Haar wavelet

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In mathematics, the Haar wavelet is a certain sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal function basis. The Haar sequence is now recognised as the first known wavelet basis and extensively used as a teaching example in the theory of wavelets.

The Haar sequence was proposed in 1909 by Alfréd Haar.[1] Haar used these functions to give an example of a countable orthonormal system for the space of square-integrable functions on the real line. The study of wavelets, and even the term "wavelet", did not come until much later. As a special case of the Daubechies wavelet, it is also known as D2.

The Haar wavelet is also the simplest possible wavelet. The technical disadvantage of the Haar wavelet is that it is not continuous, and therefore not differentiable. This property can, however, be an advantage for the analysis of signals with sudden transitions, such as monitoring of tool failure in machines.[2]

The Haar wavelet's mother wavelet function ψ(t) can be described as

and its scaling function φ(t) can be described as


Haar system

In functional analysis, the Haar systems denotes the set of Haar wavelets

\{ t \mapsto \psi_{n,k}(t)=\psi(2^n t-k) ; n \in \N, 0 \leq k < 2^n\}.

In Hilbert space terms, this constitutes a complete orthogonal system for the functions on the unit interval. There is a related Rademacher system, of sums of Haar functions, which is an orthogonal system but not complete.[3][4]

The Haar system (with the natural ordering) is further a Schauder basis for the space Lp[0,1] for 1 \leq p < +\infty. This basis is unconditional for p > 1.

Haar wavelet properties

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