Wavelet decomposition functions

Given the dependency of the wavelet coefficients, one still has to find the appropriate framework for modeling their probability density functions. A Gaussian model is not appropriate since the wavelet decomposition tends to produce a large number of small coefficients and a small number of... 

The scaling function ( ) is determined by the low-pass QMF and thus is associated with the coarse components, or approximations, of the wavelet decomposition. The wavelet function / is determined by the high-pass filter, which also produces the details of the wavelet decomposition. 

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... 

Let us now see how the theory of the wavelet-based decomposition and reconstruction of discrete-time functions can be converted into an efficient numerical algorithm for the multiscale analysis of signals. From Eq. (6b) it is easy to see that, given a discrete-time signal, FqU) we have... 

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. 

Any square-integrable signal may be represented at multiple scales by decomposition on a family of wavelets and scaling functions as shown in Fig. 1. 

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