Wavelet transforms


Wavelet transforms have been one of the important signal processing developments in the last decade, especially for the applications such as time-frequency analysis, data compression, segmentation and vision. During the past decade, several efficient implementations of wavelet transforms have been derived. The theory of wavelets has roots in quantum mechanics and the theory of functions though a unifying framework is a recent occurrence. Wavelet analysis is performed using a prototype function called a wavelet.

Wavelets are functions defined over a finite interval and having an average value of zero. The basic idea of the wavelet transform is to represent any arbitrary function f (t) as a superposition of a set of such wavelets or basis functions. These basis functions or baby wavelets are obtained from a single prototype wavelet called the mother wavelet, by dilations or contractions (scaling) and translations (shifts). Efficient implementation of the wavelet transforms has been derived based on the Fast Fourier transform and short-length 'fast-running FIR algorithms' in order to reduce the computational complexity per computed coefficient.

First of all, why do we need a transform, or what is a transform anyway?

Mathematical transformations are applied to signals to obtain further information from that signal that is not readily available in the raw signal. Now, a time-domain signal is assumed as a raw signal, and a signal that has been transformed by any available transformations as a processed signal.

There are a number of transformations that can be applied such as the Hilbert transform, short-time Fourier transform, Wigner transform, the Radon transform, among which the Fourier transform is probably the most popular transform. These mentioned transforms constitute only a small portion of a huge list of transforms that are available at engineers and mathematicians disposal. Each transformation technique has its own area of application, with advantages and disadvantages.

Importance Of The Frequency Information

Often times, the information that cannot be readily seen in the time-domain can be seen in the frequency domain. Most of the signals in practice are time-domain signals in their raw format. That is, whatever that signal is measuring, is a function of time. In other words, when we plot the signal one of the axis is time (independent variable) and the other (dependent variable) is usually the amplitude.

When we plot time-domain signals, we obtain a time-amplitude representation of the signal. This representation is not always the best representation of the signal for most signal processing related applications. In many cases, the most distinguished information is hidden in the frequency content of the signal. The frequency spectrum of a signal is basically the frequency components (spectral components) of that signal. The frequency spectrum of a signal shows what frequencies exist in the signal.