Wavelet series

These properties are desirable when representing signals through a wavelet series. In addition [44],... 

Nofe For clarification of terminology, we refer to the wavelet transform" as being the procedure for producing the wavelet coefficients. When the function f(t) is represented as a linear combination of the wavelet coefficients and wavelet basis functions this is referred to as the "wavelet series representation" or "wavelet decomposition" of f(t). This is discussed in greater detail in Section 5. 

Thus we have arrived at the wavelet series representation of f(t) (also called the wavelet decomposition of f(t)). Alternatively, one could write f(t) as a linear combination of scaling and wavelet basis functions as follows... 

As stated previously, with most applications in analytical chemistry and chemometrics, the data we wish to transform are not continuous and infinite in size but discrete and finite. We cannot simply discretise the continuous wavelet transform equations to provide us with the lattice decomposition and reconstruction equations. Furthermore it is not possible to define a MRA for discrete data. One approach taken is similar to that of the continuous Fourier transform and its associated discrete Fourier series and discrete Fourier transform. That is, we can define a discrete wavelet series by using the fact that discrete data can be viewed as a sequence of weights of a set of continuous scaling functions. This can then be extended to defining a discrete wavelet transform (over a finite interval) by equating it to one period of the data length and generating a discrete wavelet series by its infinite periodic extension. This can be conveniently done in a matrix framework. 

As stated in the previous section, most workers confine their wavelet functions in the Daubechies wavelet series only. For example, we have adopted the Daubechies wavelet function to denoise spectral data from a UV-VIS spectrophotometer [43]. In order to make use of the other available wavelet functions for chemical data analysis, Lu and Mo [44] suggested employing spline wavelets in their work for denoising UV-VIS spectra. The spline wavelet is another commonly used wavelet function in chemical studies. This function has been applied successfully in processing electrochemical signals [9,10] which will be discussed in detail in another chapter of this book. The mth order basis spline (B-spline) wavelet, Nm. is defined as follows [44] ... 

When discrete wavelets are used to transform a continuous signal, the resulting set of coefficients is called the wavelet series decomposition. For this transformation to be useful it must be invertible, and for synthesis to take place, Eq. 9.11 must be satisfied... 

The discrete parameter wavelet transform (DPWT, or wavelet series). Here the... 

Daubeohies I 1992 Wavelets (CBMS-NFS Series in Appl. Math.) (Philadelphia SIAM)... 

In order to compress the measured data through a wavelet-based technique, it is necessary to perform a series of convolutions on the data Becau.se of the finite size of the convolution filters, the data may be decomposed only after enough data has been collected so as to allow convolution and decomposition on a wavelet basis. Therefore, point-bypoint data compression as done by the boxcar or backward slope methods is not possible using wavelets. Usually, a window of data of length 2" m e Z, is collected before decomposition and selection of the appropriate... 

Percival, D.B., Walden, A.T. (2006). Wavelet Methods for Time Series Analysis, Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, New York. 

Calculation of the dynamic parameters using a ZND wave structure model do not agree with experimental measurements, mainly because the ZND structure is unstable and is never observed experimentally except under transient conditions. This disagreement is not surprising, as numerous experimental observations show that all self-sustained detonations have a three-dimensional cell structure that comes about because reacting blast wavelets collide with each other to form a series of waves which transverse to the direction of propagation. Currently, there are no suitable theories that define this three-dimensional cell structure. 

Figure 14a. Provided the wavefront is not interrupted, it moves out uniformly in a radial direction in accordance with the accepted notion of the rectilinear propagation of light. Suppose we now have a series of plane wavefronts incident on a slit with each successive wavefront being constructed from the envelope of secondary wavelets from the preceeding wave-front as shown in Figure 14b. If the width of the slit is small relative to the wavelength of light, then the portion of the wavefront at the slit may be considered as a single Huygens source radiating in all directions to produce...

To analyze the spectral composition of the pressure variations in more detail we have made use of a wavelet approach [9]. This approach, which allows us to determine instantaneous values of the frequencies and amplitudes of the various oscillatory components, is particularly useful for biological time series that often are neither homogeneous nor stationary. 

From the wavelet coefficients Ttl,[x](a, b) one can calculate the energy density Ethree-dimensional surface E(l,[x](a, t). Sections of this surface at fixed time moments t = b define the local energy spectrum Ev[x](f, t) with / = a 1. Finally, in order to obtain the mean spectral distribution of the time series x t) we may consider a so-called scalogram, i.e., the time-averaged energy spectrum. This is analogous to the classic Fourier spectrum. 

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. 

Wavelet transforms (Section 3.6.2) are a hot topic, and involve fitting a spectrum or chromatogram to a series of functions based upon a basic shape called a wavelet, of which diere are several in the literature. These transforms have the advantage that, instead of storing, for example, 1024 spectral datapoints, it may be possible to retain only a few most significant wavelets and still not lose much information. This can result in both data decompression and denoising of data. 

Seven wavelets for an eight point series are presented in Table 3.11. The smallest is a two point wavelet. It can be seen that for a series consisting of 2N points,... 

Kavacevic J, Vetterli M. Wavelets and Subband Coding. Signal Processing Series. Prentice Hall 1995. 

To handle this pulse graphically on the P-u plane, we will break the rarefaction at the back of the pulse into a series of small rarefaction wavelets, as shown in Figure 19.27. On the x-t plane, the series of wavelet interactions is shown in... 

The training problem determines the set of model parameters given above for an observed set of wavelet coefficients. In other words, one first obtains the wavelet coefficients for the time series data that we are interested in and then, the model parameters that best explain the observed data are found by using the maximum likelihood principle. The expectation maximization (EM) approach that jointly estimates the model parameters and the hidden state probabilities is used. This is essentially an upward and downward EM method, which is extended from the Baum-Welch method developed for the chain structure HMM [43, 286]. 

A trend analysis strategy is proposed that takes advantage of the wavelet-domain hidden Markov trees (HMTs) for constructing statistical models of wavelets (see Section 6.5). Figure 7.10 depicts the strategy that can be used to detect and classify faulty (abnormal) situations. As before, in the training phase, time series data collected under various conditions are... 

---