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

TheHaar sequence was proposed in 1909 byAlfréd Haar.^[1] Haar used these functions to give an example of an orthonormal system for the space ofsquare-integrable functions on theunit interval [0, 1]. The study of wavelets, and even the term "wavelet", did not come until much later. As a special case of theDaubechies wavelet, the Haar wavelet is also known asDb1.

The Haar wavelet is also the simplest possible wavelet. The technical disadvantage of the Haar wavelet is that it is notcontinuous, and therefore notdifferentiable. 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.^[2]

The Haar wavelet's mother wavelet function${\displaystyle \psi (t)}$ can be described as

Itsscaling function${\displaystyle \phi (t)}$ can be described as

For every pairn,k of integers in${\displaystyle \mathbb{Z}}$ , theHaar functionψ_n,k is defined on thereal line${\displaystyle \mathbb{R}}$  by the formula

${\displaystyle {\psi }_{n,k}(t)=2^{n/2}\psi (2^{n}t-k),t\in \mathbb{R}.}$ 

This function is supported on theright-open intervalI_n,k =[k2⁻ⁿ, (k+1)2⁻ⁿ),i.e., itvanishes outside that interval. It has integral 0 and norm 1 in theHilbert space L²(${\displaystyle \mathbb{R}}$ ),

${\displaystyle {\int }_{\mathbb{R}}{\psi }_{n,k}(t)dt=0,\Vert {\psi }_{n,k}{\Vert }_{L^2(\mathbb{R})}^2={\int }_{\mathbb{R}}{\psi }_{n,k}(t{)}^{2}dt=1.}$ 

The Haar functions are pairwiseorthogonal,

${\displaystyle {\int }_{\mathbb{R}}{\psi }_{n_1,k_1}(t){\psi }_{n_2,k_2}(t)dt={\delta }_{n_1{n}_2}{\delta }_{k_1{k}_2},}$ 

where${\displaystyle {\delta }_{ij}}$  represents theKronecker delta. Here is the reason for orthogonality: when the two supporting intervals${\displaystyle I_{n_1,k_1}}$  and${\displaystyle I_{n_2,k_2}}$  are not equal, then they are either disjoint, or else the smaller of the two supports, say${\displaystyle I_{n_1,k_1}}$ , is contained in the lower or in the upper half of the other interval, on which the function${\displaystyle {\psi }_{n_2,k_2}}$  remains constant. It follows in this case that the product of these two Haar functions is a multiple of the first Haar function, hence the product has integral 0.

TheHaar system on the real line is the set of functions

${\displaystyle \{{\psi }_{n,k}(t):n\in \mathbb{Z},k\in \mathbb{Z}\}.}$ 

It iscomplete inL²(${\displaystyle \mathbb{R}}$ ):The Haar system on the line is an orthonormal basis inL²(${\displaystyle \mathbb{R}}$ ).

The Haar wavelet has several notable properties:

In this section, the discussion is restricted to theunit interval [0, 1] and to the Haar functions that are supported on [0, 1]. The system of functions considered by Haar in 1910,^[5]called theHaar system on [0, 1] in this article, consists of the subset of Haar wavelets defined as

 

with the addition of the constant function1 on [0, 1].

InHilbert space terms, this Haar system on [0, 1] is a complete orthonormal system,i.e., anorthonormal basis, for the spaceL²([0, 1]) of square integrable functions on the unit interval.

The Haar system on [0, 1] —with the constant function1 as first element, followed with the Haar functions ordered according to thelexicographic ordering of couples(n,k)— is further amonotoneSchauder basis for the spaceL^p([0, 1]) when1 ≤p < ∞.^[6] This basis isunconditional when1 <p < ∞.^[7]

There is a relatedRademacher system consisting of sums of Haar functions,

${\displaystyle r_n(t)=2^{-n/2}\sum _{k=0}^{2^n-1}{\psi }_{n,k}(t),t\in [0,1],n\ge 0.}$ 

Notice that |r_n(t)| = 1 on [0, 1). This is an orthonormal system but it is not complete.^[8][9]In the language ofprobability theory, the Rademacher sequence is an instance of a sequence ofindependentBernoullirandom variables withmean 0. TheKhintchine inequality expresses the fact that in all the spacesL^p([0, 1]),1 ≤p < ∞, the Rademacher sequence isequivalent to the unit vector basis in ℓ².^[10] In particular, theclosed linear span of the Rademacher sequence inL^p([0, 1]),1 ≤p < ∞, isisomorphic to ℓ².

