Wavelet complex

One of the main advantages of the WT is its adaptative ability to perform time-frequency analysis [28, 29] when using complex analyzing wavelets like the Morlet s wavelet ... 

Wavelet techniques (Koscielny-Bunde et al., 1998) and detrended fluctuation analysis (DFA) (Peng et al., 1994) are among the most recently used tools along these lines. Recently, much attention has been paid to DFA, because it has already proved its usefulness in a wide variety of complex systems (e.g., Stanley, 1999 Talkner and Weber, 2000 Varotsos et al., 2003a, b, 2005 Chen et al., 2005 Varotsos and Kirk-Davidoff, 2006). More information about the DFA method is given in Section 3.6.3.2. 

Wavelet analysis is a rather new mathematical tool for the frequency analysis of nonstationary time series signals, such as ECN data. This approach simulates a complex time series by breaking up the ECN data into different frequency components or wave packets, yielding information on the amplitude of any periodic signals within the time series data and how this amplitude varies with time. This approach has been applied to the analysis of ECN data [v, vi]. Since electrochemical noise is 1/f (or flicker) noise, the new technique of -> flicker noise spectroscopy may also find increasing application. 

Recently, Tanikawa et discussed the photoconductivity of poly[7-03-N-carbazolyethyl)-L utamate] and its charge transfer complex with 2,4,7,-trinitro-9-fluorenone. The photocurrents in this polymer are about one order of magnitude smaller than those of PVK at all measured wavelet ths. The complex with TNF has a peak photocurrent at 600 nm while the absorption spectmm of the polymer... 

Figure 4. Multiscale aspects of oxide surfaces in aqueous environments, (b) represents a single trace across a hematite (001) surface undergoing biotically mediated reductive dissolution (a). Relatively clean steps 2.6 nm in size on the left of the figure give way to a complex surface morphology on the right side of the image where biotic dissolution was pervasive. The wavelet transform, to /2 levels, is shown in (c). (b) provided by Kevin Rosso, Pacific Northwest National Laboratory.

If the wavelet is symmetric in time, formation of the complex conjugate is equivalent to inversion of the time axis, so that the correlation of / with w can also be written as a convolution of / and w (cf. Sections 4.2.3 and 4.3.3). The value of the normalization factor is often chosen in practice to be /a, although different conventions are used mostly in theoretical treatments [Bar2]. 

The wavelet is the product of a complex exponential with frequency o>o and a Gaussian with variance oq. The function g(t) is necessary to enforce the admissibility condition. 

More complex wavelets have in common with the Haar wavelet that they correspond with a complementary pair of LP and HP filters that cut the... 

Fig. 1 plots the function sin(2t) which has been sampled 512 times in [—rt. tt]. The sine curve on the right has a small disturbance at t = 1.5. Below each of the sine curves are the Fourier coefficients and the wavelet coefficients. Since the Fourier coefficients are complex, the magnitude of the coeffieients are shown. A noticeable feature of the Fourier coefficients is the large coefficient at the second index which reflects the period of the sine curve. This is evident in the Fourier coefficients from both the original and disturbed signal. There is little visible difference between these sets of Fourier coefficients. The disturbance occurring at t = 1.5 has been spread across all of the Fourier coefficients. The wavelet coefficients are however able to detect the distur-... 

Wavelets are a set of basis functions that are alternatives to the complex exponential functions of Fourier transforms which appear naturally in the momentum-space representation of quantum mechanics. Pure Fourier transforms suffer from the infinite scale applicable to sine and cosine functions. A desirable transform would allow for localization (within the bounds of the Heisenberg Uncertainty Principle). A common way to localize is to left-multiply the complex exponential function with a translatable Gaussian window , in order to obtain a better transform. However, it is not suitable when <1) varies rapidly. Therefore, an even better way is to multiply with a normalized translatable and dilatable window, v /yj,(x) = a vl/([x - b]/a), called the analysing function, where b is related to position and 1/a is related to the complex momentum. vl/(x) is the continuous wavelet mother function. The transform itself is now... 

To construct parsimonious multivariate models for highly correlated spectral data, one can extract all relevant information, present in data, and eliminate the irrelevant one. This can efficiently be done in the wavelet domain, where it is easy to distinguish between significant features and features associated with noise. The latter variables can be further used for discrimination of relevant and irrelevant features for data modelling. This approach usually leads to the decrease of model complexity and to increase of its stability. 

The one-dimensional (1-D) discrete wavelet transform (DWT) defined in the first part of the book can be generalised to higher dimensions. The most general case has been studied by Lawton and Resnikoff [1]. An N-dimen-sional (N-D) DWT is described also in [2]. The separable extension of the wavelet transform (WT) to three dimensions, for example, is explained in [2, 3,4]. In this chapter, for simplicity and because of the problems studied, only the theory of the 2-D and 3-D DWT will be outlined, and only separable 2-D and 3-D wavelets will be considered. These wavelets are constructed from one-dimensional wavelets. Separable wavelets are most frequently used in practice, since they lead to significant reduction in the computational complexity. 

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