Wavelet analysis is a relatively new technique in signal processing.  It is especially suited for the analysis of non-stationary signals by providing a representation of a signal in a time-frequency plane.  A main function (mother wavelet) is used.  It is "cut" into functions (smaller waves or wavelets).  Translation (window of signal under inspection) and scale (give detailed or global view) parameters are changed for each wavelet in order to obtain a different view of a signal. By decomposing a mother wavelet into time-frequency space, both the dominant modes of variability and how the dominant modes vary with time are determined.  Thus, information regarding the time localization of the signal's components is obtained.

The main branch in mathematics leading to wavelets began in 1807 with Joseph Fourier.  His theories of frequency analysis lead to the traditional approach of measuring the frequency content of a signal by the Fourier Transform.  In this approach, a periodic function can be expressed as an infinite sum of periodic complex exponential functions.  A signal is then decomposed into complex exponential functions of different frequencies. A frequency versus amplitude representation of a signal is obtained.  This type of plot indicates how much of each frequency exists in a signal. Although the Fourier Transform provides how much of each frequency exists in a signal, it does not indicate when in time these frequency components exist.  This is a major obstacle because most signals, especially in the areas of biomedicine are non-stationary since the frequency content of the signal changes with time.  The electroactivity of the heart, brain and muscles, for example, are non-stationary signals.

An alternative approach developed in the 1930's examines the representation of functions using scale-varying basis functions.  Scale-varying basis functions analyze a function using different scales.  A signal over the domain from 0 to 1, for example, can be analyzed by a variety of step functions. One step function can be from 0 to ½ and ½ to 1. Furthermore, the original signal can be divided into additional several step functions from 0 to ¼, ¼ to ½, ½ to ¾ and ¾ to 1.  Each set of representations code the original signal with a particular scale.  This became a key concept in the development of wavelets because a signal (under wavelet analysis) is computed separately for different segments of the time-domain signal.

Fifty years later, a physicist and engineer by the names of Grossman and Morlet, respectively, defined wavelets in the context of quantum physics.  They stated that by the Heisenburg Uncertainty Principle, one couldn't know the exact time-frequency representations of a signal.  The only information one can obtain is the time intervals in which certain band frequencies exist.  Not only did this provide a way of thinking about wavelets based on physical intuition; it also indicated resolution problems that must be overcome when analyzing a signal.

In 1985, Stephane Mallat made a huge leap in the field of wavelet analysis by discovering the relationship between pyramid algorithms and orthonormal wavelet bases.  His work has become a major stepping stone of current wavelet applications.

Wavelets have vast applications ranging from seismology to fingerprinting analysis.  Their use in the fields of biomedicine, chemistry and computer graphics has enabled us to learn more about the world around us.

Author: Henry Sheen

References:
Daubchies, Ingrid.  Different Perspectives on Wavelets.  Providence, R.I.: American Mathematical Society, 1993.

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