Editing Haar wavelet (section)

{{Short description|First known wavelet basis}}
{{Use dmy dates|date=March 2024}}
[[Image:Haar wavelet.svg|thumb|right|The Haar wavelet]]

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]].<ref>see p.&nbsp;361 in {{harvtxt|Haar|1910}}.</ref> 
Haar used these functions to give an example of an orthonormal system for the space of [[square-integrable function]]s on the [[unit interval]]&nbsp;[0,&nbsp;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 function|continuous]], and therefore not [[derivative|differentiable]]. This property can, however, be an advantage for the analysis of signals with sudden transitions ([[Digital signal (signal processing)|discrete signals]]), such as monitoring of tool failure in machines.<ref>{{cite journal |first1=B. |last1=Lee |first2=Y. S. |last2=Tarng |title=Application of the discrete wavelet transform to the monitoring of tool failure in end milling using the spindle motor current |journal=International Journal of Advanced Manufacturing Technology |year=1999 |volume=15 |issue=4 |pages=238–243 |doi=10.1007/s001700050062 |s2cid=109908427 }}</ref>

The Haar wavelet's mother wavelet function <math>\psi(t)</math> can be described as
: <math>\psi(t) = \begin{cases}
  1 \quad & 0 \leq  t < \frac{1}{2},\\
 -1 & \frac{1}{2} \leq t < 1,\\
  0 &\mbox{otherwise.}
\end{cases}</math>

Its [[Father wavelets|scaling function]] <math>\varphi(t)</math> can be described as
: <math>\varphi(t) = \begin{cases}1 \quad & 0 \leq  t < 1,\\0 &\mbox{otherwise.}\end{cases}</math>