Mother 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. 

However, it is easily shown that if the mother wavelet is located in the frequency domain "around"/o (fig 8), then the wavelet a.b(t) is located around f(/a. That is to say, by the mean of the formal identification f = fata it is possible to interpret a time-scale representation as a time-frequency representation [4]. 

A measure of the time and frequency resolution of the mother wavelet is given respectively by ... 

To be more specific, given a mother wavelet with its own time and frequency properties, the small values of scale coefficient a (high frequencies) lead to high time resolution (and poor frequency resolution). Correspondingly, high values of the scale coefficient (low frequencies lead to high frequency resolution (and poor time resolution), (see figure 10)... 

Wavelet transformation (analysis) is considered as another and maybe even more powerful tool than FFT for data transformation in chemoinetrics, as well as in other fields. The core idea is to use a basis function ("mother wavelet") and investigate the time-scale properties of the incoming signal [8], As in the case of FFT, the Wavelet transformation coefficients can be used in subsequent modeling instead of the original data matrix (Figure 4-7). 

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... 

The results presented above are rather satisfactory because in this new paradigm the quantum measurement process depends, in the last instance, on the standard used. We are, in principle, free to choose the size, or the scale, of the mother wavelet Axo more suitable for the measurement precision that we want to attain. 

FIGURE 10.19 Some example mother wavelet functions. From left to right a coiflet (coif), a symlet (sym), and two Daubechies (db) wavelets. The numbers relate to the number of vanishing moments of the wavelet. 

FIGURE 10.23 The cascade of wavelet coefficient vectors output from the wavelet tree filter banks defining the discrete wavelet transform in Figure 10.22. A db-7 mother wavelet was used for the decomposition of the noisy signal in Figure 10.1. 

This transform uses the unspecified single mother wavelet ij/(s, x, x) to generate other wavelets by scaling and translation ... 

It is not a straightforward task to come up with a procedure that would lead to the best mother wavelet for a given class of signals. N-evertheless, exploiting several characteristics of the wavelet function, one can determine which family of wavelets would be more appropriate for a specific application. 

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 ... 

Mother Wavelet is a finite length waveform, which is applied in scaled and translated copies (wavelets) in a wavelet transfom. 

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

The wavelets are scaled and translated copies (known as daughter wavelets ) of a finite-length or fast-decaying oscillating waveform (known as the mother wavelet ). Wavelet transforms have advantages over traditional Fourier transforms for representing functions that have discontinuities and sharp peaks. 

The wavelet basis functions are derived by translating and dilating one basic wavelet, called a mother wavelet. The dilated and translated wavelet basis functions are called children wavelets. The wavelet coefficients are the coefficients in the expansion of the wavelet basis functions. The wavelet transform is the procedure for computing the wavelet coefficients. The wavelet coefficients convey information about the weight that a wavelet basis function contributes to the function. Since the wavelet basis functions are localised and have varying scale, the wavelet coefficients therefore provide information about the frequency-like behaviour of the function. 

The continuous wavelet transform convolves the function f(t) with translated and dilated versions of a single basis function vj/ft). The basis function v /(t) is often called a mother wavelet. The various translated and dilated versions of the mother wavelet are called children wavelets. The children wavelets have the form j/((t - b)/a), where a is the dilation parameter which squeezes or stretches the window. 

Fig. 6 indicates that by translating and dilating the mother wavelet, localised information about high and low frequency events can be obtained. It should be mentioned that we use the term frequency loosely when talking about wavelet transforms, since it is not really frequency that we are describing but rather low and high scale events. 

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