Haar wavelet

In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a <a href="/facts/Wavelet/3XcX4uIP">wavelet</a> family or basis. Wavelet analysis is similar to <a href="/facts/Fourier_analysis/sUA8JP6q">Fourier analysis</a> in that it allows a target function over an interval to be represented in terms of an <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal basis</a>. 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 <a href="/facts/Alfr%C3%A9d_Haar/FVJlnS9z">Alfréd Haar</a>. 
Haar used these functions to give an example of an orthonormal system for the space of <a href="/facts/Square-integrable_function/IASblcJJ">square-integrable functions</a> on the <a href="/facts/Unit_interval/W4dP8IJo">unit interval</a> [0, 1]. The study of wavelets, and even the term "wavelet", did not come until much later. As a special case of the <a href="/facts/Daubechies_wavelet/lhF1zmqB">Daubechies wavelet</a>, 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 <a href="/facts/Continuous_function/sKbl02pB">continuous</a>, and therefore not <a href="/facts/Derivative/7Ito70Dh">differentiable</a>. This property can, however, be an advantage for the analysis of signals with sudden transitions (<a href="/facts/Digital_signal_(signal_processing)/bCtTHxre">discrete signals</a>), 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 <a href="/facts/Father_wavelets/3XcX4uIP">scaling function</a> 
 
 
 
 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 wavelet open-in-new

Haar wavelet