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As far as applications are concerned, the DWT is used for signal coding whereas the CWT is used for signal analysis. Consequently, the DWT is commonly used in engineering and computer science and the CWT is most often used in scientific research. Wavelet transforms are now being adopted for a vast number of different applications, often replacing the conventional Fourier transform in many applications. Many areas of physics have seen this paradigm shift, including molecular dynamics, ab initio calculations , astrophysics, density-matrix localisation, seismic geophysics, optics, turbulence and quantum mechanics, as well as many other fields including image processing, blood-pressure, heart-rate and ECG analyses, DNA analysis, protein analysis, climatology, general signal processingSignal processing is the processing, amplification and interpretation of signals. Signals may come from various sources. There are various sorts of signal processing, depending on the nature of the signal, as in the following examples. Digital signal proc, speech recognitionSpeech recognition technologies allow computers equipped with a source of sound input, such as a microphone, to interpret human speech, e. for transcription or as an alternative method of interacting with a computer. Classification Such systems can be cla, computer graphicsComputer graphics (CG) is the field of visual computing, where one utilizes computers both to generate visual images synthetically and to integrate or alter visual and spatial information sampled from the real world. The first major advance in computer gr and multifractal analysisIn mathematical analysis, multifractal analysis is the process of determining the fractal dimension of a multifractal system. See also: Holder exponent. Fractals..

In historical terms, the development of wavelets can be linked to several separate trains of thought, starting with Haar's work in the early 20th century. Notable contributions to wavelet theory can be attributed to Goupillaud, Grossman and Morlet 's formulation of what is now known as the CWT (1982), Jan Olov-Strömberg's early work on discrete wavelets (1983), Daubechies ' orthogonal wavelets with compact support (1988), Mallat 's multiresolution framework (1989), Delprat 's time-frequency interpretation of the CWT (1991), Newland 's Harmonic wavelet transform and many others since.

Wavelet theory is related to several other subjects. All wavelet transforms may be considered to be forms of time-frequency representationA time-frequency representation TFR is a view of a signal (taken to be a function of time) represented over both time and frequency. Time-frequency analysis means analysis of a TFR. A signal, as a function of time, may be considered as a representation wi and are, therefore, related to the subject of harmonic analysisHarmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms. The basic waves are called ". Discrete wavelet transforms are a form of finite impulse response filter . The wavelets forming a CWT are subject to HeisenbergWerner Karl Heisenberg ( December 5, 1901 February 1, 1976) was a celebrated physicist and Nobel laureate, one of the founders of quantum mechanics. He was born in Wurzburg, Germany and died in Munich. Heisenberg was the head of Nazi Germany's nuclear ene's uncertainty principle and, equivalently, discrete wavelet bases may be considered in the context of other forms of uncertainty principle.