Wavelet decomposition

III. Wavelet Decomposition Extraction of Trends at Multiple Scales... 

For practical purposes, the wavelet decomposition can only be applied to a finite record of discrete-time signals. If N is the number of samples in the record, and t = 1, then the maximum value of the translation parameter can be found from Eq. (12), by setting u=N, and is equal to k = N/2". Consequently, the decomposition and reconstruction relations [Eqs. (5b), (11a), (6b)] take the following form ... 

Fig. 11. Wavelet decomposition (a) dyadic sampling using Daubechies-6 wavelet (b) uniform sampling using cubic spline wavelet.

Unfortunately, the requirements for translational invariance of the wavelet decomposition are difficult to satisfy. Consequently, for either discretization scheme, comparison of the wavelet coefficients for two signals may mislead us into thinking that the two trends are different, when in fact one is simply a translation of the other. 

Fig. 14. Esxtracting distinguishing features from noise pulse signal. Wavelet coefficients in shaded regions represent stable extrema, (a) Wavelet decomposition of noisy pulse signal (b) wavelet decomposition of pulse signal. (Reprinted from Bakshi and Stephanopoulos, Representation of process trends. Part III. Computers and Chemical Engineering, 18(4), p. 267, Copyright (1994), with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.)...

Fig. 18. Wavelet decomposition of a pressure signal, using Daubechies-6 wavelet.

Fio. 19. Reconstruction of compressed signal from the wavelet decomposition of Fig. 18... 

The wavelet decomposition of measured data provides a natural framework for the extraction of temporal features, which characterize operating process variables and their trends. Such characterization, local in fre-... 

Fig. 40.44, Wavelet decomposition of a 16-point signal (see text for the explanation).

Fig. 40.45. Wavelet decomposition of a signal with local features.

Usual procedures for the selection of the common best basis are based on maximum variance criteria (Walczak and Massart, 2000). For instance, the variance spectrum procedure computes at first the variance of all the variables and arranges them into a vector, which has the significance of a spectrum of the variance. The wavelet decomposition is applied onto this vector and the best basis obtained is used to transform and to compress all the objects. Instead, the variance tree procedure applies the wavelet decomposition to all of the objects, obtaining a wavelet tree for each of them. Then, the variance of each coefficient, approximation or detail, is computed, and the variance values are structured into a tree of variances. The best basis derived from this tree is used to transform and to compress all the objects. 

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. 

FIGURE 4.8 Schematic image of wavelet decomposition. The QMFs H and G are applied to the raw signal. The signal is downsampled, leading to sets of coarse and detail components of half the size. 

The descriptors are transformed by Daubechies wavelet decomposition with 20 filter coefficients (D20). 

---