In mathematics, the Haar wavelet is a 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 basis. The Haar sequence is now recognised as the first known wavelet basis and is extensively used as a teaching example. The Haar sequence was proposed in 1909 by Alfréd Haar. Haar used these functions to give an example of an orthonormal system for the space of square-integrable functions on the unit interval [0, 1]. The study of wavelets, and even the term "wavelet", did not come until much later. As a special case of the Daubechies wavelet, the Haar wavelet is also known as Db1. 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 (discrete signals), such as monitoring of tool failure in machines. The Haar wavelet's mother wavelet function ψ ( t ) {\displaystyle \psi (t)} can be described as
ψ ( t ) = { 1 0 ≤ t < 1 2 , − 1 1 2 ≤ t < 1 , 0 otherwise. {\displaystyle \psi (t)={\begin{cases}1\quad &0\leq t<{\frac {1}{2}},\\-1&{\frac {1}{2}}\leq t<1,\\0&{\mbox{otherwise.}}\end{cases}}}
Its scaling function φ ( t ) {\displaystyle \varphi (t)} can be described as
φ ( t ) = { 1 0 ≤ t < 1 , 0 otherwise. {\displaystyle \varphi (t)={\begin{cases}1\quad &0\leq t<1,\\0&{\mbox{otherwise.}}\end{cases}}}
Haar functions and Haar system For every pair n, k of integers in Z {\displaystyle \mathbb {Z} } , the Haar function ψn,k is defined on the real line R {\displaystyle \mathbb {R} } by the formula
ψ n , k ( t ) = 2 n / 2 ψ ( 2 n t − k ) , t ∈ R . {\displaystyle \psi _{n,k}(t)=2^{n/2}\psi (2^{n}t-k),\quad t\in \mathbb {R} .}
This function is supported on the right-open interval In,k = [ k2−n, (k+1)2−n), i.e., it vanishes outside that interval. It has integral 0 and norm 1 in the Hilbert space L2( R {\displaystyle \mathbb {R} } ),
∫ R ψ n , k ( t ) d t = 0 , ‖ ψ n , k ‖ L 2 ( R ) 2 = ∫ R ψ n , k ( t ) 2 d t = 1. {\displaystyle \int _{\mathbb {R} }\psi _{n,k}(t)\,dt=0,\quad \|\psi _{n,k}\|_{L^{2}(\mathbb {R} )}^{2}=\int _{\mathbb {R} }\psi _{n,k}(t)^{2}\,dt=1.}
The Haar functions are pairwise orthogonal,
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