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The Haar wavelet is the first known wavelet and was proposed in 1909 by Alfred Haar.
Note that the term wavelet was coined much later.
The Haar wavelet is also the simplest possible wavelet.
It looks like this:
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1 ****O
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0 *****O---+---*****
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-1 | ****O
0
The disadvantage of the Haar wavelet is that it is not continuous and therefore not differentiable.
The Haar Wavelet can also be described as a step function f(x) with
- <math>f(x) = \begin{cases}1 \quad & 0 \leq x < 1/2,\\
-1 & 1/2 \leq x < 1,\\0 &\mbox{otherwise.}\end{cases}<math>
Haar matrix
The 2×2 Haar matrix that is associated with the Haar wavelet is
- <math> H_2 = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}<math>
and the 4×4 Haar matrix is
- <math> H_4 = \begin{bmatrix} 1 & 1 & 1 & 0 \\ 1 & 1 & -1 & 0 \\ 1 & -1 & 0 & 1\\ 1 & -1 & 0 & -1 \end{bmatrix}<math>
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