leung et al, 2003]... 

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