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IEVref: | 103-04-11 | ID: | |

Language: | en | Status: Standard | |

Term: | wavelet | ||

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Definition: | small localized wave, represented by a function having a zero mean value and a practically finite duration Note 1 to entry: From a mother wavelet $\psi (t)$, daughter wavelets are obtained through shifting and scaling (expansion or compression): ${\psi}_{a,b}(t)=\frac{1}{\sqrt{a}}\psi \left(\frac{t-b}{a}\right)$, where Note 2 to entry: Examples (see Figures 3 and 4): - Haar wavelet: $\psi (t)=-1$ for −1/2 <
*t*< 0, $\psi (t)=1$ for 0 <*t*< 1/2, $\psi (t)=0$ outside; - Morlet wavelet: $\psi (t)={\mathrm{e}}^{-{t}^{2}/2}{\mathrm{e}}^{-\mathrm{j}\omega \text{\hspace{0.05em}}t}$ (example of exponential damping; Figure 4 gives the real part).
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Publication date: | 2009-12 | ||

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Internal notes: | 2017-02-20: Editorial revisions in accordance with the information provided in C00020 (IEV 103) - evaluation. JGO 2017-08-25: Added a <mstyle displaystyle='true'> tag. LMO | ||

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Note 1 to entry: From a mother wavelet $\psi (t)$, daughter wavelets are obtained through shifting and scaling (expansion or compression): ${\psi}_{a,b}(t)=\frac{1}{\sqrt{a}}\psi \left(\frac{t-b}{a}\right)$, where *a* is a scale parameter and *b* a position parameter.

Note 2 to entry: Examples (see Figures 3 and 4):

- Haar wavelet: $\psi (t)=-1$ for −1/2 <
*t*< 0, $\psi (t)=1$ for 0 <*t*< 1/2, $\psi (t)=0$ outside; - Morlet wavelet: $\psi (t)={\mathrm{e}}^{-{t}^{2}/2}{\mathrm{e}}^{-\mathrm{j}\omega \text{\hspace{0.05em}}t}$ (example of exponential damping; Figure 4 gives the real part).

**Figure 3 – Haar wavelet**

**Figure 3 – Ondelette de Haar**

**Figure 4 – Morlet wavelet **

**Figure 4 – Ondelette de Morlet**