TheFaber–Schauder system^[11][12][13] is the family of continuous functions on [0, 1] consisting of the constant function 1, and of multiples ofindefinite integrals of the functions in the Haar system on [0, 1], chosen to have norm 1 in themaximum norm. This system begins withs₀ = 1, thens₁(t) =t is the indefinite integral vanishing at 0 of the function 1, first element of the Haar system on [0, 1]. Next, for every integern ≥ 0, functionss_n,k are defined by the formula

 

These functionss_n,k are continuous,piecewise linear, supported by the intervalI_n,k that also supports ψ_n,k. The functions_n,k is equal to 1 at the midpointx_n,k of the interval I_n,k, linear on both halves of that interval. It takes values between 0 and 1 everywhere.

The Faber–Schauder system is aSchauder basis for the spaceC([0, 1]) of continuous functions on [0, 1].^[6] For every f inC([0, 1]), the partial sum

${\displaystyle f_{n+1}=a_0{s}_0+a_1{s}_1+\sum _{m=0}^{n-1}(\sum _{k=0}^{2^m-1}{a}_{m,k}{s}_{m,k})\in C([0,1])}$ 

of theseries expansion off in the Faber–Schauder system is the continuous piecewise linear function that agrees with f at the2ⁿ + 1 pointsk2⁻ⁿ, where 0 ≤k ≤ 2ⁿ. Next, the formula

${\displaystyle f_{n+2}-f_{n+1}=\sum _{k=0}^{2^n-1}(f(x_{n,k})-f_{n+1}(x_{n,k}))s_{n,k}=\sum _{k=0}^{2^n-1}{a}_{n,k}{s}_{n,k}}$ 

gives a way to compute the expansion off step by step. Sincef isuniformly continuous, the sequence {f_n} converges uniformly tof. It follows that the Faber–Schauder series expansion off converges inC([0, 1]), and the sum of this series is equal to f.

TheFranklin system is obtained from the Faber–Schauder system by theGram–Schmidt orthonormalization procedure.^[14][15]Since the Franklin system has the samelinear span as that of the Faber–Schauder system, this span is dense inC([0, 1]), hence inL²([0, 1]). The Franklin system is therefore an orthonormal basis forL²([0, 1]), consisting of continuous piecewise linear functions. P. Franklin proved in 1928 that this system is a Schauder basis forC([0, 1]).^[16] The Franklin system is also an unconditional Schauder basis for the spaceL^p([0, 1]) when1 <p < ∞.^[17]The Franklin system provides a Schauder basis in thedisk algebraA(D).^[17]This was proved in 1974 by Bočkarev, after the existence of a basis for the disk algebra had remained open for more than forty years.^[18]

Bočkarev's construction of a Schauder basis inA(D) goes as follows: let f be a complex valuedLipschitz function on [0, π]; then f is the sum of acosine series withabsolutely summable coefficients. Let T(f) be the element ofA(D) defined by the complexpower series with the same coefficients,

${\displaystyle \left\{f:x\in [0,\pi ]\to \sum _{n=0}^{\mathrm{\infty }}{a}_n\cos (nx)\right\}⟶\left\{T(f):z\to \sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n,|z|\le 1\right\}.}$ 

Bočkarev's basis forA(D) is formed by the images under T of the functions in the Franklin system on [0, π]. Bočkarev's equivalent description for the mapping T starts by extendingf to aneven Lipschitz function g₁ on [−π, π], identified with a Lipschitz function on theunit circle T. Next, letg₂ be theconjugate function of g₁, and defineT(f) to be the function in A(D) whose value on the boundaryT of D is equal to g₁ + ig₂.

When dealing with 1-periodic continuous functions, or rather with continuous functionsf on [0, 1] such thatf(0) =f(1), one removes the functions₁(t) =t from the Faber–Schauder system, in order to obtain theperiodic Faber–Schauder system. Theperiodic Franklin system is obtained by orthonormalization from the periodic Faber–-Schauder system.^[19]One can prove Bočkarev's result onA(D) by proving that the periodic Franklin system on [0, 2π] is a basis for a Banach spaceA_r isomorphic toA(D).^[19] The spaceA_r consists of complex continuous functions on the unit circleT whoseconjugate function is also continuous.

The 2×2 Haar matrix that is associated with the Haar wavelet is

 

Using thediscrete wavelet transform, one can transform any sequence${\displaystyle (a_0,a_1,\dots ,a_{2n},a_{2n+1})}$  of even length into a sequence of two-component-vectors${\displaystyle \left(\left(a_0,a_1\right),\left(a_2,a_3\right),\dots ,\left(a_{2n},a_{2n+1}\right)\right)}$ . If one right-multiplies each vector with the matrix${\displaystyle H_2}$ , one gets the result${\displaystyle \left(\left(s_0,d_0\right),\dots ,\left(s_n,d_n\right)\right)}$  of one stage of the fast Haar-wavelet transform. Usually one separates the sequencess andd and continues with transforming the sequences. Sequences is often referred to as theaverages part, whereasd is known as thedetails part.^[20]

If one has a sequence of length a multiple of four, one can build blocks of 4 elements and transform them in a similar manner with the 4×4 Haar matrix

 

which combines two stages of the fast Haar-wavelet transform.

