0.7 Generalizations of the basic multiresolution wavelet system

Up to this point in the book, we have developed the basic two-band wavelet system in some detail, trying to provide insight and intuition into thisnew mathematical tool. We will now develop a variety of interesting and valuable generalizations and extensions to the basic system, but in muchless detail. We hope the detail of the earlier part of the book can be transferred to these generalizations and, together with the references,will provide an introduction to the topics.

Tiling the time–frequency or time–scale plane

A qualitative descriptive presentation of the decomposition of a signal using wavelet systems or wavelet transforms consists of partitioning thetime–scale plane into tiles according to the indices $k$ and $j$ defined in [link] . That is possible for orthogonal bases (or tight frames) because ofParseval's theorem. Indeed, it is Parseval's theorem that states that the signal energy can be partitioned on the time-scale plane.The shape and location of the tiles shows the logarithmic nature of the partitioning using basic wavelets andhow the M-band systems or wavelet packets modify the basic picture. It also allows showing that the effects of time- or shift-varying waveletsystems, together with M-band and packets, can give an almost arbitrary partitioning of the plane.

The energy in a signal is given in terms of the DWT by Parseval's relation in [link] or [link] . This shows the energy is a function of the translation index $k$ and the scale index $j$ .

The wavelet transform allows analysis of a signal or parameterization of a signal that can locate energy in both the time and scale (or frequency)domain within the constraints of the uncertainty principle. The spectrogram used in speech analysis is an example of using the short-timeFourier transform to describe speech simultaneously in the time and frequency domains.

This graphical or visual description of the partitioning of energy in a signal using tiling depends on the structure of the system, not the parameters of the system. In other words, the tiling partitioning will depend on whether one uses $M=2$ or $M=3$ , whether one uses wavelet packets or time-varying wavelets, or whether one uses over-completeframe systems. It does not depend on the particular coefficients $h\left(n\right)$ or $h_i\left(n\right)$ , on the number of coefficients $N$ , or the number of zero moments. One should remember that the tiling may look as if the indices $j$ and $k$ are continuous variables, but they are not. The energy is really a function of discrete variables in the DWT domain, and theboundaries of the tiles are symbolic of the partitioning. These tiling boundaries become more literal when the continuous wavelet transform (CWT)is used as described in [link] , but even there it does not mean that the partitioned energy is literally confined to the tiles.

Nonstationary signal analysis

In many applications, one studies the decomposition of a signal in terms of basis functions. For example, stationary signals are decomposedinto the Fourier basis using the Fourier transform. For nonstationary signals (i.e., signals whose frequency characteristics are time-varyinglike music, speech, images, etc.) the Fourier basis is ill-suited because of the poor time-localization. The classical solution to this problem isto use the short-time (or windowed) Fourier transform (STFT). However, the STFT has several problems, the most severe being the fixed time-frequencyresolution of the basis functions. Wavelet techniques give a new class of (potentially signal dependent) bases that have desired time-frequencyresolution properties. The “optimal” decomposition depends on the signal (or class of signals) studied. All classical time-frequencydecompositions like the Discrete STFT (DSTFT), however, are signal independent. Each function in a basis can be considered schematically as a tile in the time-frequency plane, where most of its energy is concentrated. Orthonormality of the basis functions can beschematically captured by nonoverlapping tiles. With this assumption, the time-frequency tiles for the standard basis (i.e., delta basis) and theFourier basis (i.e., sinusoidal basis) are shown in [link] .