The subject of wavelets has received considerable attention over the last twenty years, with contributions coming from researchers in electrical engineering, mathematics and physics.
The word "wavelet" refers to a little wave, and implies functions that are reasonably localized both in the Time Domain and in the Fourier Domain.
The idea stems from the limitation posed by the Uncertainty Principle, which puts a limit on simultaneous localization in the time and frequency domains. As in the case of the Uncertainty Principle of Physics, the implications are seen more when one would like to make a microscopic analysis of signals.
In a number of signal processing situations, one does indeed need to look at local features: in fact, the requirement of simultaneous localization is far more widespread than often perceived. For example, there are many situations in audio, image and video where, for the purpose of analysis, one very often wishes to focus one's attention on a specific time/ space range and frequency range simultaneously.
A number of problems in digital communication also point to the implications of this uncertainty, and the need to address it suitably. The origin of the wavelet transform is in trying to achieve this to the best extent possible while working within the limits posed by the uncertainty principle. In fact, one may relate the idea of the wavelet transform to the use of a positional notation in the context of real numbers.
The wavelet transform allows a generalization of the positional notation for the context of functions. In fact, another aspect of the whole subject is multiresolution analysis - the process of analyzing phenomena and information with a scale/ fineness, matched to the content being analyzed.
This issue has important implications in waveform and signal synthesis and design, in data compression, in the analysis of signals coming from geophysical sources and biomedical sources, in locating and analyzing singularities in signals and functions, in interpolation and in many other areas.
The whole idea of wavelets manifests itself differently in many different disciplines although the basic principles remain the same. The aim of this course is to introduce the idea of wavelets, and the related notions of time-frequency analysis, of time-scale analysis, and to describe the manner in which technical developments related to wavelets have led to numerous applications.
A discussion on multirate filter banks will also form an important part of the course. The relation between wavelets and multirate systems will be brought out; to illustrate how wavelets may actually be realized in practice.