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The elementary building blocks in a mathematical tool for analyzing functions. The functions can be very diverse; examples are solutions of a differential equation, and one- and two-dimensional signals. The tool itself, the wavelet transform, is the result of a synthesis of ideas from many different fields, ranging from pure mathematics to quantum physics and electrical engineering.
In many practical applications, it is desirable to extract frequency information from a signal—in particular, which frequencies are present and their respective importance. An example is the decomposition into spectral lines in spectroscopy. The tool that is generally used to achieve this is the Fourier transform. Many applications, however, concern nonstationary signals, in which the makeup of the different frequency components is constantly shifting. An example is music, where this shifting nature has been recognized for centuries by the standard notation, which tells a musician which note (frequency information) to play when and how long (time information). For signals of this nature, a time-frequency representation is needed.
There exist many different mathematical tools leading to a time-frequency representation of a given signal, each with its own strengths and weaknesses. The wavelet transform is such a time-frequency analysis tool. Its strength lies in its ability to deal well with transient high-frequency phenomena, such as sudden peaks or discontinuities, as well as with the smoother portions of the signal. (An example is a crack in the sound from a damaged record, or the attack at the start of a music note.) The wavelet transform is less well adapted to harmonically oscillating parts in the signal, for which Fourier-type methods are more indicated.
Applications of wavelets include various forms of data compression (such as for images and fingerprints), data analysis (nuclear magnetic resonance, radar, seismograms, and sound), and numerical analysis (fast solvers for partial differential equations).