Compare with aWalsh matrix, which is a non-localized 1/–1 matrix.

Generally, the 2N×2N Haar matrix can be derived by the following equation.

${\displaystyle H_{2N}=\left[\begin{array}{c}{H}_N\otimes [1,1]\\ I_N\otimes [1,-1]\end{array}\right]}$ 

where  and${\displaystyle \otimes }$  is theKronecker product.

TheKronecker product of${\displaystyle A\otimes B}$ , where${\displaystyle A}$  is an m×n matrix and${\displaystyle B}$  is a p×q matrix, is expressed as

 

An un-normalized 8-point Haar matrix${\displaystyle H_8}$  is shown below

 

Note that, the above matrix is an un-normalized Haar matrix. The Haar matrix required by the Haar transform should be normalized.

From the definition of the Haar matrix${\displaystyle H}$ , one can observe that, unlike theFourier transform,${\displaystyle H}$  has only real elements (i.e., 1, -1 or 0) and is non-symmetric.

Take the 8-point Haar matrix${\displaystyle H_8}$  as an example. The first row of${\displaystyle H_8}$  measures the average value, and the second row of${\displaystyle H_8}$  measures a low frequency component of the input vector. The next two rows are sensitive to the first and second half of the input vector respectively, which corresponds to moderate frequency components. The remaining four rows are sensitive to the four section of the input vector, which corresponds to high frequency components.^[21]

TheHaar transform is the simplest of thewavelet transforms. This transform cross-multiplies a function against the Haar wavelet with various shifts and stretches, like the Fourier transform cross-multiplies a function against a sine wave with two phases and many stretches.^{[22][clarification needed]}

The Haar transform is one of the oldest transform functions, proposed in 1910 by the Hungarian mathematicianAlfréd Haar. It is found effective in applications such as signal and image compression in electrical and computer engineering as it provides a simple and computationally efficient approach for analysing the local aspects of a signal.

The Haar transform is derived from the Haar matrix. An example of a 4×4 Haar transformation matrix is shown below.

 

The Haar transform can be thought of as a sampling process in which rows of the transformation matrix act as samples of finer and finer resolution.

Compare with theWalsh transform, which is also 1/–1, but is non-localized.

The Haar transform has the following properties

1. No need for multiplications. It requires only additions and there are many elements with zero value in the Haar matrix, so the computation time is short. It is faster thanWalsh transform, whose matrix is composed of +1 and −1.
2. Input and output length are the same. However, the length should be a power of 2, i.e.${\displaystyle N=2^k,k\in \mathbb{N}}$ .
3. It can be used to analyse the localized feature of signals. Due to theorthogonal property of the Haar function, the frequency components of input signal can be analyzed.

The Haar transformy_n of an n-input functionx_n is

${\displaystyle y_n=H_n{x}_n}$ 

The Haar transform matrix is real and orthogonal. Thus, the inverse Haar transform can be derived by the following equations.

${\displaystyle H=H^{\ast },H^{-1}=H^T,\text{ i.e. }H{H}^T=I}$ 

where${\displaystyle I}$  is the identity matrix. For example, when n = 4

 

Thus, the inverse Haar transform is

${\displaystyle x_n=H^T{y}_n}$ 

The Haar transform coefficients of a n=4-point signal${\displaystyle x_4=[1,2,3,4{]}^T}$  can be found as

 

The input signal can then be perfectly reconstructed by the inverse Haar transform

 

• Dimension reduction
• Walsh matrix
• Walsh transform
• Wavelet
• Chirplet
• Signal
• Haar-like feature
• Strömberg wavelet
• Dyadic transformation

• Haar, Alfréd (1910), "Zur Theorie der orthogonalen Funktionensysteme",Mathematische Annalen,69 (3):331–371,doi:10.1007/BF01456326,hdl:2027/uc1.b2619563,S2CID 120024038
• Charles K. Chui,An Introduction to Wavelets, (1992), Academic Press, San Diego,ISBN 0-585-47090-1
• English Translation of Haar's seminal article:[1]

• "Haar system",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Free Haar wavelet filtering implementation and interactive demo
• Free Haar wavelet denoising and lossy signal compression

• Kingsbury, Nick."The Haar Transform". Archived fromthe original on 19 April 2006.
• Eck, David (31 January 2006)."Haar Transform Demo Applets".
• Ames, Greg (7 December 2002)."Image Compression"(PDF). Archived fromthe original(PDF) on 25 January 2011.
• Aaron, Anne; Hill, Michael; Srivatsa, Anand."MOSMAT 500. A photomosaic generator. 2. Theory". Archived fromthe original on 18 March 2008.
• Wang, Ruye (4 December 2008)."Haar Transform". Archived fromthe original on 21 August 2012.