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# Wavelet Concepts

Discrete Wavelet Analysis

In general when dealing with stationary signals, whose statistical properties are invariant over time, the ideal tool is the Fourier transform. The Fourier transform is an infinite linear combination of dilated cosine and sine waves. When we encounter non-stationary signals, we can represent these signals by linear combinations of atomic decompositons known as wavelets. These wavelets, or atomic decompositions, allow us to extract the simple constituents that make up a complicated structure or signal.

The atomic constituents used in signal processing can be grouped into time(space)-scale, time(space)-frequency, or a combination of both. In the time(space)-scale group we find the Grossman-Morlet and Daubechies wavelet basis defined by the functions shown below. These functions are used to obtain wavelet coefficients used in wavelet analysis and synthesis.

Grossman-Morlet wavelets:

Daubechies wavelets:

In the Grossman-Morlet wavelet function, the parameter "a" gives the scaling factor and "b" the center (location) of the function. In the case of the Daubechies wavelet function, scaling is given by the parameter and its center by the parameter "k".

In the time(space)-frequency we find the Gabor-Malvar waveletes defined by:

If we subject the Gabor-Malvar wavelets to dyadic dilations we can construct the Daubechies wavelets, thus we have the thrid group.

Wavelet transformation is a powerful tool for the analysis and synthesis of signals. Localization of signal characteristics in spatial (or time) and frequency domains can be accomplished very efficiently with wavelets. This allows us to simultaneously determine sharp transitions in the spectrum of the signal and in the position (or time) of their occurrence.

The basic structure of a wavelet transform is composed of recursive filtering and decimation, both of which are relatively easy to implement. The Kernel separability property in wavelet transform theory is an interesting feature that renders multidimensional wavelet transforms as a powerful tool for multidimensional signal processing.

The use of Wavelet Transform in numerical analysis seems to be very promising for it allows a very good representation of discontinuities, unlike the Fourier Transform. In addition, the Wavelet Transform produces sparse matrices and/or operators which can be processed with very low computational complexity.

An attractive feature of the Wavelet Transform is its relationship with Sub-band Coding Systems and Filter Banks. The only difference between the Wavelet Transform and the filters in Sub-band Coding Systems is that the former are designed to have some regularity properties (many zeros at z = 0 or z = 3.141516 (pi)). Wavelet Theory provides new ideas and insights which certainly enriches the important area of multirate filter banks.

Wavelet Transform theory can also be coupled with other techniques, like vector quantization or multiscale edges. This leads to powerful compression techniques for non-stationary signals. The fields of statistical signal processing, multiscale model of stochastic processes and analysis and synthesis of 1/f noise have shown interesting results when associated with the wavelet theory. Wavelet Packets, which correspond to arbitrary adaptive tree-structure filter banks is another very promising example.